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Publications of Yuan Gao    :chronological  alphabetical  combined listing:

%% Papers Published   
   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
   Volume = {60},
   Number = {2},
   Year = {2021},
   Month = {April},
   url = {},
   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
   Doi = {10.1007/s00526-021-01939-1},
   Key = {fds355948}

   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
   Journal = {Advances in Calculus of Variations},
   Year = {2021},
   Month = {January},
   url = {},
   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}

   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 = {},
   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
   Doi = {10.4171/ifb/451},
   Key = {fds356447}

   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 = {},
   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}

   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
   Volume = {22},
   Number = {11},
   Year = {2020},
   Month = {January},
   url = {},
   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
   Doi = {10.3934/dcdsb.2020224},
   Key = {fds358254}

   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 = {},
   Doi = {10.4310/maa.2020.v27.n2.a4},
   Key = {fds354086}

   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 = {},
   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}

   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
   Volume = {25},
   Pages = {49-49},
   Publisher = {E D P SCIENCES},
   Year = {2019},
   Month = {January},
   url = {},
   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
   Doi = {10.1051/cocv/2018037},
   Key = {fds348009}

   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
   Volume = {23},
   Number = {10},
   Pages = {4433-4453},
   Year = {2018},
   Month = {December},
   url = {},
   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
   Doi = {10.3934/dcdsb.2018170},
   Key = {fds347224}

   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
   Journal = {Calculus of Variations and Partial Differential
   Volume = {57},
   Number = {2},
   Year = {2018},
   Month = {April},
   url = {},
   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}

   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 = {},
   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}

   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}

   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 = {},
   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}

   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 = {},
   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}

   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 = {},
   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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