Parameterized Viscosity Solutions of Convex Hamiltonian Systems with Time Periodic Damping

Abstract

In this article we develop an analogue of Aubry-Mather theory for time periodic dissipative equation

$$\begin{aligned} \left\{ \begin{aligned} \dot{x}&=\partial _p H(x,p,t),\\ \dot{p}&=-\partial _x H(x,p,t)-f(t)p \end{aligned} \right. \end{aligned}$$

with $(x,p,t)\in T^*M\times \mathbb {T}$ (compact manifold M without boundary). We discuss the asymptotic behaviors of viscosity solutions for the associated Hamilton-Jacobi equation

$$\begin{aligned} \partial _t u+f(t)u+H(x,\partial _x u,t)=0,\quad (x,t)\in M\times \mathbb {T}\end{aligned}$$

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.

Appendices

Appendix A: Mather Measure of Convex Lagrangians with $[f]=0$

For a Tonelli Hamiltonian H(x, p, t), the conjugated Lagrangian L(x, v, t) can be established by 6, which is also Tonelli. On the other side, for $[f]=0$, the following Lagrangian

$$\begin{aligned} \widetilde{L}(x,v,t):=e^{F(t)}L(x,v,t),\quad (x,v,t)\in TM\times \mathbb {T}\end{aligned}$$

with

$$\begin{aligned} F(t):=\int _0^tf(s)\text{ d }s \end{aligned}$$

is still time-periodic as the case considered in [23]. Besides, the Euler-Lagrange equation associated with $\widetilde{L}$ is the same with E-L. So Mañé’s approach to get a Mather measure in [20] is still available for us. As his approach doesn’t rely on the E-L flow, that supplies us with great convenience.

Let X be a metric separable space. A probability measure on X is a nonnegative, countably additive set function $\mu$ defined on the $\sigma -$algebra $\mathscr {B}(X)$ of Borel subsets of X such that $\mu (X) = 1$. In this paper, $X=TM\times \mathbb {T}$. We say that a sequence of probability measures $\{\tilde{\mu }_n \}_{n\in \mathbb N}$ (weakly) converges to a probability measure $\tilde{\mu }$ on $TM\times \mathbb {T}$ if

$$\begin{aligned} \lim _{n\rightarrow +\infty }\int _{TM\times \mathbb {T}}h(x,v,t)\text{ d }\tilde{\mu }_n(x,v,t)=\int _{TM\times \mathbb {T}} h(x,v,t)\text{ d }\tilde{\mu }(x,v,t) \end{aligned}$$

for any $h\in C_c(TM\times \mathbb {T},\mathbb {R})$.

Definition A.1

A probability measure $\tilde{\mu }$ on $TM\times \mathbb {T}$ is called closed if it satisfies:

• $\int _{TM\times \mathbb {T}}|v|\text{ d }\tilde{\mu }(x,v,t)<+\infty$;

• $\int _{TM\times \mathbb {T}}\langle \partial _x\phi (x,t),v\rangle +\partial _t\phi (x,t) \text{ d }\tilde{\mu }(x,v,t)=0$ for every $\phi \in C^1(M\times \mathbb {T},\mathbb {R})$.

Let’s denote by ${\mathbb P}_c(TM\times \mathbb {T})$ the set of all closed measures on $TM\times \mathbb {T}$, then the following conclusion is proved in [20]:

Theorem A.2

$$\begin{aligned} \min _{\tilde{\mu }\in {\mathbb P}_c(TM\times \mathbb {T})}\int _{TM}\widetilde{L}(x,v,t)\text{ d }\tilde{\mu }(x,v,t)=-c(H). \end{aligned}$$

Moreover, the minimizer $\tilde{\mu }_{\min }$ must be a Mather measure, i.e. $\tilde{\mu }_{\min }$ is invariant w.r.t. the Euler-Lagrange flow E-L.

Proof

This conclusion is a direct adaption of Proposition 1.3 of [20] to our system $\widetilde{L}(x,v,t)$, with the c(H) already given in 13.$\square$

Appendix B: Semiconcave Functions

Here we attach a series of conclusions about the semiconcave functions which can be found in [5], for the use of Proposition 3.8.

Definition B.1

Assume S is a subset of $\mathbb {R}^n$. A function $u:S\rightarrow \mathbb {R}$ is called semiconcave, if there exists a nondecreasing upper semicontinuous function $\omega :\mathbb {R}^+\rightarrow \mathbb {R}^+$ such that $\lim _{\rho \rightarrow 0^+}\omega (\rho )=0$ and

$$\begin{aligned} \lambda u(x)-(1-\lambda )u(y)-u(\lambda x+(1-\lambda )y)\le \lambda (1-\lambda )|x-y|\omega (|x-y|),\forall \lambda \in [0,1]. \end{aligned}$$

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We call $\omega$ a modulus of semiconcavity for u in S.

Definition B.2

[Definition 3.1.1 in [5]] For any $x\in S$, the set

$$\begin{aligned} D^+u(x)=\bigg \{p\in \mathbb {R}^n| \limsup _{y\rightarrow x}\frac{u(y)-u(x)-\langle p,y-x\rangle }{|y-x|}\le 0\bigg \} \end{aligned}$$

is called the Fréchet superdifferential of u at x. We shall give some properties of $D^+u(x)$, which can be found in Chapter 3 of [5].

Proposition B.3

Assume $A\subset \mathbb {R}^n$ is open. Let $u:A\rightarrow \mathbb {R}$ be a semiconcave function with modulus $\omega$ and $x\in A$. Then,

• $D^+u(x)\not =\emptyset$.

• $D^+u(x)$ is a closed, convex set of $T_x^*A\cong \mathbb {R}^n$

• If $D^+u(x)$ is a singleton, then u is differentiable at x.

• If A is also convex, $p\in D^+u(x)$ if and only if

  $$\begin{aligned} u(y)-u(x)-\langle p, y-x\rangle \le |y-x|\omega (|y-x|) \end{aligned}$$

  for each $y\in A$.

Theorem B.4

[Theorem 3.2 in [6]] Let $u\in Lip_{loc}(\Omega \times (0,T))$ be a viscosity solution of

$$\begin{aligned} \partial _tu+G(x,\partial _xu,t,u)=0 \end{aligned}$$

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where $G\in Lip_{loc}(\Omega \times \mathbb {R}^n\times (0,T)\times \mathbb {R},\mathbb {R})$ is strictly convex in the second group of variables. Then u is locally semiconcave in $\Omega \times (0,T)$.

Theorem B.5

[Proposition 3.3.4, Theorem 3.3.6 in [5]] For any $(x,t)\in \Omega \times (0,T)$, we define the reachable derivative set of any viscosity solution u of 55 by

$$\begin{aligned}D^*u(x,t):=\big \{(p_x,p_t)=\lim _{n\rightarrow +\infty }(\partial _x u(x_n,t_n),\partial _t u(x_n,t_n))\in T^*_x\Omega \times T_t^*(0,T)\big |\\ \exists \ (x_n,t_n)_{n\in \mathbb {Z}_+}\in \Omega \times (0,T) \text {converging to ({ x},\,{ t}), at which { u} is differentiable}\big \}. \end{aligned}$$

Consequently, $D^+u(x,t)=co (D^*u(x,t))$ i.e. any superdifferential of u at (x, t) is a convex combination of elements in $D^*u(x,t)$.