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Qinbo Chen

Publications and source records attributed to Qinbo Chen.

12 recordsLinked to original sources
arXiv2026

A new selection problem for degenerate viscous Hamilton-Jacobi equations

We study a selection problem for degenerate viscous Hamilton--Jacobi equations with convex Hamiltonians, in which the approximation procedure combines a nonlinear discounted approximation with a small potential perturbation. A key question is how their simultaneous effects influence the asymptotic selection of viscosity solutions of the associated ergodic problem. Based on the nonlinear adjoint method, we establish the uniform convergence of the approximating solutions to a distinguished solution of the ergodic problem and derive a formula for the selected limit in terms of generalized Mather measures and the potential. As an application, we show that this selection principle is sufficiently flexible to realize any prescribed solution of the ergodic problem, with an explicit convergence rate.

math.AP↗
arXiv2024

The selection problem for a new class of perturbations of Hamilton-Jacobi equations and its applications

This paper studies a perturbation problem given by the equation: \begin{equation*} H(x, d_xu_λ, λu_λ(x))+λV(x,λ)=c \quad \text{in $M$}, \end{equation*} where $M$ is a closed manifold and $λ>0$ is a perturbation parameter. The Hamiltonian $H(x,p,u):T^*M\times \mathbb{R}\to \mathbb{R}$ satisfies certain convexity, superlinearity, and monotonicity conditions. $λV(\cdot,λ):M\to\mathbb{R}$ converges to zero as $λ\to 0$. First, we study the asymptotic behavior of the viscosity solution $u_λ:M\to\mathbb{R}$ as $λ$ approaches zero. This perturbation problem explores the combined effects of both the vanishing discount process and potential perturbations, leading to a new selection principle that extends beyond the classical vanishing discount approach. Additionally, we apply this principle to Hamilton-Jacobi equations with $u$-independent Hamiltonians, resulting in the introduction of a new solution operator. This operator provides new insights into the variational characterization of viscosity solutions and Mather measures.

math.AP↗
arXiv2023

Rigidity properties for some isometric extensions of partially hyperbolic actions on the torus

This paper studies local rigidity for some isometric toral extensions of partially hyperbolic $\mathbb{Z}^k$ ($k\geqslant 2$) actions on the torus. We prove a $C^\infty$ local rigidity result for such actions, provided that the smooth perturbations of the actions satisfy the intersection property. We also give a local rigidity result within a class of volume preserving actions. Our method mainly uses a generalization of the KAM iterative scheme.

math.DS↗
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

On simultaneous linearization of certain commuting nearly integrable diffeomorphisms of the cylinder

Let $\mathcal{F}$ and $\mathcal{K}$ be commuting $C^\infty$ diffeomorphisms of the cylinder $\mathbb{T}\times\mathbb{R}$ that are, respectively, close to $\mathcal{F}_0 (x, y)=(x+ω(y), y)$ and $T_α(x, y)=(x+α, y)$, where $ω(y)$ is non-degenerate and $α$ is Diophantine. Using the KAM iterative scheme for the group action we show that $\mathcal{F}$ and $\mathcal{K}$ are simultaneously $C^\infty$-linearizable if $\mathcal{F}$ has the intersection property (including the exact symplectic maps) and $\mathcal{K}$ satisfies a semi-conjugacy condition. We also provide examples showing necessity of these conditions. As a consequence, we get local rigidity of certain class of $\mathbb{Z}^2$-actions on the cylinder, generated by commuting twist maps.

math.DS↗
arXiv2021

Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems

The genericity of Arnold diffusion in the analytic category is an open problem. In this paper, we study this problem in the following a priori unstable Hamiltonian system with a time-periodic perturbation \[\mathcal{H}_\varepsilon(p,q,I,φ,t)=h(I)+\sum_{i=1}^n\pm \left(\frac{1}{2}p_i^2+V_i(q_i)\right)+\varepsilon H_1(p,q,I,φ, t), \] where $(p,q)\in \mathbb{R}^n\times\mathbb{T}^n$, $(I,φ)\in\mathbb{R}^d\times\mathbb{T}^d$ with $n, d\geq 1$, $V_i$ are Morse potentials, and $\varepsilon$ is a small non-zero parameter. The unperturbed Hamiltonian is not necessarily convex, and the induced inner dynamics does not need to satisfy a twist condition. Using geometric methods we prove that Arnold diffusion occurs for generic analytic perturbations $H_1$. Indeed, the set of admissible $H_1$ is $C^ω$ dense and $C^3$ open (a fortiori, $C^ω$ open). Our perturbative technique for the genericity is valid in the $C^k$ topology for all $k\in [3,\infty)\cup\{\infty, ω\}$.

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arXiv2021

Exponential stability of fast driven systems, with an application to celestial mechanics

We construct a normal form suited to {\it fast driven systems}. We call so systems including actions ${\rm I}$, angles {$ψ$}, and one fast coordinate $y$, moving under the action of a vector--field $N$ depending only on ${\rm I}$ and $y$ and with vanishing ${\rm I}$--components. {In absence of the coordinate $y$, such systems have been extensively investigated and it is known that, after a small perturbing term is switched on, the normalised actions ${\rm I}$ turn to have exponentially small variations compared to the size of the perturbation. We obtain the same result of the classical situation, with the additional benefit that } no trapping argument is needed, as no small denominator arises. {We use the result to prove that, in the three--body problem, the level sets of a certain function called {\it Euler integral} have exponentially small variations in a short time, closely to collisions.}

math.DS↗
arXiv2021

Convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function

Motivated by the vanishing contact problem, we study in the present paper the convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function. Let $H(x,p,u)$ be a continuous Hamiltonian which is strictly increasing in $u$, and is convex and coercive in $p$. For each parameter $λ>0$, we denote by $u^λ$ the unique viscosity solution of the H-J equation \[H( x,Du(x),λu(x) )=c.\] Under quite general assumptions, we prove that $u^λ$ converges uniformly, as $λ$ tends to zero, to a specific solution of the critical H-J equation $ H(x,Du(x),0)=c.$ We also characterize the limit solution in terms of Peierls barrier and Mather measures.

math.AP↗
arXiv2020

Gevrey genericity of Arnold diffusion in a priori unstable Hamiltonian systems

It is well known that under generic $C^r$ smooth perturbations, the phenomenon of global instability, known as Arnold diffusion, exists in a priori unstable Hamiltonian systems. In this paper, by using variational methods, we will prove that under generic Gevrey smooth perturbations, Arnold diffusion still exists in the a priori unstable Hamiltonian systems of two and a half degrees of freedom.

math.DS↗
arXiv2018

Vanishing contact structure problem and convergence of the viscosity solutions

This paper is devoted to study the vanishing contact structure problem which is a generalization of the vanishing discount problem. Let $H^λ(x,p,u)$ be a family of Hamiltonians of contact type with parameter $λ>0$ and converges to $G(x,p)$. For the contact type Hamilton-Jacobi equation with respect to $H^λ$, we prove that, under mild assumptions, the associated viscosity solution $u^λ$ converges to a specific viscosity solution $u^0$ of the vanished contact equation. As applications, we give some convergence results for the nonlinear vanishing discount problem.

math.AP↗
arXiv2016

Regular dependence of the Peierls barriers on perturbations

Let $f$ be an exact area-preserving monotone twist diffeomorphism of the infinite cylinder and $P_{ω,f}(ξ)$ be the associated Peierls barrier. In this paper, we give the Hölder regularity of $P_{ω,f}(ξ)$ with respect to the parameter $f$. In fact, we prove that if the rotation symbol $ω\in (\mathbb{R}\setminus\mathbb{Q})\bigcup(\mathbb{Q}+)\bigcup(\mathbb{Q}-)$, then $P_{ω,f}(ξ)$ is $1/3$-Hölder continuous in $f$, i.e. $$|P_{ω,f'}(ξ)-P_{ω,f}(ξ)|\leq C\|f'-f\|_{C^1}^{1/3} ,~~\forall ξ\in\mathbb{R}$$ where $C$ is a constant. Similar results also hold for the Lagrangians with one and a half degrees of freedom. As application, we give an open and dense result about the breakup of invariant circles.

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