arXiv subjects

Hung V. Tran

Publications and source records attributed to Hung V. Tran.

At least 55 records · Page 3Linked to original sources

arXiv2020

Optimal convergence rates for elliptic homogenization problems in nondivergence-form: analysis and numerical illustrations

We study optimal convergence rates in the periodic homogenization of linear elliptic equations of the form $-A(x/\varepsilon):D^2 u^{\varepsilon} = f$ subject to a homogeneous Dirichlet boundary condition. We show that the optimal rate for the convergence of $u^{\varepsilon}$ to the solution of the corresponding homogenized problem in the $W^{1,p}$-norm is $\mathcal{O}(\varepsilon)$. We further obtain optimal gradient and Hessian bounds with correction terms taken into account in the $L^p$-norm. We then provide an explicit $c$-bad diffusion matrix and use it to perform various numerical experiments, which demonstrate the optimality of the obtained rates.

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arXiv2020

Large time behavior for a Hamilton-Jacobi equation in a critical Coagulation-Fragmentation model

We study the large time behavior of the sublinear viscosity solution to a singular Hamilton-Jacobi equation that appears in a critical Coagulation-Fragmentation model with multiplicative coagulation and constant fragmentation kernels. Our results include complete characterizations of stationary solutions and optimal conditions to guarantee large time convergence. In particular, we obtain convergence results under certain natural conditions on the initial data, and a nonconvergence result when such conditions fail.

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arXiv2020

Selection problems in Large Deviations in Games under the logit choice protocol

We study large deviations in coordination games under the logit choice protocol. A major open question that [10,11] posed is whether large deviations properties under the small noise double limit and the large population double limit are identical or not. We rephrase this open question in the PDE language as some selection problems, and we provide some definitive answers to these problems.

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arXiv2020

Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel

We study a critical case of Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel. Our method is based on the study of viscosity solutions to a new singular Hamilton-Jacobi equation, which results from applying the Bernstein transform to the original Coagulation-Fragmentation equation. Our results include wellposedness, regularity and long-time behaviors of viscosity solutions to the Hamilton-Jacobi equation in certain regimes, which have implications to wellposedness and long-time behaviors of \emph{mass-conserving} solutions to the Coagulation-Fragmentation equation.

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arXiv2020

The large time profile for Hamilton--Jacobi--Bellman equations

Here, we study the large-time limit of viscosity solutions of the Cauchy problem for second-order Hamilton--Jacobi--Bellman equations with convex Hamiltonians in the torus. This large-time limit solves the corresponding stationary problem, sometimes called the ergodic problem. This problem, however, has multiple viscosity solutions and, thus, a key question is which of these solutions is selected by the limit. Here, we provide a representation for the viscosity solution to the Cauchy problem in terms of generalized holonomic measures. Then, we use this representation to characterize the large-time limit in terms of the initial data and generalized Mather measures. In addition, we establish various results on generalized Mather measures and duality theorems that are of independent interest.

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arXiv2019

Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms

We study a level-set mean curvature flow equation with driving and source terms, and establish convergence results on the asymptotic behavior of solutions as time goes to infinity under some additional assumptions. We also study the associated stationary problem in details in a particular case, and establish Alexandrov's theorem in two dimensions in the viscosity sense, which is of independent interest.

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arXiv2019

Rate of convergence in periodic homogenization of Hamilton-Jacobi equations: the convex setting

We study the rate of convergence of $u^ε$, as $ε\to 0+$, to $u$ in periodic homogenization of Hamilton-Jacobi equations. Here, $u^ε$ and $u$ are viscosity solutions to the oscillatory Hamilton-Jacobi equation and its effective equation \begin{equation*} {\rm (C)_ε} \qquad \begin{cases} u_t^ε+H\left(\frac{x}ε,Du^ε\right)=0 \qquad &\text{in} \ \mathbb{R}^n \times (0,\infty), u^ε(x,0)=g(x) \qquad &\text{on} \ \mathbb{R}^n, \end{cases} \end{equation*} and \begin{equation*} {\rm (C)} \qquad \begin{cases} u_t+\overline{H}\left(Du\right)=0 \qquad &\text{in} \ \mathbb{R}^n \times (0,\infty), u(x,0)=g(x) \qquad &\text{on} \ \mathbb{R}^n, \end{cases} \end{equation*} respectively. We assume that the Hamiltonian $H=H(y,p)$ is coercive and convex in the $p$ variable and is $\mathbb{Z}^n$-periodic in the $y$ variable, and the initial data $g$ is bounded and Lipschitz continuous.

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arXiv2019

Generalized ergodic problems: existence and uniqueness structures of solutions

We study a generalized ergodic problem (E), which is a Hamilton-Jacobi equation of contact type, in the flat $n$-dimensional torus. We first obtain existence of solutions to this problem under quite general assumptions. Various examples are presented and analyzed to show that (E) does not have unique solutions in general. We then study uniqueness structures of solutions to (E) in the convex setting by using the nonlinear adjoint method.

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arXiv2018

On obstacle problem for mean curvature flow with driving force

In this paper, we study an obstacle problem associated with the mean curvature flow with constant driving force. Our first main result concerns interior and boundary regularity of the solution. We then study in details the large time behavior of the solution and obtain the convergence result. In particular, we give full characterization of the limiting profiles in the radially symmetric setting.

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arXiv2018

Existence of asymptotic speed of solutions to birth and spread type nonlinear partial differential equations

In this paper, we prove the existence of asymptotic speed of solutions to fully nonlinear, possibly degenerate parabolic partial differential equations in a general setting. We then give some explicit examples of equations in this setting and study further properties of the asymptotic speed for each equation. Some numerical results concerning the asymptotic speed are presented.

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arXiv2018

On uniqueness sets of additive eigenvalue problems and applications

In this paper, we provide a simple way to find uniqueness sets for additive eigenvalue problems of first and second order Hamilton--Jacobi equations by using a PDE approach. An application in finding the limiting profiles for large time behaviors of first order Hamilton--Jacobi equations is also obtained.

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arXiv2017

A rigidity result for effective Hamiltonians with $3$-mode periodic potentials

We continue studying an inverse problem in the theory of periodic homogenization of Hamilton-Jacobi equations proposed in [14]. Let $V_1, V_2 \in C(\mathbb{R}^n)$ be two given potentials which are $\mathbb{Z}^n$-periodic, and $\overline{H}_1, \overline{H}_2$ be the effective Hamiltonians associated with the Hamiltonians $\frac{1}{2}|p|^2 + V_1$, $\frac{1}{2}|p|^2+V_2$, respectively. A main result in this paper is that, if the dimension $n=2$ and each of $V_1, V_2$ contains exactly $3$ mutually non-parallel Fourier modes, then $$ \overline H_1\equiv \overline H_2 \quad \iff \quad V_1(x)=V_2\left({x\over c}+x_0\right) \quad \text{ for all } x \in \mathbb{T}^2 = \mathbb{R}^2/\mathbb{Z}^2, $$ for some $c\in \mathbb{Q} \setminus\{0\}$ and $x_0 \in \mathbb{T}^2$. When $n\geq 3$, the scenario is slightly more subtle, and a complete description is provided for any dimension. These resolve partially the conjecture stated in [14]. Some other related results and open problems are also discussed.

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arXiv2017

On the Langevin equation with variable friction

We study two asymptotic problems for the Langevin equation with variable friction coefficient. The first is the small mass asymptotic behavior, known as the Smoluchowski-Kramers approximation, of the Langevin equation with strictly positive variable friction. The second result is about the limiting behavior of the solution when the friction vanishes in regions of the domain. Previous works on this subject considered one dimensional settings with the conclusions based on explicit computations.

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arXiv2017

Stochastic homogenization of viscous superquadratic Hamilton-Jacobi equations in dynamic random environment

We study the qualitative homogenization of second order viscous Hamilton-Jacobi equations in space-time stationary ergodic random environments. Assuming that the Hamiltonian is convex and superquadratic in the momentum variable (gradient) we establish a homogenization result and characterize the effective Hamiltonian for arbitrary (possibly degenerate) elliptic diffusion matrices. The result extends previous work that required uniform ellipticity and space-time homogeneity for the diffusion.

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arXiv2017

Min-max formulas and other properties of certain classes of nonconvex effective Hamiltonians

This paper is the first attempt to systematically study properties of the effective Hamiltonian $\overline{H}$ arising in the periodic homogenization of some coercive but nonconvex Hamilton-Jacobi equations. Firstly, we introduce a new and robust decomposition method to obtain min-max formulas for a class of nonconvex $\overline{H}$. Secondly, we analytically and numerically investigate other related interesting phenomena, such as "quasi-convexification" and breakdown of symmetry, of $\overline{H}$ from other typical nonconvex Hamiltonians. Finally, in the appendix, we show that our new method and those a priori formulas from the periodic setting can be used to obtain stochastic homogenization for same class of nonconvex Hamilton-Jacobi equations. Some conjectures and problems are also proposed.

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arXiv2016

The vanishing discount problem and viscosity Mather measures. Part 2: boundary value problems

In arXiv:1603.01051 (Part 1 of this series), we have introduced a variational approach to studying the vanishing discount problem for fully nonlinear, degenerate elliptic, partial differential equations in a torus. We develop this approach further here to handle boundary value problems. In particular, we establish new representation formulas for solutions of discount problems, critical values, and use them to prove convergence results for the vanishing discount problems.

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arXiv2016

The Selection problem for discounted Hamilton-Jacobi equations: some non-convex cases

Here, we study the selection problem for the vanishing discount approximation of non-convex, first-order Hamilton-Jacobi equations. While the selection problem is well understood for convex Hamiltonians, the selection problem for non-convex Hamiltonians has thus far not been studied. We begin our study by examining a generalized discounted Hamilton-Jacobi equation. Next, using an exponential transformation, we apply our methods to strictly quasi-convex and to some non-convex Hamilton-Jacobi equations. Finally, we examine a non-convex Hamiltonian with flat parts to which our results do not directly apply. In this case, we establish the convergence by a direct approach.

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arXiv2016

The vanishing discount problem and viscosity Mather measures. Part 1: the problem on a torus

We develop a variational approach to the vanishing discount problem for fully nonlinear, degenerate elliptic, partial differential equations. Under mild assumptions, we introduce viscosity Mather measures for such partial differential equations, which are natural extensions of the Mather measures. Using the viscosity Mather measures, we prove that the whole family of solutions $v^λ$ of the discount problem with the factor $λ>0$ converges to a solution of the ergodic problem as $λ\to 0$.

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