%% Papers Published
@article{fds355948,
Author = {Dong, H and Gao, Y},
Title = {Existence and uniqueness of bounded stable solutions to the
Peierls–Nabarro model for curved dislocations},
Journal = {Calculus of Variations and Partial Differential
Equations},
Volume = {60},
Number = {2},
Year = {2021},
Month = {April},
url = {http://dx.doi.org/10.1007/s00526-021-01939-1},
Abstract = {We study the well-posedness of the vector-field
Peierls–Nabarro model for curved dislocations with a
double well potential and a bi-states limit at far field.
Using the Dirichlet to Neumann map, the 3D Peierls–Nabarro
model is reduced to a nonlocal scalar Ginzburg–Landau
equation. We derive an integral formulation of the nonlocal
operator, whose kernel is anisotropic and positive when
Poisson’s ratio ν∈(-12,13). We then prove that any
bounded stable solution to this nonlocal scalar
Ginzburg–Landau equation has a 1D profile, which
corresponds to the PDE version of flatness result for
minimal surfaces with anisotropic nonlocal perimeter. Based
on this, we finally obtain that steady states to the
nonlocal scalar equation, as well as the original
Peierls–Nabarro model, can be characterized as a
one-parameter family of straight dislocation solutions to a
rescaled 1D Ginzburg–Landau equation with the half
Laplacian.},
Doi = {10.1007/s00526-021-01939-1},
Key = {fds355948}
}
@article{fds356176,
Author = {Gao, Y and Lu, XY and Wang, C},
Title = {Regularity and monotonicity for solutions to a continuum
model of epitaxial growth with nonlocal elastic
effects},
Journal = {Advances in Calculus of Variations},
Year = {2021},
Month = {January},
url = {http://dx.doi.org/10.1515/acv-2020-0114},
Abstract = {We study the following parabolic nonlocal 4-th order
degenerate equation: Ut=-[2â πâ Hâ (ux)+lnâi (uxâ
x+a)+32â (uxâ x+a)2]xâ x,u{t}=-\Bigl{[}2\pi
H(u{x})+\ln(u{XX}+a)+\frac{3}{2}(u{XX}+a){2}\Bigr{]}{% XX},
arising from the epitaxial growth on crystalline materials.
Here H denotes the Hilbert transform, and a>0{a>0} is a
given parameter. By relying on the theory of gradient flows,
we first prove the global existence of a variational
inequality solution with a general initial datum.
Furthermore, to obtain a global strong solution, the main
difficulty is the singularity of the logarithmic term when
uxâ x+a{u{XX}+a} approaches zero. Thus we show that, if the
initial datum u0{u{0}} is such that (u0)xâ
x+a{(u{0}){XX}+a} is uniformly bounded away from zero, then
such property is preserved for all positive times. Finally,
we will prove several higher regularity results for this
global strong solution. These finer properties provide a
rigorous justification for the global-in-time monotone
solution to the epitaxial growth model with nonlocal elastic
effects on vicinal surface.},
Doi = {10.1515/acv-2020-0114},
Key = {fds356176}
}
@article{fds356447,
Author = {Gao, Y and Liu, JG},
Title = {Gradient flow formulation and second order numerical method
for motion by mean curvature and contact line dynamics on
rough surface},
Journal = {Interfaces and Free Boundaries},
Volume = {23},
Number = {1},
Pages = {103-158},
Year = {2021},
Month = {January},
url = {http://dx.doi.org/10.4171/ifb/451},
Abstract = {We study the dynamics of a droplet moving on an inclined
rough surface in the absence of inertial and viscous stress
effects. In this case, the dynamics of the droplet is a
purely geometric motion in terms of the wetting domain and
the capillary surface. Using a single graph representation,
we interpret this geometric motion as a gradient flow on a
manifold. We propose unconditionally stable first/second
order numerical schemes to simulate this geometric motion of
the droplet, which is described using motion by mean
curvature coupled with moving contact lines. The schemes are
based on (i) explicit moving boundaries, which decouple the
dynamic updates of the contact lines and the capillary
surface, (ii) an arbitrary Lagrangian-Eulerian method on
moving grids and (iii) a predictor-corrector method with a
nonlinear elliptic solver up to second order accuracy. For
the case of quasi-static dynamics with continuous spatial
variable in the numerical schemes, we prove the stability
and convergence of the first/second order numerical schemes.
To demonstrate the accuracy and long-time validation of the
proposed schemes, several challenging computational examples
- including breathing droplets, droplets on inhomogeneous
rough surfaces and quasi-static Kelvin pendant droplets -
are constructed and compared with exact solutions to
quasi-static dynamics obtained by desingularized
differential-algebraic system of equations
(DAEs).},
Doi = {10.4171/ifb/451},
Key = {fds356447}
}
@article{fds354270,
Author = {Gao, Y and Liu, JG},
Title = {Large Time Behavior, Bi-Hamiltonian Structure, and Kinetic
Formulation for a Complex Burgers Equation},
Journal = {Quarterly of Applied Mathematics},
Volume = {79},
Number = {1},
Pages = {120-123},
Publisher = {American Mathematical Society (AMS)},
Year = {2020},
Month = {May},
url = {http://dx.doi.org/10.1090/QAM/1573},
Abstract = {We prove the existence and uniqueness of positive analytical
solutions with positive initial data to the mean field
equation (the Dyson equation) of the Dyson Brownian motion
through the complex Burgers equation with a force term on
the upper half complex plane. These solutions converge to a
steady state given by Wigner's semicircle law. A unique
global weak solution with nonnegative initial data to the
Dyson equation is obtained, and some explicit solutions are
given by Wigner's semicircle laws. We also construct a
bi-Hamiltonian structure for the system of real and
imaginary components of the complex Burgers equation
(coupled Burgers system). We establish a kinetic formulation
for the coupled Burgers system and prove the existence and
uniqueness of entropy solutions. The coupled Burgers system
in Lagrangian variable naturally leads to two interacting
particle systems, the Fermi–Pasta–Ulam–Tsingou model
with nearest-neighbor interactions, and the Calogero–Moser
model. These two particle systems yield the same Lagrangian
dynamics in the continuum limit.},
Doi = {10.1090/QAM/1573},
Key = {fds354270}
}
@article{fds358254,
Author = {Gao, Y and Liu, JG and Luo, T and Xiang, Y},
Title = {Revisit of the peierls-nabarro model for edge dislocations
in Hilbert space},
Journal = {Discrete and Continuous Dynamical Systems Series
B},
Volume = {22},
Number = {11},
Year = {2020},
Month = {January},
url = {http://dx.doi.org/10.3934/dcdsb.2020224},
Abstract = {In this paper, we revisit the mathematical validation of the
Peierls–Nabarro (PN) models, which are multiscale models
of dislocations that incorporate the detailed dislocation
core structure. We focus on the static and dynamic PN models
of an edge dislocation in Hilbert space. In a PN model, the
total energy includes the elastic energy in the two
half-space continua and a nonlinear potential energy, which
is always infinite, across the slip plane. We revisit the
relationship between the PN model in the full space and the
reduced problem on the slip plane in terms of both governing
equations and energy variations. The shear displacement jump
is determined only by the reduced problem on the slip plane
while the displacement fields in the two half spaces are
determined by linear elasticity. We establish the existence
and sharp regularities of classical solutions in Hilbert
space. For both the reduced problem and the full PN model,
we prove that a static solution is a global minimizer in a
perturbed sense. We also show that there is a unique
classical, global in time solution of the dynamic PN
model.},
Doi = {10.3934/dcdsb.2020224},
Key = {fds358254}
}
@article{fds354086,
Author = {Gao, Y and Liu, J-G},
Title = {Long time behavior of dynamic solution to Peierls–Nabarro
dislocation model},
Journal = {Methods and Applications of Analysis},
Volume = {27},
Number = {2},
Pages = {161-198},
Publisher = {International Press of Boston},
Year = {2020},
url = {http://dx.doi.org/10.4310/maa.2020.v27.n2.a4},
Doi = {10.4310/maa.2020.v27.n2.a4},
Key = {fds354086}
}
@article{fds347223,
Author = {Gao, Y},
Title = {Global strong solution with BV derivatives to singular
solid-on-solid model with exponential nonlinearity},
Journal = {Journal of Differential Equations},
Volume = {267},
Number = {7},
Pages = {4429-4447},
Year = {2019},
Month = {September},
url = {http://dx.doi.org/10.1016/j.jde.2019.05.011},
Abstract = {In this work, we consider the one dimensional very singular
fourth-order equation for solid-on-solid model in
attachment-detachment-limit regime with exponential
nonlinearity ht=∇⋅([Formula presented]∇e [Formula
presented])=∇⋅([Formula presented]∇e−∇⋅([Formula
presented])) where total energy E=∫|∇h| is the total
variation of h. Using a logarithmic correction for total
energy E=∫|∇h|ln|∇h|dx and gradient flow structure
with a suitable defined functional, we prove the one
dimensional evolution variational inequality solution
preserves a positive gradient hx which has upper and lower
bounds but in BV space. We also obtain the global strong
solution to the solid-on-solid model which allows an
asymmetric singularity hxx+ to happen.},
Doi = {10.1016/j.jde.2019.05.011},
Key = {fds347223}
}
@article{fds348009,
Author = {Gao, Y and Liu, JG and Lu, XY},
Title = {Gradient flow approach to an exponential thin film equation:
Global existence and latent singularity},
Journal = {Esaim: Control, Optimisation and Calculus of
Variations},
Volume = {25},
Pages = {49-49},
Publisher = {E D P SCIENCES},
Year = {2019},
Month = {January},
url = {http://dx.doi.org/10.1051/cocv/2018037},
Abstract = {In this work, we study a fourth order exponential equation,
ut = Δe-Δu derived from thin film growth on crystal
surface in multiple space dimensions. We use the gradient
flow method in metric space to characterize the latent
singularity in global strong solution, which is intrinsic
due to high degeneration. We define a suitable functional,
which reveals where the singularity happens, and then prove
the variational inequality solution under very weak
assumptions for initial data. Moreover, the existence of
global strong solution is established with regular initial
data.},
Doi = {10.1051/cocv/2018037},
Key = {fds348009}
}
@article{fds347224,
Author = {Gao, Y and Ji, H and Liu, JG and Witelski, TP},
Title = {A vicinal surface model for epitaxial growth with
logarithmic free energy},
Journal = {Discrete and Continuous Dynamical Systems Series
B},
Volume = {23},
Number = {10},
Pages = {4433-4453},
Year = {2018},
Month = {December},
url = {http://dx.doi.org/10.3934/dcdsb.2018170},
Abstract = {We study a continuum model for solid films that arises from
the modeling of one-dimensional step flows on a vicinal
surface in the attachment-detachment-limited regime. The
resulting nonlinear partial differential equation, ut =
-u2(u3 + au)hhhh, gives the evolution for the surface slope
u as a function of the local height h in a monotone step
train. Subject to periodic boundary conditions and positive
initial conditions, we prove the existence, uniqueness and
positivity of global strong solutions to this PDE using two
Lyapunov energy functions. The long time behavior of u
converging to a constant that only depends on the initial
data is also investigated both analytically and
numerically.},
Doi = {10.3934/dcdsb.2018170},
Key = {fds347224}
}
@article{fds347225,
Author = {Gao, Y and Liu, JG and Lu, XY and Xu, X},
Title = {Maximal monotone operator theory and its applications to
thin film equation in epitaxial growth on vicinal
surface},
Journal = {Calculus of Variations and Partial Differential
Equations},
Volume = {57},
Number = {2},
Year = {2018},
Month = {April},
url = {http://dx.doi.org/10.1007/s00526-018-1326-x},
Abstract = {In this work we consider (Formula presented.) which is
derived from a thin film equation for epitaxial growth on
vicinal surface. We formulate the problem as the gradient
flow of a suitably-defined convex functional in a
non-reflexive space. Then by restricting it to a Hilbert
space and proving the uniqueness of its sub-differential, we
can apply the classical maximal monotone operator theory.
The mathematical difficulty is due to the fact that w can
appear as a positive Radon measure. We prove the existence
of a global strong solution with hidden singularity. In
particular, (1) holds almost everywhere when w is replaced
by its absolutely continuous part. hh hh},
Doi = {10.1007/s00526-018-1326-x},
Key = {fds347225}
}
@article{fds346573,
Author = {Gao, Y and Liang, J and Xiao, TJ},
Title = {A new method to obtain uniform decay rates for
multidimensional wave equations with nonlinear acoustic
boundary conditions},
Journal = {Siam Journal on Control and Optimization},
Volume = {56},
Number = {2},
Pages = {1303-1320},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.1137/16M107863X},
Abstract = {In this paper, we investigate the uniform stability of a
class of nonlinear acoustic wave motions with boundary and
localized interior damping. Here the damping and potential
in the boundary displacement equation are nonlinear.
Moreover, the nonlinear system contains the localized
interior damping term, which indicates that there is a thin
absorption material and flow resistance on the endophragm of
the boundary. Since some lower-order term in the nonlinear
wave system is not below the energy level, the
“compactness-uniqueness” method is not suitable for the
problem. Our main purpose is to present a new method to
obtain uniform decay rates for these damped wave equations
with nonlinear acoustic boundary conditions.},
Doi = {10.1137/16M107863X},
Key = {fds346573}
}
@article{fds346574,
Author = {Gao, Y and Liang, J and Xiao, TJ},
Title = {Observability inequality and decay rate for wave equations
with nonlinear boundary conditions},
Journal = {Electronic Journal of Differential Equations},
Volume = {2017},
Year = {2017},
Month = {July},
Abstract = {We study a class of wave propagation problems concerning the
nonlinearity of dynamic evolution for boundary material. We
establish an observability inequality for the related linear
system, and make a connection between the linear system and
the original nonlinear coupled system. Also, we obtain the
desired energy decay rate for the original nonlinear
boundary value problem.},
Key = {fds346574}
}
@article{fds347226,
Author = {Gao, Y and Ji, H and Liu, JG and Witelski, TP},
Title = {Global existence of solutions to a tear film model with
locally elevated evaporation rates},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {350},
Pages = {13-25},
Year = {2017},
Month = {July},
url = {http://dx.doi.org/10.1016/j.physd.2017.03.005},
Abstract = {Motivated by a model proposed by Peng et al. (2014) for
break-up of tear films on human eyes, we study the dynamics
of a generalized thin film model. The governing equations
form a fourth-order coupled system of nonlinear parabolic
PDEs for the film thickness and salt concentration subject
to non-conservative effects representing evaporation. We
analytically prove the global existence of solutions to this
model with mobility exponents in several different ranges
and present numerical simulations that are in agreement with
the analytic results. We also numerically capture other
interesting dynamics of the model, including finite-time
rupture–shock phenomenon due to the instabilities caused
by locally elevated evaporation rates, convergence to
equilibrium and infinite-time thinning.},
Doi = {10.1016/j.physd.2017.03.005},
Key = {fds347226}
}
@article{fds347227,
Author = {Gao, Y and Liu, JG and Lu, J},
Title = {Continuum Limit of a Mesoscopic Model with Elasticity of
Step Motion on Vicinal Surfaces},
Journal = {Journal of Nonlinear Science},
Volume = {27},
Number = {3},
Pages = {873-926},
Year = {2017},
Month = {June},
url = {http://dx.doi.org/10.1007/s00332-016-9354-1},
Abstract = {This work considers the rigorous derivation of continuum
models of step motion starting from a mesoscopic
Burton–Cabrera–Frank-type model following the Xiang’s
work (Xiang in SIAM J Appl Math 63(1):241–258, 2002). We
prove that as the lattice parameter goes to zero, for a
finite time interval, a modified discrete model converges to
the strong solution of the limiting PDE with first-order
convergence rate.},
Doi = {10.1007/s00332-016-9354-1},
Key = {fds347227}
}
@article{fds347228,
Author = {Gao, Y and Liu, JG and Lu, J},
Title = {Weak solution of a continuum model for vicinal surface in
the attachment-detachment-limited regime},
Journal = {Siam Journal on Mathematical Analysis},
Volume = {49},
Number = {3},
Pages = {1705-1731},
Year = {2017},
Month = {January},
url = {http://dx.doi.org/10.1137/16M1094543},
Abstract = {We study in this work a continuum model derived from a
one-dimensional attachmentdetachment-limited type step flow
on a vicinal surface, ut = -u2(u3)hhhh, where u, considered
as a function of step height h, is the step slope of the
surface. We formulate a notion of a weak solution to this
continuum model and prove the existence of a global weak
solution, which is positive almost everywhere. We also study
the long time behavior of the weak solution and prove it
converges to a constant solution as time goes to infinity.
The space-time Hölder continuity of the weak solution is
also discussed as a byproduct.},
Doi = {10.1137/16M1094543},
Key = {fds347228}
}
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Mathematics Department