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Jianlu Zhang

Publications and source records attributed to Jianlu Zhang.

28 records · Page 2Linked to original sources
arXiv2020

Global Behaviors of weak KAM Solutions for exact symplectic Twist Maps

We investigated several global behaviors of the weak KAM solutions $u_c(x,t)$ parametrized by $c\in H^1(\mathbb T,\mathbb R)$. For the suspended Hamiltonian $H(x,p,t)$ of the exact symplectic twist map, we could find a family of weak KAM solutions $u_c(x,t)$ parametrized by $c(σ)\in H^1(\mathbb T,\mathbb R)$ with $c(σ)$ continuous and monotonic and \[ \partial_tu_c+H(x,\partial_x u_c+c,t)=α(c),\quad \text{a.e.\ } (x,t)\in\mathbb T^2, \] such that sequence of weak KAM solutions $\{u_c\}_{c\in H^1(\mathbb T,\mathbb R)}$ is $1/2-$Hölder continuity of parameter $σ\in \mathbb{R}$. Moreover, for each generalized characteristic (no matter regular or singular) solving \[ \left\{ \begin{aligned} &\dot{x}(s)\in \text{co} \Big[\partial_pH\Big(x(s),c+D^+u_c\big(x(s),s+t\big),s+t\Big)\Big], & \\ &x(0)=x_0,\quad (x_0,t)\in\mathbb T^2,& \end{aligned} \right. \] we evaluate it by a uniquely identified rotational number $ω(c)\in H_1(\mathbb T,\mathbb R)$. This property leads to a certain topological obstruction in the phase space and causes local transitive phenomenon of trajectories. Besides, we discussed this applies to high-dimensional cases.

math.DS↗
arXiv2020

Convergence of viscosity solutions of generalized contact Hamilton-Jacobi equations

For any compact connected manifold $M$, we consider the generalized contact Hamiltonian $H(x,p,u)$ defined on $T^*M\times\mathbb R$ which is conex in $p$ and monotonically increasing in $u$. Let $u_ε^-:M\rightarrow\mathbb R$ be the viscosity solution of the parametrized contact Hamilton-Jacobi equation \[ H(x,\partial_x u_ε^-(x),εu_ε^-(x))=c(H) \] with $c(H)$ being the Mañé Critical Value. We prove that $u_ε^-$ converges uniformly, as $ε\rightarrow 0_+$, to a specfic viscosity solution $u_0^-$ of the critical equation \[ H(x,\partial_x u_0^-(x),0)=c(H) \] which can be characterized as a minimal combination of associated Peierls barrier functions.

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arXiv2016

Generically Mane set supports uniquely ergodic measure for residual cohomology class

In this paper, we proved that for generic Tonelli Lagrangian, there always exists a residual set $\mathcal{G}\subset H^1(M,\mathbb{R})$ such that \[ \widetilde{\mathcal{M}}(c)=\widetilde{\mathcal{A}}(c)=\widetilde{\mathcal{N}}(c),\quad \forall c\in\mathcal{G} \] with $\widetilde{\mathcal{M}}(c)$ supports on a uniquely ergodic measure.

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arXiv2016

Suspension of the Billiard maps in the Lazutkin's coordinate

In this paper we proved that under the Lazutkin's coordinate, the billiard map can be interpolated by a time-1 flow of a Hamiltonian $H(x,p,t)$ which can be formally expressed by \[ H(x,p,t)=p^{3/2}+p^{5/2}V(x,p^{1/2},t),\quad(x,p,t)\in\T\times[0,+\infty)\times\T, \] where $V(\cdot,\cdot,\cdot)$ is $C^{r-5}$ smooth if the convex billiard boundary is $C^r$ smooth. Benefit from this suspension we can construct transitive trajectories between two adjacent caustics under a variational framework.

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arXiv2015

Normally Hyperbolic Invariant Laminations and diffusive behaviour for the generalized Arnold example away from resonances

In this paper we study existence of Normally Hyperbolic Invariant Laminations (NHIL) for a nearly integrable system given by the product of the pendulum and the rotator perturbed with a small coupling between the two. This example was introduced by Arnold. Using a {\it separatrix map}, introduced in a low dimensional case by Zaslavskii-Filonenko and studied in a multidimensional case by Treschev and Piftankin, for an open class of trigonometric perturbations we prove that NHIL do exist. Moreover, using a second order expansion for the separatrix map from [GKZ], we prove that the system restricted to this NHIL is a skew product of nearly integrable cylinder maps. Application of the results from [CK] about random iteration of such skew products show that in the proper $\varepsilon$-dependent time scale the push forward of a Bernoulli measure supported on this NHIL weakly converges to an Ito diffusion process on the line as $\varepsilon$ tends to zero.

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arXiv2014

Asymptotic trajectories of KAM torus

In this paper we construct a certain type of nearly integrable systems of two and a half degrees of freedom: \[H(p,q,t)=h(p)+εf(p,q,t),\quad (q,p)\in T^{*}\mathbb{T}^2,t\in \mathbb{S}^1=\mathbb{R}/\mathbb{Z}, \] with a self-similar and weak-coupled $f(p,q,t)$ and $h(p)$ strictly convex. For a given Diophantine rotation vector $\vecω$, we can find asymptotic orbits towards the KAM torus $\mathcal{T}_ω$, which persists owing to the classical KAM theory, as long as $ε\ll1$ sufficiently small and $f\in C^r(T^{*}\mathbb{T}^2\times\mathbb{S}^1,\mathbb{R})$ properly smooth. The construction bases on the new methods developed in {\it a priori} stable Arnold Diffusion problem by Chong-Qing Cheng. As an expansion of that, this paper sheds some light on the seeking of much preciser diffusion orbits.

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arXiv2013

Optimal Transportation for Generalized Lagrangian

In this paper, we study the optimal transportation for generalized Lagrangian $L=L(x, u,t)$, and consider the cost function as following: $$c(x, y)=\inf_{\substack{x(0)=x\\x(1)=y\\u\in\mathcal{U}}}\int_0^1L(x(s), u(x(s),s), s)ds.$$ Where $\mathcal{U}$ is a control set, and $x$ satisfies the following ordinary equation: $$\dot{x}(s)=f(x(s),u(x(s),s)).$$ We prove that under the condition that the initial measure $μ_0$ is absolutely continuous w.r.t. the Lebesgue measure, the Monge problem has a solution, and the optimal transport map just walks along the characteristic curves of the corresponding Hamilton-Jacobi equation: \begin{equation*} \begin{cases} V_t(t, x)+\sup_{\substack{u\in\mathcal{U}}} =0.\\ V(0,x)=ϕ_0(x) \end{cases} \end{equation*}

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