%% Papers Published
@article{fds374493,
Author = {Gong, Y and Kiselev, A},
Title = {A simple reaction-diffusion system as a possible model for
the origin of chemotaxis.},
Journal = {Journal of biological dynamics},
Volume = {17},
Number = {1},
Pages = {2260833},
Year = {2023},
Month = {December},
url = {http://dx.doi.org/10.1080/17513758.2023.2260833},
Abstract = {Chemotaxis is a directed cell movement in response to
external chemical stimuli. In this paper, we propose a
simple model for the origin of chemotaxis - namely how a
directed movement in response to an external chemical signal
may occur based on purely reaction-diffusion equations
reflecting inner working of the cells. The model is inspired
by the well-studied role of the rho-GTPase Cdc42 regulator
of cell polarity, in particular in yeast cells. We analyse
several versions of the model to better understand its
analytic properties and prove global regularity in one and
two dimensions. Using computer simulations, we demonstrate
that in the framework of this model, at least in certain
parameter regimes, the speed of the directed movement
appears to be proportional to the size of the gradient of
signalling chemical. This coincides with the form of the
chemical drift in the most studied mean field model of
chemotaxis, the Keller-Segel equation.},
Doi = {10.1080/17513758.2023.2260833},
Key = {fds374493}
}
@article{fds374494,
Author = {Kiselev, A and Luo, X},
Title = {Illposedness of C2 Vortex Patches},
Journal = {Archive for Rational Mechanics and Analysis},
Volume = {247},
Number = {3},
Year = {2023},
Month = {June},
url = {http://dx.doi.org/10.1007/s00205-023-01892-7},
Abstract = {It is well known that vortex patches are wellposed in C1,α
if 0 < α< 1 . In this paper, we prove the illposedness of
C2 vortex patches. The setup is to consider the vortex
patches in Sobolev spaces W2,p where the curvature of the
boundary is Lp integrable. In this setting, we show the
persistence of W2,p regularity when 1 < p< ∞ and construct
C2 initial patch data for which the curvature of the patch
boundary becomes unbounded immediately for t> 0 , though it
regains C2 regularity precisely at all integer times without
being time periodic. The key ingredient is the evolution
equation for the curvature, the dominant term in which turns
out to be linear and dispersive.},
Doi = {10.1007/s00205-023-01892-7},
Key = {fds374494}
}
@article{fds374495,
Author = {Kiselev, A and Luo, X},
Title = {On Nonexistence of Splash Singularities for the α -SQG
Patches},
Journal = {Journal of Nonlinear Science},
Volume = {33},
Number = {2},
Year = {2023},
Month = {April},
url = {http://dx.doi.org/10.1007/s00332-023-09893-2},
Abstract = {In this paper, we consider patch solutions to the α-SQG
equation and derive new criteria for the absence of splash
singularity where different patches or parts of the same
patch collide in finite time. Our criterion refines a result
due to Gancedo and Strain Gancedo and Strain (2014),
providing a condition on the growth of curvature of the
patch necessary for the splash and an exponential in time
lower bound on the distance between patches with bounded
curvature.},
Doi = {10.1007/s00332-023-09893-2},
Key = {fds374495}
}
@article{fds368885,
Author = {Kiselev, A and Yao, Y},
Title = {Small Scale Formations in the Incompressible Porous Media
Equation},
Journal = {Archive for Rational Mechanics and Analysis},
Volume = {247},
Number = {1},
Year = {2023},
Month = {February},
url = {http://dx.doi.org/10.1007/s00205-022-01830-z},
Abstract = {We construct examples of solutions to the incompressible
porous media (IPM) equation that must exhibit infinite in
time growth of derivatives provided they remain smooth. As
an application, this allows us to obtain nonlinear
instability for a class of stratified steady states of
IPM.},
Doi = {10.1007/s00205-022-01830-z},
Key = {fds368885}
}
@article{fds374496,
Author = {Kiselev, A and Nazarov, F and Ryzhik, L and Yao, Y},
Title = {Chemotaxis and reactions in biology},
Journal = {Journal of the European Mathematical Society},
Volume = {25},
Number = {7},
Pages = {2641-2696},
Year = {2023},
Month = {January},
url = {http://dx.doi.org/10.4171/JEMS/1247},
Abstract = {Chemotaxis plays a crucial role in a variety of processes in
biology and ecology. Quite often it acts to improve
efficiency of biological reactions. One example is the
immune system signalling, where infected tissues release
chemokines attracting monocytes to fight invading bacteria.
Another example is reproduction, where eggs release
pheromones that attract sperm. A macro scale example is
flower scent appealing to pollinators. In this paper we
consider a system of PDEs designed to model such processes.
Our interest is to quantify the effect of chemotaxis on
reaction rates compared to pure reaction-diffusion. We limit
consideration to surface chemotaxis, which is well motivated
from the point of view of many applications. Our results
provide the first insight into situations where chemotaxis
can be crucial for reaction success, and where its effect is
likely to be limited. The proofs are based on new analytical
tools; a significant part of the paper is dedicated to
building up the linear machinery that can be useful in more
general settings. In particular, we establish precise
estimates on the rates of convergence to the ground state
for a class of Fokker–Planck operators with potentials
that grow at a logarithmic rate at infinity. These estimates
are made possible by a new sharp weak weighted Poincaré
inequality.},
Doi = {10.4171/JEMS/1247},
Key = {fds374496}
}
@article{fds366478,
Author = {Kiselev, A and Tan, C},
Title = {The Flow of Polynomial Roots Under Differentiation},
Journal = {Annals of PDE},
Volume = {8},
Number = {2},
Year = {2022},
Month = {December},
url = {http://dx.doi.org/10.1007/s40818-022-00135-4},
Abstract = {The question about behavior of gaps between zeros of
polynomials under differentiation is classical and goes back
to Marcel Riesz. Recently, Stefan Steinerberger [42]
formally derived a nonlocal nonlinear partial differential
equation which models dynamics of roots of polynomials under
differentiation. In this paper, we connect rigorously
solutions of Steinerberger’s PDE and evolution of roots
under differentiation for a class of trigonometric
polynomials. Namely, we prove that the distribution of the
zeros of the derivatives of a polynomial and the
corresponding solutions of the PDE remain close for all
times. The global in time control follows from the analysis
of the propagation of errors equation, which turns out to be
a nonlinear fractional heat equation with the main term
similar to the modulated discretized fractional Laplacian (-
Δ) 1 / 2.},
Doi = {10.1007/s40818-022-00135-4},
Key = {fds366478}
}
@article{fds367156,
Author = {Gong, Y and He, S and Kiselev, A},
Title = {Random Search in Fluid Flow Aided by Chemotaxis.},
Journal = {Bulletin of mathematical biology},
Volume = {84},
Number = {7},
Pages = {71},
Year = {2022},
Month = {June},
url = {http://dx.doi.org/10.1007/s11538-022-01024-4},
Abstract = {In this paper, we consider the dynamics of a 2D
target-searching agent performing Brownian motion under the
influence of fluid shear flow and chemical attraction. The
analysis is motivated by numerous situations in biology
where these effects are present, such as broadcast spawning
of marine animals and other reproduction processes or
workings of the immune systems. We rigorously characterize
the limit of the expected hit time in the large flow
amplitude limit as corresponding to the effective
one-dimensional problem. We also perform numerical
computations to characterize the finer properties of the
expected duration of the search. The numerical experiments
show many interesting features of the process and in
particular existence of the optimal value of the shear flow
that minimizes the expected target hit time and outperforms
the large flow limit.},
Doi = {10.1007/s11538-022-01024-4},
Key = {fds367156}
}
@article{fds368303,
Author = {Chouliara, D and Gong, Y and He, S and Kiselev, A and Lim, J and Melikechi,
O and Powers, K},
Title = {Hitting time of Brownian motion subject to shear
flow},
Journal = {Involve},
Volume = {15},
Number = {1},
Pages = {131-140},
Year = {2022},
Month = {January},
url = {http://dx.doi.org/10.2140/involve.2022.15.131},
Abstract = {The 2-dimensional motion of a particle subject to Brownian
motion and ambient shear flow transportation is considered.
Numerical experiments are carried out to explore the
relation between the shear strength, box size, and the
particle’s expected first hitting time of a given target.
The simulation is motivated by biological settings such as
reproduction processes and the workings of the immune
system. As the shear strength grows, the expected first
hitting time converges to the expected first hitting time of
the 1-dimensional Brownian motion. The dependence of the
hitting time on the shearing rate is monotone, and only the
form of the shear flow close to the target appears to play a
role. Numerical experiments also show that the expected
hitting time drops significantly even for quite small values
of shear rate near the target.},
Doi = {10.2140/involve.2022.15.131},
Key = {fds368303}
}
@article{fds368304,
Author = {Kiselev, A and Tan, C},
Title = {GLOBAL REGULARITY FOR A NONLOCAL PDE DESCRIBING EVOLUTION OF
POLYNOMIAL ROOTS UNDER DIFFERENTIATION},
Journal = {SIAM Journal on Mathematical Analysis},
Volume = {54},
Number = {3},
Pages = {3161-3191},
Year = {2022},
Month = {January},
url = {http://dx.doi.org/10.1137/21M1422859},
Abstract = {In this paper, we analyze a nonlocal nonlinear partial
differential equation formally derived by Steinerberger
[Proc. Amer. Math. Soc., 147 (2019), pp. 4733-4744] to model
dynamics of roots of polynomials under differentiation. This
partial differential equation is critical and bears striking
resemblance to hydrodynamic models used to describe
collective behavior of agents (such as birds, fish, or
robots) in mathematical biology. We consider a periodic
setting and show global regularity and exponential in time
convergence to uniform density for solutions corresponding
to strictly positive smooth initial data.},
Doi = {10.1137/21M1422859},
Key = {fds368304}
}
@article{fds361462,
Author = {Kiselev, A and Luo, X},
Title = {On nonexistence of splash singularities for the $α$-SQG
patches},
Year = {2021},
Month = {November},
Abstract = {In this paper, we consider patch solutions to the
$\alpha$-SQG equation and derive new criteria for the
absence of splash singularity where different patches or
parts of the same patch collide in finite time. Our
criterion refines a result due to Gancedo and Strain
\cite{GS}, providing a condition on the growth of curvature
of the patch necessary for the splash and an exponential in
time lower bound on the distance between patches with
bounded curvature.},
Key = {fds361462}
}
@article{fds358295,
Author = {He, S and Kiselev, A},
Title = {Boundary layer models of the Hou-Luo scenario},
Journal = {Journal of Differential Equations},
Volume = {298},
Pages = {182-204},
Year = {2021},
Month = {October},
url = {http://dx.doi.org/10.1016/j.jde.2021.07.007},
Abstract = {Finite time blow up vs global regularity question for 3D
Euler equation of fluid mechanics is a major open problem.
Several years ago, Luo and Hou [16] proposed a new finite
time blow up scenario based on extensive numerical
simulations. The scenario is axi-symmetric and features fast
growth of vorticity near a ring of hyperbolic points of the
flow located at the boundary of a cylinder containing the
fluid. An important role is played by a small boundary layer
where intense growth is observed. Several simplified models
of the scenario have been considered, all leading to finite
time blow up [3,2,9,13,11,15]. In this paper, we propose two
models that are designed specifically to gain insight in the
evolution of fluid near the hyperbolic stagnation point of
the flow located at the boundary. One model focuses on
analysis of the depletion of nonlinearity effect present in
the problem. Solutions to this model are shown to be
globally regular. The second model can be seen as an attempt
to capture the velocity field near the boundary to the next
order of accuracy compared with the one-dimensional models
such as [3,2]. Solutions to this model blow up in finite
time.},
Doi = {10.1016/j.jde.2021.07.007},
Key = {fds358295}
}
@article{fds361650,
Author = {Gong, Y and He, S and Kiselev, A},
Title = {Random search in fluid flow aided by chemotaxis},
Year = {2021},
Month = {July},
Abstract = {In this paper, we consider the dynamics of a 2D
target-searching agent performing Brownian motion under the
influence of fluid shear flow and chemical attraction. The
analysis is motivated by numerous situations in biology
where these effects are present, such as broadcast spawning
of marine animals and other reproduction processes or
workings of the immune systems. We rigorously characterize
the limit of the expected hit time in the large flow
amplitude limit as corresponding to the effective
one-dimensional problem. We also perform numerical
computations to characterize the finer properties of the
expected duration of the search. The numerical experiments
show many interesting features of the process, and in
particular existence of the optimal value of the shear flow
that minimizes the expected target hit time and outperforms
the large flow limit.},
Key = {fds361650}
}
@article{fds361651,
Author = {Gong, Y and Kiselev, A},
Title = {Chemotactic Reaction Enhancement in One Dimension},
Year = {2021},
Month = {March},
Abstract = {Chemotaxis, the directional locomotion of cells towards a
source of a chemical gradient, is an integral part of many
biological processes - for example, bacteria motion,
single-cell or multicellular organisms development, immune
response, etc. Chemotaxis directs bacteria's movement to
find food (e.g., glucose) by swimming toward the highest
concentration of food molecules. In multicellular organisms,
chemotaxis is critical to early development (e.g., movement
of sperm towards the egg during fertilization). Chemotaxis
also helps mobilize phagocytic and immune cells at sites of
infection, tissue injury, and thus facilitates immune
reactions. In this paper, we study a PDE system that
describes such biological processes in one dimension, which
may correspond to a thin channel, the setting relevant in
many applications: for example, spermatozoa progression to
the ovum inside a Fallopian tube or immune response in a
blood vessel.},
Key = {fds361651}
}
@article{fds361348,
Author = {Kiselev, A and Yao, Y},
Title = {Small scale formations in the incompressible porous media
equation},
Year = {2021},
Month = {February},
Abstract = {We construct examples of solutions to the incompressible
porous media (IPM) equation that must exhibit infinite in
time growth of derivatives provided they remain smooth. As
an application, this allows us to obtain nonlinear
instability for a class of stratified steady states of
IPM.},
Key = {fds361348}
}
@article{fds356022,
Author = {He, S and Kiselev, A},
Title = {Small-scale creation for solutions of the sqg
equation},
Journal = {Duke Mathematical Journal},
Volume = {170},
Number = {5},
Pages = {1027-1041},
Publisher = {Duke University Press},
Year = {2021},
Month = {January},
url = {http://dx.doi.org/10.1215/00127094-2020-0064},
Abstract = {We construct examples of solutions to the conservative
surface quasigeostrophic (SQG) equation that must either
exhibit infinite-in-Time growth of derivatives or blow up in
finite time.},
Doi = {10.1215/00127094-2020-0064},
Key = {fds356022}
}
@article{fds361594,
Author = {Kiselev, A and Tan, C},
Title = {The Flow of Polynomial Roots Under Differentiation},
Year = {2020},
Month = {December},
Abstract = {The question about the behavior of gaps between zeros of
polynomials under differentiation is classical and goes back
to Marcel Riesz. In this paper, we analyze a nonlocal
nonlinear partial differential equation formally derived by
Stefan Steinerberger to model dynamics of roots of
polynomials under differentiation. Interestingly, the same
equation has also been recently obtained formally by Dimitri
Shlyakhtenko and Terence Tao as the evolution equation for
free fractional convolution of a measure - an object in free
probability that is also related to minor processes for
random matrices. The partial differential equation bears
striking resemblance to hydrodynamic models used to describe
the collective behavior of agents (such as birds, fish or
robots) in mathematical biology. We consider periodic
setting and show global regularity and exponential in time
convergence to uniform density for solutions corresponding
to strictly positive smooth initial data. In the second part
of the paper we connect rigorously solutions of the
Steinerberger's PDE and evolution of roots under
differentiation for a class of trigonometric polynomials.
Namely, we prove that the distribution of the zeros of the
derivatives of a polynomial and the corresponding solutions
of the PDE remain close for all times. The global in time
control follows from the analysis of the propagation of
errors equation, which turns out to be a nonlinear
fractional heat equation with the main term similar to the
modulated discretized fractional Laplacian
$(-\Delta)^{1/2}$.},
Key = {fds361594}
}
@article{fds361463,
Author = {Kiselev, A and Nazarov, F and Ryzhik, L and Yao, Y},
Title = {Chemotaxis and Reactions in Biology},
Year = {2020},
Month = {April},
Abstract = {Chemotaxis plays a crucial role in a variety of processes in
biology and ecology. Quite often it acts to improve
efficiency of biological reactions. One example is the
immune system signalling, where infected tissues release
chemokines attracting monocytes to fight invading bacteria.
Another example is reproduction, where eggs release
pheromones that attract sperm. A macro scale example is
flower scent appealing to pollinators. In this paper we
consider a system of PDE designed to model such processes.
Our interest is to quantify the effect of chemotaxis on
reaction rates compared to pure reaction-diffusion. We limit
consideration to surface chemotaxis, which is well motivated
from the point of view of many applications. Our results
provide the first insight into situations where chemotaxis
can be crucial for reaction success, and where its effect is
likely to be limited. The proofs are based on new analytical
tools; a significant part of the paper is dedicated to
building up the linear machinery that can be useful in more
general settings. In particular we establish precise
estimates on the rates of convergence to ground state for a
class of Fokker-Planck operators with potentials that grow
at a logarithmic rate at infinity. These estimates are made
possible by a new sharp weak weighted Poincar\'e inequality
improving in particular a result of Bobkov and
Ledoux.},
Key = {fds361463}
}
@article{fds376401,
Author = {Kiselev, AA},
Title = {Small Scale Creation in Active Scalars},
Volume = {2272},
Pages = {125-161},
Booktitle = {Lecture Notes in Mathematics},
Year = {2020},
Month = {January},
url = {http://dx.doi.org/10.1007/978-3-030-54899-5_4},
Abstract = {The focus of the course is on small scale formation in
solutions of the incompressible Euler equation of fluid
dynamics and associated models. We first review the
regularity results and examples of small scale growth in two
dimensions. Then we discuss a specific singular scenario for
the three-dimensional Euler equation discovered by Hou and
Luo, and analyze some associated models. Finally, we will
also talk about the surface quasi-geostrophic (SQG)
equation, and construct an example of singularity formation
in the modified SQG patch solutions as well as examples of
unbounded growth of derivatives for the smooth
solutions.},
Doi = {10.1007/978-3-030-54899-5_4},
Key = {fds376401}
}
@article{fds357911,
Author = {Kiselev, AA},
Title = {Small Scale Creation in Active Scalars},
Journal = {PROGRESS IN MATHEMATICAL FLUID DYNAMICS},
Volume = {2272},
Pages = {123-159},
Booktitle = {Lecture Notes in Mathematics},
Year = {2020},
ISBN = {978-3-030-54898-8},
url = {http://dx.doi.org/10.1007/978-3-030-54899-5_4},
Abstract = {The focus of the course is on small scale formation in
solutions of the incompressible Euler equation of fluid
dynamics and associated models. We first review the
regularity results and examples of small scale growth in two
dimensions. Then we discuss a specific singular scenario for
the three-dimensional Euler equation discovered by Hou and
Luo, and analyze some associated models. Finally, we will
also talk about the surface quasi-geostrophic (SQG)
equation, and construct an example of singularity formation
in the modified SQG patch solutions as well as examples of
unbounded growth of derivatives for the smooth
solutions.},
Doi = {10.1007/978-3-030-54899-5_4},
Key = {fds357911}
}
@article{fds341002,
Author = {Kiselev, A and Li, C},
Title = {Global regularity and fast small-scale formation for Euler
patch equation in a smooth domain},
Journal = {Communications in Partial Differential Equations},
Volume = {44},
Number = {4},
Pages = {279-308},
Year = {2019},
Month = {April},
url = {http://dx.doi.org/10.1080/03605302.2018.1546318},
Abstract = {It is well known that the Euler vortex patch in R 2 will
remain regular if it is regular enough initially. In bounded
domains, the regularity theory for patch solutions is less
complete. In this article, we study Euler vortex patches in
a general smooth bounded domain. We prove global in time
regularity by providing an upper bound on the growth of
curvature of the patch boundary. For a special symmetric
scenario, we construct an example of double exponential
curvature growth, showing that our upper bound is
qualitatively sharp.},
Doi = {10.1080/03605302.2018.1546318},
Key = {fds341002}
}
@article{fds340353,
Author = {Do, T and Kiselev, A and Xu, X},
Title = {Stability of Blowup for a 1D Model of Axisymmetric 3D Euler
Equation},
Journal = {Journal of Nonlinear Science},
Volume = {28},
Number = {6},
Pages = {2127-2152},
Publisher = {Springer Nature America, Inc},
Year = {2018},
Month = {December},
url = {http://dx.doi.org/10.1007/s00332-016-9340-7},
Abstract = {The question of the global regularity versus finite- time
blowup in solutions of the 3D incompressible Euler equation
is a major open problem of modern applied analysis. In this
paper, we study a class of one-dimensional models of the
axisymmetric hyperbolic boundary blow-up scenario for the 3D
Euler equation proposed by Hou and Luo (Multiscale Model
Simul 12:1722–1776, 2014) based on extensive numerical
simulations. These models generalize the 1D Hou–Luo model
suggested in Hou and Luo Luo and Hou (2014), for which
finite-time blowup has been established in Choi et al.
(arXiv preprint. arXiv:1407.4776, 2014). The main new
aspects of this work are twofold. First, we establish
finite-time blowup for a model that is a closer
approximation of the three-dimensional case than the
original Hou–Luo model, in the sense that it contains
relevant lower-order terms in the Biot–Savart law that
have been discarded in Hou and Luo Choi et al. (2014).
Secondly, we show that the blow-up mechanism is quite
robust, by considering a broader family of models with the
same main term as in the Hou–Luo model. Such blow-up
stability result may be useful in further work on
understanding the 3D hyperbolic blow-up scenario.},
Doi = {10.1007/s00332-016-9340-7},
Key = {fds340353}
}
@article{fds340371,
Author = {Kiselev, A},
Title = {Special Issue Editorial: Small Scales and Singularity
Formation in Fluid Dynamics},
Journal = {Journal of Nonlinear Science},
Volume = {28},
Number = {6},
Pages = {2047-2050},
Publisher = {Springer Nature America, Inc},
Year = {2018},
Month = {December},
url = {http://dx.doi.org/10.1007/s00332-018-9452-3},
Doi = {10.1007/s00332-018-9452-3},
Key = {fds340371}
}
@article{fds335539,
Author = {Do, T and Kiselev, A and Ryzhik, L and Tan, C},
Title = {Global Regularity for the Fractional Euler Alignment
System},
Journal = {Archive for Rational Mechanics and Analysis},
Volume = {228},
Number = {1},
Pages = {1-37},
Publisher = {Springer Nature},
Year = {2018},
Month = {April},
url = {http://dx.doi.org/10.1007/s00205-017-1184-2},
Abstract = {We study a pressureless Euler system with a non-linear
density-dependent alignment term, originating in the
Cucker–Smale swarming models. The alignment term is
dissipative in the sense that it tends to equilibrate the
velocities. Its density dependence is natural: the alignment
rate increases in the areas of high density due to species
discomfort. The diffusive term has the order of a fractional
Laplacian (-∂xx)α/2,α∈(0,1). The corresponding Burgers
equation with a linear dissipation of this type develops
shocks in a finite time. We show that the alignment
nonlinearity enhances the dissipation, and the solutions are
globally regular for all α∈ (0 , 1). To the best of our
knowledge, this is the first example of such regularization
due to the non-local nonlinear modulation of
dissipation.},
Doi = {10.1007/s00205-017-1184-2},
Key = {fds335539}
}
@article{fds330278,
Author = {Kiselev, A and Tan, C},
Title = {Finite time blow up in the hyperbolic Boussinesq
system},
Journal = {Advances in Mathematics},
Volume = {325},
Pages = {34-55},
Publisher = {Elsevier BV},
Year = {2018},
Month = {February},
url = {http://dx.doi.org/10.1016/j.aim.2017.11.019},
Abstract = {In recent work of Luo and Hou [10], a new scenario for
finite time blow up in solutions of 3D Euler equation has
been proposed. The scenario involves a ring of hyperbolic
points of the flow located at the boundary of a cylinder. In
this paper, we propose a two dimensional model that we call
“hyperbolic Boussinesq system”. This model is designed
to provide insight into the hyperbolic point blow up
scenario. The model features an incompressible velocity
vector field, a simplified Biot–Savart law, and a
simplified term modeling buoyancy. We prove that finite time
blow up happens for a natural class of initial
data.},
Doi = {10.1016/j.aim.2017.11.019},
Key = {fds330278}
}
@article{fds340825,
Author = {Kiselev, A and Tan, C},
Title = {Global regularity for 1D eulerian dynamics with singular
interaction forces},
Journal = {SIAM Journal on Mathematical Analysis},
Volume = {50},
Number = {6},
Pages = {6208-6229},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.1137/17M1141515},
Abstract = {The Euler-Poisson-alignment (EPA) system appears in
mathematical biology and is used to model, in a hydrodynamic
limit, a set of agents interacting through mutual
attraction/repulsion as well as alignment forces. We
consider one-dimensional EPA system with a class of singular
alignment terms as well as natural attraction/repulsion
terms. The singularity of the alignment kernel produces an
interesting effect regularizing the solutions of the
equation and leading to global regularity for wide range of
initial data. This was recently observed in [Do et al.,
Arch. Ration. Mech. Anal., 228(2018), pp. 1-37]. Our goal in
this paper is to generalize the result and to incorporate
the attractive/repulsive potential. We prove that global
regularity persists for these more general
models.},
Doi = {10.1137/17M1141515},
Key = {fds340825}
}
@article{fds330279,
Author = {Choi, K and Hou, TY and Kiselev, A and Luo, G and Sverak, V and Yao,
Y},
Title = {On the Finite-Time Blowup of a One-Dimensional Model for the
Three-Dimensional Axisymmetric Euler Equations},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {70},
Number = {11},
Pages = {2218-2243},
Publisher = {WILEY},
Year = {2017},
Month = {November},
url = {http://dx.doi.org/10.1002/cpa.21697},
Abstract = {In connection with the recent proposal for possible
singularity formation at the boundary for solutions of
three-dimensional axisymmetric incompressible Euler's
equations (Luo and Hou, Proc. Natl. Acad. Sci. USA (2014)),
we study models for the dynamics at the boundary and show
that they exhibit a finite-time blowup from smooth data. ©
2017 Wiley Periodicals, Inc.},
Doi = {10.1002/cpa.21697},
Key = {fds330279}
}
@article{fds330280,
Author = {Kiselev, A and Yao, Y and Zlatoš, A},
Title = {Local Regularity for the Modified SQG Patch
Equation},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {70},
Number = {7},
Pages = {1253-1315},
Publisher = {WILEY},
Year = {2017},
Month = {July},
url = {http://dx.doi.org/10.1002/cpa.21677},
Abstract = {We study the patch dynamics on the whole plane and on the
half-plane for a family of active scalars called modified
surface quasi-geostrophic (SQG) equations. These involve a
parameter α that appears in the power of the kernel in
their Biot-Savart laws and describes the degree of
regularity of the equation. The values α=0 and α=½
correspond to the two-dimensional Euler and SQG equations,
respectively. We establish here local-in-time regularity for
these models, for all α ∊ (0,½) on the whole plane and
for all small α > 0 on the half-plane. We use the latter
result in [16], where we show existence of regular initial
data on the half-plane that lead to a finite-time
singularity.© 2016 Wiley Periodicals, Inc.},
Doi = {10.1002/cpa.21677},
Key = {fds330280}
}
@article{fds330282,
Author = {Kiselev, A and Xu, X},
Title = {Suppression of Chemotactic Explosion by Mixing},
Journal = {Archive for Rational Mechanics and Analysis},
Volume = {222},
Number = {2},
Pages = {1077-1112},
Publisher = {Springer Nature},
Year = {2016},
Month = {November},
url = {http://dx.doi.org/10.1007/s00205-016-1017-8},
Abstract = {Chemotaxis plays a crucial role in a variety of processes in
biology and ecology. In many instances, processes involving
chemical attraction take place in fluids. One of the most
studied PDE models of chemotaxis is given by the
Keller–Segel equation, which describes a population
density of bacteria or mold which is attracted chemically to
substance they secrete. Solutions of the Keller–Segel
equation can exhibit dramatic collapsing behavior, where
density concentrates positive mass in a measure zero region.
A natural question is whether the presence of fluid flow can
affect singularity formation by mixing the bacteria thus
making concentration harder to achieve. In this paper, we
consider the parabolic-elliptic Keller–Segel equation in
two and three dimensions with an additional advection term
modeling ambient fluid flow. We prove that for any initial
data, there exist incompressible fluid flows such that the
solution to the equation stays globally regular. On the
other hand, it is well known that when the fluid flow is
absent, there exists initial data leading to finite time
blow up. Thus the presence of fluid flow can prevent the
singularity formation. We discuss two classes of flows that
have the explosion arresting property. Both classes are
known as very efficient mixers. The first class are the
relaxation enhancing (RE) flows of (Ann Math:643–674,
2008). These flows are stationary. The second class of flows
are the Yao–Zlatos near-optimal mixing flows (Mixing and
un-mixing by incompressible flows. arXiv:1407.4163, 2014),
which are time dependent. The proof is based on the
nonlinear version of the relaxation enhancement construction
of (Ann Math:643–674, 2008), and on some variations of the
global regularity estimate for the Keller–Segel
model.},
Doi = {10.1007/s00205-016-1017-8},
Key = {fds330282}
}
@article{fds330283,
Author = {Popov, IY and Kurasov, PA and Naboko, SN and Kiselev, AA and Ryzhkov,
AE and Yafyasov, AM and Miroshnichenko, GP and Karpeshina, YE and Kruglov, VI and Pankratova, TF and Popov, AI},
Title = {A distinguished mathematical physicist Boris S.
Pavlov},
Journal = {Nanosystems: Physics, Chemistry, Mathematics},
Pages = {782-788},
Publisher = {ITMO University},
Year = {2016},
Month = {October},
url = {http://dx.doi.org/10.17586/2220-8054-2016-7-5-782-788},
Doi = {10.17586/2220-8054-2016-7-5-782-788},
Key = {fds330283}
}
@article{fds330281,
Author = {Kiselev, A and Ryzhik, L and Yao, Y and Zlatoš, A},
Title = {Finite time singularity for the modified SQG patch
equation},
Journal = {Annals of Mathematics},
Volume = {184},
Number = {3},
Pages = {909-948},
Publisher = {Annals of Mathematics, Princeton U},
Year = {2016},
Month = {January},
url = {http://dx.doi.org/10.4007/annals.2016.184.3.7},
Abstract = {It is well known that the incompressible Euler equations in
two dimensions have globally regular solutions. The inviscid
surface quasi-geostrophic (SQG) equation has a Biot-Savart
law that is one derivative less regular than in the Euler
case, and the question of global regularity for its
solutions is still open. We study here the patch dynamics in
the half-plane for a family of active scalars that
interpolates between these two equations, via a parameter α
ε [0, 1/2] appearing in the kernels of their Biot-Savart
laws. The values α = 0 and α = 1/2 correspond to the 2D
Euler and SQG cases, respectively. We prove global in time
regularity for the 2D Euler patch model, even if the patches
initially touch the boundary of the half-plane. On the other
hand, for any suffciently small α > 0, we exhibit initial
data that lead to a singularity in finite time. Thus, these
results show a phase transition in the behavior of solutions
to these equations and provide a rigorous foundation for
classifying the 2D Euler equations as critical.},
Doi = {10.4007/annals.2016.184.3.7},
Key = {fds330281}
}
@article{fds330284,
Author = {Kiselev, A and Zlatoš, A},
Title = {Blow up for the 2D Euler equation on some bounded
domains},
Journal = {Journal of Differential Equations},
Volume = {259},
Number = {7},
Pages = {3490-3494},
Publisher = {Elsevier BV},
Year = {2015},
Month = {October},
url = {http://dx.doi.org/10.1016/j.jde.2015.04.027},
Abstract = {We find a smooth solution of the 2D Euler equation on a
bounded domain which exists and is unique in a natural class
locally in time, but blows up in finite time in the sense of
its vorticity losing continuity. The domain's boundary is
smooth except at two points, which are interior
cusps.},
Doi = {10.1016/j.jde.2015.04.027},
Key = {fds330284}
}
@article{fds330285,
Author = {Choi, K and Kiselev, A and Yao, Y},
Title = {Finite Time Blow Up for a 1D Model of 2D Boussinesq
System},
Journal = {Communications in Mathematical Physics},
Volume = {334},
Number = {3},
Pages = {1667-1679},
Publisher = {Springer Nature},
Year = {2015},
Month = {March},
url = {http://dx.doi.org/10.1007/s00220-014-2146-2},
Abstract = {The 2D conservative Boussinesq system describes inviscid,
incompressible, buoyant fluid flow in a gravity field. The
possibility of finite time blow up for solutions of this
system is a classical problem of mathematical hydrodynamics.
We consider a 1D model of the 2D Boussinesq system motivated
by a particular finite time blow up scenario. We prove that
finite time blow up is possible for the solutions to the
model system.},
Doi = {10.1007/s00220-014-2146-2},
Key = {fds330285}
}
@article{fds361422,
Author = {Choi, K and Hou, TY and Kiselev, A and Luo, G and Sverak, V and Yao,
Y},
Title = {On the Finite-Time Blowup of a 1D Model for the 3D
Axisymmetric Euler Equations},
Year = {2014},
Month = {July},
Abstract = {In connection with the recent proposal for possible
singularity formation at the boundary for solutions of 3d
axi-symmetric incompressible Euler's equations (Luo and Hou,
2013), we study models for the dynamics at the boundary and
show that they exhibit a finite-time blow-up from smooth
data.},
Key = {fds361422}
}
@article{fds330286,
Author = {Kiselev, A and Šverák, V},
Title = {Small scale creation for solutions of the incompressible
two-dimensional Euler equation},
Journal = {Annals of Mathematics},
Volume = {180},
Number = {3},
Pages = {1205-1220},
Publisher = {Annals of Mathematics, Princeton U},
Year = {2014},
Month = {January},
url = {http://dx.doi.org/10.4007/annals.2014.180.3.9},
Abstract = {We construct an initial data for the two-dimensional Euler
equation in a disk for which the gradient of vorticity
exhibits double exponential growth in time for all times.
This estimate is known to be sharp - the double exponential
growth is the fastest possible growth rate. © 2014
Department of Mathematics, Princeton University.},
Doi = {10.4007/annals.2014.180.3.9},
Key = {fds330286}
}
@article{fds330287,
Author = {Iyer, G and Kiselev, A and Xu, X},
Title = {Lower bounds on the mix norm of passive scalars advected by
incompressible enstrophy-constrained flows},
Journal = {Nonlinearity},
Volume = {27},
Number = {5},
Pages = {973-985},
Publisher = {IOP Publishing},
Year = {2014},
Month = {January},
url = {http://dx.doi.org/10.1088/0951-7715/27/5/973},
Abstract = {Consider a diffusion-free passive scalar θ being mixed by
an incompressible flow u on the torus d. Our aim is to study
how well this scalar can be mixed under an enstrophy
constraint on the advecting velocity field. Our main result
shows that the mix-norm (∥ θ (t)∥H-1) is bounded below
by an exponential function of time. The exponential decay
rate we obtain is not universal and depends on the size of
the support of the initial data. We also perform numerical
simulations and confirm that the numerically observed decay
rate scales similarly to the rigorous lower bound, at least
for a significant initial period of time. The main idea
behind our proof is to use the recent work of Crippa and De
Lellis (2008 J. Reine Angew. Math. 616 15-46) making
progress towards the resolution of Bressan's rearrangement
cost conjecture. © 2014 IOP Publishing Ltd & London
Mathematical Society.},
Doi = {10.1088/0951-7715/27/5/973},
Key = {fds330287}
}
@article{fds330288,
Author = {Dabkowski, M and Kiselev, A and Silvestre, L and Vicol,
V},
Title = {Global well-posedness of slightly supercritical active
scalar equations},
Journal = {Analysis and PDE},
Volume = {7},
Number = {1},
Pages = {43-72},
Publisher = {Mathematical Sciences Publishers},
Year = {2014},
Month = {January},
url = {http://dx.doi.org/10.2140/apde.2014.7.43},
Abstract = {The paper is devoted to the study of slightly supercritical
active scalars with nonlocal diffusion. We prove global
regularity for the surface quasigeostrophic (SQG) and
Burgers equations, when the diffusion term is supercritical
by a symbol with roughly logarithmic behavior at infinity.
We show that the result is sharp for the Burgers equation.
We also prove global regularity for a slightly supercritical
two-dimensional Euler equation. Our main tool is a nonlocal
maximum principle which controls a certain modulus of
continuity of the solutions. ©2014 Mathematical Sciences
Publishers.},
Doi = {10.2140/apde.2014.7.43},
Key = {fds330288}
}
@article{fds330289,
Author = {Kiselev, A and Nazarov, F},
Title = {A simple energy pump for the surface quasi-geostrophic
equation},
Journal = {Nonlinear Partial Differential Equations: The Abel Symposium
2010},
Pages = {175-179},
Publisher = {Springer Berlin Heidelberg},
Year = {2012},
Month = {December},
url = {http://dx.doi.org/10.1007/978-3-642-25361-4_9},
Abstract = {We consider the question of growth of high order Sobolev
norms of solutions of the conservative surface
quasi-geostrophic equation. We show that if s > 0 is large
then for every given A there exists initial data with a norm
that is small in Hs such that the Hs norm of corresponding
solution at some time exceeds A. The idea of the
construction is quasilinear. We use a small perturbation of
a stable shear flow. The shear flow can be shown to create
small scales in the perturbation part of the flow. The
control is lost once the nonlinear effects become too large.
© Springer-Verlag Berlin Heidelberg 2012.},
Doi = {10.1007/978-3-642-25361-4_9},
Key = {fds330289}
}
@article{fds330290,
Author = {Kiselev, A and Ryzhik, L},
Title = {Biomixing by chemotaxis and efficiency of biological
reactions: The critical reaction case},
Journal = {Journal of Mathematical Physics},
Volume = {53},
Number = {11},
Pages = {115609-115609},
Publisher = {AIP Publishing},
Year = {2012},
Month = {November},
url = {http://dx.doi.org/10.1063/1.4742858},
Abstract = {Many phenomena in biology involve both reactions and
chemotaxis. These processes can clearly influence each
other, and chemotaxis can play an important role in
sustaining and speeding up the reaction. In continuation of
our work [A. Kiselev and L. Ryzhik, "Biomixing by chemotaxis
and enhancement of biological reactions," Comm. Partial
Differential Equations37, 298-318 (2012)]10.1080/03605302.2011.589879,
we consider a model with a single density function involving
diffusion, advection, chemotaxis, and absorbing reaction.
The model is motivated, in particular, by the studies of
coral broadcast spawning, where experimental observations of
the efficiency of fertilization rates significantly exceed
the data obtained from numerical models that do not take
chemotaxis (attraction of sperm gametes by a chemical
secreted by egg gametes) into account. We consider the case
of the weakly coupled quadratic reaction term, which is the
most natural from the biological point of view and was left
open in Kiselev and Ryzhik ["Biomixing by chemotaxis and
enhancement of biological reactions," Comm. Partial
Differential Equations37, 298-318 (2012)]10.1080/03605302.2011.589879.
The result is that similarly to Kiselev and Ryzhik
["Biomixing by chemotaxis and enhancement of biological
reactions," Comm. Partial Differential Equations37, 298-318
(2012)]10.1080/03605302.2011.589879, the chemotaxis plays a
crucial role in ensuring efficiency of reaction. However,
mathematically, the picture is quite different in the
quadratic reaction case and is more subtle. The reaction is
now complete even in the absence of chemotaxis, but the
timescales are very different. Without chemotaxis, the
reaction is very slow, especially for the weak reaction
coupling. With chemotaxis, the timescale and efficiency of
reaction are independent of the coupling parameter. © 2012
American Institute of Physics.},
Doi = {10.1063/1.4742858},
Key = {fds330290}
}
@article{fds330291,
Author = {Dabkowski, M and Kiselev, A and Vicol, V},
Title = {Global well-posedness for a slightly supercritical surface
quasi-geostrophic equation},
Journal = {Nonlinearity},
Volume = {25},
Number = {5},
Pages = {1525-1535},
Publisher = {IOP Publishing},
Year = {2012},
Month = {May},
url = {http://dx.doi.org/10.1088/0951-7715/25/5/1525},
Abstract = {We use a non-local maximum principle to prove the global
existence of smooth solutions for a slightly supercritical
surface quasi-geostrophic equation. By this we mean that the
velocity field u is obtained from the active scalar by a
Fourier multiplier with symbol iζ ⊥|ζ| -1m(|ζ|), where
m is a smooth increasing function that grows slower than log
log|ζ| as |ζ| → ∞. © 2012 IOP Publishing Ltd & London
Mathematical Society.},
Doi = {10.1088/0951-7715/25/5/1525},
Key = {fds330291}
}
@article{fds330292,
Author = {Kiselev, A and Ryzhik, L},
Title = {Biomixing by Chemotaxis and Enhancement of Biological
Reactions},
Journal = {Communications in Partial Differential Equations},
Volume = {37},
Number = {2},
Pages = {298-318},
Publisher = {Informa UK Limited},
Year = {2012},
Month = {February},
url = {http://dx.doi.org/10.1080/03605302.2011.589879},
Abstract = {Many phenomena in biology involve both reactions and
chemotaxis. These processes can clearly influence each
other, and chemotaxis can play an important role in
sustaining and speeding up the reaction. However, to the
best of our knowledge, the question of reaction enhancement
by chemotaxis has not yet received extensive treatment
either analytically or numerically. We consider a model with
a single density function involving diffusion, advection,
chemotaxis, and absorbing reaction. The model is motivated,
in particular, by studies of coral broadcast spawning, where
experimental observations of the efficiency of fertilization
rates significantly exceed the data obtained from numerical
models that do not take chemotaxis (attraction of sperm
gametes by a chemical secreted by egg gametes) into account.
We prove that in the framework of our model, chemotaxis
plays a crucial role. There is a rigid limit to how much the
fertilization efficiency can be enhanced if there is no
chemotaxis but only advection and diffusion. On the other
hand, when chemotaxis is present, the fertilization rate can
be arbitrarily close to being complete provided that the
chemotactic attraction is sufficiently strong. Moreover, an
interesting feature of the estimates on fertilization rate
and timescales in the chemotactic case is that they do not
depend on the amplitude of the reaction term. © 2012
Copyright Taylor and Francis Group, LLC.},
Doi = {10.1080/03605302.2011.589879},
Key = {fds330292}
}
@article{fds330293,
Author = {Kiselev, A},
Title = {Nonlocal maximum principles for active scalars},
Journal = {Advances in Mathematics},
Volume = {227},
Number = {5},
Pages = {1806-1826},
Publisher = {Elsevier BV},
Year = {2011},
Month = {August},
url = {http://dx.doi.org/10.1016/j.aim.2011.03.019},
Abstract = {Active scalars appear in many problems of fluid dynamics.
The most common examples of active scalar equations are 2D
Euler, Burgers, and 2D surface quasi-geostrophic equations.
Many questions about regularity and properties of solutions
of these equations remain open. We develop the idea of
nonlocal maximum principle introduced in Kiselev, Nazarov
and Volberg (2007) [19], formulating a more general
criterion and providing new applications. The most
interesting application is finite time regularization of
weak solutions in the supercritical regime. © 2011 Elsevier
Inc.},
Doi = {10.1016/j.aim.2011.03.019},
Key = {fds330293}
}
@article{fds330294,
Author = {Kiselev, A and Nazarov, F},
Title = {Variation on a theme of caffarelli and vasseur},
Journal = {Journal of Mathematical Sciences},
Volume = {166},
Number = {1},
Pages = {31-39},
Publisher = {Springer Nature},
Year = {2010},
Month = {March},
url = {http://dx.doi.org/10.1007/s10958-010-9842-z},
Abstract = {Recently, using DiGiorgi-type techniques, Caffarelli and
Vasseur have shown that a certain class of weak solutions to
the drift diffusion equation with initial data in L2 gain
Ḧolder continuity, provided that the BMO norm of the drift
velocity is bounded uniformly in time. We show a related
result: a uniform bound on the BMO norm of a smooth velocity
implies a uniform bound on the Cβ norm of the solution for
some β > 0. We apply elementary tools involving the control
of Ḧolder norms by using test functions. In particular,
our approach offers a third proof of the global regularity
for the critical surface quasigeostrophic (SQG) equation in
addition to the two proofs obtained earlier. Bibliography: 6
titles. © 2010 Springer Science+Business Media,
Inc.},
Doi = {10.1007/s10958-010-9842-z},
Key = {fds330294}
}
@article{fds330295,
Author = {Kiselev, A and Nazarov, F},
Title = {Global regularity for the critical dispersive dissipative
surface quasi-geostrophic equation},
Journal = {Nonlinearity},
Volume = {23},
Number = {3},
Pages = {549-554},
Publisher = {IOP Publishing},
Year = {2010},
Month = {February},
url = {http://dx.doi.org/10.1088/0951-7715/23/3/006},
Abstract = {We consider the surface quasi-geostrophic equation with
dispersive forcing and critical dissipation. We prove the
global existence of smooth solutions given sufficiently
smooth initial data. This is done using a maximum principle
for the solutions involving conservation of a certain family
of moduli of continuity. © 2010 IOP Publishing Ltd and
London Mathematical Society.},
Doi = {10.1088/0951-7715/23/3/006},
Key = {fds330295}
}
@article{fds330296,
Author = {Berestycki, H and Kiselev, A and Novikov, A and Ryzhik,
L},
Title = {The explosion problem in a flow},
Journal = {Journal d'Analyse Mathematique},
Volume = {110},
Number = {1},
Pages = {31-65},
Publisher = {Springer Nature},
Year = {2010},
Month = {January},
url = {http://dx.doi.org/10.1007/s11854-010-0002-7},
Abstract = {We consider the explosion problem in an incompressible flow
introduced in [5]. We use a novel Lp - L∞ estimate for
elliptic advection-diffusion problems to show that the
explosion threshold obeys a positive lower bound which is
uniform in the advecting flow. We also identify the flows
for which the explosion threshold tends to infinity as their
amplitude grows and obtain an effective description of the
explosion threshold in the strong flow asymptotics in
two-dimensional cellular flows. © 2010 Hebrew University
Magnes Press.},
Doi = {10.1007/s11854-010-0002-7},
Key = {fds330296}
}
@article{fds330297,
Author = {Kiselev, A},
Title = {Regularity and blow up for active scalars},
Journal = {Mathematical Modelling of Natural Phenomena},
Volume = {5},
Number = {4},
Pages = {225-255},
Publisher = {E D P SCIENCES},
Year = {2010},
Month = {January},
url = {http://dx.doi.org/10.1051/mmnp/20105410},
Abstract = {We review some recent results for a class of fluid mechanics
equations called active scalars, with fractional
dissipation. Our main examples are the surface
quasi-geostrophic equation, the Burgers equation, and the
Cordoba-Cordoba-Fontelos model. We discuss nonlocal maximum
principle methods which allow to prove existence of global
regular solutions for the critical dissipation. We also
recall what is known about the possibility of finite time
blow up in the supercritical regime. © EDP Sciences,
2010.},
Doi = {10.1051/mmnp/20105410},
Key = {fds330297}
}
@article{fds330298,
Author = {Kim, A and Kiselev, A},
Title = {Absolutely continuous spectrum of discrete Schrödinger
operators with slowly oscillating potentials},
Journal = {Mathematische Nachrichten},
Volume = {282},
Number = {4},
Pages = {552-568},
Publisher = {WILEY},
Year = {2009},
Month = {April},
url = {http://dx.doi.org/10.1002/mana.200810754},
Abstract = {We show that when a potential bn of a discrete Schrödinger
operator, defined on l2(Z{double-struck}+), slowly
oscillates satisfying the conditions bn ∈ l∞ and ∂bn =
bn+1 - bn ∈ lp, p < 2, then all solutions of the equation
Ju = Eu are bounded near infinity at almost every E ∈ [-2
+ lim supn→∞ bn, 2 + lim supn→∞ bn] ∩ [-2 + lim
infn→∞bn, 2 + lim infn→∞bn]. We derive an asymptotic
formula for generalized eigenfunctions in this case. As a
corollary, the absolutely continuous spectrum of the
corresponding Jacobi operator is essentially supported on
the same interval of E. © 2009 WILEY-VCH Verlag GmbH & Co.
KGaA, Weinheim.},
Doi = {10.1002/mana.200810754},
Key = {fds330298}
}
@article{fds331091,
Author = {Kiselev, A},
Title = {Some recent results on the critical surface
quasi-geostrophic equation: A review},
Journal = {HYPERBOLIC PROBLEMS: THEORY, NUMERICS AND APPLICATIONS, PART
1},
Volume = {67},
Pages = {105-122},
Publisher = {AMER MATHEMATICAL SOC},
Editor = {Tadmor, E and Liu, J and Tzavaras, A},
Year = {2009},
Month = {January},
ISBN = {978-0-8218-4729-9},
Key = {fds331091}
}
@article{fds331092,
Author = {Kiselev, A},
Title = {Diffusion and Mixing in Fluid Flow: A Review},
Pages = {357-369},
Publisher = {Springer Netherlands},
Year = {2009},
ISBN = {9789048128099},
url = {http://dx.doi.org/10.1007/978-90-481-2810-5_24},
Doi = {10.1007/978-90-481-2810-5_24},
Key = {fds331092}
}
@article{fds330300,
Author = {Kiselev, A and Shterenberg, R and Zlatoš, A},
Title = {Relaxation enhancement by time-periodic flows},
Journal = {Indiana University Mathematics Journal},
Volume = {57},
Number = {5},
Pages = {2137-2152},
Publisher = {Indiana University Mathematics Journal},
Year = {2008},
Month = {December},
url = {http://dx.doi.org/10.1512/iumj.2008.57.3349},
Abstract = {We study enhancement of diffusive mixing by fast
incompressible time-periodic flows. The class of
relaxation-enhancing flows that are especially efficient in
speeding up mixing has been introduced in [2]. The
relaxation-enhancing property of a flow has been shown to be
intimately related to the properties of the dynamical system
it generates. In particular, time-independent flows u such
that the operator u · ▽ has sufficiently smooth
eigenfunctions are not relaxation-enhancing. Here we extend
results of [2] to time-periodic flows u(x, t) and, in
particular, show that there exist flows such that for each
fixed time the flow is Hamiltonian, but the resulting
time-dependent flow is relaxation-enhancing. Thus we confirm
the physical intuition that time dependence of a flow may
aid mixing. We also provide an extension of our results to
the case of a nonlinear diffusion model. The proofs are
based on a general criterion for the decay of a semigroup
generated by an operator of the form Γ + iAL(t) with a
negative unbounded self-adjoint operator Γ, a time-periodic
self-adjoint operator-valued function L(t), and a parameter
A ≫ 1.},
Doi = {10.1512/iumj.2008.57.3349},
Key = {fds330300}
}
@article{fds330299,
Author = {Kiselev, A and Nazarov, F and Shterenberg, R},
Title = {Blow up and regularity for fractal burgers
equation},
Journal = {Dynamics of Partial Differential Equations},
Volume = {5},
Number = {3},
Pages = {211-240},
Publisher = {International Press of Boston},
Year = {2008},
Month = {January},
url = {http://dx.doi.org/10.4310/DPDE.2008.v5.n3.a2},
Abstract = {The paper is a comprehensive study of the existence,
uniqueness, blow up and regularity properties of solutions
of the Burgers equation with fractional dissipation. We
prove existence of the finite time blow up for the power of
Laplacian α < 1/2, and global existence as well as
analyticity of solution for α ≥ 1/2. We also prove the
existence of solutions with very rough initial data uo ∈
Lp, 1 < p < ∞. Many of the results can be extended to a
more general class of equations, including the surface
quasi-geostrophic equation. ©2008 International
Press.},
Doi = {10.4310/DPDE.2008.v5.n3.a2},
Key = {fds330299}
}
@article{fds331093,
Author = {Constantin, P and Kiselev, A and Ryzhik, L and Zlatoš,
A},
Title = {Diusion and mixing in fluid flow},
Journal = {Annals of Mathematics},
Volume = {168},
Number = {2},
Pages = {643-674},
Publisher = {Annals of Mathematics, Princeton U},
Year = {2008},
Month = {January},
url = {http://dx.doi.org/10.4007/annals.2008.168.643},
Abstract = {We study enhancement of diffusive mixing on a compact
Riemannian manifold by a fast incompressible flow. Our main
result is a sharp description of the class of flows that
make the deviation of the solution from its average
arbitrarily small in an arbitrarily short time, provided
that the flow amplitude is large enough. The necessary and
suffcient condition on such flows is expressed naturally in
terms of the spectral properties of the dynamical system
associated with the flow. In particular, we find that weakly
mixing flows always enhance dissipation in this sense. The
proofs are based on a general criterion for the decay of the
semigroup generated by an operator of the form Γ + iAL with
a negative unbounded self-adjoint operator Γ, a
self-adjoint operator L, and parameter A » 1. In
particular, they employ the RAGE theorem describing
evolution of a quantum state belonging to the continuous
spectral subspace of the hamiltonian (related to a classical
theorem of Wiener on Fourier transforms of measures).
Applications to quenching in reaction-diffusion equations
are also considered.},
Doi = {10.4007/annals.2008.168.643},
Key = {fds331093}
}
@article{fds330301,
Author = {Kiselev, A and Nazarov, F and Volberg, A},
Title = {Global well-posedness for the critical 2D dissipative
quasi-geostrophic equation},
Journal = {Inventiones Mathematicae},
Volume = {167},
Number = {3},
Pages = {445-453},
Publisher = {Springer Nature},
Year = {2007},
Month = {March},
url = {http://dx.doi.org/10.1007/s00222-006-0020-3},
Abstract = {We give an elementary proof of the global well-posedness for
the critical 2D dissipative quasi-geostrophic equation. The
argument is based on a non-local maximum principle involving
appropriate moduli of continuity. © 2006
Springer-Verlag.},
Doi = {10.1007/s00222-006-0020-3},
Key = {fds330301}
}
@article{fds331946,
Author = {Denisov, SA and Kiselev, A},
Title = {Spectral properties of schrodinger operators with decaying
potentials},
Journal = {SPECTRAL THEORY AND MATHEMATICAL PHYSICS: A FESTSCHRIFT IN
HONOR OF BARRY SIMON'S 60TH BIRTHDAY},
Volume = {76},
Pages = {565-589},
Publisher = {AMER MATHEMATICAL SOC},
Editor = {Gesztesy, F and Deift, P and Galvez, C and Perry, P and Schlag,
W},
Year = {2007},
Month = {January},
Key = {fds331946}
}
@article{fds330302,
Author = {Kiselev, A and Zlatoš, A},
Title = {Quenching of combustion by shear flows},
Journal = {Duke Mathematical Journal},
Volume = {132},
Number = {1},
Pages = {49-72},
Publisher = {Duke University Press},
Year = {2006},
Month = {March},
url = {http://dx.doi.org/10.1215/S0012-7094-06-13212-X},
Abstract = {We consider a model describing premixed combustion in the
presence of fluid flow: a reaction-diffusion equation with
passive advection and ignition-type nonlinearity. What kinds
of velocity profiles are capable of quenching (suppressing)
any given flame, provided the velocity's amplitude is
adequately large? Even for shear flows, the solution turns
out to be surprisingly subtle. In this article we provide a
sharp characterization of quenching for shear flows; the
flow can quench any initial data if and only if the velocity
profile does not have an interval larger than a certain
critical size where it is identically constant. The
efficiency of quenching depends strongly on the geometry and
scaling of the flow. We discuss the cases of slowly and
quickly varying flows, proving rigorously scaling laws that
have been observed earlier in numerical experiments. The
results require new estimates on the behavior of the
solutions to the advection-enhanced diffusion equation (also
known as passive scalar in physical literature), a classical
model describing a wealth of phenomena in nature. The
technique involves probabilistic and partial-differential-equation
(PDE) estimates, in particular, applications of Malliavin
calculus and the central limit theorem for
martingales.},
Doi = {10.1215/S0012-7094-06-13212-X},
Key = {fds330302}
}
@article{fds330303,
Author = {Fannjiang, A and Kiselev, A and Ryzhik, L},
Title = {Quenching of reaction by cellular flows},
Journal = {Geometric and Functional Analysis},
Volume = {16},
Number = {1},
Pages = {40-69},
Publisher = {Springer Nature},
Year = {2006},
Month = {February},
url = {http://dx.doi.org/10.1007/s00039-006-0554-y},
Abstract = {We consider a reaction-diffusion equation in a cellular
flow. We prove that in the strong flow regime there are two
possible scenarios for the initial data that is compactly
supported and the size of the support is large enough. If
the flow cells are large compared to the reaction length
scale, propagating fronts will always form. For small cell
size, any finitely supported initial data will be quenched
by a sufficiently strong flow. We estimate that the flow
amplitude required to quench the initial data of support L 0
is A > CL 04 ln(L 0). The essence of the problem is the
question about the decay of the L ∞-norm of a solution to
the advection-diffusion equation, and the relation between
this rate of decay and the properties of the Hamiltonian
system generated by the two-dimensional incompressible fluid
flow. © Birkhäuser Verlag.},
Doi = {10.1007/s00039-006-0554-y},
Key = {fds330303}
}
@article{fds330304,
Author = {Andrzejewski, D and Butzlaff, E and Kiselev, A and Markely,
LRA},
Title = {Enhancement of combustion by drift in a coupled
reaction-diffusion model},
Journal = {Communications in Mathematical Sciences},
Volume = {4},
Number = {1},
Pages = {213-225},
Publisher = {International Press of Boston},
Year = {2006},
url = {http://dx.doi.org/10.4310/cms.2006.v4.n1.a8},
Doi = {10.4310/cms.2006.v4.n1.a8},
Key = {fds330304}
}
@article{fds330305,
Author = {Berestycki, H and Hamel, F and Kiselev, A and Ryzhik,
L},
Title = {Quenching and propagation in KPP reaction-diffusion
equations with a heat loss},
Journal = {Archive for Rational Mechanics and Analysis},
Volume = {178},
Number = {1},
Pages = {57-80},
Publisher = {Springer Nature},
Year = {2005},
Month = {October},
url = {http://dx.doi.org/10.1007/s00205-005-0367-4},
Abstract = {We consider a reaction-diffusion system of KPP type in a
shear flow and with a non-zero heat-loss parameter. We
establish criteria for the flame blow-off and propagation,
and identify the propagation speed in terms of the
exponential decay of the initial data. We prove the
existence of travelling fronts for all speeds c>max(0,c*) in
the case Le=1, where c* ∈ ℝ. This seems to be one of the
first non-perturbative results on the existence of fronts
for the thermo-diffusive system in higher dimensions. ©
Springer-Verlag (2005).},
Doi = {10.1007/s00205-005-0367-4},
Key = {fds330305}
}
@article{fds330306,
Author = {Kiselev, A and Zlatoš, A},
Title = {On discrete models of the Euler equation},
Journal = {International Mathematics Research Notices},
Number = {38},
Pages = {2315-2339},
Year = {2005},
Month = {August},
url = {http://dx.doi.org/10.1155/imrn.2005.2315},
Doi = {10.1155/imrn.2005.2315},
Key = {fds330306}
}
@article{fds330307,
Author = {Kiselev, A},
Title = {Imbedded singular continuous spectrum for Schrödinger
operators},
Journal = {Journal of the American Mathematical Society},
Volume = {18},
Number = {3},
Pages = {571-603},
Year = {2005},
Month = {July},
url = {http://dx.doi.org/10.1090/S0894-0347-05-00489-3},
Doi = {10.1090/S0894-0347-05-00489-3},
Key = {fds330307}
}
@article{fds330308,
Author = {Germinet, F and Kiselev, A and Tcheremchantsev,
S},
Title = {Transfer matrices and transport for Schrödinger
operators},
Journal = {Annales de l’institut Fourier},
Volume = {54},
Number = {3},
Pages = {787-830},
Publisher = {Cellule MathDoc/CEDRAM},
Year = {2004},
url = {http://dx.doi.org/10.5802/aif.2034},
Doi = {10.5802/aif.2034},
Key = {fds330308}
}
@article{fds330309,
Author = {Constantin, P and Kiselev, A and Ryzhik, L},
Title = {Fronts in Reactive Convection: Bounds, Stability, and
Instability},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {56},
Number = {12},
Pages = {1781-1803},
Publisher = {WILEY},
Year = {2003},
Month = {December},
url = {http://dx.doi.org/10.1002/cpa.10110},
Abstract = {This paper examines a simplified active combustion model in
which the reaction influences the flow. We consider front
propagation in a reactive Boussinesq system in an infinite
vertical strip. Nonlinear stability of planar fronts is
established for narrow domains when the Rayleigh number is
not too large. Planar fronts are shown to be linearly
unstable with respect to long-wavelength perturbations if
the Rayleigh number is sufficiently large. We also prove
uniform bounds on the bulk burning rate and the Nusselt
number in the KPP reaction case. © 2003 Wiley Periodicals,
Inc.},
Doi = {10.1002/cpa.10110},
Key = {fds330309}
}
@article{fds330310,
Author = {Vladimirova, N and Constantin, P and Kiselev, A and Ruchayskiy, O and Ryzhik, L},
Title = {Flame enhancement and quenching in fluid
flows},
Journal = {Combustion Theory and Modelling},
Volume = {7},
Number = {3},
Pages = {487-508},
Publisher = {Informa UK Limited},
Year = {2003},
Month = {September},
url = {http://dx.doi.org/10.1088/1364-7830/7/3/303},
Abstract = {We perform direct numerical simulations of an advected
scalar field which diffuses and reacts according to a
nonlinear reaction law. The objective is to study how the
bulk burning rate of the reaction is affected by an imposed
flow. In particular, we are interested in comparing the
numerical results with recently predicted analytical upper
and lower bounds. We focus on the reaction enhancement and
quenching phenomena for two classes of imposed model flows
with different geometries: periodic shear flow and cellular
flow. We are primarily interested in the fast advection
regime. We find that the bulk burning rate v in a shear flow
satisfies v ∼ aU + b where U is the typical flow velocity
and a is a constant depending on the relationship between
the oscillation length scale of the flow and laminar front
thickness. For cellular flow, we obtain v ∼ U1/4. We also
study the flame extinction (quenching) for an ignition-type
reaction law and compactly supported initial data for the
scalar field. We find that in a shear flow the flame of size
W can be typically quenched by a flow with amplitude U ∼
αW. The constant α depends on the geometry of the flow and
tends to infinity if the flow profile has a plateau larger
than a critical size. In a cellular flow, we find that the
advection strength required for quenching is U ∼ W4 if the
cell size is smaller than a critical value.},
Doi = {10.1088/1364-7830/7/3/303},
Key = {fds330310}
}
@article{fds330312,
Author = {Kiselev, A and Last, Y and Simon, B},
Title = {Stability of singular spectral types under decaying
pertubations},
Journal = {Journal of Functional Analysis},
Volume = {198},
Number = {1},
Pages = {1-27},
Publisher = {Elsevier BV},
Year = {2003},
Month = {February},
url = {http://dx.doi.org/10.1016/S0022-1236(02)00053-8},
Abstract = {We look at invariance of a.e. boundary condition spectral
behavior under perturbations, W , of half-line, continuum or
discrete Schrödinger operators. We extend the results of
del Rio, Simon, Stolz from compactly supported W's to
suitable short-range W. We also discuss invariance of the
local Hausdroff dimension of spectral measures under such
pertubations. © 2002 Elsevier Science (USA). All rights
reserved.},
Doi = {10.1016/S0022-1236(02)00053-8},
Key = {fds330312}
}
@article{fds330311,
Author = {Christ, M and Kiselev, A},
Title = {Absolutely continuous spectrum of Stark operators},
Journal = {Arkiv for Matematik},
Volume = {41},
Number = {1},
Pages = {1-33},
Publisher = {International Press of Boston},
Year = {2003},
Month = {January},
url = {http://dx.doi.org/10.1007/BF02384565},
Abstract = {We prove several new results on the absolutely continuous
spectra of perturbed one-dimensional Stark operators. First,
we find new classes of perturbations, characterized mainly
by smoothness conditions, which preserve purely absolutely
continuous spectrum. Then we establish stability of the
absolutely continuous spectrum in more general situations,
where imbedded singular spectrum may occur. We present two
kinds of optimal conditions for the stability of absolutely
continuous spectrum: decay and smoothness. In the decay
direction, we show that a sufficient (in the power scale)
condition is |q(x)| ≤ C(1 + |x|)-1/4-ε; in the smoothness
direction, a sufficient condition in Holder classes is q ∈
C1/2+ε(R). On the other hand, we show that there exist
potentials which both satisfy |q(x)| ≤ C(1 + |x|)-1/4 and
belong to C1/2(R) for which the spectrum becomes purely
singular on the whole real axis, so that the above results
are optimal within the scales considered.},
Doi = {10.1007/BF02384565},
Key = {fds330311}
}
@article{fds330313,
Author = {Killip, R and Kiselev, A and Last, Y},
Title = {Dynamical upper bounds on wavepacket spreading},
Journal = {American Journal of Mathematics},
Volume = {125},
Number = {5},
Pages = {1165-1198},
Publisher = {Johns Hopkins University Press},
Year = {2003},
Month = {January},
url = {http://dx.doi.org/10.1353/ajm.2003.0031},
Abstract = {We derive a general upper bound on the spreading rate of
wavepackets in the framework of Sohrödinger time evolution.
Our result consists of showing that a portion of the
wavepacket cannot escape outside a ball whose size grows
dynamically in time, where the rate of this growth is
determined by properties of the spectral measure and by
spatial properties of solutions of an associated
time-independent Schrödinger equation. We also derive a new
lower bound on the spreading rate, which is strongly
connected with our upper bound. We apply these new bounds to
the Fibonacci Hamiltonian - the most studied one-dimensional
model of quasicrystals. As a result, we obtain for this
model upper and lower dynamical bounds establishing
wavepacket spreading rates which are intermediate between
ballistic transport and localization. The bounds have the
same qualitative behavior in the limit of large
coupling.},
Doi = {10.1353/ajm.2003.0031},
Key = {fds330313}
}
@article{fds330315,
Author = {Gesztesy, F and Kiselev, A and Makarov, KA},
Title = {Uniqueness results for matrix-valued Schrödinger, Jacobi,
and Dirac-type operators},
Journal = {Mathematische Nachrichten},
Volume = {239-240},
Number = {1},
Pages = {103-145},
Publisher = {WILEY},
Year = {2002},
Month = {August},
url = {http://dx.doi.org/10.1002/1522-2616(200206)239:1<103::AID-MANA103>3.0.CO;2-F},
Abstract = {Let g(z,x) denote the diagonal Green's matrix of a
self-adjoint m × m matrix-valued Schrödinger operator H =
-d2/dx2Im + Q in L2(ℝ)m, m ∈ ℕ. One of the principal
results proven in this paper states that for a fixed x0 ∈
ℝ and z ∈ ℂ+, g(z,x0) and g′(z,x0) uniquely
determine the matrix-valued m × m potential Q(x) for a.e. x
∈ ℝ. We also prove the following local version of this
result. Let gj(z,x), j = 1, 2 be the diagonal Green's
matrices of the self-adjoint Schrödinger operators Hj =
-d2/dx2Im + Qj in L2(ℝ)m. Suppose that for fixed a > 0 and
x0 ∈ ℝ, ∥g1(z,x0) - g2(z,x0)∥ℂm×m +
∥g′1(z,x0) - g′2(z,x0)∥ℂm×m = |z|→∞
O(e-2Im(z1/2)a) for z inside a cone along the imaginary axis
with vertex zero and opening angle less than π/2, excluding
the real axis. Then Q1(x) = Q2(x) for a.e. x ∈ [x0 - a,x0
+ a]. Analogous results are proved for matrix-valued Jacobi
and Dirac-type operators.},
Doi = {10.1002/1522-2616(200206)239:1<103::AID-MANA103>3.0.CO;2-F},
Key = {fds330315}
}
@article{fds330314,
Author = {Christ, M and Kiselev, A},
Title = {Scattering and wave operators for one-dimensional
Schrödinger operators with slowly decaying nonsmooth
potentials},
Journal = {Geometric and Functional Analysis},
Volume = {12},
Number = {6},
Pages = {1174-1234},
Publisher = {Springer Nature},
Year = {2002},
Month = {January},
url = {http://dx.doi.org/10.1007/s00039-002-1174-9},
Abstract = {We prove existence of modified wave operators for
one-dimensional Schrödinger equations with potential in
LP(ℝ). p < 2. If in addition the potential is
conditionally integrable, then the usual Möller wave
operators exist. We also prove asymptotic completeness of
these wave operators for some classes of random potentials,
and for almost every boundary condition for any given
potential.},
Doi = {10.1007/s00039-002-1174-9},
Key = {fds330314}
}
@article{fds330316,
Author = {Constantin, P and Kiselev, A and Ryzhik, L},
Title = {Quenching of flames by fluid advection},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {54},
Number = {11},
Pages = {1320-1342},
Publisher = {WILEY},
Year = {2001},
Month = {November},
url = {http://dx.doi.org/10.1002/cpa.3000},
Abstract = {We consider a simple scalar reaction-advection-diffusion
equation with ignition-type nonlinearity and discuss the
following question: What kinds of velocity profiles are
capable of quenching any given flame, provided the
velocity's amplitude is adequately large? Even for shear
flows, the answer turns out to be surprisingly subtle. If
the velocity profile changes in space so that it is nowhere
identically constant (or if it is identically constant only
in a region of small measure), then the flow can quench any
initial data. But if the velocity profile is identically
constant in a sizable region, then the ensuing flow is
incapable of quenching large enough flames, no matter how
much larger the amplitude of this velocity is. The constancy
region must be wider across than a couple of laminar
propagating front widths. The proof uses a linear PDE
associated to the nonlinear problem, and quenching follows
when the PDE is hypoelliptic. The techniques used allow the
derivation of new, nearly optimal bounds on the speed of
traveling-wave solutions. © 2001 John Wiley & Sons,
Inc.},
Doi = {10.1002/cpa.3000},
Key = {fds330316}
}
@article{fds330317,
Author = {Kiselev, A and Ryzhik, L},
Title = {An upper bound for the bulk burning rate for
systems},
Journal = {Nonlinearity},
Volume = {14},
Number = {5},
Pages = {1297-1310},
Publisher = {IOP Publishing},
Year = {2001},
Month = {September},
url = {http://dx.doi.org/10.1088/0951-7715/14/5/319},
Abstract = {We consider a system of reaction-diffusion equations with
passive advection term and Lewis number Le not equal to one.
Such systems are used to describe chemical reactions in a
flow in a situation where temperature and material
diffusivities are not equal. It is expected that the fluid
advection will distort the reaction front, increasing the
area of reaction and thus speeding up the reaction process.
While a variety of estimates on the influence of the flow on
reaction are available for a single reaction-diffusion
equation (corresponding to the case of Lewis number equal to
one), the case of the system is largely open. We prove a
general upper bound on the reaction rate in such systems in
terms of the reaction rate for a single reaction-diffusion
equation, showing that the long-time average of reaction
rate with Le ≠ 1 does not exceed the Le = 1 case. Thus the
upper estimates derived for Le = 1 apply to the systems.
Both front-like and compact initial data (hot blob) are
considered.},
Doi = {10.1088/0951-7715/14/5/319},
Key = {fds330317}
}
@article{fds330320,
Author = {Christ, M and Kiselev, A},
Title = {WKB asymptotic behavior of almost all generalized
eigenfunctions for one-dimensional Schrödinger operators
with slowly decaying potentials},
Journal = {Journal of Functional Analysis},
Volume = {179},
Number = {2},
Pages = {426-447},
Publisher = {Elsevier BV},
Year = {2001},
Month = {February},
url = {http://dx.doi.org/10.1006/jfan.2000.3688},
Abstract = {We prove the WKB asymptotic behavior of solutions of the
differential equation -d2u/dx2+V(x)u=Eu for a.e. E>A where
V=V1+V2, V1∈Lp(R), and V2 is bounded from above with
A=limsupx→∞V(x), while V′2(x)∈Lp(R), 1≤p<2. These
results imply that Schrödinger operators with such
potentials have absolutely continuous spectrum on (A, ∞).
We also establish WKB asymptotic behavior of solutions for
some energy-dependent potentials. © 2001 Academic
Press.},
Doi = {10.1006/jfan.2000.3688},
Key = {fds330320}
}
@article{fds330321,
Author = {Christ, M and Kiselev, A},
Title = {Maximal functions associated to filtrations},
Journal = {Journal of Functional Analysis},
Volume = {179},
Number = {2},
Pages = {409-425},
Publisher = {Elsevier BV},
Year = {2001},
Month = {February},
url = {http://dx.doi.org/10.1006/jfan.2000.3687},
Abstract = {Let T be a bounded linear, or sublinear, operator from Lp(Y)
to Lq(X). A maximal operator T*f(x)=supjT(f·χYj)(x) is
associated to any sequence of subsets Yj of Y. Under the
hypotheses that q>p and the sets Yj are nested, we prove
that T* is also bounded. Classical theorems of Menshov and
Zygmund are obtained as corollaries. Multilinear
generalizations of this theorem are also established. These
results are motivated by applications to the spectral
analysis of Schrödinger operators. © 2001 Academic
Press.},
Doi = {10.1006/jfan.2000.3687},
Key = {fds330321}
}
@article{fds330318,
Author = {Kiselev, A and Ryzhik, L},
Title = {Enhancement of the traveling front speeds in
reaction-diffusion equations with advection},
Journal = {Annales de l'Institut Henri Poincare (C) Analyse Non
Lineaire},
Volume = {18},
Number = {3},
Pages = {309-358},
Publisher = {Elsevier BV},
Year = {2001},
Month = {January},
url = {http://dx.doi.org/10.1016/S0294-1449(01)00068-3},
Abstract = {We establish rigorous lower bounds on the speed of traveling
fronts and on the bulk burning rate in reaction-diffusion
equation with passive advection. The non-linearity is
assumed to be of either KPP or ignition type. We consider
two main classes of flows. Percolating flows, which are
characterized by the presence of long tubes of streamlines
mixing hot and cold material, lead to strong speed-up of
burning which is linear in the amplitude of the flow, U. On
the other hand the cellular flows, which have closed
streamlines, are shown to produce weaker increase in
reaction. For such flows we get a lower bound which grows as
U1/5 for a large amplitude of the flow. © 2001 Éditions
scientifiques et médicales Elsevier SAS.},
Doi = {10.1016/S0294-1449(01)00068-3},
Key = {fds330318}
}
@article{fds330319,
Author = {Christ, M and Kiselev, A},
Title = {WKB and spectral analysis of one-dimensional Schrödinger
operators with slowly varying potentials},
Journal = {Communications in Mathematical Physics},
Volume = {218},
Number = {2},
Pages = {245-262},
Publisher = {Springer Nature},
Year = {2001},
Month = {January},
url = {http://dx.doi.org/10.1007/PL00005556},
Abstract = {Consider a Schrödinger operator on L2 of the line, or of a
half line with appropriate boundary conditions. If the
potential tends to zero and is a finite sum of terms, each
of which has a derivative of some order in L1 + Lp for some
exponent p < 2, then an essential support of the the
absolutely continuous spectrum equals ℝ+. Almost every
generalized eigenfunction is bounded, and satisfies certain
WKB-type asymptotics at infinity. If moreover these
derivatives belong to Lp with respect to a weight |x|γ with
γ > 0, then the Hausdorff dimension of the singular
component of the spectral measure is strictly less than
one.},
Doi = {10.1007/PL00005556},
Key = {fds330319}
}
@article{fds330322,
Author = {Kiselev, A},
Title = {Absolutely continuous spectrum of perturbed stark
operators},
Journal = {Transactions of the American Mathematical
Society},
Volume = {352},
Number = {1},
Pages = {243-256},
Year = {2000},
Month = {January},
url = {http://dx.doi.org/10.1090/s0002-9947-99-02450-2},
Abstract = {We prove new results on the stability of the absolutely
continuous spectrum for perturbed Stark operators with
decaying or satisfying certain smoothness assumption
perturbation. We show that the absolutely continuous
spectrum of the Stark operator is stable if the perturbing
potential decays at the rate (1 + x)-1/3-ε or if it is
continuously differentiate with derivative from the Holder
space Ca(R), with any α > 0. © 1999 American Mathematical
Society.},
Doi = {10.1090/s0002-9947-99-02450-2},
Key = {fds330322}
}
@article{fds330323,
Author = {Constantin, P and Kiselev, A and Oberman, A and Ryzhik,
L},
Title = {Bulk burning rate in passive-reactive diffusion},
Journal = {Archive for Rational Mechanics and Analysis},
Volume = {154},
Number = {1},
Pages = {53-91},
Publisher = {Springer Nature},
Year = {2000},
Month = {January},
url = {http://dx.doi.org/10.1007/s002050000090},
Abstract = {We consider a passive scalar that is advected by a
prescribed mean zero divergence-free velocity field,
diffuses, and reacts according to a KPP-type nonlinear
reaction. We introduce a quantity, the bulk burning rate,
that makes both mathematical and physical sense in general
situations and extends the often ill-defined notion of front
speed. We establish rigorous lower bounds for the bulk
burning rate that are linear in the amplitude of the
advecting velocity for a large class of flows. These
"percolating" flows are characterized by the presence of
tubes of streamlines connecting distant regions of burned
and unburned material and generalize shear flows. The bound
contains geometric information on the velocity streamlines
and degenerates when these oscillate on scales that are
finer than the width of the laminar burning region. We give
also examples of very different kind of flows, cellular
flows with closed streamlines, and rigorously prove that
these can produce only sub-linear enhancement of the bulk
burning rate.},
Doi = {10.1007/s002050000090},
Key = {fds330323}
}
@article{fds330324,
Author = {Kiselev, A and Last, Y},
Title = {Solutions, spectrum, and dynamics for schrödinger operators
on infinite domains},
Journal = {Duke Mathematical Journal},
Volume = {102},
Number = {1},
Pages = {125-150},
Publisher = {Duke University Press},
Year = {2000},
Month = {January},
url = {http://dx.doi.org/10.1215/S0012-7094-00-10215-3},
Doi = {10.1215/S0012-7094-00-10215-3},
Key = {fds330324}
}
@article{fds330326,
Author = {Kiselev, A and Remling, C and Simon, B},
Title = {Effective perturbation methods for one-dimensional
Schrödinger operators},
Journal = {Journal of Differential Equations},
Volume = {151},
Number = {2},
Pages = {290-312},
Publisher = {Elsevier BV},
Year = {1999},
Month = {January},
url = {http://dx.doi.org/10.1006/jdeq.1998.3514},
Doi = {10.1006/jdeq.1998.3514},
Key = {fds330326}
}
@article{fds330325,
Author = {Kiselev, A},
Title = {An interpolation theorem related to the A.E. convergence of
integral operators},
Journal = {Proceedings of the American Mathematical
Society},
Volume = {127},
Number = {6},
Pages = {1781-1785},
Year = {1999},
Month = {January},
url = {http://dx.doi.org/10.1090/s0002-9939-99-04681-x},
Abstract = {We show that for integral operators of general form the norm
bounds in Lorentz spaces imply certain norm bounds for the
maximal function. As a consequence, the a.e. convergence for
the integral operators on Lorentz spaces follows from the
appropriate norm estimates. ©1999 American Mathematical
Society.},
Doi = {10.1090/s0002-9939-99-04681-x},
Key = {fds330325}
}
@article{fds330327,
Author = {Christ, M and Kiselev, A},
Title = {Absolutely continuous spectrum for one-dimensional
Schrödinger operators with slowly decaying potentials: Some
optimal results},
Journal = {Journal of the American Mathematical Society},
Volume = {11},
Number = {4},
Pages = {771-797},
Year = {1998},
Month = {January},
url = {http://dx.doi.org/10.1090/s0894-0347-98-00276-8},
Doi = {10.1090/s0894-0347-98-00276-8},
Key = {fds330327}
}
@article{fds330328,
Author = {Kiselev, A},
Title = {Stability of the absolutely continuous spectrum of the
Schrödinger equation under slowly decaying perturbations
and A.E. convergence of integral operators},
Journal = {Duke Mathematical Journal},
Volume = {94},
Number = {3},
Pages = {619-646},
Publisher = {Duke University Press},
Year = {1998},
Month = {January},
url = {http://dx.doi.org/10.1215/S0012-7094-98-09425-X},
Doi = {10.1215/S0012-7094-98-09425-X},
Key = {fds330328}
}
@article{fds330329,
Author = {Kiselev, A and Last, Y and Simon, B},
Title = {Modified prüfer and EFGP transforms and the spectral
analysis of one dimensional schrödinger
operators},
Journal = {Communications in Mathematical Physics},
Volume = {194},
Number = {1},
Pages = {1-45},
Publisher = {Springer Nature},
Year = {1998},
Month = {January},
url = {http://dx.doi.org/10.1007/s002200050346},
Doi = {10.1007/s002200050346},
Key = {fds330329}
}
@article{fds330330,
Author = {Kiselev, A},
Title = {Some examples in one-dimensional "geometric" scattering on
manifolds},
Journal = {Journal of Mathematical Analysis and Applications},
Volume = {212},
Number = {1},
Pages = {263-280},
Publisher = {Elsevier BV},
Year = {1997},
Month = {August},
url = {http://dx.doi.org/10.1006/jmaa.1997.5497},
Abstract = {We consider "geometric" scattering for a Laplace-Beltrami
operator on a compact Riemannian manifold inserted between
"wires," that is, two half-lines. We discuss applicability
and correctness of this model. With an example, we show that
such a scattering problem may exhibit unusual properties:
the transition coefficient has a sequence of sharp peaks
which become more and more distant at high energy and
otherwise turns to zero. © 1997 Academic
Press.},
Doi = {10.1006/jmaa.1997.5497},
Key = {fds330330}
}
@article{fds330331,
Author = {Christ, M and Kiselev, A and Remling, C},
Title = {The absolutely continuous spectrum of one-dimensional
Schrödinger operators with decaying potentials},
Journal = {Mathematical Research Letters},
Volume = {4},
Number = {5},
Pages = {719-723},
Publisher = {International Press of Boston},
Year = {1997},
Month = {January},
url = {http://dx.doi.org/10.4310/MRL.1997.v4.n5.a9},
Doi = {10.4310/MRL.1997.v4.n5.a9},
Key = {fds330331}
}
@article{fds330332,
Author = {Kiselev, A},
Title = {Absolutely continuous spectrum of one-dimensional
Schrödinger operators and Jacobi matrices with slowly
decreasing potentials},
Journal = {Communications in Mathematical Physics},
Volume = {179},
Number = {2},
Pages = {377-399},
Publisher = {Springer Nature},
Year = {1996},
Month = {January},
url = {http://dx.doi.org/10.1007/bf02102594},
Abstract = {We prove that for any one-dimensional Schrödinger operator
with potential V(x) satisfying decay condition |V(x)| ≦
Cx-3/4-ε, the absolutely continuous spectrum fills the
whole positive semi-axis. The description of the set in ℝ+
on which the singular part of the spectral measure might be
supported is also given. Analogous results hold for Jacobi
matrices.},
Doi = {10.1007/bf02102594},
Key = {fds330332}
}
@article{fds330333,
Author = {Kiselev, AA and Popov, IY},
Title = {Indefinite metric and scattering by a domain with a small
hole},
Journal = {Mathematical Notes},
Volume = {58},
Number = {6},
Pages = {1276-1285},
Publisher = {Springer Nature},
Year = {1995},
Month = {January},
url = {http://dx.doi.org/10.1007/BF02304886},
Abstract = {For the problem of plane waves scattered by a domain with a
small hole, we suggest a model based on the theory of
self-adjoint extensions of symmetric operators in a space
with indefinite metric. For two-dimensional problems of
scattering on a line with a hole and on a semi-ellipse
connected by a hole with a half-plane, we justify the choice
of extension that guarantees the coincidence of the model
solution with the solution of the “actual” problem in
the far zone with a high degree of accuracy. © 1996, Plenum
Publishing Corporation. All rights reserved.},
Doi = {10.1007/BF02304886},
Key = {fds330333}
}
@article{fds330334,
Author = {Kiselev, A and Simon, B},
Title = {Rank one perturbations with infinitesimal
coupling},
Journal = {Journal of Functional Analysis},
Volume = {130},
Number = {2},
Pages = {345-356},
Publisher = {Elsevier BV},
Year = {1995},
Month = {January},
url = {http://dx.doi.org/10.1006/jfan.1995.1074},
Abstract = {We consider a positive self-adjoint operator A and formal
rank one perturbations B = A + α(φ, ·)φ, where φ ∈
H-2(A) but φ ∉ H-1 (A), with Hs(A) the usual scale of
spaces. We show that B can be defined for such φ and what
are essentially negative infinitesimal values of α. In a
sense we will make precise, every rank one perturbation is
one of three forms: (i) φ ∈ H-1(A), α ∈ R; (ii) φ ∈
H-1, α = ∞; or (iii) the new type we consider here. ©
1995 Academic Press Limited.},
Doi = {10.1006/jfan.1995.1074},
Key = {fds330334}
}
@article{fds357912,
Author = {Kiselev, AA and Pavlov, BS},
Title = {Eigenfrequencies and eigenfunctions of the Laplacian for
Neumann boundary conditions in a system of two coupled
cavities},
Journal = {Theoretical and Mathematical Physics},
Volume = {100},
Number = {3},
Pages = {1065-1074},
Year = {1994},
Month = {September},
url = {http://dx.doi.org/10.1007/BF01018571},
Abstract = {A model Laplacian with Neumann boundary conditions (Neumann
problem) in a system of two cavities joined by a thin
channel is investigated. An expression is obtained for the
resolvent and also the first terms in the asymptotic
expansions of the eigenvalues and eigenfunctions with
respect to the small coupling parameter. © 1995 Plenum
Publishing Corporation.},
Doi = {10.1007/BF01018571},
Key = {fds357912}
}
@article{fds357913,
Author = {Kiselev, AA and Pavlov, BS},
Title = {Essential spectrum of the Laplacian for the Neumann problem
in a model region of complicated structure},
Journal = {Theoretical and Mathematical Physics},
Volume = {99},
Number = {1},
Pages = {383-395},
Year = {1994},
Month = {April},
url = {http://dx.doi.org/10.1007/BF01018792},
Abstract = {A class of regions in which the Laplacian for the Neumann
problem has an essential spectrum is considered. The
connection between the geometrical characteristics of the
region and spectral properties of the Laplacian for the
Neumann problem is studied in specific examples. © 1994
Plenum Publishing Corporation.},
Doi = {10.1007/BF01018792},
Key = {fds357913}
}
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Mathematics Department