Global well-posedness and blow-up of solutions for the Camassa–Holm equations with fractional dissipation

Abstract

Consideration in this paper is the effect of varying fractional dissipation with the dissipative operator power $\gamma \ge 0$ on the well-posedness of the Camassa–Holm equations with fractional dissipation. It is shown that the zero-filter limit ($\alpha \rightarrow 0$) of the Camassa–Holm equation with fractional dissipation is the fractal Burgers equation. It is known that in the supercritical case $\gamma \in [0, 1)$, the fractal Burgers equation blows up in finite time in $H^s({\mathbb {R}})$ with $s>\frac{3}{2}-\gamma$. It is established here that the dissipative Camassa–Holm equation is globally well-posed in the critical Sobolev space $H^{\frac{3}{2}-\gamma }({\mathbb {R}})$ with the fractional parameter $\gamma \in [\frac{1}{2}, 1)$. Moreover, it is also demonstrated that the solution of the dissipative Camassa–Holm equation blows up in finite time in the particular supercritical case when $\gamma = 0$.

Appendix: Littlewood–Paley analysis

The proofs of Theorems 1.1–1.5 require the Littlewood–Paley decomposition, or a dyadic decomposition of the Fourier variables, which may be explained how it may be built in the case $x\in {\mathbb {R}}^d$ (see e.g. [2, 9, 28]) as follows.

We recall the following proposition about the smoothing effect of the operator $\partial _t+\Lambda ^{\gamma }$ with $t>0$ and $\gamma >0$, which was obtained from [10] (up to a slight modification).

Proposition 6.1

([10]) Let ${\mathcal {C}}$ be an annulus. Positive constants c and C exist such that for any p in $[1,+\infty ]$ and any couple $(t, \lambda )$ of positive real numbers, we have

$$\begin{aligned} \mathop {\mathrm{Supp}}\nolimits \,\widehat{u} \subset \lambda {\mathcal {C}} \, \Rightarrow \, \Vert e^{-t \Lambda ^{\gamma }} u\Vert _{L^p} \le C e^{-ct\lambda ^{\gamma }}\Vert u\Vert _{L^p}. \end{aligned}$$

Let us now recall a dyadic partition of unity, which may be found in [2]. Let us first define by ${\mathcal {C}}$ the ring of center 0, of small radius 3 / 4 and great radius 8 / 3. It exists two radial functions $\chi$ and $\varphi$ the values of which are in the interval [0, 1], belonging respectively to ${\mathcal {D}}(B(0,4/3))$ and to ${\mathcal {D}}({\mathcal {C}})$ such that

$$\begin{aligned}&\chi (\xi ) + \sum _{j\ge 0} \varphi (2^{-j}\xi ) = 1 \quad (\forall \xi \in {\mathbb {R}}^d), \qquad \sum _{j\in {\mathbb {Z}}} \varphi \left( 2^{-j}\xi \right) = 1 \quad \left( \forall \xi \in {\mathbb {R}}^d\setminus \{0\}\right) ,\nonumber \\&|j-j'|\ge 2 \Rightarrow \mathop {\mathrm{Supp}}\nolimits \,\varphi (2^{-j}\cdot )\cap \mathop {\mathrm{Supp}}\nolimits \,\varphi (2^{-j'}\cdot )=\emptyset ,\\&j\ge 1 \Rightarrow \mathop {\mathrm{Supp}}\nolimits \,\chi \cap \mathop {\mathrm{Supp}}\nolimits \,\varphi (2^{-j}\cdot ) = \emptyset .\nonumber \end{aligned}$$

(6.1)

Setting $h\mathop {=}\limits ^{\mathrm{def}}{{{\mathcal {F}}}}^{-1}\varphi ,$ $\widetilde{h}\mathop {=}\limits ^{\mathrm{def}}{{{\mathcal {F}}}}^{-1}\chi$. The inhomogeneous dyadic blocks ${\Delta }_j$ and the inhomogeneous low-frequency cut-off operator ${S}_j$ are defined for all $j \in {\mathbb {N}}\cup \{-1\}$ by

$$\begin{aligned} \begin{aligned}&{\Delta }_{j} f\mathop {=}\limits ^{\mathrm{def}}\varphi (2^{-j}D)f=2^{j d}\int _{{\mathbb {R}}^{d}}h(2^j y)f(x-y)dy \quad (\forall \quad j \ge 0),\\&{\Delta }_{-1} f\mathop {=}\limits ^{\mathrm{def}}\chi (D)f=\int _{{\mathbb {R}}^{d}}\widetilde{h}( y)f(x-y)dy,\quad {\Delta }_{j} f\mathop {=}\limits ^{\mathrm{def}}0 \quad (\forall \quad j \le -2),\\&{S}_{j} f\mathop {=}\limits ^{\mathrm{def}}\sum _{ j^{\prime } \le j-1}{\Delta }_{j^{\prime }} f=\chi (2^{-j}D)f=2^{j d}\int _{{\mathbb {R}}^{d}}\widetilde{h}(2^j y)f(x-y)dy. \end{aligned} \end{aligned}$$

(6.2)

The homogeneous dyadic blocks $\dot{\Delta }_j$ and the homogeneous low-frequency cut-off operators $\dot{S}_j$ are defined for all $j \in {\mathbb {Z}}$ by

$$\begin{aligned} \begin{aligned}&\dot{\Delta }_{j} f\mathop {=}\limits ^{\mathrm{def}}\varphi (2^{-j}D)f=2^{j d}\int _{{\mathbb {R}}^{d}}h(2^j y)f(x-y)dy, \quad \dot{S}_{j} f\mathop {=}\limits ^{\mathrm{def}}\sum _{ j^{\prime } \le j-1}\dot{\Delta }_{j^{\prime }} f. \end{aligned} \end{aligned}$$

(6.3)

We should point out that all the above operators $\Delta _j$, $\dot{\Delta }_j$, $S_{j}$, and $\dot{S}_{j}$ map $L^p$ into $L^p$ with norms which do not depend on j.

With above notations in hand, we are in a position to define Besov spaces. Let $s\in {\mathbb {R}},\, 1\le p,\,r\le \infty .$ The inhomogenous (or homogeneous) Besov space ${B}^s_{p,r}({\mathbb {R}}^{d})$ (or $\dot{B}^s_{p,r}({\mathbb {R}}^{d})$) is defined by

$$\begin{aligned}&{B}^s_{p,r}({\mathbb {R}}^{d})\mathop {=}\limits ^{\mathrm{def}}\{f\in {\mathcal {S}}^{\prime }({\mathbb {R}}^{d})| \, \Vert f\Vert _{{B}^s_{p,r}}<\infty \}\,\bigg (\text{ or }\quad \dot{B}^s_{p,r}({\mathbb {R}}^{d})\mathop {=}\limits ^{\mathrm{def}}\{f\in {\mathcal S}^{\prime }_{h}({\mathbb {R}}^{d})| \, \Vert f\Vert _{\dot{B}^s_{p,r}}<\infty \}\bigg ) \, \text{ with }\\&\Vert f\Vert _{{B}^s_{p,r}}\mathop {=}\limits ^{\mathrm{def}}{\left\{ \begin{array}{ll} \bigg (\sum _{j \in {\mathbb {Z}}} 2^{j s r}\Vert {\Delta }_j f\Vert _{L^p}^r\bigg )^{\frac{1}{r}},\quad \hbox {if}\quad r<\infty ,\\ \sup _{j \in {\mathbb {Z}}}2^{j s}\Vert {\Delta }_j f\Vert _{L^p}, \quad \quad \ \quad \hbox {if} \quad r=\infty , \end{array}\right. } (\text{ or }\\&\Vert f\Vert _{\dot{B}^s_{p,r}}\mathop {=}\limits ^{\mathrm{def}}{\left\{ \begin{array}{ll} \bigg (\sum _{j \in {\mathbb {Z}}} 2^{j s r}\Vert \dot{\Delta }_j f\Vert _{L^p}^r\bigg )^{\frac{1}{r}},\quad \hbox {for}\quad r<\infty ,\\ \sup _{j \in {\mathbb {Z}}}2^{j s}\Vert \dot{\Delta }_j f\Vert _{L^p}, \quad \quad \ \quad \hbox { for} \quad r=\infty , \end{array}\right. }\text{ and }\\&{\mathcal {S}}^{\prime }_{h}({\mathbb {R}}^{d}) \mathop {=}\limits ^{\mathrm{def}}\{f \in {\mathcal S}^{\prime }({\mathbb {R}}^{d})| \, \lim _{j \rightarrow -\infty }\dot{S}_{j} f =0 \quad \hbox {in} \quad {\mathcal {S}}^{\prime }({\mathbb {R}}^{d}) \}). \end{aligned}$$

If $s=\infty$, $B^{\infty }_{p, r} \mathop {=}\limits ^{\mathrm{def}}\bigcap _{s \in {\mathbb {R}}} B^{s}_{p, r} .$

Remark 6.1

• (i)   Let $s\in {\mathbb {R}}, \,1\le p,\,r\le \infty$, and $u \in {\mathcal S}'_{h}.$ Then u belongs to $\dot{B}^{s}_{p, r}$ if and only if there exists $\{c_{j, r}\}_{j \in {\mathbb {Z}}}$ such that $\Vert c_{j, r}\Vert _{\ell ^{r}} =1$ and $\Vert \dot{\Delta }_{j}u\Vert _{L^{p}}\le C c_{j, r} 2^{-j s } \Vert u\Vert _{\dot{B}^{s}_{p, r}}.$

• (ii)   Let $s\in {\mathbb {R}},\, 1\le p,\,r\le \infty$. Then u belongs to ${B}^{s}_{p, r}$ if and only if there exists $\{c_{j, r}\}_{j \in {\mathbb {N}} \cup \{-1\}}$ such that $\Vert c_{j, r}\Vert _{\ell ^{r}} =1$ and $\Vert {\Delta }_{j}u\Vert _{L^{p}}\le C c_{j, r} 2^{-j s } \Vert u\Vert _{{B}^{s}_{p, r}}.$

We should point out that if $s>0$ then $B^s_{p,r}({\mathbb {R}}^d)=\dot{B}^s_{p,r}({\mathbb {R}}^d)\cap L^p({\mathbb {R}}^d)$ and $\Vert u\Vert _{B^s_{p,r}}\approx \Vert u\Vert _{\dot{B}^s_{p,r}}+\Vert u\Vert _{L^p}.$ It is easy to verify that the homogeneous Besov space $\dot{B}^s_{2,2}({\mathbb {R}}^d)$ coincides with the classical homogeneous Sobolev space $\dot{H}^{s}({\mathbb {R}}^d)$. Similar assertions may be easily understood for the inhomogeneous Besov spaces.

In order to obtain a better description of the regularizing effect of the transport-diffusion equation, we will use Chemin–Lerner type spaces $\widetilde{L}^{\lambda }_T(B^s_{p,r}({\mathbb {R}}^d))$ from [10].

Definition 6.1

Let $s\in {\mathbb {R}}$, $(r,\lambda ,p)\in [1,\,+\infty ]^3$ and $T\in (0,\,+\infty ]$. We define $\widetilde{L}^{\lambda }_T( B^s_{p\,r}({\mathbb {R}}^d))$ as the completion of $C([0,T],{\mathcal {S}}({\mathbb {R}}^d))$ by the norm $\Vert f\Vert _{\widetilde{L}^{\lambda }_T( B^s_{p,r})} \mathop {=}\limits ^{\mathrm{def}}\Big (\sum _{q\in {{\mathbb {Z}}}}2^{qrs} \Big (\int _0^T\Vert \Delta _q\,f(t) \Vert _{L^p}^{\lambda }\, dt\Big )^{\frac{r}{\lambda }}\Big )^{\frac{1}{r}} <\infty$ with the usual change if $r=\infty .$ For short, we just denote this space by $\widetilde{L}^{\lambda }_T( B^s_{p,r}).$ In the particular case when $p=r=2,$ we denote this space by $\widetilde{L}^\lambda _T({H}^s).$ We also denote the space ${\mathcal {C}}([0, T]; B^s_{p,r}) \cap \widetilde{L}^{\infty }_T(B^s_{p,r})$ by $\widetilde{{\mathcal {C}}}([0, T]; B^s_{p,r})$.

It is easy to observe that for $\theta \in [0,1],$ we have $\Vert u\Vert _{\widetilde{L}^{\lambda }_T( B^s_{p,r})} \le \Vert u\Vert _{\widetilde{L}^{\lambda _1}_T( B^{s_1}_{p,r})}^{\theta } \Vert u\Vert _{\widetilde{L}^{\lambda _2}_T(B^{s_2}_{p,r})}^{1-\theta }$ with $\frac{1}{\lambda }=\frac{\theta }{\lambda _1} +\frac{1-\theta }{\lambda _2}$ and $s=\theta s_1+(1-\theta )s_2.$ Moreover, Minkowski inequality implies that

$$\begin{aligned} \Vert u\Vert _{\widetilde{L}^{\lambda }_T(B^s_{p,r})} \le \Vert u\Vert _{L^{\lambda }_T(B^s_{p,r})} \quad \text{ if }\quad \lambda \le r \quad \hbox {and}\quad \Vert u\Vert _{L^{\lambda }_T( B^s_{p,r})} \le \Vert u\Vert _{\widetilde{L}^{\lambda }_T(B^s_{p,r})} \quad \text{ if }\quad r\le \lambda . \end{aligned}$$

Similar definitions and the above properties may be easily understood for the homogeneous version of $\widetilde{L}^{\lambda }_T(\dot{B}^s_{p,r}({\mathbb {R}}^d))$.

In what follows, we shall frequently use Bony’s decomposition [6] in the both homogeneous and inhomogeneous context:

$$\begin{aligned} \begin{aligned}&uv=\dot{T}_u v+\dot{R}(u,v)=\dot{T}_u v+\dot{T}_v u+\dot{{\mathcal {R}}}(u,v)\qquad \text{ and } \\&uv=T_u v+R(u,v)=T_u v+T_v u+{\mathcal {R}}(u,v), \end{aligned} \end{aligned}$$

(6.4)

where

$$\begin{aligned} \begin{aligned}&\dot{T}_u v\mathop {=}\limits ^{\mathrm{def}}\sum _{q \in {\mathbb {Z}}}\dot{S}_{q-1}u\dot{\Delta }_q v,\qquad \dot{R}(u,v)\mathop {=}\limits ^{\mathrm{def}}\sum _{q\in {{\mathbb {Z}}}}\dot{\Delta }_q u \dot{S}_{q+2}v,\\&\dot{{\mathcal {R}}}(u,v)\mathop {=}\limits ^{\mathrm{def}}\sum _{q\in {{\mathbb {Z}}}}\dot{\Delta }_q u \widetilde{\dot{\Delta }}_{q}v\quad \text{ and }\quad \widetilde{\dot{\Delta }}_{q}v\mathop {=}\limits ^{\mathrm{def}}\sum _{|q'-q|\le 1}\dot{\Delta }_{q'}v, \end{aligned} \end{aligned}$$

and similar definitions for the inhomogeneous version of $T_uv$, R(u, v), and ${\mathcal {R}}(u,v).$

From this, we readily get the following product laws in Besov spaces.

Lemma 6.1

Assume that $1 \le p, \, r \le +\infty ,$ the following estimates hold:

• (i). for $s>0$, $\Vert fg\Vert _{B^{s}_{p, r}}\le C (\Vert f\Vert _{B^{s}_{p, r}}\Vert g\Vert _{L^{\infty }}+ \Vert g\Vert _{B^{s}_{p, r}}\Vert f\Vert _{L^{\infty }});$

• (ii). for $|s| < \frac{d}{2}$, $\Vert fg\Vert _{H^{s}({\mathbb {R}}^d)}\le C \Vert f\Vert _{B^{\frac{d}{2}}_{2,1}({\mathbb {R}}^d)}\Vert g\Vert _{H^{s}({\mathbb {R}}^d)};$

• (iii). for $s_1 \le \frac{d}{p},\, s_2>\frac{d}{p}$ ($s_2\ge \frac{d}{p}$ if $r=1$) and $s_1+s_2>0,$

  $$\begin{aligned} \Vert fg\Vert _{B^{s_1}_{p, r}({\mathbb {R}}^d)}\le C \Vert f\Vert _{B^{s_1}_{p, r}({\mathbb {R}}^d)}\Vert g\Vert _{B^{s_2}_{p, r}({\mathbb {R}}^d)}; \end{aligned}$$

• (iv). for any $(s_1, s_2) \in (-d/2, d/2)$ and $s_1+s_2>0$,

  $$\begin{aligned} \Vert fg\Vert _{\dot{B}^{s_1+s_2-\frac{d}{2}}_{2, 1}({\mathbb {R}}^d)}\le C \Vert f\Vert _{\dot{H}^{s_1}({\mathbb {R}}^d)}\Vert g\Vert _{\dot{H}^{s_2}({\mathbb {R}}^d)}, \end{aligned}$$

where C’s are constants independent of f and g.

The following basic lemma will be of constant use in this paper.

Lemma 6.2

(Lemma 2.97 in [2]) (Commutator estimates) Let $(p, q, r) \in [1, \infty ]^3$, $\theta$ be a $C^1$ function on ${\mathbb {R}}^{d}$ such that $(1+|\cdot |) \hat{\theta } \in L^1$. There exists a constant C such that for any Lipschitz function a with gradient in $L^p$ and any function b in $L^q$, we have, for any positive $\lambda$,

$$\begin{aligned} \Vert [\theta (\lambda ^{-1} D), a]b\Vert _{L^r} \le C \lambda ^{-1}\Vert \nabla a\Vert _{L^p}\Vert b\Vert _{L^q} \quad \text{ with } \quad \frac{1}{p}+ \frac{1}{q}= \frac{1}{r}. \end{aligned}$$

(6.5)

From this, we may get the following important corollary (see Lemma 2.100 in [2] up to a slight modification).

Proposition 6.2

(Lemma 2.100 in [2]) Let $\sigma \in {\mathbb {R}}$, $1 \le r\le \infty$, and $1 \le p \le p_1\le \infty$, $1 \le p_2 \le \infty$. Let v be a vector field over ${\mathbb {R}}^{d}$. Assume that

$$\begin{aligned} \sigma >-d\min \Bigl \{\frac{1}{p_1},\frac{1}{p'}\Bigr \} \quad \text{ or }\quad \sigma >-1-d\min \Bigl \{\frac{1}{p_1},\frac{1}{p'}\Bigr \}\quad if\quad \text{ div }\, v=0. \end{aligned}$$

(6.6)

For $j\in {\mathbb {Z}},$ denote $R_j:=[v\cdot \nabla , \dot{\Delta }_j]f$ (or $R_j(u,v):=\text{ div }\,[v,\dot{\Delta }_j]f$ if $\text{ div }\, v=0$). There exists a constant C, depending continuously on p, $p_1$, $\sigma$, and d, such that

$$\begin{aligned} \bigg \Vert \bigg (2^{j\sigma }\Vert R_j\Vert _{L^{p}}\bigg )_j\bigg \Vert _{\ell ^r}\lesssim {\left\{ \begin{array}{ll} \Vert \nabla v\Vert _{\dot{B}^{\frac{d}{p_1}}_{p_1,\infty }\cap L^{\infty }}\Vert f\Vert _{\dot{B}^{\sigma }_{p,r}} \quad \text{ if }\quad \sigma < 1+\frac{d}{p_1},\\ \Vert \nabla v\Vert _{\dot{B}^{\frac{d}{p_1}}_{p_1,1}}\Vert f\Vert _{\dot{B}^{\sigma }_{p,r}} \quad {if}\quad \sigma = 1+\frac{d}{p_1} \quad {and}\quad r=1. \end{array}\right. } \end{aligned}$$

(6.7)

Further, if $f=v$, $\sigma >0$ (or $\sigma >-1$ if $\text{ div }\, v=0$), then

$$\begin{aligned} \bigg \Vert \bigg (2^{j\sigma }\Vert R_j\Vert _{L^{p}}\bigg )_j\bigg \Vert _{\ell ^r}\lesssim \Vert \nabla v\Vert _{ L^{\infty }}\Vert f\Vert _{\dot{B}^{\sigma }_{p,r}}. \end{aligned}$$

(6.8)

In the limit case $\sigma =-d\min \Bigl \{\frac{1}{p_1},\frac{1}{p'}\Bigr \}$ or $\sigma =-1-d\min \Bigl \{\frac{1}{p_1},\frac{1}{p'}\Bigr \}$ if $\text{ div }\, v=0$, we have The following limit cases also hold true: $\sup _{j \ge -1} 2^{j\sigma }\Vert R_j\Vert _{L^{p}} \lesssim \Vert \nabla v\Vert _{\dot{B}^{\frac{d}{p_1}}_{p_1,1}}\Vert f\Vert _{\dot{B}^{\sigma }_{p,\infty }}.$

We now consider the transport-diffusion equation

$$\begin{aligned} (TD_{\nu }) {\left\{ \begin{array}{ll} \partial _t f+v\cdot \nabla f+\nu \Lambda ^{\gamma } f=g, \quad (t, x) \in {\mathbb {R}}^{+}\times {\mathbb {R}}^{d},\\ f|_{t=0}=f_0, \end{array}\right. } \end{aligned}$$

(6.9)

where $\nu \ge 0$, $\gamma \in (0, 2)$, $f_0$, g, and v stand for given initial data, external force, and vector field, respectively. We aim to state a priori estimates which apply for all possible values of $f_0$ and Lipschitz vector fields v.

From Proposition 6.2, we may get the following lemma, which proof is similar to the one of Theorem 3.14 in [2] and we omit the details.

Lemma 6.3

Let $2\le p_1 \le \infty$ and $r \in \{1, \, 2\}$. Let $\sigma \in {\mathbb {R}}$ satisfy

$$\begin{aligned} s > -d \min \left\{ \frac{1}{p_1}, \frac{1}{2}\right\} \quad \text{ or } \quad s > -1-d \min \left\{ \frac{1}{p_1}, \frac{1}{2}\right\} \quad \text{ if } \quad \text{ div }\, v=0. \end{aligned}$$

(6.10)

There exists a constant $C>0$ depending only on $d, \, s,\, r,\,$ and $s- 1- \frac{d}{p_1}$, such that for any smooth solution f of $(TD_{\nu })$ with $\nu > 0$, and $\rho \in [\rho _1, \infty ]$, we have the following a priori estimates:

$$\begin{aligned}&\begin{aligned} \Vert f\Vert _{\widetilde{L}^{\infty }_t(\dot{B}^{s}_{2, r})}\le \Vert u_0\Vert _{\dot{B}^{s}_{2, r}}+C\Vert g\Vert _{\widetilde{L}^{1}_t(\dot{B}^{s}_{2, r})} + C\int _0^t V'_{p_1}(\tau )\Vert f\Vert _{\dot{B}^{s}_{2, r}}\,d\tau \quad \text{ and } \quad \end{aligned}\\&\begin{aligned} \nu \, \Vert f\Vert _{\widetilde{L}^{1}_t(\dot{B}^{s+\gamma }_{2, r})}\lesssim&\bigg \Vert 2^{qs}(1-e^{-c_1 \nu 2^{q\gamma }t} ) \Vert \dot{\Delta }_q f_0\Vert _{L^2}\bigg \Vert _{\ell ^{r}({\mathbb {Z}})}+ \Vert g\Vert _{\widetilde{L}^{1}_t(\dot{B}^{s}_{2, r})} + \int _0^t V'_{p_1}(\tau )\Vert f\Vert _{\dot{B}^{s}_{2, r}}\,d\tau , \end{aligned} \end{aligned}$$

with, if the inequality is strict in (6.10),

$$\begin{aligned} V_{p_1}(t) \mathop {=}\limits ^{\mathrm{def}}{\left\{ \begin{array}{ll} \int _0^{t} \Vert \nabla v(\tau )\Vert _{\dot{B}^{\frac{d}{p_1}}_{p_1, \infty }\cap L^{\infty }}\, d\tau \quad &{}\text{ if } \quad s < 1+ \frac{d}{p_1},\\ \int _0^{t} \Vert \nabla v(\tau )\Vert _{\dot{B}^{\frac{d}{p_1}}_{p_1, 1}}\, d\tau \quad &{}\text{ if } \quad s =1+ \frac{d}{p_1}\,\, \text{ and } \quad r=1,\\ \int _0^{t} \Vert \nabla v(\tau )\Vert _{L^{\infty }}\, d\tau \quad &{}\text{ if } \quad f=v \, \text{ and } \, s>0 (\text{ or } \,\, s>-1\,\, \text{ if } \quad \text{ div }\, v=0). \end{array}\right. } \end{aligned}$$