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.

1 Introduction

For a smooth compact Riemannian manifold M without boundary, the Hamiltonian H is usually characterized as a $C^{r\ge 2}-$smooth function on the cotangent bundle $T^*M$, with the associated Hamilton’s equation defined by

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

(1)

for each initial point $(x,p)\in T^*M$. From the physical aspect, the Hamilton’s equation describes the movement of particles with conservative energy, since the Hamiltonian H(x, p) verifies to be a First Integral of (1). In particular, if the potential periodically depends on the time t (for systems with periodic propulsion or procession), we can introduce an augmented Hamiltonian

$$\begin{aligned} \widetilde{H}(x,p,t,I)=I+H(x,p,t),\quad \quad (x,p,t,I)\in T^*M\times T^*\mathbb {T}\end{aligned}$$

(2)

such that the associated Hamilton’s equation

$$\begin{aligned} \mathsf{(Conservative)\quad }\left\{ \begin{aligned} \dot{x}&=\partial _p H(x,p,t)\\ \dot{p}&=-\partial _x H(x,p,t)\\ \dot{t}&=1\\ \dot{I}&=-\partial _t H(x,p,t) \end{aligned} \right. \end{aligned}$$

(3)

still preserves $\widetilde{H}$.

However, the realistic motion of the masses inevitably sustains a dissipation of energy, due to the friction from the environment, e.g. the wind, the fluid, interface, etc. That urges us to make rational modifications of previous equations. In the current paper, the damping is assumed to be time-periodically proportional to the momentum. Precisely, we modify (3) into

$$\begin{aligned} \mathsf{(Dissipative)\quad }\left\{ \begin{aligned} \dot{x}&=\partial _p H(x,p,t)\\ \dot{p}&=-\partial _x H(x,p,t)-f(t)p\\ \dot{t}&=1\\ \dot{I}&=-\partial _t H(x,p,t)-f'(t)u-f(t)I\\ \dot{u}&=\langle H_p,p\rangle -H+\alpha -f(t)u \end{aligned} \right. \end{aligned}$$

(4)

with $\alpha \in \mathbb {R}$ being a constant of initial energy and $f\in C^{r}(\mathbb {T}:=\mathbb {R}/[0,1], \mathbb {R})$. Notice that the former three equations of (4) is decoupled with the latter two, so we can denote the flow of the former three equations in (4) by $\varphi _H^t$ and by $\widehat{\varphi }_{ H}^t$ the flow of the whole (4). The following individual cases of f(t) will be considered:

• (${\textbf {H0}}^-$) $[f]:=\int _0^1f(t)\text{ d }t>0$

• (${\textbf {H0}}^+$) $[f]<0$

• (${\textbf {H0}}^0)$ $[f]=0$

Besides, we propose the following standing assumptions for the Hamiltonian:

• (H1) [Smoothness] $H:TM\times \mathbb {T}\rightarrow \mathbb {R}$ is $C^{r}$ smooth;

• (H2) [Convexity] For any $(x,t)\in M\times \mathbb {T}$, $H(x,\cdot ,t)$ is strictly convex on $T_x^*M$;

• (H3) [Superlinearity] For any $(x,t)\in M\times \mathbb {T}$, $\lim _{|p|_x\rightarrow +\infty }H(x,p,t)/|p|_x=+\infty$ where $|\cdot |_x$ is the norm deduced from the Riemannian metric.

• (H4) [Completeness] For any $(x,p,\theta )\in T^*M\times \mathbb {T}$, the flow $\varphi _H^t(x,p,\theta )$ exists for all $t\in \mathbb {R}$.

Remark 1.1

• i) As we can see, the three different cases of (H0) respectively leads to a dissipation, acceleration and periodic conservation of energy along $\widehat{\varphi }_H^t$ in the forward time, if we take

  $$\begin{aligned} \widehat{H}(x,p,t,I, u)=\widetilde{H}(x,p,t,I)+f(t)u-\alpha . \end{aligned}$$

  (5)

  See (40) of Section 4 for the proof.

• ii) (H1-H3) are usually called Tonelli conditions. As for (H4), the completeness of $\varphi _H^t$ is actually equivalent to the completeness of $\widehat{\varphi }_H^t$. A sufficient condition to (H4) is the following:

  $$\begin{aligned} |H_x|\le \kappa (1+|p|_x) \text { for all }(x,p,t)\in T^*M\times \mathbb {T}\end{aligned}$$

  for some constant $\kappa$.

• iii) Observe that the time-1 map $\varphi _H^1:\{(x,p,t (\textrm{mod}\; 1)=0)\}\rightarrow \{(x,p,t (\textrm{mod}\; 1)=0)\}$ is conformally symplectic, i.e.

  $$\begin{aligned} (\varphi _H^1)^*\text{ d }p\wedge \text{ d }x=e^{[f]}\text{ d }p\wedge \text{ d }x. \end{aligned}$$

  Such maps have wide applications in astronomy [9], optimal transport [32], biological physics [8] and economics [1], etc (see Sect. 2 for more details).

• iv) For $f\equiv 0$, (4) degenerates to (3), and that is just the case concerned by Mather’s variational theory [23]. Besides, Maro and Sorrentino considered another special setting: $f\equiv \lambda >0$ in [21]. They established an analogue of Aubry-Mather theory under such a setting. Simultaneously, [4, 29, 30] explored the variational properties for contact Hamiltonian systems which was originally proposed by Lions in [18]. In earlier works [2, 11], the authors considered a time-periodic Lagrangian system, which can be considered as a special case of equation (4) with $f\equiv 0$. As the extension of their works, in the current paper we focus on the dynamical mechanisms of damping (or accelerating) phenomena and obtain the attractive (or repulsive) invariant sets for the case $[f]\ne 0$. Such a consideration is quite different.

1.1 Variational Principle and Hamilton-Jacobi Equation

As the dual of the Hamiltonian, the Lagrangian can be defined as the following

$$\begin{aligned} L(x,v,t):=\max _{p\in T^*_xM} \langle p,v\rangle -H(x,p,t),\quad (x,v,t)\in TM\times \mathbb {T}. \end{aligned}$$

(6)

of which the maximum is achieved at $v=H_p(x,p,t)\in T_xM$, once (H1-H3) are assumed. Therefore, the Legendre transformation

$$\begin{aligned} \mathcal {L}: T^*M\times \mathbb {T}\rightarrow TM\times \mathbb {T}, \quad \text {via } (x,p,t)\rightarrow (x, H_p(x,p,t),t) \end{aligned}$$

(7)

is a diffeomorphism. Notice that the Lagrangian $L:TM\times \mathbb {T}\rightarrow \mathbb {R}$ is also $C^r-$smooth and convex, superlinear in $p\in T_x M$, so by a slight abuse of notions we say it satisfies (H1-H3) as well. As a conjugation of $\varphi ^t_H$, the Euler-Lagrangian flow $\varphi _L^t$ is defined by

$$\begin{aligned} \left\{ \begin{array}{l}\dot{x}=\upsilon ,\\ \frac{d}{dt}{L}_{\upsilon }\left( x,\upsilon ,t\right) ={L}_{x}\left( x,\upsilon ,t\right) -f\left( t\right) {L}_{\upsilon }\left( x,\upsilon ,t\right) \\ \dot{t}=1.\end{array}\right. \end{aligned}$$

(E-L)

It has equivalently effective in exploring the dynamics of (4), and the completeness of $\varphi _L^t$ is equivalent to the completeness of $\varphi _H^t$. To present (E-L) a variational characterization, we introduce a minimal variation on the absolutely continuous curves with fixed endpoints

$$\begin{aligned} h_\alpha ^{s,t}(x,y)=\inf _{\begin{array}{c} \gamma \in C^{ac}([s,t],M) \\ \gamma (s)=x,\gamma (t)=y \end{array}}\int ^t_se^{F(\tau )}(L(\gamma ,\dot{\gamma },\tau )+\alpha )\text{ d }\tau , \end{aligned}$$

where $F(t)=\int ^t_0f(\tau )\text{ d }\tau$ and $\alpha \in \mathbb {R}$. It is a classical result in the calculus of variations that the infimum is always available for all $s<t\in \mathbb {R}$, which is actually $C^r-$ smooth and satisfies (E-L), once (H4) is assumed (due to the Weierstrass Theorem in [23] or Theorem 3.7.1 of [15]).

Theorem 1.2

(main 1). For f(t) satisfying (${\textbf {H0}}^-$), H(x, p, t) satisfying (H1-H4) and any $\alpha \in \mathbb {R}$, the following

$$\begin{aligned} u_\alpha ^-(x,t):=\inf _{\begin{array}{c} \gamma \in C^{ac}((-\infty ,t],M)\\ \gamma (t)=x \end{array}}\int ^t_{-\infty }e^{F(s)-F(t)}(L(\gamma (s),\dot{\gamma }(s),s)+\alpha )\text{ d }s \end{aligned}$$

is well defined for $(x,t)\in M\times \mathbb {R}$ and satisfies

• (1) (Periodicity) $u_\alpha ^-(x,t+1)=u_\alpha ^-(x,t)$ for any $x\in M$ and $t\in \mathbb {R}$. By taking $\bar{t}\in [0,1)$ with $t\equiv \bar{t}\ (mod\ 1)$ for any $t\in \mathbb {R}$, we can interpret $u_\alpha ^-$ as a function on $M\times \mathbb {T}$.

• (2) (Lipschitzness) $u_\alpha ^-:M\times \mathbb {T}\rightarrow \mathbb {R}$ is Lipschitz, with the Lipschitz constant depending on L and f;

• (3) (Domination¹) For any absolutely continuous curve $\gamma :[s,t]\rightarrow M$ connecting $(x,\bar{s})\in M\times \mathbb {T}$ and $(y,\bar{t})\in M\times \mathbb {T}$, we have

  $$\begin{aligned} e^{F(t)}u_\alpha ^-(y,\bar{t})-e^{F(s)}u_\alpha ^-(x,\bar{s})\le \int _s^t e^{F(\tau )}\Big (L(\gamma ,\dot{\gamma },\tau )+\alpha \Big )\text{ d }\tau . \end{aligned}$$

  (8)

• (4) (Calibration) For any $(x,\theta )\in M\times \mathbb {T}$, there exists a backward calibrated curve $\gamma _{x,\theta }^-:(-\infty ,\theta ]\rightarrow M$, $C^r-$smooth and $\gamma _{x,\theta }^-(\theta )=x$, such that for all $s\le t\le \theta$, we have

  $$\begin{aligned}{} & {} e^{F(t)}u_\alpha ^-(\gamma _{x,\theta }^-(t),\bar{t})-e^{F( s)}u_\alpha ^-(\gamma _{x,\theta }^-(s),\bar{s})\nonumber \\= & {} \int _s^t e^{F(\tau )}\Big (L(\gamma _{x,\theta }^-,\dot{\gamma }_{x,\theta }^-,\tau )+\alpha \Big )\text{ d }\tau . \end{aligned}$$

  (9)

• (5) (Viscosity) $u_\alpha ^-:M\times \mathbb {T}\rightarrow \mathbb {R}$ is a viscosity solution of the following Stationary Hamilton-Jacobi equation (with time periodic damping):

Theorem 1.3

(main 1’). For f(t) satisfying (${\textbf {H0}}^0$) and H(x, p, t) satisfying (H1-H4), there exists a unique $c(H)\in \mathbb {R}$ (Mañé Critical Value) such that

$$\begin{aligned} u^-_{z,\bar{\varsigma }}(x,\bar{t}):=\varliminf _{\begin{array}{c} \bar{\varsigma }\equiv \varsigma , \bar{t}\equiv t(mod\; 1) \\ t-\varsigma \rightarrow +\infty \end{array}}\bigg (\inf _{\begin{array}{c} \gamma \in C^{ac}([\varsigma ,t],M) \\ \gamma (\varsigma )=z,\gamma (t)=x \end{array}}\int ^t_{\varsigma }e^{F(\tau )-F(t)}\big (L(\gamma ,\dot{\gamma },\tau )+c(H)\big )\text{ d }\tau \bigg ) \end{aligned}$$

(10)

is well defined on $M\times \mathbb {T}$ (for any fixed $(z,\bar{\varsigma })\in M\times \mathbb {T}$) and satisfies

• (1) (Lipschitzness) $u_{z,\bar{\varsigma }}^-:M\times \mathbb {T}\rightarrow \mathbb {R}$ is Lipschitz.

• (2) (Domination) For any absolutely continuous curve $\gamma :[s,t]\rightarrow M$ connecting $(x,\bar{s})\in M\times \mathbb {T}$ and $(y,\bar{t})\in M\times \mathbb {T}$, we have

  $$\begin{aligned} e^{F(t)}u_{z,\bar{\varsigma }}^-(y,\bar{t})-e^{F(s)}u_{z,\bar{\varsigma }}^-(x,\bar{s}) \le \int _s^t e^{F(\tau )}\Big (L(\gamma ,\dot{\gamma },\tau )+c(H)\Big )\text{ d }\tau . \end{aligned}$$

  (11)

  Namely, $u_{z,\bar{\varsigma }}^-\prec _f L+c(H)$.

• (3) (Calibration) For any $(x,\theta )\in M\times \mathbb {T}$, there exists a $C^r$ curve $\gamma _{x,\theta }^-:(-\infty ,\theta ]\rightarrow M$ with $\gamma _{x,\theta }^-(\theta )=x$, such that for all $s\le t\le \theta$, we have

  $$\begin{aligned}{} & {} e^{F(t)}u_{z,\bar{\varsigma }}^-(\gamma _{x,\theta }^-(t),\bar{t})-e^{F( s)}u_{z,\bar{\varsigma }}^-(\gamma _{x,\theta }^-(s),\bar{s})\nonumber \\= & {} \int _s^t e^{F(\tau )}\Big (L(\gamma _{x,\theta }^-,\dot{\gamma }_{x,\theta }^-,\tau )+c(H)\Big )\text{ d }\tau . \end{aligned}$$

  (12)

• (4) (Viscosity) $u_{z,\bar{\varsigma }}^-$ is a viscosity solution of

Remark 1.4

• i) For f satisfying (${\textbf {H0}}^+$), we can similarly define

  $$\begin{aligned} u_{\alpha }^+(x,t):=\sup _{\begin{array}{c} \gamma \in C^{ac}([t,+\infty ),M) \\ \gamma (t)=x \end{array}}\int _t^{+\infty }-e^{F(s)-F(t)}(L(\gamma (s),\dot{\gamma }(s),s)+\alpha )\text{ d }s, \end{aligned}$$

  and verify similar properties (1)-(4) as in Theorem (1.2) for it. Moreover, $-u^+_\alpha (x,-t)$ is a viscosity solution of the following reverse equation of ($\text {HJ}_+$):

  $$\begin{aligned} \partial _t u-f(-t)u+H(x,-\partial _xu,-t)=\alpha . \end{aligned}$$

• ii) Following the terminologies in [15, 21], it’s appropriate to call the function given in Theorem 1.2 (resp. Theorem 1.3) a (backward) weak KAM solution. Such a solution can be used to pick up different types of invariant sets with variational meanings of (4):

Theorem 1.5

(main 2). For f(t) satisfying (${\textbf {H0}}^-$), H(x, p, t) satisfying (H1-H4) and any $\alpha \in \mathbb {R}$, we can get the following sets:

• (Aubry Set) $\gamma :\mathbb {R}\rightarrow M$ is called globally calibrated, if for any $s<t\in \mathbb {R}$, (9) holds on [s, t]. There exists a $\varphi _L^t-$invariant set defined by

  $$\begin{aligned} \widetilde{\mathcal {A}}:=\{(\gamma (t),\dot{\gamma }(t),\bar{t})\in TM\times \mathbb {T}|\gamma \text { is globally calibrated}\} \end{aligned}$$

  with the following properties:

  • $\widetilde{\mathcal {A}}$ is a Lipschitz graph over the projected Aubry set $\mathcal {A}:=\pi \widetilde{\mathcal {A}}\subset M\times {\mathbb {T}}$, where $\pi :T^*M\times \mathbb {T}\rightarrow M\times \mathbb {T}$ is the standard projection.

  • $\widetilde{\mathcal {A}}$ is upper semicontinuous w.r.t. $L:TM\times \mathbb {T}\rightarrow \mathbb {R}$.

  • $u_\alpha ^-$ is differentiable on $\mathcal {A}$.

• (Mather Set) Suppose $\mathfrak M_{L}$ is the set of all $\varphi _L^t-$invariant probability measure, then $\tilde{\mu }\in \mathfrak M_L$ is called a Mather measure if it minimizes

  $$\begin{aligned} \min _{\tilde{\nu }\in \mathfrak M_L}\int _{TM\times \mathbb {T}}L+\alpha - f(t)u_\alpha ^-\text{ d }\tilde{\nu }. \end{aligned}$$

  Let’s denote by $\mathfrak M_m$ the set of all Mather measures. Accordingly, the Mather set is defined by

  $$\begin{aligned} \widetilde{\mathcal {M}}:=\overline{\bigcup \left\{ supp \widetilde{\mu }\left| \widetilde{\mu }\right. \in {\mathfrak {M}}_{m}\right\} } \end{aligned}$$

  which satisfies

  • (1) $\widetilde{\mathcal {M}}\ne \varnothing and \widetilde{\mathcal {M}}\subset \widetilde{\mathcal {A}}.$.

  • (2) $\widetilde{\mathcal {M}}$ is a Lipschitz graph over the projected Mather set $\mathcal {M}:=\pi \widetilde{\mathcal {M}}\subset M\times {\mathbb {T}}.$.

• (Maximal Global Attractor) Define

  $$\begin{aligned}\widehat{\Sigma }_H^-:= & {} \big \{(x,p,\bar{s},\alpha -f(s)u-H(x,p,s), u)\in T^*M\times T^*\mathbb {T}\times \mathbb {R}\big |\\{} & {} \quad u> u_\alpha ^-(x,s)\big \} \end{aligned}$$

  and

  $$\begin{aligned}\widehat{\Sigma }_H^0:= & {} \big \{(x,p,\bar{s},\alpha -f(s)u-H(x,p,s), u)\in T^*M\times T^*\mathbb {T}\times \mathbb {R}\big |\\{} & {} \quad u= u_\alpha ^-(x,s)\big \}, \end{aligned}$$

  then $\Omega :=\bigcap _{t\ge 0}\widehat{\varphi }_{ H}^t( \widehat{\Sigma }_H^-\cup \widehat{\Sigma }_H^0)$ is the maximal $\widehat{\varphi }_{ H}^t-$invariant set, which satisfies:

  • (1) If the $p-$component of $\Omega$ is bounded, then the $u-$ and $I-$component of $\Omega$ are also bounded.

  • (2) If $\Omega$ is compact, it has to be a global attractor in the sense that for any point $(x,p,\bar{s},I,u)\in T^*M\times T^*\mathbb {T}\times \mathbb {R}$ and any open neighborhood $\mathcal {U}\supseteq \Omega$, there exists a $T_{\Omega }(\mathcal {U})$ such that for all $t\ge T_{\Omega }(\mathcal {U})$, $\widehat{\varphi }_{ H}^t(x,p,\bar{s},I,u)\in \mathcal {U}$. Besides, the followings hold:

    • $\Omega$ is a maximal attractor, i.e. it isn’t strictly contained in any other global attractor;

    • $\widetilde{\mathcal {A}}$ is the maximal invariant set contained in ${\sum }_{H}^{0},$ where

      $$\begin{aligned}\widetilde{\mathcal {A}}:=\left\{ \left( \mathcal {L}\left( x,{\partial }_{x}{u}_{\alpha }^{-}\left( x,s\right) ,\overline{s}\right) ,{\partial }_{t}{u}_{\alpha }^{-}\left( x,s\right) ,{u}_{\alpha }^{-}\left( x,s\right) \right) \in TM\times {T}^{*}{\mathbb {T}}\times {\mathbb {R}}\left| \left( x,\overline{s}\right) \in \mathcal {A}\right. \right\} . \end{aligned}$$

Remark 1.6

For f(t) satisfying (${\textbf {H0}}^+$), we can also define the associated Aubry sets and Mather sets and the maximal global attractor by using the function $u^+_\alpha (x,t)$ in Remark 1.4. The procedure is similar with the proof of Theorem 1.5 (see Sect. 4 for more details).

Theorem 1.7

(main 2’). For f(t) satisfying (${\textbf {H0}}^0$) and H(x, p, t) satisfying (H1-H4), the Mañé Critical Value c(H) has an alternative expression

$$\begin{aligned} -c(H)=\dfrac{\inf _{\tilde{\mu }\in \mathfrak M_{ L}}\int _{TM\times \mathbb {T}} e^{F(t)}L(x,v,t)\text{ d }\tilde{\mu }}{\int _0^1e^{F(t)}\text{ d }t}. \end{aligned}$$

(13)

Moreover, the minimizer achieving the right side of (13) has to be a Mather measure. Similarly we can define the Mather set $\widetilde{\mathcal {M}}$ as the union of the support sets of all the Mather measures, which is Lipschitz-graphic over the projected Mather set $\mathcal {M}:=\pi \widetilde{\mathcal {M}}$.

1.2 Parametrized Viscosity Solutions and Asymptotic Dynamics

In this section we deal with two kinds of parametrized viscosity solutions with practical meanings (see Sect. 2 for more physical viewpoints). The first case corresponds to a Hamiltonian

$$\begin{aligned} \widehat{H}_{\delta }(x,p,t,I, u):=I+ H(x,p,t)+f_\delta (t)u, \end{aligned}$$

(14)

with $(x,p,\bar{t},I,u)\in T^*M\times T^*\mathbb {T}\times \mathbb {R}$ and $f_\delta \in C^r(\mathbb {T},\mathbb {R})$ continuous of $\delta \in \mathbb {R}$. For suitable $\alpha \in \mathbb {R}$, we can seek the weak KAM solution of

$$\begin{aligned} \partial _tu_{\delta }(x,t)+H(x,\partial _x u_{\delta },t)+f_\delta (t) u_{\delta }=\alpha \end{aligned}$$

(15)

as we did in previous theorems. Consequently, it’s natural to explore the convergence of viscosity solutions w.r.t. the parameter $\delta$:

Theorem 1.8

(main 3). Suppose $f_\delta$ converges to $f_0$ w.r.t. the uniform norm as $\delta \rightarrow 0_+$ such that $[f_0]=0$ and the right derivative of $f_\delta$ w.r.t. $\delta$ exists at 0, i.e.

$$\begin{aligned} f_1(t):=\lim _{\delta \rightarrow 0_+}\frac{f_\delta (t)-f_0(t)}{\delta }>0. \end{aligned}$$

(16)

If H(x, p, t) satisfies (H1-H4), then there exists a unique $c(H)\in \mathbb {R}$ given by (13) and a $\delta _0>0$, such that the weak KAM solution $u^-_\delta (x,t)$ of (15) associated with $f_\delta$ and $\alpha _{\delta }\equiv c(H)$ for all $\delta \in (0,\delta _0]$ converges to a uniquely identified viscosity solution of

$$\begin{aligned} \partial _tu(x,t)+H(x,\partial _x u,t)+f_0(t) u=c(H), \end{aligned}$$

(17)

which equals

$$\begin{aligned} \sup \Big \{u\prec _{f_0}L+c(H)\Big |\int _{TM\times \mathbb {T}}e^{F_0(t)} f_1(t)\cdot u(x,t)\text{ d } \tilde{\mu }\le 0,\ \forall \; \tilde{\mu }\in \mathfrak M_m(\delta =0)\Big \} \end{aligned}$$

with $F_0(t)=\int _0^tf_0(\tau )\text{ d }\tau$ and $\mathfrak M_m(0)$ being the set of Mather measures for the system with $\delta =0$.

Remark 1.9

The convergence of the viscosity solutions for several kinds of $1^{st}$ order PDEs was recently explored in [10, 13, 31, 33]. Such a viscous approximation problem was initiated by Lions, Papanicolaou and Varadhan in [19]. Usually in such a problem the Comparison principle is necessarily used to guarantee the uniqueness of viscosity solution for (15). However, in our case $f_\delta$ could be negative, which invalidates this principle and brings new difficulties to prove the equi-boundedness and equi-Lipschitzness of $\{u_\delta ^-\}_{\delta >0}$. Nonetheless, by analyzing the properties of the Lax-Oleinik semigroups associated with $\widehat{H}_\delta$ systems, these difficulties can be overcome (see Sect. 5 for more details).

The second parametrized problem we concern takes $M=\mathbb {T}$ and a mechanical H(x, p, t). Let $H^1(\mathbb {T},\mathbb {R})$ be the first order cohomology group of manifold $\mathbb {T}$. We can involve a cohomology parameter $c\in H^1(\mathbb {T},\mathbb {R})$ to

$$\begin{aligned} \widehat{H}(x,p,t,I,u)=I+\underbrace{\frac{1}{2}(p+c)^2+V(x,t)}_{H(x,p,t)}+f(t)u \end{aligned}$$

(18)

of which H(x, p, t) surely satisfies (H1-H4), then (4) becomes

$$\begin{aligned} \mathsf{(Dissipative)\quad } \left\{ \begin{aligned} \dot{x}&=p+c\\ \dot{p}&=-V_x-f(t)p\\ \dot{t}&=1\\ \dot{I}&=-V_t-f'(t)u-f(t)I\\ \dot{u}&=\frac{1}{2}(p^2-c^2)-V(x,t)-f(t)u. \end{aligned} \right. \end{aligned}$$

(19)

In physical models, the former three equations of (19) is usually condensed into a single equation

$$\begin{aligned} \ddot{x}+V_x(x,t)+f(t)(\dot{x}-c)=0,\quad (x,t)\in M\times \mathbb {T}. \end{aligned}$$

(20)

Theorem 1.10

(main 4). For f(t) satisfying (${\textbf {H0}}^-$), the following conclusions hold for equation (20):

• For any $c\in H^1(\mathbb {T},\mathbb {R})$, there exists a unified rotation number of $\widetilde{\mathcal {A}}(c)$, which is defined by

  $$\begin{aligned} \rho (c):=\lim _{T\rightarrow +\infty }\frac{1}{T}\int _0^T\text{ d }\gamma ,\quad \forall \ \text {globally calibrated curve } \gamma . \end{aligned}$$

• $\rho (c)$ is continuous of $c\in H^1(\mathbb {T},\mathbb {R})$. Moreover, we have

  $$\begin{aligned} |\rho (c)-c|\le \varsigma ([f])\cdot \Vert V(x,t)\Vert _{C^1} \end{aligned}$$

  (21)

  for some constant $\varsigma$ depending only on [f]. Consequently, for any $p/q\in \mathbb {Q}$ irreducible, there always exists a $c_{p/q}$ such that $\rho (c_{p/q})=p/q$.

• There exists an compact maximal global attractor $\Omega \subset T^*\mathbb {T}\times T^*\mathbb {T}\times \mathbb {R}$ of the flow $\widehat{\varphi }_{ H}^t$.

Organization of the article The paper is organized as follows: In Sect. 2, we exhibit a list of physical models with time periodic damping. For these models, we state some notable dynamic phenomena and show how these phenomena can be linked to our main conclusions. In Sect. 3, we prove Theorems 1.2 and 1.3. In Sect. 4, we get an analogue of Aubry-Mather theory for systems satisfying condition (${\textbf {H0}}^-$), and prove Theorem 1.5. Besides, we also prove Theorem 1.7 for systems satisfying condition (H0$^0$). In Sect. 5, we discuss the parametrized viscosity solutions of (15), and prove the convergence of them. In Sect. 6, for 1-D mechanical systems with time periodic damping, we prove Theorem 1.10, which is related to the dynamic phenomena of the models in Sect. 2. For the consistency of the proof, parts of preliminary conclusions are postponed to the Appendix.

2 Zoo of Practical Models

In this section we display a bunch of physical models with time-periodic damping, and introduce some practical problems (related with our main conclusions) around them.

2.1 Conformally Symplectic Systems

For $f(t)\equiv \lambda >0$ being constant, we get a so called conformally symplectic system (or discount system). The associated ODE becomes

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

(22)

This kind of systems has been considered in [3, 13, 21], although earlier results on Aubry-Mather sets have been discussed by Le Calvez [17] and Casdagli [7] for $M=\mathbb {T}$. Besides, we need to specify that the Duffing equation with viscous damping also conforms to this case, which concerns all kinds of oscillations widely found in electromagnetics [22] and elastomechanics [24].

A significant property this kind of systems possess is that

$$\begin{aligned} (\varphi _H^1)^*\text{ d }p\wedge \text{ d }x= e^{\lambda }\text{ d }p\wedge \text{ d }x. \end{aligned}$$

When H(x, p, t) is mechanical, the equation usually describes the low velocity oscillation of a solid in a fluid medium (see Fig. 1), which can be formally expressed as

$$\begin{aligned} \ddot{x}+\lambda \dot{x}+\partial _x V(x,t)=0, \quad x\in \mathbb {T},\;\lambda >0. \end{aligned}$$

(23)

Chaos and bifurcations topics of this setting has ever been rather popular in 1970 s [16].

Fig. 1

A dissipative pendulum with $\lambda =1/5$ and $V(t,x)=1-\cos x$

2.2 Tidal Torque Model

The tidal torque model was firstly introduced by [26], describing the motion of a rigid satellite S under the gravitational influence of a point-mass planet P. Due to the internal non-rigidity of the body, a tidal torque will cause a time-periodic dissipative to the motion of S, which can be formalized by

$$\begin{aligned} \ddot{x} +\varepsilon V_x(x,e,t)+\kappa \eta (e,t)(\dot{x}-c(e))=0,\quad (x,t)\in \mathbb {T}^2, \end{aligned}$$

(24)

with the parameter e is the eccentricity of the elliptic motion S around P. Due to the astronomical observation, $\varepsilon$ is the equatorial ellipticity of the satellite and

$$\begin{aligned} \kappa \propto \frac{1}{a^3}\cdot \frac{m_P}{m_S}, \end{aligned}$$

with a being the semi-major and $m_P$ (resp. $m_S$) being the mass respectively.

Fig. 2

A tidal torque model for Moon-Earth and Mercury-Sun

Although this model might seem very special, there are several examples in the solar system for which such a model yields a good description of the motion, at least in a first approximation, and anyhow represents a first step toward the understanding of the problem. For instance, in the pairs Moon-Earth, Enceladus-Saturn, Dione-Saturn, Rhea-Saturn even Mercury-Sun this model is available. Besides, we need to specify that usually $\kappa \ll \varepsilon$ in all these occasions.

A few interesting phenomena have been explained by numerical approaches, e.g. the 1 : 1 resonance for Moon-Earth system which make the people can only see one side of the moon from the earth. However, the Mercury-Sun model shows a different 3 : 2 resonance because of the large eccentricity, see Fig. 2.

Due to Theorems 1.5 and 1.8, such a resonance seems to be explained by the following aspect: any trajectory within the global attractor $\Omega$ of (24) has a longtime stability of velocity, namely, the average velocity is close to certain rotation number, or even asymptotic to it. In Sect. 6 we will show that variational minimal trajectories indeed match this description.

Fig. 3

A simulation of the pumping of the swing

Remark 2.1

As a further simplification, a spin-orbit model with $\eta (e)$ being a constant is also widely concerned, which is actually a conformally symplectic system. In [3] they further discussed the existence of KAM torus for this model and proved the local attraction of the KAM torus.

2.3 Pumping of the Swing

The pumping of a swing is usually modeled as a rigid object forced to rotate back and forth at the lower ends of supporting ropes. After a series of approximations and reasonable simplifications, the pumping of the swing can be characterized as a harmonic oscillator with driving and parametric terms [8]. Therefore, this model has a typical meaning in understanding the dynamics of motors.

As shown in Fig. 3, the length of the rope supporting the swinger is l, and s is the distance between the center of mass of the swinger to the lower ends of the rope. The angle of the supporting rope to the vertical position is denoted by $\phi$, and the angle between the symmetric axis of the swinger and the rope is $\theta$, which varies as $\theta =\theta _0\cos \omega t$. So we get the equation of the motion by

$$\begin{aligned} (l^2-2ls\cos \theta +s^2+R^2)\ddot{\phi }= & {} -gl\sin \phi +gs\sin (\phi +\theta )-ls\sin \theta \dot{\theta }^2\\{} & {} +(ls\cos \theta -s^2-R^2)\ddot{\theta }-2ls\sin \theta \dot{\theta }\dot{\phi },\quad \phi \in \mathbb {T}\nonumber \end{aligned}$$

(25)

where g is the gravity index and $mR^2$ is the moment of inertia of the center (m is the mass of swinger). We can see that by reasonable adjustment of $\varvec{l,s,\omega }$ parameters, this system can be dissipative, accelerative or critical.

Notice that numerical research of this equation for $|\phi |\ll 1$ has been done by numerical experts in a bunch of papers, see [27] for a survey of that. Those results successfully simulate the swinging at small to modest amplitudes. As the amplitude grows, these results become less and less accurate, and that’s why we resort to a theoretical analysis in this paper.

3 Weak KAM Solution of (${\textbf {HJ}}_+$)

Due to the superlinearity of L(x, v, t), for each $k\ge 0$, there exists $C(k)\ge 0$, such that

$$\begin{aligned} L(x,v,t)\ge k|v|-C(k), k>0,x\in M. \end{aligned}$$

Moreover, the compactness of M implies that for each $k>0$, there exists $C_k>0$ such that

$$\begin{aligned} \max _{\begin{array}{c} (x,t)\in M\times \mathbb {T}\\ |v|\le k \end{array}}L(x,v,t)\le C_k. \end{aligned}$$

3.1 Weak KAM Solution of (${\textbf {HJ}}_+$) in the Condition (${\textbf {H0}}^-$)

Note that $[f]>0$. The following conclusion can be easily checked.

Lemma 3.1

Assume $t>s$, then

• (1) $F(s)-F(t)\le 2k_0-(t-s-1)[f];$

• (2) $\int ^t_se^{F(\tau )-F(t)}\text{ d }\tau \le \frac{e^{2k_0+[f]}}{[f]}\big (1-e^{-(t-s)[f]}\big );$

• (3) $\int ^t_{-\infty }e^{F(\tau )-F(t)}\text{ d }\tau \le \frac{e^{2k_0+[f]}}{[f]},$

where $k_0 =\max _{s\in [0,2]}\big |\int ^s_0f(\tau )\text{ d }\tau \big |$.

Now we define a function $u_{\alpha }^-:M\times \mathbb {R}\rightarrow \mathbb {R}$ by

$$\begin{aligned} u_\alpha ^-(x,t):= & {} \inf \int ^t_{-\infty }e^{F(s)-F(t)}(L(\gamma (s),\dot{\gamma }(s),s)+\alpha )\text{ d }s \end{aligned}$$

(26)

where the infimum is taken for all $\gamma \in C^{ac}((-\infty ,t],M)$² with $\gamma (t)=x$. We can easily prove this function is bounded, since

$$\begin{aligned} -|C(k=0)-\alpha |\cdot \frac{e^{2k_0+[f]}}{[f]}\le u_{\alpha }^-(x,t)\le |C_{k=0}+\alpha |\frac{e^{2k_0+[f]}}{[f]}, \end{aligned}$$

where C(0) and $C_0$ have been defined in the beginning of Sect. 3. Consequently, (1) and (3) of Theorem 1.2 can be easily achieved.

Lemma 3.2

For each $(x,t)\in M\times \mathbb {R}$ and $s<t$, it holds

$$\begin{aligned}{} & {} e^{F(t)}u_{\alpha }^-(x,t)\\= & {} \inf _{\begin{array}{c} \gamma \in C^{ac}([s,t],M)\\ \gamma (t)=x \end{array}}\bigg \{ e^{F(s)}u_{\alpha }^-(\gamma (s),s)+\int ^t_se^{F(\tau )}(L(\gamma (\tau ),\dot{\gamma }(\tau ),\tau )+\alpha )\text{ d }\tau \bigg \}.\nonumber \end{aligned}$$

(27)

Moreover, the infimum in (27) can be achieved by a $C^r$ smooth minimizer.

Proof

Equality (27) can be proved by a same method as employed in the proof of Proposition 6.5 in [13]. Therefore, we can find a sequence of absolutely continuous curve $\{\gamma _n\}$ with $\gamma _n(t)=x$ such that

$$\begin{aligned} e^{F(t)}u_{\alpha }^-(x,t)=\lim _{n\rightarrow \infty }\bigg \{e^{F(s)}u_{\alpha }^-(\gamma _n(s),s)+\int ^t_se^{F(\tau )}(L(\gamma _n,\dot{\gamma }_n,\tau )+\alpha )\text{ d }\tau \bigg \}. \end{aligned}$$

Hence, there exists a constant c independent of n, such that

$$\begin{aligned} \int ^t_se^{F(\tau )}(L(\gamma _n(\tau ),\dot{\gamma }_n(\tau ),\tau )+\alpha )\text{ d }\tau \le c. \end{aligned}$$

(28)

Due to Dunford-Petti Theorem (Theorem 6.4 in [13]), there exists a subsequence $\{\gamma _{n_k}\}$ converging to a curve $\gamma _*$ in the space $C^{ac}([s,t],M)$ endowed with metric $d_0(\gamma _1,\gamma _2)=\sup \{\text{ d }(\gamma _1(\sigma ),\gamma _2(\sigma )): \sigma \in [s,t]\}$, where $d(\cdot ,\cdot )$ is the distance function induced by the Riemannian metric on M, such that

$$\begin{aligned} \int ^t_se^{F(\tau )}(L(\gamma _*,\dot{\gamma }_*,\tau )+\alpha )\text{ d }\tau \le \varliminf _{k\rightarrow \infty }\int ^t_se^{F(\tau )}(L(\gamma _{n_k},\dot{\gamma }_{n_k},\tau )+\alpha )\text{ d }\tau , \end{aligned}$$

(29)

Hence, the infimum in (27) can be achieved at $\gamma _*: [s,t]\rightarrow M$, which definitely solves the Euler-Lagrange equation (E-L). Due to the Weierstrass Theorem in [23], $\gamma _*$ is $C^r$ smooth.$\square$

For any $n\in \mathbb {Z}_+$, we apply Lemma 3.2 on the inerval $[t-n,t]\subset \mathbb {R}$ and achieve the infimum curve $\gamma _n$. By Dunford-Petti Theorem and a diagonal argument, the uniform limit of $\gamma _n$ (up extraction of a subsequence) exists as a $C^r$ calibrated curve on $(-\infty ,t]$.

Lemma 3.3

[(4) of Theorem 1.2] For each $\alpha \in \mathbb {R}$ and $(x,t)\in M\times \mathbb {R}$, there exists a $C^r$ curve $\gamma _{x,t}^-:(-\infty ,t]\rightarrow M$ with $\gamma _{x,t}^-(t)=x$ such that for each $t_1<t_2\le t$,

$$\begin{aligned}{} & {} e^{F(t_2)}u_{\alpha }^-(\gamma _{x,t}^-(t_2),t_2)-e^{F(t_1)}u_{\alpha }^-(\gamma _{x,t}^-(t_1),t_1)\\= & {} \int ^{t_2}_{t_1}e^{F(\tau )}(L(\gamma _{x,t}^-(\tau ),\dot{\gamma }_{x,t}^-(\tau ),\tau )+\alpha )\text{ d }\tau .\nonumber \end{aligned}$$

(30)

Remark 3.4

Due to the boundedness of $u_\alpha ^-$, by making $t_1\rightarrow -\infty$ in (30) we instantly get

$$\begin{aligned} u_{\alpha }^-(x,t)=\int ^t_{-\infty }e^{F(\tau )-F(t)}(L(\gamma _{x,t}^-(\tau ),\dot{\gamma }_{x,t}^-(\tau ),\tau )+\alpha )\text{ d }\tau , \end{aligned}$$

i.e. the infimum in (26) is achieved at $\gamma _{x,t}^-:(-\infty ,t]\rightarrow M$.

Lemma 3.5

Suppose $\gamma _{x,\theta }^-:(-\infty ,\theta ]\rightarrow M$ is a backward calibrated curve ending with x of $u_{\alpha }^-(x,\theta )$, then

$$\begin{aligned} |\dot{\gamma }_{x,\theta }^-(\tau )|\le \kappa _0,\quad \forall \ (x,\theta )\in M\times \mathbb {T}, \tau <\theta . \end{aligned}$$

for a constant $\kappa _0$ depending on L and $\alpha$. That implies $\gamma _{x,\theta }^-$ is Lipschitz on $(-\infty ,\theta ]$.

Proof

Let $s_1,s_2\le \theta$ and $s_2-s_1=1$. Due to Lemma 3.3,

$$\begin{aligned}&e^{F(s_2)}u^-_{\alpha }(\gamma ^-_{x,\theta }(s_2),s_2)-e^{F(s_1)}u^-_{\alpha }(\gamma ^-_{x,\theta }(s_1),s_1)\\&=\int ^{s_2}_{s_1}e^{F(\tau )}(L(\gamma ^-_{x,\theta }(\tau ),\dot{\gamma }^-_{x,\theta }(\tau ),\tau )+\alpha )\text{ d }\tau \\&\ge \int ^{s_2}_{s_1}e^{F(\tau )}(|\dot{\gamma }^-_{x,\theta }(\tau )|-C(1)+\alpha )\text{ d }\tau . \end{aligned}$$

On the other hand, let $\beta :[s_1,s_2]\rightarrow M$ be a geodesic satisfying $\beta (s_1)=\gamma ^-_{x,\theta }(s_1),\beta (s_2)=\gamma ^-_{x,\theta }(s_2)$, and $|\dot{\beta }(\tau )|\le \text{ diam(M) }=:k_1$. Then

$$\begin{aligned}&e^{F(s_2)}u^-_{\alpha }(\gamma ^-_{x,\theta }(s_2),s_2)-e^{F(s_1)}u^-_{\alpha }(\gamma ^-_{x,\theta }(s_1),s_1)\\&\le \int ^{s_2}_{s_1}e^{F(\tau )}(L(\beta (\tau ),\dot{\beta }(\tau ),\tau )+\alpha )\text{ d }\tau \\&\le \int ^{s_2}_{s_1}e^{F(\tau )}(C_{k_1}+\alpha )\text{ d }\tau . \end{aligned}$$

Hence,

$$\begin{aligned} \int ^{s_2}_{s_1}e^{F(\tau )}|\dot{\gamma }^-_{x,\theta }(\tau )|\text{ d }\tau \le \int ^{s_2}_{s_1}e^{F(\tau )}(C_{k_1}+C(1))\text{ d }\tau . \end{aligned}$$

Due to the continuity of $\dot{\gamma }^-_{x,\theta }(\tau )$, there exists $s_0\in (s_1,s_2)$ such that

$$\begin{aligned} |\dot{\gamma }^-_{x,\theta }(s_0)|\le C_{k_1}+C(1). \end{aligned}$$

(31)

Note that $\gamma ^-_{x,\theta }$ solves (E-L), so $|\dot{\gamma }^-_{x,\theta }(\tau )|$ is uniformly bounded for $(x,\theta )\in M\times \mathbb {T}$ and $\tau \in (-\infty ,\theta ]$.$\square$

Lemma 3.6

[(2) of Theorem 1.2] For each $\alpha \in \mathbb {R}$, $u_{\alpha }^-$ is Lipschitz on $M\times \mathbb {T}$.

Proof

First of all, we prove $u_\alpha ^-(\cdot ,\theta ):M\rightarrow \mathbb {R}$ is uniformly Lipschitz w.r.t. $\theta \in \mathbb {T}$. Let $x,y\in M$, $\Delta t=d(x,y)$, and $\gamma ^-_{x,\theta }:(-\infty ,\theta ]\rightarrow M$ be a minimizer of $u^-_{\alpha }(x,\theta )$. Define $\tilde{\gamma }:(-\infty ,\theta ]\rightarrow M$ by

$$\begin{aligned} \tilde{\gamma }(s)= {\left\{ \begin{array}{ll} \gamma ^-_{x,\theta }(s),s\in (-\infty ,\theta -\Delta t),\\ \beta (s),s\in [\theta -\Delta t,\theta ], \end{array}\right. } \end{aligned}$$

where $\beta :[\theta -\Delta t,\theta ]\rightarrow M$ is a geodesic satisfying $\beta (\theta -\Delta t)=\gamma _{x,\theta }^-(\theta -\Delta t),\beta (\theta )=y$, and

$$\begin{aligned} |\dot{\beta }(s)|\equiv \frac{d(\gamma ^-_{x,\theta }(\theta -\Delta t),y)}{\Delta t}\le \frac{d(\gamma _{x,\theta }^-(\theta -\Delta t),x)}{\Delta t}+1\le \kappa _0+1. \end{aligned}$$

Then,

$$\begin{aligned} u^-_{\alpha }(y,\theta )-u^-_{\alpha }(x,\theta )&\le \int ^\theta _{\theta -\Delta t}e^{F(\tau )-F(\theta )}(L(\beta ,\dot{\beta },\tau )-L(\gamma _{x,\theta }^-,\dot{\gamma }_{x,\theta }^-,\tau ))\text{ d }\tau \\&\le (C_{\kappa _0+1}+C(0))\int ^\theta _{\theta -\Delta t}e^{F(\tau )-F(\theta )}\text{ d }\tau \\&\le (C_{\kappa _0+1}+C(0))e^{2k_0+[f]}\cdot d(x,y). \end{aligned}$$

By a similar approach, we derive the opposite inequality holds. Hence,

$$\begin{aligned} |u^-_{\alpha }(y,\theta )-u^-_{\alpha }(x,\theta )|\le \rho _*\cdot d(x,y), \end{aligned}$$

(32)

where $\rho _*=(C_{\kappa _0+1}+C(0))e^{2k_0+[f]}$.

Next, we prove $u^-_{\alpha }(x,\cdot )$ is uniformly Lipschitz continuous for $x\in M$. Let $\bar{t},\bar{t}'\in \mathbb {T}, d(\bar{t},\bar{t}')=t'-t$, and $t\in [0,1)$. Then, $t'\in [0,2]$. A curve $\eta :(-\infty ,t']\rightarrow M$ is defined by

$$\begin{aligned} \eta (s)= {\left\{ \begin{array}{ll} \gamma ^-_{x,t}(s),s\in (-\infty ,t],\\ x,\ \ s\in (t,t']. \end{array}\right. } \end{aligned}$$

Then,

$$\begin{aligned}&\ \ \ \ e^{F(t')}u^-_{\alpha }(x,t')-e^{F(t)}u^-_{\alpha }(x,t)\\&\le \int ^{t'}_{-\infty }e^{F(\tau )}(L(\eta ,\dot{\eta },\tau )+\alpha )\text{ d }\tau -\int ^t_{-\infty }e^{F(\tau )}(L(\gamma ^-_{x,t},\dot{\gamma }^-_{x,t},\tau )+\alpha )\text{ d }\tau \\&\le \int ^{t'}_te^{F(\tau )}(C_0+\alpha )\text{ d }\tau \le (C_0+|\alpha |)\max _{\tau \in [0,2]}e^{F(\tau )}\cdot |t'-t|. \end{aligned}$$

On the other hand, we write $\Delta t=d(\bar{t}',\bar{t})$ and define $\eta _1\in C^{ac}((-\infty ,t],M)$ by

$$\begin{aligned} \eta _1(s)= {\left\{ \begin{array}{ll} \gamma ^-_{x,t'}(s),s\in (-\infty ,t-\Delta t],\\ \gamma ^-_{x,t'}(2(s-t)+t'),s\in (t-\Delta t,t]. \end{array}\right. } \end{aligned}$$

It is easy to check $\eta _1(t)=x$, and $|\dot{\eta }_1(\tau )|\le 2\kappa _0$, where $\kappa _0$ is a Lipschitz constant of $\gamma _{x,t'}^-$.

$$\begin{aligned} e^{F(t)}u^-_{\alpha }(x,t)&\le \int ^{t}_{-\infty }e^{F(\tau )}(L(\eta _1(\tau ),\dot{\eta }_1(\tau ),\tau )+\alpha )\text{ d }\tau \\&\le \int ^{t}_{t-\Delta t}e^{F(\tau )}(L(\eta _1(\tau ),\dot{\eta }_1(\tau ),\tau )+\alpha )\text{ d }\tau \\&\;\; +\int ^{t-\Delta t}_{-\infty }e^{F(\tau )}(L(\gamma ^-_{x,t'}(\tau ),\dot{\gamma }^-_{x,t'}(\tau ),\tau )+\alpha )\text{ d }\tau . \end{aligned}$$

Note that $\gamma ^-_{x,t'}$ is a minimizer of $u^-_\alpha (x,t')$. We derive that

$$\begin{aligned}&\ \ \ e^{F(t)}u^-_{\alpha }(x,t)-e^{F(t')}u^-_\alpha (x,t')\\&\le \int ^t_{t-\Delta t}e^{F(\tau )}(L(\eta _1(\tau ),\dot{\eta }_1(\tau ),\tau )+\alpha )\text{ d }\tau \\&-\int ^{t'}_{t-\Delta t}e^{F(\tau )}(L(\gamma ^-_{x,t'}(\tau ),\dot{\gamma }^-_{x,t'}(\tau ),\tau )+\alpha )\text{ d }\tau \\&\le (C_{2\kappa _0}+2C(0)+|\alpha |)\max _{\tau \in [0,2]} e^{F(\tau )}\cdot d(\bar{t}',\bar{t}). \end{aligned}$$

We have proved the map $t\longmapsto e^{F(t)}u^-_{\alpha }(x,t)$ is uniformly Lipschitz for $x\in M$, with Lipschitz constant depends only on L, f and $\alpha$. Note that F(t) is $C^{r+1}$ and $F'(t)=f(t)$ is 1-periodic. We derive $u^-_\alpha (x,\cdot )$ is uniformly Lipschitz for $x\in M$ with Lipschitz constant $\rho ^*_0$ depending on L, f, and $\alpha$. It follows that

$$\begin{aligned} |u^-_{\alpha }(x',\theta ')-u^-_{\alpha }(x,\theta )|&\le |u^-_{\alpha }(x',\theta ')-u^-_{\alpha }(x,\theta ')|+|u^-_{\alpha }(x,\theta ')-u^-_{\alpha }(x,\theta )|\\&\le \rho _*d(x',x)+\rho ^*_0d(\theta ',\theta ) \end{aligned}$$

so we finish the proof. $\square$

Lemma 3.7

[(5) of Theorem 1.2] The function $u_{\alpha }^-(x,t)$ defined by (26) is a viscosity solution of ($HJ_+$).

Proof

By a standard argument as in Proposition 7.2.7 in[15] (or Proposition 6.3 in[10]), $u^-_\alpha (x,t)$ is viscosity solution can be derived from the weak KAM properties of $u_\alpha ^-(x,t)$. $\square$

As a complement, the following result, which is similar to Proposition 6 of [21], will be useful in the following sections:

Proposition 3.8

The weak KAM solution $u_\alpha ^-$ of ($HJ_+$) is differentiable at $(\gamma _{x,t}^-(s),\bar{s})$ for any $\mathbb {R}\ni s<t$, where $\gamma _{x,t}^-: (-\infty ,t]\rightarrow M$ is a backward calibrated curve ending with x. In other words, we have

$$\begin{aligned} \partial _tu^-_\alpha (\gamma _{x,t}^-(s),s)+H(\gamma _{x,t}^-(s),\partial _xu^-_\alpha (\gamma _{x,t}^-(s),s),s)+f(s)u^-_\alpha (\gamma _{x,t}^-(s),s)=\alpha \end{aligned}$$

and

$$\begin{aligned} (\gamma _{x,t}^-(s),\dot{\gamma }_{x,t}^-(s),\bar{s})=\mathcal {L}\Big (\gamma _{x,t}^-(s),\partial _xu^-_\alpha (\gamma _{x,t}^-(s),s),\bar{s}\Big ) \end{aligned}$$

(33)

for all $\mathbb {R}\ni s<t$.

Proof

By Theorem B.4, we derive $u^-_\alpha (x,t)$ is semiconcave. Let $s\in (-\infty ,t)$ and $\tilde{p}=(p_x,p_t)\in D^+u^-_{\alpha }(\gamma _{x,t}^-(s),s)$. For $\Delta s>0$,

$$\begin{aligned}{} & {} \frac{e^{F(s+\Delta s)}u^-_\alpha (\gamma _{x,t}^-(s+\Delta s),s+\Delta s)-e^{F(s)}u^-_{\alpha }(\gamma ^-_{x,t}(s),s)}{\Delta s}\\= & {} \frac{1}{\Delta s}\int ^{s+\Delta s}_{s}e^{F(\tau )}(L(\gamma _{x,t}^-(\tau ),\dot{\gamma }_{x,t}^-(\tau ),\tau )+\alpha )\text{ d }\tau . \end{aligned}$$

Then

$$\begin{aligned}{} & {} \lim _{\Delta s\rightarrow 0^+}\frac{u^-_{\alpha }(\gamma _{x,t}^-(s+\Delta s),s+\Delta s)-u^-_{\alpha }(\gamma _{x,t}^-(s),s)}{\Delta s}\\= & {} L(\gamma _{x,t}^-(s),\dot{\gamma }_{x,t}^-(s),s)+\alpha -f(s)u^-_{\alpha }(\gamma _{x,t}^-(s),s). \end{aligned}$$

By Proposition B.3.,

$$\begin{aligned} \lim _{\Delta s\rightarrow 0^+}\frac{u^-_{\alpha }(\gamma _{x,t}^-(s+\Delta s),s+\Delta s)-u^-_{\alpha }(\gamma _{x,t}^-(s),s)}{\Delta s}\le p_x\cdot \dot{\gamma }_{x,t}^-(s)+p_t, \end{aligned}$$

which implies

$$\begin{aligned}{} & {} p_t+H(\gamma _{x,t}^-(s),p_x,s)+f(s)u^-_{\alpha }(\gamma _{x,t}^-(s),s)\ge \alpha . \end{aligned}$$

On the other hand, $u^-_\alpha$ is a viscosity solution of ($\text {HJ}_+$). Hence, for each $(p_x,p_t)\in D^+u^-_{\alpha }(\gamma _{x,t}^-(s),s)$,

$$\begin{aligned} p_t+H(\gamma _{x,t}^-(s),p_x,s)+f(s)u^-_{\alpha }(\gamma _{x,t}^-(s),s)=\alpha . \end{aligned}$$

(34)

Note that H(x, p, t) is strictly convex with respect to p. By (34), We derive $D^+u^-_{\alpha }(\gamma _{x,t}^-(s),s)$ is a singleton. By Proposition B.3., $u^-_{\alpha }(x,t)$ is differentiable at $(\gamma _{x,t}^-(s),s)$. $\square$

3.2 Weak KAM Solution of (${\textbf {HJ}}_0$) in the Condition (H0 $^0$)

Now $[f]=0$, so $F(t):=\int ^t_0 f(\tau )\text{ d }\tau$ is 1-periodic. Let $\ell =\int ^1_0e^{F(\tau )}\text{ d }\tau$, then we define a new Lagrangian ${\textbf {L}}:TM\times \mathbb {T}\rightarrow \mathbb {R}$ by

$$\begin{aligned} {\textbf {L}}(x,v,t)=e^{F(t)}L(x,v,t). \end{aligned}$$

For such a ${\textbf {L}}$, the Peierls Barrier ${\textbf {h}}^\infty _\alpha :M\times \mathbb {T}\times M\times \mathbb {T}\rightarrow \mathbb {R}$

$$\begin{aligned} {\textbf {h}}^\infty _\alpha (x,\bar{s},y,\bar{t})=\liminf _{\begin{array}{c} t\equiv \bar{t},s\equiv \bar{s} (mod\; 1)\\ t-s\rightarrow +\infty \end{array}}\inf _{\begin{array}{c} \gamma \in C^{ac}([s,t],M) \\ \gamma (s)=x,\gamma (t)=y \end{array}}\int ^t_s{\textbf {L}}(\gamma ,\dot{\gamma },\tau )+\alpha \cdot \ell \text{ d }\tau , \end{aligned}$$

is well-defined, once $\alpha$ is uniquely established by

$$\begin{aligned} c(H)=\inf \{\alpha \in \mathbb {R}|\int ^t_s{\textbf {L}}(\gamma ,\dot{\gamma },\tau )+\alpha \cdot \ell \text{ d }\tau \ge 0,\ \forall \gamma \in \mathcal {C} \} \end{aligned}$$

(35)

with $\mathcal {C}=\{\gamma \in C^{ac}([s,t],M)|\gamma (s)=\gamma (t) \text{ and } t-s\in \mathbb {Z}_+\}$, due to Proposition 2 of [11]. Moreover, the following properties were proved in [11]:

Proposition 3.9

• (i): If $\alpha <c(H)$, ${\textbf {{h}}}^\infty _\alpha \equiv -\infty$.

• (ii): If $\alpha >c(H)$, ${\textbf {{h}}}^\infty _\alpha \equiv +\infty$.

• (iii): ${\textbf {{h}}}^\infty _{c(H)}$ is finite.

• (iv): ${\textbf {{h}}}^\infty _{c(H)}$ is Lipschitz.

• (v): For each $\gamma \in C^{ac}([s,t],M)$ with $\gamma (s)=x,\gamma (t)=y$,

$$\begin{aligned} {\textbf {h }}^{\infty }_{c(H)}(z,\bar{\varsigma },y,\bar{t})-{\textbf {h }}^\infty _{c(H)}(z,\bar{\varsigma },x,\bar{s})\le \int ^t_s{\textbf {L }}(\gamma ,\dot{\gamma },\tau )+c(H)\cdot \ell d \tau . \end{aligned}$$

Consequently, for any $(z,\bar{\varsigma })\in M\times \mathbb {T}$ fixed, we construct a function $u^-_{z,\bar{\varsigma }}:M\times \mathbb {T}\rightarrow \mathbb {R}$ by

$$\begin{aligned} u^-_{z,\bar{\varsigma }}(x,\bar{t})=e^{-F(\bar{t})}\bigg ({\textbf {h}}^\infty _{c(H)}(z,\bar{\varsigma },x,\bar{t})+c(H)\cdot \int ^t_\varsigma e^{F(\tau )}-\ell \text{ d }\tau \bigg ). \end{aligned}$$

(36)

Proof of Theorem 1.3:

• (1) Due to (iv) of Proposition 3.9, $u^-_{z,\bar{\varsigma }}$ is also Lipschitz.

• (2) The domination property of $u^-_{z,\bar{\varsigma }}$ can be achieved immediately by (v) of Proposition 3.9.

• (3) By Tonelli Theorem and the definition of $u^-_{z,\bar{\varsigma }}$, there exists a sequence ${\varsigma _{k}}$ tending to $-\infty$ and a sequence $\gamma _k\in C^{ac}([\varsigma _k,\theta ],M)$ with $\gamma _k(\varsigma _k)=z,\gamma _k(\theta )=x$, such that $\gamma _k$ minimizes the action function

  $$\begin{aligned} \mathcal {F}(\beta )= \inf _{\begin{array}{c} \beta \in C^{ac}([\varsigma _k,\theta ]) \\ \beta (\varsigma _k)=z,\beta (\theta )=x \end{array}}\int ^\theta _{\varsigma _k}e^{F(\tau )}(L(\beta ,\dot{\beta },\tau )+c(H))\text{ d }\tau \end{aligned}$$

  and

  $$\begin{aligned} e^{F(\theta )}u^-_{z,\bar{\varsigma }}(x,\theta )=\lim _{k\rightarrow +\infty }\int ^\theta _{\varsigma _k}e^{F(\tau )}(L(\gamma _k,\dot{\gamma }_k,\tau )+c(H))\text{ d }\tau . \end{aligned}$$

  Since each $\gamma _k$ solves (E-L), which implies $\gamma _k$ is $C^r$. By a standard way, there exists $\kappa _0$ independent of the choice of k, such that $|\dot{\gamma }_k|\le \kappa _0$, when $\theta -\varsigma _k\ge 1$. By Ascoli Theorem, there exists a subsequence of $\{\gamma _k\}$ (denoted still by $\gamma _k$) and an absolutely continuous curve $\gamma ^-_{x,\theta }:(-\infty ,\theta ]\rightarrow M$ such that $\gamma _k$ converges uniformly to $\gamma ^-_{x,\theta }$ on each compact subset of $(-\infty ,\theta ]$ and $\gamma ^-_{x,\theta }(\theta )=x$. Then, for each $s<\theta$,

  $$\begin{aligned}e^{F(\theta )}u^-_{z,\bar{\varsigma }}(x,\theta )= & {} \lim _{k\rightarrow +\infty }\bigg (\int ^s_{\varsigma _k}e^{F(\tau )}(L(\gamma _k,\dot{\gamma }_k,\tau )+c(H))\text{ d }\tau \\{} & {} +\int ^{\theta }_se^{F(\tau )}(L(\gamma _k,\dot{\gamma }_k,\tau )+c(H))\text{ d }\tau \bigg )\\\ge & {} \liminf _{k\rightarrow +\infty }\int ^s_{\varsigma _k}e^{F(\tau )}(L(\gamma _k,\dot{\gamma }_k,\tau )+c(H))\text{ d }\tau \\{} & {} +\liminf _{k\rightarrow +\infty }\int ^\theta _se^{F(\tau )}(L(\gamma _k,\dot{\gamma }_k,\tau )+c(H))\text{ d }\tau \\\ge & {} e^{F(s)}u^-_{z,\bar{\varsigma }}(\gamma ^-_{x,\theta }(s),s)+\int ^\theta _se^{F(\tau )}(L(\gamma ^-_{x,\theta },\dot{\gamma }^-_{x,\theta },\tau )+c(H))\text{ d }\tau \end{aligned}$$

  which implies $\gamma ^-_{x,\theta }$ is a calibrated curve by $u^-_{z,\bar{\varsigma }}$.

• (4) By a similar approach of the proof of Lemma 3.7, we derive $u^-_{z,\bar{\varsigma }}$ is also a viscosity solution of ($\text {HJ}_0$).$\square$

4 Various Properties of Variational Invariant Sets

4.1 Aubry Set in the Condition (${\textbf {H0}}^-$)

Due to Theorem 1.2 and Proposition 3.8, for any $(x,\bar{s})\in M\times \mathbb {T}$ we can find a backward calibrated curve

$$\begin{aligned} \widetilde{\gamma }_{x,s}^-:=\begin{pmatrix} \gamma _{x,s}^-(t) \\ \bar{t} \end{pmatrix}:t\in (-\infty ,s]\rightarrow M\times \mathbb {T}\end{aligned}$$

(37)

ending with it, such that the associated backward orbit $\varphi _L^{t-s}(\gamma _{x,s}^-(s),\dot{\gamma }_{x,s}^-(s), s)$ has an $\alpha -$limit set $\widetilde{\mathcal {A}}_{x,s}\subset TM\times \mathbb {T}$, which is invariant and graphic over $\mathcal {A}_{x,s}:=\pi \widetilde{\mathcal {A}}_{x,s}$. Therefore, any critical curve $\widetilde{\gamma }_{x,s}^\infty$ in $\mathcal {A}_{x,s}$ has to be a globally calibrated curve, namely

$$\begin{aligned} \widetilde{\mathcal {A}}_{x,s}\subset \widetilde{\mathcal {A}},\quad (\text {resp. } {\mathcal {A}}_{x,s}\subset {\mathcal {A}}). \end{aligned}$$

So $\widetilde{\mathcal {A}}\ne \emptyset$.

Recall that any critical curve in $\mathcal {A}$ is globally calibrated, then due to Proposition 3.8, that implies for any $(x,\bar{s})\in {\mathcal {A}}$, the critical curve $\widetilde{\gamma }_{x,s}$ passing it is unique. In other words, $\pi ^{-1}:{\mathcal {A}}\rightarrow \widetilde{\mathcal {A}}$ is a graph, and

$$\begin{aligned} \dot{\gamma }_{x,s}(t)=\partial _p H(\text{ d }u^-(\gamma _{x,s}(t),t),t),\quad \forall \ t\in \mathbb {R}. \end{aligned}$$

That indicates that $\text{ d }u^-:{\mathcal {A}}\rightarrow TM$ coincides with $\partial _v L\circ (\pi |_{\widetilde{\mathcal {A}}})^{-1}$. On the other side, $\Vert \dot{\widetilde{\gamma }}_{x,s}(t)\Vert \le A<+\infty$ for all $t\in \mathbb {R}$ due to Lemma 3.5, so $\partial _v L\circ (\pi |_{\widetilde{\mathcal {A}}})^{-1}$ has to be Lipschitz. So $\widetilde{\mathcal {A}}$ is Lipschitz over $\mathcal {A}$. This is an analogue of Theorem 4.11.5 of [15] and a.4) of [21], which is known as Mather’s graph theorem in more earlier works [23] for conservative Hamiltonian systems.

Lemma 4.1

$\widetilde{\mathcal {A}}$ has an equivalent expression

$$\begin{aligned} \widetilde{\mathcal {A}}:=\{(\gamma (t),\dot{\gamma }(t),\bar{t})\in TM\times \mathbb {T}|\;\forall \; a<b\in \mathbb {R}, \gamma \text {achieves} h_{\alpha }^{a,b}(\gamma (a),\gamma (b))\}.\quad \end{aligned}$$

(38)

Proof

Let $\gamma :\mathbb {R}\rightarrow M$ be a globally calibrated curve by $u^-_\alpha$. Due to (3) and (4) of Theorem 1.2, for $a<b\in \mathbb {R}$,

$$\begin{aligned} \int ^b_ae^{F(\tau )}(L(\gamma ,\dot{\gamma },\tau )+\alpha )\text{ d }\tau&= e^{F(b)}u_\alpha ^-(\gamma (b),b)-e^{F(a)}u_\alpha ^-(\gamma (a),a)\\&\le h^{a,b}_\alpha (\gamma (a),\gamma (b)). \end{aligned}$$

Due to the definition of $h^{a,b}_\alpha (\gamma (a),\gamma (b))$, we derive $\gamma$ achieves $h^{a,b}_\alpha (\gamma (a),\gamma (b))$ for all $a<b\in \mathbb {R}$.

To prove the lemma, it suffices to show any curve $\gamma :\mathbb {R}\rightarrow M$ achieving $h^{a,b}_\alpha (\gamma (a),\gamma (b))$ for all $a<b\in \mathbb {R}$ is a calibrated curve by $u^-_\alpha$. We claim

$$\begin{aligned} \lim _{s\rightarrow -\infty }h^{s,t}_\alpha (z,x)=e^{F(t)}u^-_\alpha (x,t), \ \ \forall x,z\in M,t\in \mathbb {R}. \end{aligned}$$

(39)

Due to (3) of Theorem 1.2, for $s<t$,

$$\begin{aligned} e^{F(t)}u_\alpha ^-(x,t)-h^{s,t}_\alpha (z,x)\le e^{F(s)}u_\alpha ^-(z,s)\rightarrow 0, s\rightarrow -\infty . \end{aligned}$$

On the other hand, we assume $\gamma _{x,t}$ is a globally calibrated curve by $u^-_\alpha$ with $\gamma _{x,t}(t)=x$ and $s+1<t$. Let $\beta :[s,s+1]\rightarrow M$ be a geodesic with $\beta (s)=z,\beta (s+1)=\gamma _{x,t}(s+1)$ satisfying $|\dot{\beta }|\le k_1:=\text{ diam }(M)$. Then,

$$\begin{aligned} h^{s,t}_\alpha (z,x)&\le \int ^{s+1}_se^{F(\tau )}(L(\beta ,\dot{\beta },\tau )+\alpha )\text{ d }\tau +\int ^t_{s+1}e^{F(\tau )}(L(\gamma _{x,t},\dot{\gamma }_{x,t},\tau )+\alpha )\text{ d }\tau \\&\le (C_{k_1}+\alpha )e^{\max f+[f][s]}+e^{F(t)}u^-_\alpha (x,t)-e^{F(s+1)}u^-_\alpha (\gamma _{x,t}(s+1),s+1). \end{aligned}$$

Hence,

$$\begin{aligned} h^{s,t}_\alpha (z,x)-e^{F(t)}u^-_\alpha (x,t)\le (C_{k_1}+\alpha )e^{\max f+[f][s]}-e^{F(s+1)}u^-_\alpha (\gamma _{x,t}(s+1),s+1). \end{aligned}$$

From $[f]>0$, it follows that the right side of the inequality above tending to 0, as $s\rightarrow -\infty$. Hence, (39) holds. Actually, the limit in (39) is uniform for $x,z\in M$ and $t\in \mathbb {R}$.

If $\gamma$ achieves $h^{a,b}_\alpha (\gamma (a),\gamma (b))$ for $a<b\in \mathbb {R}$, then

$$\begin{aligned} h^{s,b}_\alpha (\gamma (s),\gamma (b))-h^{s,a}_\alpha (\gamma (s),\gamma (a))=\int ^b_ae^{F(\tau )}(L(\gamma ,\dot{\gamma },\tau )+\alpha )\text{ d }\tau ,\forall s<a. \end{aligned}$$

Taking $s\rightarrow -\infty$, we derive $\gamma$ is also a calibrated curve by $u^-_\alpha$. $\square$

With the help of (38), the following Lemma can be proved:

Lemma 4.2

(Upper Semi-continuity). The set valued function

$$\begin{aligned} L\in \underbrace{C^{r\ge 2}(TM\times \mathbb {T},\mathbb {R})}_{\Vert \cdot \Vert _{C^r}}\longrightarrow \widetilde{\mathcal {A}}\subset \underbrace{TM\times \mathbb {T}}_{d_{\mathcal {H}}(\cdot ,\cdot )} \end{aligned}$$

is upper semi-continuous. Here $\Vert \cdot \Vert _{C^r}$ is the $C^r-$norm and $d_{\mathcal {H}}$ is the Hausdorff distance.

Proof

It suffices to prove that for any $L_n\rightarrow L$ w.r.t. $\Vert \cdot \Vert _{C^r}-$norm, the accumulating curve of any sequence of curves $\widetilde{\gamma }_n$ in $\mathcal {A}(L_n)$ should lie in $\mathcal {A}(L)$. Due to Lemma 3.5, for any $n\in \mathbb {Z}_+$ such that $\Vert L_n-L\Vert _{C^r}\le 1$, $\widetilde{\mathcal {A}}(L_n)$ is uniformly compact in the phase space. Therefore, for any sequence $\{\widetilde{\gamma }_n\}$ each of which is globally minimal, the accumulating curve $\widetilde{\gamma }_*$ satisfies

$$\begin{aligned}\int _t^se^{F(\tau )}\big (L(\gamma _*,\dot{\gamma }_*,\tau )+\alpha \big )\text{ d }\tau\le & {} \lim _{n\rightarrow +\infty }\int _t^se^{F(\tau )}\big (L_n(\gamma _n,\dot{\gamma }_n,\tau )+\alpha \big )\text{ d }\tau \\\le & {} \lim _{n\rightarrow +\infty }\int _t^se^{F(\tau )}\big (L_n(\eta _n,\dot{\eta }_n,\tau )+\alpha \big )\text{ d }\tau \end{aligned}$$

for any absolutely continuous $\eta _n:[t,s]\rightarrow M$ ending with $\gamma _n(t)$ and $\gamma _n(s)$. Since for any absolutely continuous $\eta :[t,s]\rightarrow M$ ending with $\gamma _*(t)$ and $\gamma _*(s)$, we can find such a sequence $\eta _n:[t,s]\rightarrow M$ converging to $\eta$ uniformly, then we get

$$\begin{aligned} \int _t^se^{F(\tau )}(L(\gamma _*,\dot{\gamma }_*,\tau )+\alpha )\text{ d }\tau \le \inf _{\begin{array}{c} \eta \in C^{ac}([t,s],M)\\ \eta (t)=\gamma _*(t)\\ \eta (s)=\gamma _*(s) \end{array}}\int _t^se^{F(\tau )}(L(\eta , \dot{\eta },\tau )+\alpha )\text{ d }\tau \end{aligned}$$

for any $t<s\in \mathbb {R}$, which implies $\gamma _*$ satisfies the Euler-Lagrange equation. Due to Theorem 1.2, the weak KAM solution $u_*^-$ associated with L is unique, so $\gamma _*$ is globally minimal, then globally calibrated by $u_*^-$, i.e. $\widetilde{\gamma }_*\in \mathcal {A}(L)$.$\square$

4.2 Mather Set in the Condition (H0 $^-$)

For any globally calibrated curve $\widetilde{\gamma }$, we can always find a sequence $T_n>0$, such that a $\varphi _L^t-$invariant measure $\widetilde{\mu }$ can be found by

$$\begin{aligned} \int _{TM\times \mathbb {T}}f(x,v,t)\text{ d }\widetilde{\mu }=\lim _{n\rightarrow +\infty }\frac{1}{T_n}\int _0^{T_n}f(\gamma ,\dot{\gamma }, t)\text{ d }t,\quad \forall f\in C_c(TM\times \mathbb {T},\mathbb {R}). \end{aligned}$$

So the set of $\varphi _L^t-$invariant measures ${\mathfrak M}_L$ is not empty.

Proposition 4.3

For all $\widetilde{\nu }\in {\mathfrak M}_L$ and $\alpha \in \mathbb {R}$, we have

$$\begin{aligned} \int _{TM\times \mathbb {T}}L+\alpha -f(t)u_\alpha ^-d\widetilde{\nu }\ge 0. \end{aligned}$$

Besides,

$$\begin{aligned} \int _{TM\times \mathbb {T}}L+\alpha -f(t)u_\alpha ^-d \widetilde{\nu }= 0 \quad \Longleftrightarrow \quad \text {supp}(\widetilde{\nu })\subset \widetilde{\mathcal {A}} \end{aligned}$$

Proof

For any Euler-Lagrange curve $\gamma :\mathbb {R}\rightarrow M$ contained in $\pi _x$³$\text {supp}(\widetilde{\nu })$, we have

$$\begin{aligned}{} & {} \int _{TM\times \mathbb {T}} f(t) u_\alpha ^-(x,t)\text{ d }\widetilde{\nu }\\= & {} \lim _{T\rightarrow +\infty }\frac{1}{T}\int _0^Tf(t)u_\alpha ^-(\gamma (t),t)\text{ d }t\\\le & {} \lim _{T\rightarrow +\infty }\frac{1}{T}\int _0^Tf(t)\int _{-\infty }^t e^{F(s)-F(t)} [L(\gamma (s),\dot{\gamma }(s),s)+\alpha ] \text{ d }s \text{ d }t\\= & {} \lim _{T\rightarrow +\infty }\frac{1}{T}\int _0^Tf(t)e^{-F(t)}\int _{-\infty }^te^{F(s)} [L(\gamma (s),\dot{\gamma }(s),s) +\alpha ]\text{ d }s\text{ d }t\\= & {} \lim _{T\rightarrow +\infty }-\frac{1}{T}\int _0^T\Big ( \int _{-\infty }^te^{F(s)} [L(\gamma (s),\dot{\gamma }(s),s)+\alpha ] \text{ d }s\Big )\text{ d } e^{-F(t)}\\= & {} \lim _{T\rightarrow +\infty }-\frac{1}{T}\Big (e^{-F(t)}\int _{-\infty }^te^{F(s)} [L(\gamma (s),\dot{\gamma }(s),s) +\alpha ]\text{ d }s\Big |_0^T\Big )\\{} & {} +\lim _{T\rightarrow +\infty }\frac{1}{T}\int _0^TL(\gamma (t),\dot{\gamma }(t),t)+\alpha \text{ d }t\\= & {} \int _{TM\times \mathbb {T}}L(x,v,t)+\alpha \text{ d } \widetilde{\nu },\nonumber \end{aligned}$$

which is an equality only when $\gamma$ is a backward calibrated curve of $(-\infty ,t]$ for all $t\in \mathbb {R}$, which implies $\gamma$ is globally calibrated. $\square$

Due to this Proposition we can easily show that $\emptyset \ne \widetilde{\mathcal {M}}\subset \widetilde{\mathcal {A}}$. Moreover, as we did for the Aubry set, we can similarly get that $\pi ^{-1}:\mathcal {M}\rightarrow \widetilde{\mathcal {M}}$ is a Lipschitz function.

4.3 Maximal Global Attractor in the Condition (H0 $^-$)

Note that $\hat{H}(x,p,t,I,u)=H(x,p,t)+I+f(t)u-\alpha$ and equation(4). We derive

$$\begin{aligned} \frac{\text{ d }}{\text{ d }t}\hat{H}&=\frac{\text{ d }}{\text{ d }t}\big (H(x,p,t)+I+f(t)u-\alpha \big )\nonumber \\&=\partial _xH\cdot \dot{x}+\partial _pH\cdot \dot{p}+\partial _tH+\dot{I}+f'(t)u+f(t)\dot{u}\nonumber \\&=\partial _xH\cdot \partial _pH+\partial _pH\cdot (-\partial _xH-f(t)p)+\partial _tH+(-\partial _tH-f(t)I)\\&\ \ \ \ \ \ +f'(t)u+f(t)(p\cdot \partial _pH-H+\alpha -f(t)u)\nonumber \\&=-f(t)(H+I+f(t)u-\alpha )\nonumber \\&=-f(t)\hat{H}.\nonumber \end{aligned}$$

(40)

From Remark 1.1 and $[f]>0$, it follows that for any initial point (x, p, s, I, u), the $\omega -$limit of trajectory $\widehat{\varphi }_{ H}^t(x,p,\bar{s},I,u)$ lies in

$$\begin{aligned} \widehat{\Sigma }_{ H}:=\{{\widehat{H}}(x,p,\bar{s},I,u)=0\}\subset T^*M\times T^*\mathbb {T}\times \mathbb {R}. \end{aligned}$$

(41)

Lemma 4.4

For any point $Z:=\big (x,p,\bar{s},\alpha -f(s)u-H(x,p,s),u\big )\in \widehat{\Sigma }_{ H}$ with $u\le u^-_\alpha (x,s)$, if

$$\begin{aligned} \liminf _{t\rightarrow -\infty }e^{F(t)}\big |\pi _u\widehat{\varphi }_{ H}^t(Z)\big |=0, \end{aligned}$$

then $\pi _x\widehat{\varphi }_{ H}^t(Z)$ is a backward calibrated curve for $t\le 0$.

Proof

From the equation $\dot{u}=\langle H_p,p\rangle -H+\alpha -f(t)u$, we derive

$$\begin{aligned}e^{F(s)}\pi _uZ= & {} \int _{-\infty }^0 \frac{d}{dt}e^{F(t+s)}\pi _u\widehat{\varphi }_{ H}^t(Z)\text{ d }t\\= & {} \int _{-\infty }^s e^{F(t)}\big (L(\mathcal {L}( \varphi _{ H}^{t-s}(x,p,\bar{s})))+\alpha \big )\text{ d }t\le u_\alpha ^-(x,s), \end{aligned}$$

then due to the expression of $u^-_\alpha$ in (26), $\pi _x\widehat{\varphi }_{ H}^t(Z)$ is a backward calibrated curve for $t\le 0$.$\square$

This Lemma inspires us to decompose $\widehat{\Sigma }_{ H}$ further:

$$\begin{aligned} \left\{ \begin{aligned} \widehat{\Sigma }_{ H}^-:= \big \{(x,p,\bar{s},\alpha -f(s)u-H(x,p,s), u)\big | u> u^-_\alpha (x,s)\big \},\\ \widehat{\Sigma }_{ H}^0:=\big \{(x,p,\bar{s},\alpha -f(s)u-H(x,p,s), u)\big | u= u^-_\alpha (x,s)\big \},\\ \widehat{\Sigma }_{ H}^+:=\big \{(x,p,\bar{s},\alpha -f(s)u-H(x,p,s), u)\big | u< u^-_\alpha (x,s)\big \}. \end{aligned} \right. \end{aligned}$$

Lemma 4.5

For any $Z=\big (x,p,\bar{s},\alpha -f(s)u-H(x,p,s),u\big )\in \widehat{\Sigma }_{ H}$, we have

$$\begin{aligned}{} & {} \partial _t^+\Big (u_\alpha ^-(\pi _{x,t}\widehat{\varphi }_{ H}^t(Z))-\pi _u\widehat{\varphi }_{ H}^t(Z)\Big )\\\le & {} -f(t+s)\Big (u_\alpha ^-(\pi _{x,t}\widehat{\varphi }_{ H}^t(Z))-\pi _u\widehat{\varphi }_{ H}^t(Z)\Big ).\nonumber \end{aligned}$$

(42)

Consequently, $\lim _{t\rightarrow +\infty }\widehat{\varphi }_{H}^t(Z)\in \widehat{\Sigma }_{ H}^-\cup \widehat{\Sigma }_{ H}^0$.

Proof

As $\widehat{\varphi }_{ H}^t(Z)=\big (x(t),p(t),\overline{t+s},-f(s+t)u(t)-H(x(t),p(t),s+t),u(t)\big )$, then

$$\begin{aligned}{} & {} \partial _t^+\big [u_\alpha ^-(x(t),s+t)-u(t)\big ]\\\le & {} \max \big \langle \partial _x^* u_\alpha ^-(x(t),s+t),\dot{x}(t)\big \rangle +\partial _t^*u_\alpha ^-(x(t),s+t)-\dot{u}(t)\\\le & {} \max H(x(t),\partial _x^* u_\alpha ^-(x(t),s+t),s+t)+L(x(t),\dot{x}(t),s+t)\\{} & {} +\partial ^*_tu_\alpha ^-(x(t),s+t)-\langle H_p(x(t),p(t),t+s),p(t)\rangle \\{} & {} +f(t+s)u(t)+H(x(t),p(t),s+t)-\alpha \\= & {} \max H(x(t),\partial _x^* u_\alpha ^-(x(t),s+t),s+t)+\partial ^*_tu^-_\alpha (x(t),s+t)\\{} & {} +f(t+s)u(t)-\alpha \\\le & {} f(t+s)[u(t)-u_\alpha ^-(x(t),t+s)] \end{aligned}$$

where the ‘max’ is about all the element $(\partial _x^* u_\alpha ^-(x(t),s+t), \partial _t^* u_\alpha ^-(x(t),s+t))$ in $D^*u_\alpha ^-(x(t),s+t)$ (see Theorem B.5 for the definition). So $\lim _{t\rightarrow +\infty }\widehat{\varphi }_{ H}^{t}(Z)\in \widehat{\Sigma }_{ H}^-\cup \widehat{\Sigma }_{ H}^0$.$\square$

Proposition 4.6

$\Omega :=\bigcap _{t\ge 0} \widehat{\varphi }_{ H}^t(\widehat{\Sigma }_{ H}^-\cup \widehat{\Sigma }_{ H}^0)$ is the maximal invariant set contained in $\widehat{\Sigma }_{H}^-\cup \widehat{\Sigma }_{ H}^0$.

Proof

Due to (42), $\widehat{\Sigma }_{ H}^-\cup \widehat{\Sigma }_{ H}^0$ is forward invariant. Besides, any invariant set in $\widehat{\Sigma }_{ H}$ has to lie in $\widehat{\Sigma }_{ H}^-\cup \widehat{\Sigma }_{ H}^0$. So $\Omega$ is the maximal invariant set in $\widehat{\Sigma }_{ H}^-\cup \widehat{\Sigma }_{ H}^0$.$\square$

Lemma 4.7

If the $p-$component of $\Omega$ is bounded, then the ${u,I}-$components of $\Omega$ are also bounded.

Proof

It suffices to prove that for any $(x_0,p_0,\bar{t}_0,I_0,u_0)\in T^*M\times T^*\mathbb {T}\times \mathbb {R}$, there exists a time $T(x_0,p_0,\bar{t}_0,I_0,u_0)>0$ such that for any $t\ge T$,

$$\begin{aligned} \big \Vert \pi _{u,I}\widehat{\varphi }_{ H}^t(x_0,p_0,\bar{t}_0,I_0,u_0)\big \Vert \le C \end{aligned}$$

(*)

for a uniform constant $C=C(\pi _{p}\Omega )$. Since $\pi _{p}\Omega$ is bounded, due to the definition of $\Omega$, for any $(x_0,p_0,\bar{t}_0,I_0,u_0)\in T^*M\times T^*\mathbb {T}\times \mathbb {R}$, there always exists a time $T'(x_0,p_0,\bar{t}_0,I_0,u_0)>0$ such that for any $t\ge T'$,

$$\begin{aligned} \big \Vert \pi _p\widehat{\varphi }_{ H}^t(x_0,p_0,\bar{t}_0,I_0,u_0)\big \Vert \le C'=\frac{3}{2} \text {diam}(\pi _{p}\Omega ). \end{aligned}$$

On the other side, the $u-$equation of (4) implies that for any $t> 0$,

$$\begin{aligned}{} & {} \big \Vert \pi _u\widehat{\varphi }_{ H}^{t+T'}(x_0,p_0,\bar{t}_0,I_0,u_0)\big \Vert \\\le & {} e^{F(t_0+T')-F(t+T'+t_0)}|\pi _u\widehat{\varphi }_{ H}^{T'}(x_0,p_0,\bar{t}_0,I_0,u_0)|\\{} & {} +\int _0^te^{F(s+t_0+T')-F(t+t_0+T')}\Big |\langle H_p,p\rangle -H\Big |_{\widehat{\varphi }_{ H}^{s+T'}(x_0,p_0,\bar{t}_0,I_0,u_0)}\text{ d }s \end{aligned}$$

where the first term of the right hand side will tend to zero as $t\rightarrow +\infty$, and the second term has a uniform bound depending only on $[f], C'$. Therefore, there exists a time $T''(x_0,p_0,\bar{t}_0,I_0,u_0)$ such that for any $t\ge T'+T''$, there exists a constant $C''=C''(C',[f])$ such that

$$\begin{aligned} \big \Vert \pi _u\widehat{\varphi }_{ H}^{t}(x_0,p_0,\bar{t}_0,I_0,u_0)\big \Vert \le C''. \end{aligned}$$

Benefiting from the boundedness of $u-$component, we can repeat aforementioned scheme to the $I-$equation of (4), then prove (*).$\square$

Once $\Omega$ is compact, it has to be the maximal global attractor of $\widehat{\varphi }_{ H}^t$ in the whole phase space $T^*M\times T^*\mathbb {T}\times \mathbb {R}$. Then due to Proposition 3.8, any backward calibrated curve $\gamma _{x,s}^-:(-\infty ,s]\rightarrow M$ decides a unique trajectory

$$\begin{aligned}\widehat{\varphi }_{ H}^t&\Big (&\mathcal {L}^{-1}(x,\lim _{\varsigma \rightarrow s_-}\dot{\gamma }_{x,s}^-(\varsigma ),s),\alpha -f(s)u_\alpha ^-(x,s)\\{} & {} -H\big (\mathcal {L}^{-1}(x,\lim _{\varsigma \rightarrow s_-}\dot{\gamma }_{x,s}^-(\varsigma ),s)\big ),u_\alpha ^-(x,s)\Big ) \end{aligned}$$

for $t\in \mathbb {R}$, which lies in $\widehat{\Sigma }_{ H}$. Furthermore,

$$\begin{aligned} \widehat{\mathcal {A}}:=\Big \{\Big (\mathcal {L}^{-1}(x,\partial _x u_\alpha ^-(x,t),t),\partial _t u_\alpha ^-(x,t),u_\alpha ^-(x,t)\Big )\Big |(x,t)\in {\mathcal {A}}\Big \}\subset \Omega \end{aligned}$$

because $\Omega$ is the maximal invariant set in $\widehat{\Sigma }_{ H}$.

Lemma 4.8

$\widehat{\mathcal {A}}$ is the maximal invariant set contained in $\widehat{\Sigma }_{H}^0$.

Proof

If $\mathcal {I}$ is an invariant set contained in $\widehat{\Sigma }_{ H}^0$, then $\pi _u(\widehat{\varphi }_{ H}^t(\mathcal {I}))$ is always bounded. Due to Lemma 4.4, any trajectory in $\mathcal {I}$ has to be backward calibrated. As $\mathcal {I}$ is invariant, any trajectory in it has to be contained in $\widehat{\mathcal {A}}$.

Proof of Theorem 1.7: Let $\tilde{\mu }\in \mathfrak {M}_L$ be ergodic, then we can find $(x_0,v_0,t_0)\in TM\times \mathbb {T}$ such that

$$\begin{aligned}\int _{TM\times \mathbb {T}}e^{F(t)}(L(x,v,t)+c(H))\text{ d }\tilde{\mu }=\lim _{T\rightarrow +\infty }\frac{1}{T} \int ^0_{-T}e^{F(\tau )}(L(\varphi _L^\tau (x_0,v_0,t_0))+c(H))\text{ d }\tau . \end{aligned}$$

Therefore, for any weak KAM solution $u_c^-:M\times \mathbb {T}\rightarrow \mathbb {R}$ of ($\text {HJ}_0$), we have

$$\begin{aligned}{} & {} e^{F(0)}u_c^-(x_0,t_0)-e^{F(-T)}u_c^-(\pi _{x,t}\varphi _L^{-T}(x_0,v_0,t_0))\\\le & {} \int ^0_{-T}e^{F(\tau )}(L(\varphi _L^\tau (x_0,v_0,t_0))+c(H))\text{ d }\tau , \end{aligned}$$

which implies

$$\begin{aligned}{} & {} \lim _{T\rightarrow +\infty }\frac{1}{T}\int ^0_{-T}e^{F(\tau )}(L(\varphi _L^\tau (x_0,v_0,t_0))+c(H))\text{ d }\tau \\= & {} \lim _{T\rightarrow +\infty }\frac{1}{T} (e^{F(0)}u_c^-(x_0,t_0)-e^{F(-T)}u_c^-(\pi _{x,t}\varphi _L^{-T}(x_0,v_0,t_0))=0. \end{aligned}$$

Hence,

$$\begin{aligned} \int _{TM\times \mathbb {T}}e^{F(t)}(L(x,v,t)+c(H))\text{ d }{\tilde{\mu }}\ge 0. \end{aligned}$$

That further implies

$$\begin{aligned} \frac{\inf _{\mu \in \mathfrak M_L}\int _{TM\times \mathbb {T}}e^{F(t)}L(x,v,t)\text{ d }\tilde{\mu }}{ \int _{0}^1e^{F(\tau )}\text{ d }\tau }\ge -c(H). \end{aligned}$$

On the other side, for any $(x,0)\in M\times \mathbb {T}$ fixed, the backward calibrated curve $\gamma ^-_{x,0}:(-\infty ,0]\rightarrow M$ satisfies

$$\begin{aligned} e^{F(0)}u_c(\gamma ^-_{x,0}(0),0)&-e^{F(-n)}u_c(\gamma ^-_{x,0}(-n),-n)=\int ^{0}_{-n}e^{F(\tau )}(L(\gamma ^-_{x,0}(\tau ),\dot{\gamma }^-_{x,0}(\tau ),\tau )+ c)\text{ d }\tau \end{aligned}$$

for any $n\in \mathbb {Z}_+$. By the Resize Representation Theorem, the time average w.r.t. $\gamma ^-_{x,0}|_{[-n,0]}:[-n,0]\rightarrow M$ decides a sequence of Borel probability measures ${\mu }_n$. Due to Lemma 3.5, we can always find a subsequence $\{{\tilde{\mu }}_{n_k}\}$ converging to a unique Borel probability measure ${\tilde{\mu }}^*$, i.e.

$$\begin{aligned}\int _{TM\times \mathbb {T}}g(x,v,t)\text{ d }{\tilde{\mu }}^*= & {} \lim _{k\rightarrow \infty }\int _{TM\times \mathbb {T}}g(x,v,t)\text{ d }{\tilde{\mu }}_{n_k}\\= & {} \lim _{k\rightarrow \infty }\frac{1}{n_k}\int ^0_{-n_k}g(\gamma ^-_{x,0}(\tau ),\dot{\gamma }^-_{x,0}(\tau ),\bar{\tau })\text{ d }\tau \end{aligned}$$

for any $g\in C_c(TM\times \mathbb {T},\mathbb {R})$. Besides, we can easily prove that $\tilde{\mu }^*\in \mathfrak M_L$ and

$$\begin{aligned}{} & {} \int _{TM\times \mathbb {T}}e^{F(t)}(L(x,v,t)+c(H))\text{ d }{\tilde{\mu }}^*\\= & {} \lim _{k\rightarrow \infty }\frac{1}{n_k}\int ^{0}_{-n_k}e^{F(\tau )}(L(\gamma ^-_{x,0}(\tau ),\dot{\gamma }^-_{x,0}(\tau ),\tau )+ c(H))\text{ d }\tau \\= & {} \lim _{k\rightarrow \infty }\frac{1}{n_k}\bigg (u^-_c(\gamma ^-_{x,0}(0),0)-u^-_c(\gamma ^-_{x,0}(-n_k),-n_k)\bigg )=0. \end{aligned}$$

Then,

$$\begin{aligned} -c(H)=\frac{\inf _{ \tilde{\mu }\in \mathfrak M_L}\int _{TM\times \mathbb {T}}e^{F(t)}L(x,v,t)\text{ d }{\tilde{\mu }}}{\int _0^1e^{F(\tau )}\text{ d }\tau }. \end{aligned}$$

Gathering all the infimum of the right side of previous equality, we get a set of Mather measures $\mathfrak M_m$. Due to the Crossing Lemma in [23], the Mather set

$$\begin{aligned} \widetilde{\mathcal {M}}:=\overline{\bigcup _{\tilde{\mu }\in \mathfrak M_m} \text {supp}(\tilde{\mu })} \end{aligned}$$

is a Lipschitz graph over $\mathcal {M}:=\pi \widetilde{\mathcal {M}}$. $\square$

5 Convergence of Parameterized Viscosity Solutions

In this section we deal with the convergence of weak KAM solution $u_\delta ^-$ for system (14) as $\delta \rightarrow 0_+$. Recall that $[f_0]=0$ and

$$\begin{aligned} f_1(t):=\lim _{\delta \rightarrow 0_+}\dfrac{f_\delta (t)-f_0(t)}{\delta }>0, \end{aligned}$$

there must exist a $\delta _0>0$ such that

$$\begin{aligned} f_{\delta }(t)>f_0(t),\quad \forall \ t\in \mathbb {T}\end{aligned}$$

for all $\delta \in [0,\delta _0]$. Due to Theorem 1.3 there exists a unique c(H), such that the weak KAM solutions $u_0^-$ of (17) with $\alpha =c(H)$ exist. For each $(x,t)\in M\times \mathbb {R}$ and $s<t$, the Lax-Oleinik operator

$$\begin{aligned} T_s^{\delta ,-}(x,t)=\inf _{\begin{array}{c} \gamma \in C^{ac}([s,t],M)\\ \gamma (t)=x \end{array}}\int ^t_se^{F_\delta (\tau )-F_\delta (t)}\big (L(\gamma (\tau ),\dot{\gamma }(\tau ),\tau )+c(H)\big )\text{ d }\tau \end{aligned}$$

is well defined, of which the following Lemma holds:

Lemma 5.1

For each $\delta \ge 0$ and $T_s^{\delta ,-}(x,t)$ converges uniformly to $u^-_{\delta }(x,t)$ on each compact subset of $M\times \mathbb {R}$ as $s\rightarrow -\infty$.

Proof

Let $\gamma ^-_{\delta ,x,t}:(-\infty ,t]\rightarrow M$ be a calibrated curve of $u_\delta ^-(x,t)$. Then,

$$\begin{aligned} e^{F_\delta (t)}u_\delta ^-(x,t)=e^{F_\delta (s)}u_\delta ^-(\gamma ^-_{\delta ,x,t}(s),s)+\int ^t_se^{F_\delta (\tau )}(L(\gamma ^-_{\delta ,x,t},\dot{\gamma }^-_{\delta ,x,t},\tau )+c(H))\text{ d }\tau \end{aligned}$$

and

$$\begin{aligned} e^{F_\delta (t)}T_s^{\delta ,-}(x,t)\le \int ^t_se^{F_\delta (\tau )}(L(\gamma ^-_{\delta ,x,t},\dot{\gamma }^-_{\delta ,x,t},\tau )+c(H))\text{ d }\tau . \end{aligned}$$

Then,

$$\begin{aligned} T_s^{\delta ,-}(x,t)-u_\delta ^-(x,t)\le -e^{F_\delta (s)-F_\delta (t)}u^-_\delta (\gamma ^-_{\delta ,x,t}(s),s). \end{aligned}$$

(43)

On the other hand, let $\gamma _0:[s,t]\rightarrow M$ be a minimizer of $T_s^{\delta ,-}(x,t)$. Then,

$$\begin{aligned} e^{F_\delta (t)}T_s^{\delta ,-}(x,t)=\int ^t_se^{F_\delta (\tau )}(L(\gamma _0(\tau ),\dot{\gamma }_0(\tau ),\tau )+c(H))\text{ d }\tau \end{aligned}$$

and

$$\begin{aligned} e^{F_\delta (t)}u^-_{\delta }(x,t)-e^{F_\delta (s)}u^-_\delta (\gamma _0(s),s)\le \int ^t_se^{F_\delta (\tau )}(L(\gamma _0,\dot{\gamma }_0,\tau )+c(H))\text{ d }\tau . \end{aligned}$$

Hence,

$$\begin{aligned} u^-_\delta (x,t)-T_s^{\delta ,-}(x,t)\le e^{F_\delta (s)-F_\delta (t)}u_\delta ^-(\gamma _0(s),s). \end{aligned}$$

(44)

From (43) and (44), it follows

$$\begin{aligned} |u^-_\delta (x,t)-T_s^{\delta ,-}(x,t)|\le e^{F_\delta (s)-F_\delta (t)}\max u^-_\delta , \end{aligned}$$

which means $T_s^{\delta ,-}(x,t)$ converges uniformly to $u_\delta ^-(x,t)$ on each compact subset of $M\times \mathbb {R}$. $\square$

Lemma 5.2

$u_\delta ^-:M\times \mathbb {T}\rightarrow \mathbb {R}$ are equi-bounded and equi-Lipschitz w.r.t. $\delta \in (0,\delta _0]$.

Proof

To show $u_{\delta }^-$ are equi-bounded from below, it suffices to show

$$\begin{aligned} \{T_s^{\delta ,-}(x,t)|(x,t)\in M\times [0,1],s\le 0,\delta \in (0,\delta _0]\} \end{aligned}$$

is bounded from below. Let $\gamma _0:[s,t]\rightarrow M$ be a minimizer of $T_s^{\delta ,-}(x,t)$, $u_\delta (\tau ):=T_s^{\delta ,-}(\gamma _0(\tau ),\tau )$, and $\tilde{u}_\delta (\tau ):=e^{F_\delta (\tau )}u_\delta (\tau ), \tau \in [s,t]$. Then,

$$\begin{aligned} \frac{\text{ d }\tilde{u}_\delta (\tau )}{\text{ d }\tau }=e^{F_\delta (\tau )}(L(\gamma _0(\tau ),\dot{\gamma }_0(\tau ),\tau )+c(H)). \end{aligned}$$

Hence,

$$\begin{aligned} \frac{\text{ d }u_\delta (\tau )}{\text{ d }\tau }=L(\gamma _0(\tau ),\dot{\gamma }_0(\tau ),\tau )+c(H)-f_\delta (\tau )u_\delta (\tau ). \end{aligned}$$

We could assume $T_s^{\delta ,-}(x,t)<0$ for some $\delta \in (0,\delta _0]$, $(x,t)\in M\times [0,1],s\le 0$, otherwise 0 is a uniform lower bound of $\{T_s^{\delta ,-}(x,t)|(x,t)\in M\times [0,1],s\le 0,\delta \in (0,\delta _0]\}$. Note that $u_\delta (\cdot )$ are continuous and $u_\delta (s)=0$. There exists $s_0\in [s,t)$ such that $u_\delta (s_0)=0$ and $u_\delta (\tau )<0,\tau \in (s_0,t]$. From $f_\delta >f_0$, it follows that

$$\begin{aligned} \frac{\text{ d }u_\delta (\tau )}{\text{ d }\tau }\ge L(\gamma _0(\tau ),\dot{\gamma }_0(\tau ),\tau )+c(H)-f_0(\tau )u_\delta (\tau ),\tau \in [s_0,t]. \end{aligned}$$

Hence,

$$\begin{aligned} \frac{\text{ d }}{\text{ d }\tau }\big (e^{F_0(\tau )}u_\delta (\tau )\big )\ge e^{F_0(\tau )}(L(\gamma _0(\tau ),\dot{\gamma }_0(\tau ),\tau )+c(H)), \end{aligned}$$

where $F_0(\tau )=\int ^\tau _0f_0(\sigma )\text{ d }\sigma$. Integrating on $[s_0,t]$, it holds that

$$\begin{aligned} e^{F_0(t)}\cdot u_\delta (t)\ge \int ^t_{s_0}e^{F_0(\tau )}(L(\gamma _0(\tau ),\dot{\gamma }_0(\tau ),\tau )+c(H))\text{ d }\tau . \end{aligned}$$

(45)

Let $\beta :[t,t+2-\overline{t-s_0}]\rightarrow M$ be a geodesic with $\beta (t)=\gamma _0(t),\beta (t+2-\overline{t-s_0})=\gamma _0(s_0)$, and

$$\begin{aligned} |\dot{\beta }(\tau )|=\frac{d(\gamma _0(s_0),\gamma _0(t))}{2-\overline{t-s_0}}\le \text{ diam }(M)=:k_1. \end{aligned}$$

Due to the definition of c(H) in (35), we derive

$$\begin{aligned}&\int ^t_{s_0}e^{F_0(\tau )}(L(\gamma _0(\tau ),\dot{\gamma }_0(\tau ),\tau )+c(H))\text{ d }\tau \\&+\int ^{t+2-\overline{t-s_0}}_te^{F_0(\tau )}(L(\beta (\tau ),\dot{\beta }(\tau ),\tau )+c(H))\text{ d }\tau \ge 0. \end{aligned}$$

Note that

$$\begin{aligned}{} & {} \int ^{t+2-\overline{t-s_0}}_te^{F_0(\tau )}(L(\beta (\tau ),\dot{\beta }(\tau ),\tau )+c(H))\text{ d }\tau \\\le & {} \int ^{t+2-\overline{t-s_0}}_te^{F_0(\tau )}(C_{k_1}+c(H))\text{ d }\tau \le 2(C_{k_1}+c(H))e^{\max _{t\in \mathbb {T}} F_0(t)}. \end{aligned}$$

Hence,

$$\begin{aligned} \int ^t_{s_0}e^{F_0(\tau )}(L(\gamma _0(\tau ),\dot{\gamma }_0(\tau ),\tau )+c(H))\text{ d }\tau \ge -2(C_{k_1}+c(H))e^{\max F_0}. \end{aligned}$$

Combining (45), we derive

$$\begin{aligned} u_\delta (t)\ge -2 |C_{k_1}+c(H)| e^{\max F_0-\min F_0}. \end{aligned}$$

Next, we prove $u_{\delta }^-(x,t)$ are equi-bounded from above. It suffices to show $\{T^{\delta ,-}_s(x,t)|(x,t)\in M\times [0,1],s\le 0,\delta \in (0,\delta _0]\}$ is bounded from above. We could assume $T^{\delta ,-}_s(x,t)>0$ for some $\delta \in (0,\delta _0]$, $(x,t)\in M\times [0,1],s\le 0$, otherwise 0 is a uniform upper bound of $\{T^{\delta ,-}_s(x,t)|(x,t)\in M\times [0,1],s\le 0,\delta \in (0,\delta _0]\}$.

Let $u_0^-(x,t)$ be a weak KAM solution of

$$\begin{aligned} \partial _tu+H(x,\partial _xu,t)+f_0(t)u=c(H), \end{aligned}$$

and $\gamma ^-_{x,t}:(-\infty ,t]\rightarrow M$ be a calibrated curve of $u_0^-(x,t)$. Let

$$\begin{aligned} v_\delta (\tau ):=T^{\delta ,-}_s(\gamma ^-_{x,t}(\tau ),\tau ),\tau \in [s,t]. \end{aligned}$$

Then

$$\begin{aligned}\frac{e^{F_\delta (\tau +\Delta \tau )}v_\delta (\tau +\Delta \tau )-e^{F_\delta (\tau )}v_\delta (\tau )}{\Delta \tau }\le \frac{1}{\Delta \tau }\int ^{\tau +\Delta \tau }_{\tau }e^{F_\delta (\sigma )}(L(\gamma ^-_{x,t}(\sigma ),\dot{\gamma }^-_{x,t}(\sigma ),\sigma )+c(H))\text{ d }\sigma . \end{aligned}$$

Note that

$$\begin{aligned}&\varlimsup _{\Delta \tau \rightarrow 0}\frac{e^{F_\delta (\tau +\Delta \tau )}v_\delta (\tau +\Delta \tau )-e^{F_\delta (\tau )}v_\delta (\tau )}{\Delta \tau }\\&=\varlimsup _{\Delta \tau \rightarrow 0}\frac{e^{F_\delta (\tau +\Delta \tau )}v_\delta (\tau +\Delta \tau )-e^{F_\delta (\tau +\Delta \tau )}v_\delta (\tau )+e^{F_\delta (\tau +\Delta \tau )}v_\delta (\tau )-e^{F_\delta (\tau )}v_\delta (\tau )}{\Delta \tau }\\&=e^{F_\delta (\tau )}\varlimsup _{\Delta \tau \rightarrow 0}\bigg (\frac{v_\delta (\tau +\Delta \tau )-v_\delta (\tau )}{\Delta \tau }\bigg )+e^{F_\delta (\tau )}f_\delta (\tau )v_\delta (\tau ). \end{aligned}$$

Hence,

$$\begin{aligned} \varlimsup _{\Delta \tau \rightarrow 0}\bigg (\frac{v_\delta (\tau +\Delta \tau )-v_\delta (\tau )}{\Delta \tau }\bigg )\le L(\gamma ^-_{x,t}(\tau ),\dot{\gamma }^-_{x,t}(\tau ),\tau )+c(H)-f_\delta (\tau )v_\delta (\tau ). \end{aligned}$$

Since $v_\delta (s)=0$ and $v_\delta (\tau )$ is continuous, there exists $s_1\in [s,t)$ such that $v_\delta (s_1)=0$ and $v_\delta (\tau )>0,\tau \in (s_1,t]$.

For $\tau \in (s_1,t]$,

$$\begin{aligned} \varlimsup _{\Delta \tau \rightarrow 0}\bigg (\frac{v_\delta (\tau +\Delta \tau )-v_\delta (\tau )}{\Delta \tau }\bigg )&\le L(\gamma ^-_{x,t}(\tau ),\dot{\gamma }^-_{x,t}(\tau ),\tau )+c(H)-f_\delta (\tau )v_\delta (\tau )\\&\le L(\gamma ^-_{x,t}(\tau ),\dot{\gamma }^-_{x,t}(\tau ),\tau )+c(H)-f_0(\tau )v_\delta (\tau ). \end{aligned}$$

Then,

$$\begin{aligned}\varlimsup _{\Delta \tau \rightarrow 0}\bigg (\frac{e^{F_0(\tau +\Delta \tau )}v_\delta (\tau +\Delta \tau )-e^{F_0(\tau )}v_\delta (\tau )}{\Delta \tau }\bigg )\le e^{F_0(\tau )}(L(\gamma ^-_{x,t}(\tau ),\dot{\gamma }^-_{x,t}(\tau ),\tau )+c(H)). \end{aligned}$$

From $v_\delta (s_1)=0$, it follows that

$$\begin{aligned} e^{F_0(t)}v_\delta (t)&\le \int ^t_{s_1}e^{F_0(\tau )}(L(\gamma ^-_{x,t},\dot{\gamma }^-_{x,t},\tau )+c(H))\text{ d }\tau \\&=e^{F_0(t)}u_0^-(x,t)-e^{F_0(s_1)}u_0^-(\gamma ^-_{x,t}(s_1),s_1). \end{aligned}$$

Then,

$$\begin{aligned} v_\delta (t)\le 2\max |u_0^-|\cdot e^{\max F_0-\min F_0}. \end{aligned}$$

Note that $u_\delta ^-(x,t)$ are equi-bounded. By a similar approach of the proof of Lemma 3.6, we derive that $u_\delta ^-$ are equi-Lipschitz. $\square$

Lemma 5.3

For any $\delta \in (0,\delta _0]$ and any $(x,\bar{s})\in M\times \mathbb {T}$, the backward calibrated curve $\gamma _{\delta ,x,s}^-:(-\infty ,s]\rightarrow M$ associated with $u_\delta ^-$ has a uniformly bounded velocity, i.e. there exists a constant $K>0$, such that

$$\begin{aligned} |\dot{\gamma }_{\delta ,x,s}^-(t)|\le K, \quad \forall \delta \in (0,1] \text{ and } t\in (-\infty ,s). \end{aligned}$$

Proof

By a similar way in the proof of Lemma 3.5, there exists $s_0$ in each interval with length 1, such that

$$\begin{aligned} |\dot{\gamma }^-_{\delta ,x,s}(s_0)|\le C_{k_1}+C(1), \end{aligned}$$

where $k_1=\text{ diam }(M)$. Note that $f_{\delta }$ depends continuously on $\delta$ and is 1-periodic. We derive the Lagrangian flow $(\gamma ^-_{\delta ,x,s}(\tau ),\dot{\gamma }^-_{\delta ,x,s}(\tau ),\tau )$ is 1-periodic and depends continuously on the parameter $\delta$. Hence, there exists $K>0$ depends only on $L, k_1$, and $\delta _0$, such that $|\dot{\gamma }^-_{\delta ,x,s}|<K$.$\square$

Proposition 5.4

For any ergodic measure $\tilde{\mu }\in \mathfrak M_m(0)$ and any $0<\delta \le \delta _0$, we have

$$\begin{aligned} \int _{TM\times \mathbb {T}}e^{F_0(t)}\frac{f_\delta (t)-f_0(t)}{\delta }u_\delta ^-(x,t)\textrm{d}\tilde{\mu }(x,v,t)\le 0. \end{aligned}$$

(46)

Proof

Since $\{u_\delta ^-\}_{\delta \in (0,\delta _0]}$ is uniformly bounded and $[f_0]=0$, then

$$\begin{aligned} \lim _{T\rightarrow +\infty }\frac{1}{T}\int _0^Tu_\delta ^-(\gamma (t),t)\text{ d } e^{F_0(t)}=\int _{TM\times \mathbb {T}}u_\delta ^-(x,t)f_0(t)e^{F_0(t)}\text{ d }\tilde{\mu }(x,v,t) \end{aligned}$$

for any regular curve $\widetilde{\gamma }(t)=(\gamma (t),\bar{t}):t\in \mathbb {R}\rightarrow M\times \mathbb {T}$ contained in $\mathcal {M}(\delta )$. Due to Proposition 3.8,

$$\begin{aligned}\frac{1}{T}\int _0^Tu_\delta ^-(\gamma (t),t)\text{ d } e^{F_0(t)}= & {} \frac{1}{T}u_\delta ^-(\gamma (t),t) e^{F_0(t)}\Big |_0^T-\frac{1}{T}\int _0^T e^{F_0(t)}\big [\partial _t u_\delta ^-(\gamma (t),t)\\+ & {} \langle \dot{\gamma }(t),\partial _xu_\delta ^-(\gamma (t),t)\rangle \big ]\text{ d }t \end{aligned}$$

and

$$\begin{aligned}{} & {} \frac{1}{T}\int _0^T e^{F_0(t)}\big [\partial _t u_\delta ^-(\gamma (t),t)+\langle \dot{\gamma }(t),\partial _xu_\delta ^-(\gamma (t),t)\rangle \big ]\text{ d }t\\\le & {} \frac{1}{T}\int _0^T e^{F_0(t)}\big [L(\gamma ,\dot{\gamma },t)+H(\gamma (t),\partial _x u_\delta ^-(\gamma (t),t),t)+\partial _t u_\delta ^-(\gamma (t),t)\big ]\text{ d }t\\\le & {} \frac{1}{T}\int _0^T e^{F_0(t)}\big [L(\gamma ,\dot{\gamma },t)+c(H)-f_\delta (t)u_\delta ^-(\gamma (t),t) \big ]\text{ d }t, \end{aligned}$$

by taking $T\rightarrow +\infty$ and dividing both sides by $\delta$ we get the conclusion.

Definition 5.5

Let’s denote by $\mathcal {F}_-$ the set of all viscosity subsolutions $\omega :M\times \mathbb {T}\rightarrow \mathbb {R}$ of (15) with $\delta =0$ such that

$$\begin{aligned} \int _{TM\times \mathbb {T}} f_1(t)\omega (x,t)e^{F_0(t)}\text{ d }\tilde{\mu }\le 0,\quad \forall \ \tilde{\mu }\in \mathfrak M_m(0). \end{aligned}$$

(47)

Lemma 5.6

The set $\mathcal {F}_-$ is uniformly bounded from above, i.e.

$$\begin{aligned} \sup \{u(x)|\ \forall \ x\in M,\ u\in \mathcal {F}_-\}<+\infty . \end{aligned}$$

Proof

By an analogy of Lemma 10 of [11], all the functions in the set

$$\begin{aligned} \Big \{ e^{F_0(t)}\omega :M\times \mathbb {T}\rightarrow \mathbb {R}\Big |\omega \prec _{f_0} L+c(H)\Big \} \end{aligned}$$

are uniformly Lipschitz with a Lipschitz constant $\kappa >0$. For any $\tilde{\mu }\in \mathfrak {M}_m(0)$ and $u\in \mathcal {F}_-$

$$\begin{aligned}\min _{(x,t)\in M\times \mathbb {T}}u(x,t)e^{F_0(t)}= & {} \frac{\int _{TM\times \mathbb {T}} f_1(t)\min _{(x,t)\in M\times \mathbb {T}} u(x,t) e^{F_0(t)}\text{ d }\tilde{\mu }}{\int _{TM\times \mathbb {T}}f_1(t) \text{ d }\tilde{\mu } }\\= & {} \frac{\int _{TM\times \mathbb {T}} f_1(t) \min _{(x,t)\in M\times \mathbb {T}} u(x,t)e^{F_0(t)} \text{ d }\tilde{\mu }}{\int _0^1f_1(t)\text{ d }t }\\\le & {} \dfrac{\int _{M\times \mathbb {T}}f_1(t) u(x,t)e^{F_0(t)} \text{ d }\tilde{\mu }}{\int _0^1 f_1(t) \text{ d }t }\le 0. \end{aligned}$$

Then,

$$\begin{aligned}\max _{(x,t)\in M\times \mathbb {T}}u(x,t) e^{F_0(t)}\le & {} \max _{(x,t)\in M\times \mathbb {T}}u(x,t) e^{F_0(t)}-\min _{(x,t)\in M\times \mathbb {T}}u(x,t) e^{F_0(t)}\\\le & {} \kappa \ \text {diam}(M\times \mathbb {T})<+\infty . \end{aligned}$$

As a result,

$$\begin{aligned} \max _{(x,t)\in M\times \mathbb {T}} u(x,t)\le \frac{\max _{(x,t)\in M\times \mathbb {T}}u(x,t) e^{F_0(t)}}{\min _{t\in \mathbb {T}} e^{F_0(t)}}<+\infty \end{aligned}$$

so we finish the proof. $\square$

As $\mathcal {F}_-$ is now upper bounded, we can define a supreme subsolution by

$$\begin{aligned} u_0^*:=\sup _{u\in \mathcal {F}_-} u. \end{aligned}$$

(48)

Later we will see that this is indeed a viscosity solution of (15) for $\delta =0,\alpha =c(H)$ and is the unique accumulating function of $u_\delta ^-$ as $\delta \rightarrow 0_+$.

Proposition 5.7

For any $\delta >0$, any viscosity subsolution $\omega :M\times \mathbb {T}\rightarrow \mathbb {R}$ of (15) with $\delta =0,\alpha =c(H)$ and any point $(x,s)\in M\times \mathbb {T}$, there exists a $\varphi _{L}^t-$backward invariant finite measure $\tilde{\mu }_{x,s}^\delta :TM\times \mathbb {T}\rightarrow \mathbb {R}$ such that

$$\begin{aligned} u_\delta ^-(x,s)\ge \omega (x,s)-\int _{TM\times \mathbb {T}}\omega (y,t)e^{F_0(t)}f_1(t)d\tilde{\mu }_{x,s}^\delta (y,v_y,t) \end{aligned}$$

(49)

where

$$\begin{aligned}{} & {} \int _{TM\times \mathbb {T}}g(y,t)d\tilde{\mu }_{x,s}^\delta (y,v_y,t)\\:= & {} \int _{-\infty }^s\frac{g(\gamma _{\delta ,x,s}^-(t),t)\cdot \frac{\text{ d }}{\text{ d }t}(e^{F_\delta (t)}-e^{F_0(t)})}{f_1(t)}\text{ d }t,\ \forall g\in C(M\times \mathbb {T},\mathbb {R}). \end{aligned}$$

Proof

For any $(x,\bar{s})\in M\times \mathbb {T}$ and any $\delta \in (0,\delta _0]$, there exists a backward calibrated curve $\gamma _{\delta ,x,s}^-:(-\infty ,s]\rightarrow M$ ending with x, such that the viscosity solution $u_\delta ^-$ is differentiable along $(\gamma _{\delta ,x,s}^-(t),\overline{t})$ for all $t\in (-\infty ,s)$ due to Proposition 3.8. Precisely, for all $t\in (-\infty ,s)$

$$\begin{aligned} \frac{\text{ d }}{\text{ d }t}\big (e^{F_\delta }(t)u_\delta ^-(\gamma ^-_{\delta ,x,s}(t),t)\big )=e^{F_\delta (t)}\big (L(\gamma _{\delta ,x,x}(t),\dot{\gamma }_{\delta ,x,s}(t),t)+c(H)\big ). \end{aligned}$$

Integrating on $[s,-T]$,

$$\begin{aligned}e^{F_\delta (s)}u_\delta ^-(x,s)-e^{F_\delta (-T)}u_\delta ^-(\gamma _{\delta ,x,s}^-(-T),-T)= & {} \int _{-T}^se^{F_\delta (t)}\Big [L\Big (\gamma _{\delta ,x,s}^-(t),\dot{\gamma }_{\delta ,x,s}^-(t),t\Big )\\+ & {} c(H)\Big ]\text{ d }t \end{aligned}$$

for any $T>0$, where $F_\delta (t):=\int _0^tf_\delta (\tau )\text{ d }\tau$. On the other side,

$$\begin{aligned} \partial _t\omega (x,t)+H(x,\partial _x\omega (x),t)+f_0(t)\omega (x,t)\le c(H),\quad a.e.\ (x,\bar{t})\in M\times \mathbb {T}\end{aligned}$$

since $\omega$ is also a subsolution of (15) (with $\delta =0$), then

$$\begin{aligned}{} & {} e^{F_\delta (s)}u_\delta ^-(x,s)-e^{F_\delta (-T)}u_\delta ^-(\gamma _{\delta ,x,s}^-(-T),-T)\\\ge & {} \int _{-T}^se^{F_\delta (t)}\Big [L\Big (\gamma _{\delta ,x,s}^-(t),\dot{\gamma }_{\delta ,x,s}^-(t),t\Big )+H\Big (\gamma _{\delta ,x,s}^-(t),\partial _x\omega (\gamma _{\delta ,x,s}^-(t), t),t\Big )\\ {}{} & {} +\partial _t\omega (\gamma _{\delta ,x,s}^-(t),t)+f_0(t)\omega (\gamma _{\delta ,x,s}^-(t),t)\Big ]\text{ d }t\\\ge & {} \int _{-T}^se^{F_\delta (t)}\Big [\frac{d}{dt}\omega (\gamma _{\delta ,x,s}^-(t),t)+f_0(t)\omega (\gamma _{\delta ,x,s}^-(t),t)\Big ]\text{ d }t\\\ge & {} e^{F_\delta (s)} \omega (x,s)-e^{F_\delta (-T)}\omega (\gamma _{\delta ,x,s}^-(-T),-T)-\int _{-T}^s\omega (\gamma _{\delta ,x,s}^-(t),t) e^{F_\delta (t)}\Big (f_\delta (t)-f_0(t)\Big )\text{ d }t. \end{aligned}$$

By taking $T\rightarrow +\infty$ we finally get

$$\begin{aligned}e^{F_\delta (s)}u_\delta ^-(x,s)-e^{F_\delta (s)}\omega (x,s)\ge -\int _{-\infty }^s\omega (\gamma _{\delta ,x,s}^-(t),t) e^{F_\delta (t)}\Big (f_\delta (t)-f_0(t)\Big )\text{ d }t. \end{aligned}$$

By a suitable transformation,

$$\begin{aligned}u_\delta ^-(x,s)\ge & {} \omega (x,s)-\int _{-\infty }^s\omega (\gamma _{\delta ,x,s}^-(t),t)e^{F_0(t)} e^{F_\delta (t)-F_0(t)}\Big (f_\delta (t)-f_0(t)\Big )\text{ d }t\\= & {} \omega (x,s)-\int _{-\infty }^s\omega (\gamma _{\delta ,x,s}^-(t),t) e^{F_0(t)}\text{ d }e^{F_\delta (t)-F_0(t)}\\= & {} \omega (x,s)-\int _{-\infty }^s\omega (\gamma _{\delta ,x,s}^-(t),t)e^{F_0(t)}f_1(t)\frac{\text{ d }e^{F_\delta (t)-F_0(t)}}{f_1(t)}. \end{aligned}$$

Then for any $g\in C(M\times \mathbb {T},\mathbb {R})$, the measure $\tilde{\mu }_{x,s}^\delta$ defined by

$$\begin{aligned} \int _{TM\times \mathbb {T}}g(y,\tau )\text{ d }\tilde{\mu }_{x,s}^\delta (y,v_y,\tau ):=\int _{-\infty }^sg(\gamma _{\delta ,x,s}^-(t),t) \frac{\text{ d }e^{F_\delta (t)-F_0(t)}}{f_1(t)} \end{aligned}$$

is just the desired one. $\square$

Lemma 5.8

Any weak limit of the normalized measure

$$\begin{aligned} \widehat{\mu }_{x,s}^\delta :=\frac{\tilde{\mu }_{x,s}^\delta }{\int _{TM\times \mathbb {T}}\text{ d }\tilde{\mu }_{x,s}^\delta } \end{aligned}$$

(50)

as $\delta \rightarrow 0_+$ is contained in $\mathfrak M_m(0)$, i.e. a Mather measure.

Proof

As is proved in Proposition 5.7, $\tilde{\mu }_{x,s}^\delta$ are uniformly bounded w.r.t. $\delta \in (0,\delta _0]$. Therefore, it suffices to prove that any weak limit $\tilde{\mu }_{x,s}$ of $\tilde{\mu }_{x,s}^{\delta }$ as $\delta \rightarrow 0_+$ satisfies the following two conclusions:

First, we show $\tilde{\mu }_{x,s}$ is a closed measure. It is equivalent to show that for any $\phi (\cdot )\in C^1(M\times \mathbb {T},\mathbb {R})$,

$$\begin{aligned} \lim _{\delta \rightarrow 0_+}\int _{-\infty }^s\frac{\text{ d }}{\text{ d }t}\phi (\gamma _{\delta ,x,s}^-(t),t)\frac{\text{ d }e^{F_\delta (t)-F_0(t)}}{f_1(t)}=0. \end{aligned}$$

Indeed, we have

$$\begin{aligned}{} & {} \lim _{\delta \rightarrow 0_+}\int _{-\infty }^s\frac{\text{ d }}{\text{ d }t}\phi (\gamma _{\delta ,x,s}^-(t),t)\frac{\text{ d }e^{F_\delta (t)-F_0(t)}}{f_1(t)}\\= & {} \lim _{\delta \rightarrow 0_+}\int _{-\infty }^s e^{F_\delta (t)-F_0(t)} \frac{f_\delta (t)-f_0(t)}{f_1(t)}\text{ d }\phi (\gamma _{\delta ,x,s}^-(t),t)\\= & {} \lim _{\delta \rightarrow 0_+}\frac{f_\delta (t)-f_0(t)}{f_1(t)}e^{F_\delta (t)-F_0(t)}\phi (\gamma _{\delta ,x,s}^-(t),t)\Bigg |_{-\infty }^s\\{} & {} -\lim _{\delta \rightarrow 0_+}\int _{-\infty }^s\phi (\gamma _{\delta ,x,s}^-(t),t)\cdot \text{ d }\Big (\frac{f_\delta (t)-f_0(t)}{f_1(t)}e^{F_\delta (t)-F_0(t)}\Big )=0 \end{aligned}$$

because $f_\delta \rightarrow f_0$ uniformly as $\delta \rightarrow 0_+$.

Next, we can show that

$$\begin{aligned}\lim _{\delta \rightarrow 0_+}\int _{-\infty }^se^{F_\delta (t)}\Big [L\Big (\gamma _{\delta ,x,s}^-(t),\dot{\gamma }_{\delta ,x,s}^-(t),t\Big )+c(H)\Big ]\frac{\text{ d }e^{F_\delta (t)-F_0(t)}}{f_1(t)}=0. \end{aligned}$$

Note that

$$\begin{aligned} \frac{\text{ d }}{\text{ d }t}\big (e^{F_\delta (t)}u^-_\delta (\gamma ^-_{\delta ,x,s}(t),t)\big )= e^{F_\delta (t)}\big (L(\gamma ^-_{\delta ,x,s}(t),\dot{\gamma }_{\delta ,x,x}^-(t),t)+c(H)\big ). \end{aligned}$$

We derive

$$\begin{aligned}{} & {} \lim _{\delta \rightarrow 0_+}\int _{-\infty }^se^{F_\delta (t)}\Big [L\Big (\gamma _{\delta ,x,s}^-(t),\dot{\gamma }_{\delta ,x,s}^-(t),t\Big )+c(H)\Big ]\frac{\text{ d }e^{F_\delta (t)-F_0(t)}}{f_1(t)}\\= & {} \lim _{\delta \rightarrow 0_+}\int _{-\infty }^s\frac{\text{ d }}{\text{ d }t}\Big ( e^{F_\delta (t)}u_\delta ^-(\gamma _{\delta ,x,s}^-(t),t)\Big )\frac{\text{ d }e^{F_\delta (t)-F_0(t)}}{f_1(t)}=0, \end{aligned}$$

since $u_\delta ^-$ is differentiable along $(\gamma _{\delta ,x,s}^-(t),\bar{t})$ for all $t\in (-\infty ,s)$ and $\tilde{\mu }_{x,s}$ is closed. So we finish the proof.

Proof of Theorem 1.8: Due to the stability of viscosity solution (see Theorem 1.4 in [12]), any accumulating function $u_0^-$ of $u_\delta ^-$ as $\delta \rightarrow 0_+$ is a viscosity solution of (15) with $\delta =0$. Therefore, Proposition 5.4 indicates $u_0^-\in \mathcal {F}_-$, so $u_0^-\le u_0^*$. On the other side, Proposition 5.7 implies $u_0^-\ge \omega$ for any $\omega \in \mathcal {F}_-$ as $\delta \rightarrow 0_+$, since any weak limit of $\widehat{\mu }_{x,s}^\delta$ as $\delta \rightarrow 0_+$ proves to be a Mather measure in Lemma 5.8. So we have $u_0^-\ge u_0^*$.$\square$

6 Asymptotic Behaviors of Trajectories of 1-D Mechanical Systems

Lemma 6.1

For system (20), $\rho (c)$ is continuous of $c\in H^1(\mathbb {T},\mathbb {R})$.

Proof

Firstly, all the orbits in $\widetilde{\mathcal {A}}(c)$ should have the unified rotation number. This is because $\pi ^{-1}:{\mathcal {A}}(c)\rightarrow \widetilde{\mathcal {A}}(c)$ is a Lipschitz graph and dim$(M)=1$. Secondly, $\varlimsup _{c'\rightarrow c}\widetilde{\mathcal {A}}(c')\subset \widetilde{\mathcal {A}}(c)$ due to Lemma 4.2. That further indicates $\lim _{c'\rightarrow c}\rho (c')=\rho (c)$. $\square$

Lemma 6.2

For system (20), the rotation number $\rho (c)$ can be dominated by

$$\begin{aligned} -\Vert V\Vert _{C^1}\cdot \varsigma -c\le \rho (c)\le \Vert V\Vert _{C^1}\cdot \varsigma -c \end{aligned}$$

(51)

where $\varsigma =\varsigma ([f])>0$ tends to infinity as $[f]\rightarrow 0_+$.

Proof

Recall that

$$\begin{aligned} \dot{p}=-V_x(x,t)-f(t)p, \end{aligned}$$

then starting from any point $(x_0,p_0,\bar{t}_0)\in T^*M\times \mathbb {T}$, we get

$$\begin{aligned} p(t)=e^{-F(t)}p_0-e^{-F(t)}\int _0^te^{F(s)}V_x(x(s),s)\text{ d }s, \quad t>0. \end{aligned}$$

As $t\rightarrow +\infty$, we have

$$\begin{aligned} \lim _{t\rightarrow +\infty }|p(t)|\le & {} \Vert V\Vert _{C^1}\cdot \limsup _{t\rightarrow +\infty }e^{-F(t)}\int _0^te^{F(s)}\text{ d }s\nonumber \\\le & {} \varsigma (\Vert f\Vert ) \cdot \Vert V\Vert _{C^1} \end{aligned}$$

(52)

for a constant $\varsigma (\Vert f\Vert )>0$ depending only on f. As a consequence,

$$\begin{aligned} -\Vert V\Vert _{C^1}\cdot \varsigma \le \pi _p\widetilde{\mathcal {A}}(c)\le \Vert V\Vert _{C^1}\cdot \varsigma \end{aligned}$$

(53)

dominates the $p-$component of $\widetilde{\mathcal {A}}(c)$. $\square$

Proof of Theorem 1.10: The first two items have been proved in previous Lemmas 6.1 and 6.2. As for the third item, Lemma 6.2 has shown the boundedness of $p-$component of $\Omega$, then due to Theorem 1.5, we get the compactness of $\Omega$.$\square$

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

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

(54)

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

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

(55)

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 $D^*u(x,t)$.