%% Papers Published
@article{fds370952,
Author = {Kiselev, A and Luo, X},
Title = {Illposedness of C2 Vortex Patches},
Journal = {Archive for Rational Mechanics and Analysis},
Volume = {247},
Number = {3},
Publisher = {Springer Science and Business Media LLC},
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 = {fds370952}
}
@article{fds369692,
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},
Publisher = {Springer Science and Business Media LLC},
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 = {fds369692}
}
@article{fds365618,
Author = {Cheskidov, A and Luo, X},
Title = {Sharp nonuniqueness for the Navier–Stokes
equations},
Journal = {Inventiones Mathematicae},
Volume = {229},
Number = {3},
Pages = {987-1054},
Year = {2022},
Month = {September},
url = {http://dx.doi.org/10.1007/s00222-022-01116-x},
Abstract = {In this paper, we prove a sharp nonuniqueness result for the
incompressible Navier–Stokes equations in the periodic
setting. In any dimension d≥ 2 and given any p< 2 , we
show the nonuniqueness of weak solutions in the class
LtpL∞, which is sharp in view of the classical
Ladyzhenskaya–Prodi–Serrin criteria. The proof is based
on the construction of a class of non-Leray–Hopf weak
solutions. More specifically, for any p< 2 , q< ∞, and ε>
0 , we construct non-Leray–Hopf weak solutions
u∈LtpL∞∩Lt1W1,q that are smooth outside a set of
singular times with Hausdorff dimension less than ε. As a
byproduct, examples of anomalous dissipation in the class
Lt3/2-εC1/3 are given in both the viscous and inviscid
case.},
Doi = {10.1007/s00222-022-01116-x},
Key = {fds365618}
}
@article{fds365620,
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 = {fds365620}
}
@article{fds354611,
Author = {Cheskidov, A and Luo, X},
Title = {Nonuniqueness of Weak Solutions for the Transport Equation
at Critical Space Regularity},
Journal = {Annals of Pde},
Volume = {7},
Number = {1},
Publisher = {Springer Science and Business Media LLC},
Year = {2021},
Month = {June},
url = {http://dx.doi.org/10.1007/s40818-020-00091-x},
Abstract = {We consider the linear transport equations driven by an
incompressible flow in dimensions d≥ 3. For
divergence-free vector fields u∈Lt1W1,q, the celebrated
DiPerna-Lions theory of the renormalized solutions
established the uniqueness of the weak solution in the class
Lt∞Lp when 1p+1q≤1. For such vector fields, we show that
in the regime 1p+1q>1, weak solutions are not unique in the
class Lt1Lp. One crucial ingredient in the proof is the use
of both temporal intermittency and oscillation in the convex
integration scheme.},
Doi = {10.1007/s40818-020-00091-x},
Key = {fds354611}
}
@article{fds359732,
Author = {CHESKIDOV, A and LUO, X},
Title = {Anomalous dissipation, anomalous work, and energy balance
for the navier-stokes equations},
Journal = {Siam Journal on Mathematical Analysis},
Volume = {53},
Number = {4},
Pages = {3856-3887},
Year = {2021},
Month = {January},
url = {http://dx.doi.org/10.1137/20M1354076},
Abstract = {In this paper, we study the energy balance for a class of
solutions of the Navier-Stokes equations with external
forces in dimensions three and above. The solution and force
are smooth on (0, T) and the total dissipation and work of
the force are both finite. We show that a possible failure
of the energy balance stems from two effects. The first is
the anomalous dissipation of the solution, which has been
studied in many contexts. The second is what we call the
anomalous work done by the force, a phenomenon that has not
been analyzed before. There are numerous examples of
solutions exhibiting anomalous work, for which even a
continuous energy profile does not rule out the anomalous
dissipation, but only implies the balance of the strengths
of these two effects, which we confirm in explicit
constructions. More importantly, we show that there exist
solutions exhibiting anomalous dissipation with zero
anomalous work. Hence the violation of the energy balance
results from the nonlinearity of the solution instead of
artifacts of the force. Such examples exist in the class u
∈ L3t B 1 3 3,7infty; and f ∈ L2 t H 1, which implies
the sharpness of many existing conditions on the energy
balance.},
Doi = {10.1137/20M1354076},
Key = {fds359732}
}
@article{fds351396,
Author = {Cheskidov, A and Luo, X},
Title = {Energy equality for the Navier–Stokes equations in
weak-in-time Onsager spaces},
Journal = {Nonlinearity},
Volume = {33},
Number = {4},
Pages = {1388-1403},
Publisher = {IOP Publishing},
Year = {2020},
Month = {April},
url = {http://dx.doi.org/10.1088/1361-6544/ab60d3},
Doi = {10.1088/1361-6544/ab60d3},
Key = {fds351396}
}
@article{fds351397,
Author = {Luo, X},
Title = {On the possible time singularities for the 3D
Navier–Stokes equations},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {395},
Pages = {37-42},
Publisher = {Elsevier BV},
Year = {2019},
Month = {August},
url = {http://dx.doi.org/10.1016/j.physd.2019.02.008},
Doi = {10.1016/j.physd.2019.02.008},
Key = {fds351397}
}
@article{fds351398,
Author = {Luo, X},
Title = {Stationary Solutions and Nonuniqueness of Weak Solutions for
the Navier–Stokes Equations in High Dimensions},
Journal = {Archive for Rational Mechanics and Analysis},
Volume = {233},
Number = {2},
Pages = {701-747},
Publisher = {Springer Science and Business Media LLC},
Year = {2019},
Month = {August},
url = {http://dx.doi.org/10.1007/s00205-019-01366-9},
Doi = {10.1007/s00205-019-01366-9},
Key = {fds351398}
}
@article{fds351399,
Author = {Luo, X},
Title = {A Beale–Kato–Majda Criterion with Optimal Frequency and
Temporal Localization},
Journal = {Journal of Mathematical Fluid Mechanics},
Volume = {21},
Number = {1},
Publisher = {Springer Science and Business Media LLC},
Year = {2019},
Month = {March},
url = {http://dx.doi.org/10.1007/s00021-019-0411-z},
Doi = {10.1007/s00021-019-0411-z},
Key = {fds351399}
}
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Mathematics Department