Searcharxiv⌕ Search

arXiv subjects

Jianlu Zhang

Publications and source records attributed to Jianlu Zhang.

At least 19 recordsLinked to original sources
arXiv2026

Convergence Rate of Birkhoff Average for Toral Quasi-Periodic Rotations and Applications

In this paper, we establish quantitative Denjoy--Koksma type estimates for higher-dimensional quasi-periodic torus rotations. For Diophantine frequency vectors, we establish quantitative estimates on the discrepancy between Birkhoff averages and spatial averages for observables with various Besov-type regularities. By means of suitable Sobolev embeddings, these estimates yield, to the best of our knowledge, the sharpest currently available convergence rates for Hölder continuous observables. As applications, we obtain substantially improved quantitative homogenization results for Hamilton--Jacobi equations in spatially quasi-periodic settings, as well as nearly optimal statistical regularity estimates for invariant measures under perturbations.

math.AP↗
arXiv2026

Quantitative homogenization for static contact Hamilton-Jacobi equations

We characterize possible pairs $(u_\varepsilon,c)\in C(\mathbb{R}^n\backslash\varepsilon\mathbb{Z}^n,\mathbb{R})\times\mathbb{R}$ addressing the homogenization problem for Hamilton--Jacobi equations $$ H\left(\frac{x}{\varepsilon}, d u_\varepsilon, u_\varepsilon\right)=c, \quad \left({\mathrm resp.} \quad H\left(\frac{x}{\varepsilon}, d u_\varepsilon, u_\varepsilon\right)=\varepsilonΔu_\varepsilon+c \right) $$ for all $\varepsilon>0$. Under a (not necessarily strict) monotonicity assumption on the Hamiltonian, we proposed certain criteria (based on the structure of Mather measures), under which all possible solutions $u_\varepsilon$ converge to a uniquely identified limit $u\in C(\mathbb{R}^n,\mathbb{R})$ solving the effective equation \[ \overline H( du,u)=c,\quad ({\mathrm resp.}\quad \overline H(du,u)=Δu+c) \] as $\varepsilon\rightarrow 0_+$ with a uniform rate $\mathcal{O}(\varepsilon)$.

math.AP↗
arXiv2026

Statistical regularity and linear response of Mather measures for Tonelli Lagrangian systems

We study the statistical regularity of Mather measures associated with $C^1$ perturbations of a Tonelli Lagrangian. When the unperturbed Mather measure is supported on a quasi-periodic torus with a Diophantine frequency, we establish Hölder continuity of the perturbed Mather measure with respect to the perturbation parameter. The Hölder exponent is shown to depend explicitly on the Diophantine index of the frequency. We also discuss the possibility of achieving Lipschitz regularity using KAM theory.

math.DS↗
arXiv2025

Generalized comparison principle for contact Hamilton-Jacobi equations

In this paper, we discuss all the possible pairs $(u,c)\in C(M,\mathbb R)\times\mathbb R$ solving (in the sense of viscosity) the contact Hamilton-Jacobi equation \[ H (x, d_xu, u) = c,\quad x\in M \] of which $M$ is a closed manifold and the continuous Hamiltonian $H: (x,p,u)\in T^*M\times\mathbb R\rightarrow\mathbb R$ is convex, coercive in $p$ but merely non-decreasing in $u$. Firstly, we propose a comparison principle for solutions by using the dynamical information of Mather measures. We then describe the structure of $\mathfrak C$ containing all the $c\in\mathbb R$ makes previous equation solvable. We also propose examples to verify the optimality of our approach.

math.DS↗
arXiv2025

On the vanishing viscosity limit of Hamilton-Jacobi equations with nearly optimal discount

In this paper, we establish the convergence of solutions to the viscous Hamilton-Jacobi equation (with a Tonelli Hamiltonian): \[ λu +H(x, du)=\varepsilon(λ)Δu,\quad λ>0 \] as $λ\rightarrow 0_+$, once the modulus $\varepsilon(λ)$ satisfies $\varlimsup_{λ\rightarrow 0_+}\varepsilon(λ)/λ=0$. Such an exponent of $\varepsilon(λ)$ is nearly optimal in the convergence.

math.AP↗
arXiv2025

Vanishing discount limits for first-order fully nonlinear Hamilton-Jacobi equations on noncompact domains

We study the asymptotic behavior of solutions to the fully nonlinear Hamilton-Jacobi equation $H(x, Du, λu) = 0$ in $\mathbb{R}^n$ as $λ\to 0^+$. Under the assumption that the Aubry set is localized, we employ a variational approach to derive limiting Mather-type measures and formulate a selection principle. Central to our analysis is a modified variational formula that bridges global and local state-constraint solutions, thereby extending localization techniques to the nonlinear framework.

math.AP↗
arXiv2024

Polynomial Convergence Rate for Quasi-Periodic Homogenization of Hamilton-Jacobi Equations and Application to Ergodic Estimates

In this paper, we demonstrate a polynomial convergence rate for homogenization of Hamilton-Jacobi equations with quasi-periodic potentials. We establish a connection between the convergence rate of homogenization and the regularity of the effective Hamiltonian, by using a new quantitative ergodic estimate for bounded quasi-periodic functions with Diophantine frequencies. As an application, we also study the convergent rate for Birkhoff average of unbounded quasi-periodic functions.

math.AP↗
arXiv2024

Smooth subsolutions of the discounted Hamilton-Jacobi equations

For the discounted Hamilton-Jacobi equation,$$λu+H(x,d_x u)=0, \ x \in M, $$we construct $C^{1,1}$ subsolutions which are indeed solutions on the projected Aubry set. The smoothness of such subsolutions can be improved under additional hyperbolicity assumptions. As applications, we can use such subsolutions to identify the maximal global attractor of the associated conformally symplectic flow and to control the convergent speed of the Lax-Oleinik semigroups

math.DS↗
arXiv2024

Generalized convergence of solutions for nonlinear Hamilton-Jacobi equations with state-constraint

For a continuous Hamiltonian $H : (x, p, u) \in T^*\mathbb{R}^n \times \mathbb{R}\rightarrow \mathbb{R}$, we consider the asymptotic behavior of associated Hamilton--Jacobi equations with state-constraint $H(x, Du, λu) \leq C_λ$ in $Ω_λ\subset \mathbb{R}^n$ and $H(x, Du, λu) \geq C_λ$ on $\overlineΩ_λ\subset \mathbb{R}^n$ a $λ\rightarrow 0^+$. When $H$ satisfies certain convex, coercive, and monotone conditions, the domain $Ω_λ:=(1+r(λ))Ω$ keeps bounded, star-shaped for all $λ>0$ with $\lim_{λ\rightarrow 0^+}r(λ)=0$, and $\lim_{λ\rightarrow 0^+}C_λ=c(H)$ equals the ergodic constant of $H(\cdot,\cdot,0)$, we prove the convergence of solutions $u_λ$ to a specific solution of the critical equation $H(x, Du, 0)\leq c(H) $ in $Ω$ and $H(x, Du, 0)\geq c(H) $ on $\overlineΩ$. We also discuss the generalization of such a convergence for equations with more general $C_λ$ and $Ω_λ$.

math.AP↗
arXiv2023

Convergence of the solutions of the nonlinear discounted Hamilton-Jacobi equation: The central role of Mather measures

Given a continuous Hamiltonian $H : (x,p,u) \mapsto H(x,p,u)$ defined on $ T^*M \times \mathbb R $, where $M$ is a closed connected manifold, we study viscosity solutions, $u_λ: M\to \mathbb R$, of discounted equations: $ H(x, d_x u_λ, λu_λ(x))=c$ in $M$, where $λ>0$ is called a discount factor and $c$ is the critical value of $H(\cdot, \cdot , 0)$. When $H$ is convex and superlinear in $p$ and non--decreasing in $u$, under an additional non--degeneracy condition, we obtain existence and uniqueness (with comparison principles) results of solutions and we prove that the family of solutions $(u_λ)_{λ>0}$ converges to a specific solution $u_0$ of $ H(x, d_x u_0, 0)=c$ in $M$. Our degeneracy condition requires $H$ to be increasing (in $u$) on localized regions linked to the support of Mather measures, whereas usual similar results are obtained for Hamiltonians that are everywhere increasing in $u$.

math.AP↗
arXiv2022

Limit of solutions for semilinear Hamilton-Jacobi equations with degenerate viscosity

In the paper we prove the convergence of viscosity solutions $u_λ$ as $λ\rightarrow0_+$ for the parametrized degenerate viscous Hamilton-Jacobi equation \[ H(x,d_x u, λu)=α(x)Δu,\quad α(x)\geq 0,\quad x\in \mathbb T^n \] under suitable convex and monotonic conditions on $H: T^*M\times\mathbb R\rightarrow\mathbb R$. Such a limit can be characterized in terms of stochastic Mather measures associated with the critical equation \[ H(x,d_x u,0)=α(x)Δu. \]

math.AP↗
arXiv2022

Variational attraction of the KAM torus for the conformally symplectic system

For the conformally symplectic system \[ \left\{ \begin{aligned} \dot{q}&=H_p(q,p),\quad(q,p)\in T^*\mathbb{T}^n\\ \dot p&=-H_q(q,p)-λp, \quad λ>0 \end{aligned} \right. \] with a positive definite Hamiltonian, we discuss the variational significance of invariant Lagrangian graphs and explain how the KAM torus impacts the $W^{1,\infty}-$convergence speed of the Lax-Oleinik semigroup.

math.DS↗
arXiv2021

On the negative limit of viscosity solutions for discounted Hamilton-Jacobi equations

Suppose $M$ is a closed Riemannian manifold. For a $C^2$ generic (in the sense of Mañé) Tonelli Hamiltonian $H: T^*M\rightarrow\mathbb{R}$, the minimal viscosity solution $u_λ^-:M\rightarrow \mathbb{R}$ of the negative discounted equation \[-λu+H(x,d_xu)=c(H),\quad x\in M,\ λ>0 \] with the Mañé's critical value $c(H)$ converges to a uniquely established viscosity solution $u_0^-$ of the critical Hamilton-Jacobi equation \[ H(x,d_x u)=c(H),\quad x\in M \] as $λ\rightarrow 0_+$. We also propose a dynamical interpretation of $u_0^-$.

math.AP↗
arXiv2021

Parameterized viscosity solutions of convex Hamiltonian systems with time periodic damping

In this article we develop an analogue of Aubry Mather theory for time periodic dissipative equation \[ \left\{ \begin{aligned} \dot x&=\partial_p H(x,p,t),\\ \dot p&=-\partial_x H(x,p,t)-f(t)p \end{aligned} \right. \] with $(x,p,t)\in T^*M\times\mathbb T$ (compact manifold $M$ without boundary). We discuss the asymptotic behaviors of viscosity solutions of associated Hamilton-Jacobi equation \[ \partial_t u+f(t)u+H(x,\partial_x u,t)=0,\quad(x,t)\in M\times\mathbb T \] w.r.t. certain parameters, and analyze the meanings in controlling the global dynamics. We also discuss the prospect of applying our conclusions to many physical models.

math.DS↗
arXiv2021

Essential forward weak KAM solution for the convex Hamilton-Jacobi equation

For a convex, coercive continuous Hamiltonian on a compact closed Riemannian manifold $M$, we construct a unique forward weak KAM solution of \[ H(x, d_x u)=c(H) \] by a vanishing discount approach, where $c(H)$ is the Mañé critical value. We also discuss the dynamical significance of such a special solution.

math.DS↗
arXiv2020

Oscillatory orbits in the Restricted Planar 4 Body Problem

The restricted planar four body problem describes the motion of a massless body under the Newtonian gravitational force of other three bodies (the primaries), of which the motion gives us general solutions of the three body problem. A trajectory is called {\it oscillatory} if it goes arbitrarily faraway but returns infinitely many times to the same bounded region. We prove the existence of such type of trajectories provided the primaries evolve in suitable periodic orbits.

math.DS↗