The Attractors of Camassa–Holm Equation in Unbounded Domains

Abstract

In this paper, we deal with the existence of global mild solutions and asymptotic behavior to the viscous Camassa–Holm equation in the locally uniform spaces. First we establish the global well-posedness for the Cauchy problem of viscous Camassa–Holm equation in ${\mathbb {R}}^1$ for any initial data $u_0\in {\dot{H}}^1_U({\mathbb {R}}^1).$ Then we study the long time dynamical behavior of non-autonomous viscous Camassa–Holm equation on ${\mathbb {R}}^1$ with a new class of external forces and show the existence of $(H^1_U({\mathbb {R}}^1),H^1_\phi ({\mathbb {R}}^1))$-uniform(w.r.t. $g\in \mathcal {H}_{L^2_U({\mathbb {R}}^1)}(g_0)$) attractor $\mathcal {A}_{\mathcal {H}_{L^2_U({\mathbb {R}}^1)}(g_0)}$ with locally uniform external forces being translation uniform bounded but not translation compact in $L_b^2({\mathbb {R}};L^2_U({\mathbb {R}}^1))$.

Appendix: Some Properties of Uniform Normal External Forces

In this section we present some result concerning the locally uniform spaces $L_{loc}^p({\mathbb {R}};L^q_U({\mathbb {R}}^N))$ with $1<p$ and $1<q<N.$ Recall that a function $\sigma (s)\in L_{loc}^p({\mathbb {R}};L^q_U({\mathbb {R}}^N))$ is said to be translation compact(tr.c.) if the closure of $\{\sigma (s+h)|h\in {\mathbb {R}}\}$ is compact in $L_{loc}^p({\mathbb {R}};L^q_U({\mathbb {R}}^N)).$ The set of all tr.c. functions in $L_{loc}^p({\mathbb {R}};L^q_U({\mathbb {R}}^N))$ is denoted by $L_c^p({\mathbb {R}};L^q_U({\mathbb {R}}^N)).$

We first recall two propositions [11] that gives the compactness criterion in $L^p(t_1,t_2;L^q_U({\mathbb {R}}^N)).$

Proposition A.1

A set $\Sigma \subset L_{loc}^p({\mathbb {R}};L^q_U({\mathbb {R}}^N))$ is precompact in $L_{loc}^p({\mathbb {R}};L^q_U({\mathbb {R}}^N))$ if and only if the set $\Sigma |_\Omega$ is precompact in $L^p_{loc}({\mathbb {R}};L^q(\Omega ))$ for every bounded subset $\Omega \subset {\mathbb {R}}^N.$ Here $\Sigma |_{\Omega }$ denotes the restriction of the set $\Sigma$ to the subset $\Omega .$

According to the definition of translation compact function in $L_{loc}^p({\mathbb {R}};L^q_U({\mathbb {R}}^N))$ and this proposition, a function $g\in L_{loc}^p({\mathbb {R}};L^q_U({\mathbb {R}}^N))$ is tr.c. if and only if

$$\begin{aligned} \{T(h)g_0(t)|h\in {\mathbb {R}},\;t\in [0,1]\}\quad \text {is precompact in}\;\; L^p([0,1];L^q(\Omega )). \end{aligned}$$

The following proposition gives a compactness criterion in $L_{loc}^p({\mathbb {R}};L^q_U({\mathbb {R}}^N)),$ whose proof is essentially given in Proposition 3.2 of [11].

Proposition A.2

Let $\Omega \subset {\mathbb {R}}^N$ be a bounded domain. Then a function $\psi (s)$ is tr.c. in $L_{loc}^p({\mathbb {R}};L^q(\Omega ))$ if and only if

• (i) for any fixed $h\in {\mathbb {R}}$ the set $\big \{\int _t^{t+h}\psi (s)ds|t\in {\mathbb {R}}\big \}$ is precompact in $L^q(\Omega );$

• (ii) there exists a positive function $\xi (s)$ with $\xi (s)\rightarrow 0^+$ as $s\rightarrow 0^+$ such that

  $$\begin{aligned} \int _t^{t+1} \Vert \psi (s)-\psi (s+l)\Vert ^p_{L^q(\Omega )}ds\le \xi (|l|)\quad \text {for all}\;\;t\in {\mathbb {R}}. \end{aligned}$$

  (A.1)

The following relationship between tr.c. function space $L_c^p({\mathbb {R}};L^q_U({\mathbb {R}}^N)),$ uniform normal function space $L_{un}^p({\mathbb {R}};L^q_U({\mathbb {R}}^N))$ and $L_b^p({\mathbb {R}};L^q_U({\mathbb {R}}^N))$ has been essentially established for the bounded domain of ${\mathbb {R}}^N$ in [25]. According to Propositions A.1, A.2 and the definition of topology of these spaces, we can prove the following conclusion by using the method of [25]. Hence, we omit details here.

Theorem A.3

Let $p,q\ge 1,$ then

• (i) $L_c^p({\mathbb {R}};L^q_U({\mathbb {R}}^N))$ is a closed subspace of $L_b^p({\mathbb {R}};L^q_U({\mathbb {R}}^N));$

• (ii) $L_c^p({\mathbb {R}};L^q_U({\mathbb {R}}^N))\subset L_{un}^p({\mathbb {R}};L^q_U({\mathbb {R}}^N));$

• (iii) $L_{un}^p({\mathbb {R}};L^q_U({\mathbb {R}}^N))$ is a closed subspace of $L_b^p({\mathbb {R}};L^q_U({\mathbb {R}}^N)).$

In particular, if the external force g(x, t) is given by $g(x,t)=m(t)n(x),$ $n(x)\in L^q_U({\mathbb {R}}^N),$ and let $m(t)\in L^p_{loc}({\mathbb {R}})$ be the uniform normal, i.e., $\sup \limits _{t\in {\mathbb {R}}}\int _t^{t+\eta }\Vert m(s)\Vert ^pds\le \varepsilon ,$ then

$$\begin{aligned} \sup \limits _{t\in {\mathbb {R}}}\int _t^{t+\eta }\Vert g(s)\Vert _{L^q_U({\mathbb {R}}^N)}^pds&= \sup \limits _{t\in {\mathbb {R}}}\int _t^{t+\eta }\Vert m(t)n(x)\Vert _{L^q_U({\mathbb {R}}^N)}^pds\\&\le \Vert n(x)\Vert _{L^q_U({\mathbb {R}}^N)}^p\sup \limits _{t\in {\mathbb {R}}}\int _t^{t+\eta }|m(s)|^pds\\&\le \varepsilon \Vert n(x)\Vert _{L^q_U({\mathbb {R}}^N)}^p. \end{aligned}$$

Now, we will give some examples to explain the inclusion relationship of these spaces given by Theorem A.3.

Example A.4

For $k_1,k_2=1,2,\ldots ,$ we take

$$\begin{aligned} n(x)= {\left\{ \begin{array}{ll} k_1,\quad |x|\in \left[ k_1,k_1+\frac{1}{k_1^q}\right] ,\\ 0,\quad \text {otherwise}, \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} m(s)= {\left\{ \begin{array}{ll} k_2^{\frac{1}{p}},\quad s\in \left[ k_2,k_2+\frac{1}{k_2}\right] ,\\ 0,\quad \text {otherwise}, \end{array}\right. } \end{aligned}$$

it follows that

$$\begin{aligned} \Vert g(x,s)\Vert ^p_{L_b^p({\mathbb {R}};L^q_U({\mathbb {R}}^N))}&=\Vert m(s)n(x)\Vert ^p_{L_b^p({\mathbb {R}};L^q_U({\mathbb {R}}^N))}\\&= \sup \limits _{t\in {\mathbb {R}}}\int _t^{t+1}\Vert m(s)n(x)\Vert _{L^q_U({\mathbb {R}}^N))}^pds\le C. \end{aligned}$$

On the other hand, for any $\eta >0$ there exist $k_2$ such that $\frac{1}{k_2}\le \eta ,$ set $t=k_2,$ then

$$\begin{aligned} \sup \limits _{t\in {\mathbb {R}}}\int _t^{t+\eta }\Vert m(s)n(x)\Vert _{L^q_U({\mathbb {R}}^N))}^pds\ge \int _{k_2}^{k_2+\frac{1}{k_2}}\Vert m(s)\Vert ^pds=1, \end{aligned}$$

which implies g(x, t) belonging to $L_b^p({\mathbb {R}};L^q_U({\mathbb {R}}^N))$ but not $L_{un}^p({\mathbb {R}};L^q_U({\mathbb {R}}^N)).$

Example A.5

For $k_1,k_2=1,2,\ldots ,$ we take

$$\begin{aligned} n(x)= {\left\{ \begin{array}{ll} k_1,\quad |x|\in \left[ k_1,k_1+\frac{1}{k_1^q}\right] ,\\ 0,\quad \text {otherwise}, \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} m(s)= {\left\{ \begin{array}{ll} k_2^{\frac{1}{p}},\quad s\in \left[ k_2+\frac{i}{k_2},k_2+\frac{i}{k_2}+\frac{1}{k_2^p}\right] ,\\ i=0,1,\ldots ,k_2^{p-1}-1,\\ 0,\quad \;\text {otherwise}, \end{array}\right. } \end{aligned}$$

it yields

$$\begin{aligned}&\int _0^{\eta }\Vert m(s)n(x)\Vert _{L^q_U({\mathbb {R}}^N))}^pds\\&\quad \le {\left\{ \begin{array}{ll} (i+1)\frac{1}{k_2^p}(k_2^{\frac{1}{p}})^p\le \frac{2i}{k_2^{p-1}}\le 2\eta ^{p-1}, &{}\text {if}\;\frac{i}{k_2}\le \eta<\frac{i+1}{k_2},i\ge 1,\\ \frac{1}{k_2^p}(k_2^{\frac{1}{p}})^p\le \frac{1}{k_2^{p-1}}\le \eta ^{\frac{p-1}{p}}, &{}\text {if}\;\frac{1}{k_2^p}\le \eta<\frac{1}{k_2},\\ \eta (k_2^{\frac{1}{p}})^p\le \eta \left( \frac{1}{\eta }\right) ^{\frac{1}{p}}=\eta ^{\frac{p-1}{p}}, &{}\text {if}\;0\le \eta <\frac{1}{k_2^p}. \end{array}\right. } \end{aligned}$$

Thus, for any $0<\varepsilon <\frac{1}{2^{\frac{1}{p-1}}},$ let $\eta =\varepsilon ^{\frac{p}{p-1}},$ and we have

$$\begin{aligned} \sup \limits _{t\in {\mathbb {R}}}\int _t^{t+\eta }\Vert m(s)n(x)\Vert _{L^q_U({\mathbb {R}}^N))}^pds&= \sup \limits _{t\in {\mathbb {R}}}\int _0^{\eta }\Vert m(t+s)n(x)\Vert _{L^q_U({\mathbb {R}}^N))}^pds\\ {}&= \sup \limits _{k_2\in N}\int _0^{\eta }\Vert m(s+k_2)n(x)\Vert _{L^q_U({\mathbb {R}}^N))}^pds\le \varepsilon . \end{aligned}$$

However, it is easy to see that for any $\frac{1}{k_2^p}\le l<\frac{1}{k_2}-\frac{1}{k_2^p}$ satisfying $k_2>2^{\frac{1}{p-1}},$

$$\begin{aligned} \int _0^1\Vert m(s+k_2)n(x)-m(s+k_2-l)n(x)\Vert _{L^q_U({\mathbb {R}}^N))}^pds=\int _0^1k_2 \frac{1}{k_2^p}(2k_2^{p-1})2ds=4, \end{aligned}$$

which implies from Proposition A.2 that $g(x,t)=m(t)n(x)$ dose not belong to $L_c^p({\mathbb {R}};L^q_U({\mathbb {R}}^N))$ but in $L_{un}^p({\mathbb {R}};L^q_U({\mathbb {R}}^N))$.