arXiv subjects
Publications and source records attributed to Yuming Paul Zhang.
We investigate a Hele-Shaw type free boundary problem in general spatial dimension. We establish a comparison principle and well-posedness for the problem using a notion of viscosity flows. An important feature of our work is that our hypotheses allow for the free boundary velocity to take negative values.
We study quantitative large-time averages for Hamilton--Jacobi equations in a dynamic random environment that is stationary ergodic and has unit-range dependence in time. Our motivation comes from stochastic growth models related to the tensionless (inviscid) KPZ equation, which can be formulated as Hamilton--Jacobi equations with random forcing. Understanding the large-time behavior of solutions is closely connected to fundamental questions concerning fluctuations and scaling in such growth processes. In this article, we establish, up to slowly varying factors, convergence rates with exponent $1/2$ for the large-time averages of both the solutions and the associated metric problem toward their ergodic limits. Our proof relies crucially on a new almost-Lipschitz regularity theory for the metric problem, which is of independent interest.
This paper is a continuation of the work in \cite{kimzhang2024} concerning Hele-Shaw flow with both drift and source terms. We prove that, in a local neighborhood, if the free boundary is Lipschitz continuous with a sufficiently small Lipschitz constant, then the free boundary is $C^{1}$. As a corollary, we also consider the 2D vertical Hele-Shaw (or one-phase Muskat) problem with an advection term. We show that, provided the initial data and the advection term are small and the propagation speed is large, the free boundary becomes uniformly $C^1$ after a finite time.
This paper establishes a rigorous connection between regularized discrete-time reinforcement learning (RL) and continuous-time stochastic optimal control. Specifically, classical RL algorithms are typically solving a regularized discrete-time Bellman equation. We study the discretization error, namely, the gap between the optimal policy induced by the regularized discrete-time Bellman equation and the true optimal feedback control of the underlying continuous-time stochastic control problem. By deriving quantitative convergence rates for this gap, we provide a rigorous foundation for understanding the stability and implementation of exploratory RL policies in stochastic continuous-time environments.
This paper investigates the spreading properties of globally defined bounded positive solutions of a chemotaxis system featuring a logistic source and consumption: \[ \left\{ \begin{aligned} &\partial_tu=Δu - χ\nabla\cdot(u\nabla v)+ u(a-bu),\quad &(t,x)\in [0,\infty)\times\mathbb{R}^N, \\ &{τ\partial_tv}=Δv-uv,\quad & (t,x)\in [0,\infty)\times\mathbb{R}^N, \end{aligned} \right. \] where $u(t,x)$ represents the population density of a biological species, and $v(t,x)$ denotes the density of a chemical substance. Key findings of this study include: (i) the species spreads at least at the speed $c^*=2\sqrt a$ (equalling the speed when $v\equiv 0$), suggesting that the chemical substance does not hinder the spreading; (ii) the chemical substance does not induce infinitely fast spreading of $u$; (iii) the spreading speed remains unaffected under conditions that $v(0,\cdot)$ decays spatially or $0<-χ\ll 1$ and $τ=1$. Additionally, our numerical simulations reveal a noteworthy phase transition in $χ$: for $v(0, \cdot)$ uniformly distributed across space, the spreading speed accelerates only when $χ$ surpasses a critical positive value.
This series of papers is concerned with the global solvability, boundedness, regularity, and uniqueness of weak solutions to the following parabolic-parabolic chemotaxis system with a logistic source and chemical consumption: \begin{equation*} \begin{cases} u_t = m\nabla\cdot \left((\eps+u)^{m-1}\nabla u\right) - χ\nabla \cdot (u \nabla v) + u(a - b u), & \text{ in } (0,\infty)\times\mathbb{R}^N, \\ v_t = Δv - uv, & \text{ in } (0,\infty)\times\mathbb{R}^N, \end{cases} \end{equation*} where $m > 1$ and $\eps \geq 0$. The present paper focuses on the global solvability and boundedness of weak solutions. For general bounded initial data, which may be non-integrable, we prove the existence of global weak solutions that remain uniformly bounded for all times. The proof relies on deriving local $L^p$ estimates that are uniform in time via a new continuity-type argument and obtaining $L^\infty$ bounds using Moser's iteration; all of these estimates are uniform as $\eps\to0$. In part II, we will study the regularity and uniqueness of weak solutions.
We investigate a Hele-Shaw type free boundary problem in one spatial dimension, where heterogeneities appear both on the free boundary and within the interior of the positivity set. Our contributions are twofold. First, we establish well-posedness and a comparison principle for the problem by introducing a novel notion of viscosity flows. Second, under the assumption that the coefficients are stationary ergodic, we prove a stochastic homogenization result. Our results are new even in the periodic setting. To derive the effective free boundary velocity, we use a new approximation that accounts for both interior homogenization and free boundary propagation.
We study the policy iteration algorithm (PIA) for entropy-regularized stochastic control problems on an infinite time horizon with a large discount rate, focusing on two main scenarios. First, we analyze PIA with bounded coefficients where the controls applied to the diffusion term satisfy a smallness condition. We demonstrate the convergence of PIA based on a uniform $\mathcal{C}^{2,α}$ estimate for the value sequence generated by PIA, and provide a quantitative convergence analysis for this scenario. Second, we investigate PIA with unbounded coefficients but no control over the diffusion term. In this scenario, we first provide the well-posedness of the exploratory Hamilton--Jacobi--Bellman equation with linear growth coefficients and polynomial growth reward function. By such a well-posedess result we achieve PIA's convergence by establishing a quantitative locally uniform $\mathcal{C}^{1,α}$ estimates for the generated value sequence.
This paper is concerned with the convergence rate of policy iteration for (deterministic) optimal control problems in continuous time. To overcome the problem of ill-posedness due to lack of regularity, we consider a semi-discrete scheme by adding a viscosity term via finite differences in space. We prove that PI for the semi-discrete scheme converges exponentially fast, and provide a bound on the error induced by the semi-discrete scheme. We also consider the discrete space-time scheme, where both space and time are discretized. Convergence rate of PI and the discretization error are studied.
We study the convergence rates of policy iteration (PI) for nonconvex viscous Hamilton--Jacobi equations using a discrete space-time scheme, where both space and time variables are discretized. We analyze the case with an uncontrolled diffusion term, which corresponds to a possibly degenerate viscous Hamilton--Jacobi equation. We first obtain an exponential convergent result of PI for the discrete space-time schemes. We then investigate the discretization error.
In this paper we study the regularity property of Hele-Shaw flow, where source and drift are present in the evolution. More specifically we consider Hölder continuous source and Lipschitz continuous drift. We show that if the free boundary of the solution is locally close to a Lipschitz graph, then it is indeed Lipschitz, given that the Lipschitz constant is small. When there is no drift, our result establishes $C^{1,γ}$ regularity of the free boundary by combining our result with the obstacle problem theory. In general, when the source and drift are both smooth, we prove that the solution is non-degenerate, indicating higher regularity of the free boudary.
We investigate the general Porous Medium Equations with drift and source terms that model tumor growth. Incompressible limit of such models has been well-studied in the literature, where convergence of the density and pressure variables are established, while it remains unclear whether the free boundaries of the solutions exhibit convergence as well. In this paper, we provide an affirmative result by showing that the free boundaries converge in the Hausdorff distance in the incompressible limit. To achieve this, we quantify the relation between the free boundary motion and spatial average of the pressure, and establish a uniform-in-$m$ strict expansion property of the pressure supports. As a corollary, we derive upper bounds for the Hausdorff dimensions of the free boundaries and show that the limiting free boundary has finite $(d-1)$-dimensional Hausdorff measure.
While much literature on chemotaxis systems focuses on bounded domains, this paper emphasizes the global existence of classical solutions for three primary chemotaxis systems with a logistic source on $\mathbb{R}^n$. We present a unified proof demonstrating global existence of solutions can be deduced from their locally uniform boundedness in $L^p(\mathbb{R}^n)$ for some $p>\max\{1,\frac{n}{2}\}$. We then provide sufficient conditions for the global existence and boundedness of classical solutions. Notably, our findings even improve several existing results for bounded domains.
This paper is concerned with the problem of budget control in a large particle system modeled by stochastic differential equations involving hitting times, which arises from considerations of systemic risk in a regional financial network. Motivated by Tang and Tsai (Ann. Probab., 46(2018), pp. 1597{1650), we focus on the number or proportion of surviving entities that never default to measure the systemic robustness. First we show that both the mean-field particle system and its limiting McKean-Vlasov equation are well-posed by virtue of the notion of minimal solutions. We then establish a connection between the proportion of surviving entities in the large particle system and the probability of default in the limiting McKean-Vlasov equation as the size of the interacting particle system N tends to infinity. Finally, we study the asymptotic efficiency of budget control in different economy regimes: the expected number of surviving entities is of constant order in a negative economy; it is of order of the square root of N in a neutral economy; and it is of order N in a positive economy where the budget's effect is negligible.
We study the reaction-fractional-diffusion equation $u_t+(-Δ)^{s} u=f(u)$ with ignition and monostable reactions $f$, and $s\in(0,1)$. We obtain the first optimal bounds on the propagation of front-like solutions in the cases where no traveling fronts exist. Our results cover most of these cases, and also apply to propagation from localized initial data.
Motivated by numerical challenges in first-order mean field games (MFGs) and the weak noise theory for the Kardar-Parisi-Zhang equation, we consider the problem of vanishing viscosity approximations for MFGs. We provide the first results on the convergence rate to the vanishing viscosity limit in mean field games, with a focus on the dimension dependence of the rate exponent. Two cases are studied: MFGs with a local coupling and those with a nonlocal, regularizing coupling. In the former case, we use a duality approach and our results suggest that there may be a phase transition in the dimension dependence of vanishing viscosity approximations in terms of the growth of the Hamiltonian and the local coupling. In the latter case, we rely on the regularity analysis of the solution, and derive a faster rate compared to MFGs with a local coupling. A list of open problems are presented.
We prove time-dependent versions of Kingman's subadditive ergodic theorem, which can be used to study stochastic processes as well as propagation of solutions to PDE in time-dependent environments.
We prove stochastic homogenization for reaction-advection-diffusion equations with random space-time-dependent KPP reactions with temporal correlations that are decaying in an appropriate sense. We show that the limiting homogenized dynamic has the simple form of spreading with some deterministic direction-dependent speeds from the support of the initial datum. We obtain analogous results for G-equations with random flame speeds and incompressible background advections. Important ingredients in our proofs are a non-autonomous subadditive theorem and the principle of virtual linearity for KPP reactions from the companion papers [30, 35].