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arXiv · 2105.10667

Parameterized viscosity solutions of convex Hamiltonian systems with time periodic damping

Ya-Nan Wang·Jun Yan·Jianlu Zhang
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Abstract

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.

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Ya-Nan Wang, Jun Yan, Jianlu Zhang. 2021-05-22. Parameterized viscosity solutions of convex Hamiltonian systems with time periodic damping. https://arxiv.org/abs/2105.10667

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