A single exponential BKM type estimate for the 3D incompressible ideal MHD equations

Abstract

In this paper, we give a Beale-Kato-Majda type criterion of strong solutions to the incompressible ideal MHD equations. Instead of double exponential estimates, we get a single exponential bound on ${\parallel (u,h)\parallel }_{H^s}$ ($s>\frac{5}{2}$). It can be applied to a system of an ideal viscoelastic flow.

MSC:35B65, 76W05.

1 Introduction

In this paper, we will get the Beale-Kato-Majda type criterion for the breakdown of smooth solutions to the incompressible ideal MHD equations in ${\mathbb{R}}^3$ as follows:

$$\{\begin{array}{c}{u}_t+u\cdot \mathrm{\nabla }u+\mathrm{\nabla }(p+\frac{1}{2}{|h|}^2)=h\cdot \mathrm{\nabla }h,\hfill \\ h_t+u\cdot \mathrm{\nabla }h=h\cdot \mathrm{\nabla }u,\hfill \\ \mathrm{\nabla }\cdot u=0,\mathrm{\nabla }\cdot h=0,\hfill \\ t=0\text{:}u=u_0,h=h_0,\hfill \end{array}$$

(1)

where $x\in {\mathbb{R}}^3$, $t\ge 0$, u is the flow velocity, h is the magnetic field, p is the pressure, while $u_0$ and $h_0$ are, respectively, the given initial velocity and initial magnetic field satisfying $\mathrm{\nabla }\cdot u_0=0$, $\mathrm{\nabla }\cdot h_0=0$.

Using the standard energy method [1], it is well known that for $(u_0,h_0)\in H^s({\mathbb{R}}^3)$, $s\ge 3$, there exists a $T>0$ such that the Cauchy problem (1) has a unique smooth solution $(u,h)$ on $[0,T]$ satisfying

$$(u(t,x),h(t,x))\in C([0,T];H^s)\cap C^1([0,T];H^{s-1}).$$

(2)

Recently, Caflisch et al. [2] extended the well-known result of Beale et al. [3] to the 3D ideal MHD equations. More precisely, they showed that if the smooth solution $(u,h)$ satisfies the following condition:

$${\int }_0^T{\parallel \mathrm{\nabla }\times u\parallel }_{L^{\mathrm{\infty }}}dt<\mathrm{\infty }\text{and}{\int }_0^T{\parallel \mathrm{\nabla }\times h\parallel }_{L^{\mathrm{\infty }}}dt<\mathrm{\infty },$$

(3)

then the solution $(u,h)$ can be extended beyond $t=T$, namely, for some $T^{\star }>T$, $(u,h)\in C([0,T^{\star });H^s({\mathbb{R}}^3))\cap C^1([0,T^{\star });H^{s-1}({\mathbb{R}}^3))$. Many authors also considered the blow-up criterion of the ideal MHD equations in other spaces; see [4–6] and references therein. More recently, for the following incompressible Euler equations:

$$\{\begin{array}{c}{u}_t+u\cdot \mathrm{\nabla }u+\mathrm{\nabla }p=0,\hfill \\ \mathrm{\nabla }\cdot u=0,\hfill \\ t=0\text{:}u=u_0,\hfill \end{array}$$

(4)

with $\mathrm{\nabla }\cdot u_0=0$, Chen and Pavlovic [7] showed that if the solution u to (4) satisfies

$${\int }_0^T{({\iota }_{\gamma }(\tau ))}^{-\frac{5}{2}}d\tau <\mathrm{\infty },$$

(5)

where ${\iota }_{\gamma }(t)=min\{L,{(\frac{{\parallel \omega (t)\parallel }_{C^{\gamma }}}{{\parallel u_0\parallel }_{L^2}})}^{-\frac{2}{2\gamma +5}}\}$, $\omega =\mathrm{\nabla }\times u$ and ${\parallel \omega \parallel }_{C^{\gamma }}={sup}_{|x-y|<L}\frac{|\omega (x)-\omega (y)|}{{|x-y|}^{\gamma }}$, then the solution u can be extended beyond $t=T$. The quantity ${\iota }_{\gamma }(t)$ was introduced by Constantin in [8] (see also the work of Constantin et al. [9]). For the blow-up criterion of incompressible Euler equations, we refer to [7, 10] and references therein.

2 Main results

In this short note, we develop these ideas further and establish an analogous blow-up criterion for solutions of the 3D ideal MHD equations (1). More precisely, we can get the following theorem.

Theorem 2.1 Let $(u,h)$ be a solution to (1) in the class (2) for $s=\frac{5}{2}+\gamma$. Assume that

$${\int }_0^T{(l_{\gamma }(\tau ))}^{-\frac{5}{2}}d\tau <\mathrm{\infty },$$

(6)

where $l_{\gamma }(t)=min\{L,{(\frac{{\parallel \omega (t)\parallel }_{C^{\gamma }}}{{\parallel u_0\parallel }_{L^2}})}^{-\frac{2}{2\gamma +5}},{(\frac{{\parallel \mathrm{\Omega }(t)\parallel }_{C^{\gamma }}}{{\parallel h_0\parallel }_{L^2}})}^{-\frac{2}{2\gamma +5}}\}$, $\omega =\mathrm{\nabla }\times u$, $\mathrm{\Omega }=\mathrm{\nabla }\times h$ and the definition of $C^{\gamma }$ as above. Then there exists a finite positive constant $C_{\gamma }=O({\gamma }^{-1})$ independent of $(u,h)$ and t such that

$${\parallel (u,h)\parallel }_{H^s}\le {\parallel (u_0,h_0)\parallel }_{H^s}exp\left\{C_{\gamma }{\parallel (u_0,h_0)\parallel }_{L^2}{\int }_0^t{(l_{\gamma }(\tau ))}^{-\frac{5}{2}}d\tau \right\}$$

(7)

holds for $0\le t\le T$.

Remark 2.1 Using a similar method, we also can get the blow-up criterion result about ideal viscoelastic flow

$$\{\begin{array}{c}{\partial }_{t}u+u\cdot \mathrm{\nabla }u+\mathrm{\nabla }p={\sum }_{k=1}^3(F_k\cdot \mathrm{\nabla })F_k,\hfill \\ {\partial }_t{F}_k+u\cdot \mathrm{\nabla }{F}_k=(F_k\cdot \mathrm{\nabla })u,\hfill \\ \mathrm{\nabla }\cdot u=0,\mathrm{\nabla }\cdot F_k=0,\hfill \\ t=0\text{:}u=u_0(x),F_k=F_{k0}(x)\hfill \end{array}$$

(8)

with $\mathrm{\nabla }\cdot u_0=0$, $\mathrm{\nabla }\cdot F_{k0}=0$.

Theorem 2.2 Let $(u,F)$ be a solution to (8) in the class (2) for $s=\frac{5}{2}+\gamma$. Assume that $L_{\gamma }(t)$ is defined as above, and that

$${\int }_0^T{(L_{\gamma }(\tau ))}^{-\frac{5}{2}}d\tau <\mathrm{\infty },$$

(9)

where $L_{\gamma }(t)=min\{L,{(\frac{{\parallel (\mathrm{\nabla }\times u)(t)\parallel }_{C^{\gamma }}}{{\parallel u_0\parallel }_{L^2}})}^{-\frac{2}{2\gamma +5}},{(\frac{{\parallel (\mathrm{\nabla }\times F)(t)\parallel }_{C^{\gamma }}}{{\parallel F_0\parallel }_{L^2}})}^{-\frac{2}{2\gamma +5}}\}$. Then there exists a finite positive constant $C_{\gamma }=O({\gamma }^{-1})$ independent of $(u,F)$ and t such that

$${\parallel (u,F)\parallel }_{H^s}\le {\parallel (u_0,F_0)\parallel }_{H^s}exp\left\{C_{\gamma }{\parallel (u_0,F_0)\parallel }_{L^2}{\int }_0^t{(L_{\gamma }(\tau ))}^{-\frac{5}{2}}d\tau \right\}$$

(10)

holds for $0\le t\le T$.

This system arises in the Oldroyd model for an ideal viscoelastic flow, i.e. a viscoelastic fluid whose elastic properties dominate its behavior. Here $F=F(x,t)\in {\mathbb{R}}^{3\times 3}$ represents the local deformation gradient of the fluid. The blow-up criterion of the ideal viscoelastic system can be found in [11] and references therein.

3 Proof of Theorem 2.1

For the proof of our main result, firstly we give some properties about the gradient of velocity. Recall that the full gradient of the velocity, ∇u, can be decomposed into symmetric and antisymmetric parts,

$$\mathrm{\nabla }u=D{u}^{+}+D{u}^{-},$$

(11)

where

$$D{u}^{\pm }=\frac{1}{2}(\mathrm{\nabla }u\pm \mathrm{\nabla }{u}^T),$$

(12)

$D{u}^{+}$ is called the deformation tensor.

In the following lemmas, we recall some important properties of $D{u}^{+}$ and $D{u}^{-}$ without proof [7, 8].

Lemma 3.1 For both the symmetric and the antisymmetric parts $D{u}^{+}$, $D{u}^{-}$ of ∇u, the $L^2$ bound

$${\parallel D{u}^{\pm }\parallel }_{L^2}\le C{\parallel \omega \parallel }_{L^2}$$

(13)

holds.

The antisymmetric part $D{u}^{-}$ satisfies

$$D{u}^{-}v=\frac{1}{2}\omega \wedge v$$

(14)

for any vector $v\in {\mathbb{R}}^3$. The vorticity ω satisfies the identity

$$\omega (x)=\frac{1}{4}\mathit{P}.\mathit{V}.\int \sigma (\hat{y})\omega (x+y)\frac{dy}{{|y|}^3},$$

(15)

(‘$\mathit{P}.\mathit{V}.$’ denotes principal value) where $\sigma (\hat{y})=3\hat{y}\otimes \hat{y}-1$, with $\hat{y}=\frac{y}{|y|}$. Notably,

$${\int }_{S^2}\sigma (\hat{y})d{\mu }_{S^2}(y)=0,$$

(16)

where $d{\mu }_{S^2}$ denotes the standard measure on the sphere $S^2$.

The matrix components of the symmetric part have the form

$$D{u}_{ij}^{+}=\sum _k{T}_{ij}^k({\omega }_k)=\sum _k{\mathrm{\Gamma }}_{ij}^k\ast {\omega }_k,$$

(17)

where ${\omega }_l$ are the vector components of ω, and where the integral kernels ${\mathrm{\Gamma }}_{ij}^k$ have the properties

$${\mathrm{\Gamma }}_{ij}^k(y)={\sigma }_{ij}^k(\hat{y}){|y|}^{-3},$$

(18)

$${\parallel {\sigma }_{ij}^k\parallel }_{C^1(S^2)}\le C,$$

(19)

$${\int }_{S^2}{\sigma }_{ij}^k(\hat{y})d{\mu }_{S^2}(y)=0.$$

(20)

Thus, in particular, $T_{ij}^k$ is a Calderon-Zygmund operator, for every $i,j,k\in \{1,2,3\}$.

We can also give the following useful lemma to provide an upper bound of singular integral operator for the incompressible Euler equations in [7].

Lemma 3.2 For $L>0$ fixed, and $\gamma >0$, let ${\iota }_{\gamma }(t)$ be defined as above. Moreover, let ${\omega }_k$ ($k=1,2,3$) denote the components of the vorticity vector $\omega (t)$. Then any singular integral operator

$$T{\omega }_k(x)=\frac{1}{4\pi }\mathit{P}.\mathit{V}.\int {\sigma }_T(\hat{y}){\omega }_k(x+y)\frac{dy}{{|y|}^3},$$

(21)

with

$${\int }_{S^2}{\sigma }_T(\hat{y})d{\mu }_{S^2}(y)=0,{\parallel {\sigma }_T\parallel }_{C^1(S^2)}<C,$$

(22)

satisfies

$${\parallel T{\omega }_k\parallel }_{L^{\mathrm{\infty }}}\le C{\parallel u_0\parallel }_{L^2}{\iota }_{\gamma }^{\frac{5}{2}}(t)$$

(23)

for $k\in \{1,2,3\}$ and the constant C independent of u and t.

Now we are ready to give a proof of Theorem 2.1, which is based on combining an energy estimate for ideal MHD equations with the estimate of $({\parallel \mathrm{\nabla }u\parallel }_{L^{\mathrm{\infty }}}+{\parallel \mathrm{\nabla }h\parallel }_{L^{\mathrm{\infty }}})$.

For $s>\frac{5}{2}$, we recall the definitions of the homogeneous and inhomogeneous Besov norms for $1\le p,q\le \mathrm{\infty }$,

$${\parallel f\parallel }_{{\dot{B}}_{p,q}^s}={\left(\sum _{j\in \mathbb{Z}}{2}^{jqs}{\parallel f_j\parallel }_{L^p}^q\right)}^{\frac{1}{q}}$$

(24)

and

$${\parallel f\parallel }_{B_{p,q}^s}={\parallel f\parallel }_{L^p}+{\left(\sum _{j\in \mathbb{Z}}{2}^{jqs}{\parallel f_j\parallel }_{L^p}^q\right)}^{\frac{1}{q}},$$

(25)

where $f_j=P_{j}f$ is the Paley-Littlewood projection of f of scale j. We take the $B_{2,2}^s$ Besov norm of $u(t)$ and $h(t)$; then

$$\begin{array}{c}\frac{d}{dt}({\parallel u(t)\parallel }_{B_{2,2}^s}^2+{\parallel h(t)\parallel }_{B_{2,2}^s}^2)\hfill \\ \le C({\parallel \mathrm{\nabla }u\parallel }_{\mathrm{\infty }}+{\parallel \mathrm{\nabla }h\parallel }_{L^{\mathrm{\infty }}})({\parallel u(\cdot ,t)\parallel }_{B_{2,2}^s}^2+{\parallel h(\cdot ,t)\parallel }_{B_{2,2}^s}^2).\hfill \end{array}$$

(26)

Therefore,

$$\begin{array}{c}\frac{d}{dt}({\parallel u(t)\parallel }_{B_{2,2}^s}+{\parallel h(t)\parallel }_{B_{2,2}^s})\hfill \\ \le C({\parallel \mathrm{\nabla }u\parallel }_{\mathrm{\infty }}+{\parallel \mathrm{\nabla }h\parallel }_{L^{\mathrm{\infty }}})({\parallel u(\cdot ,t)\parallel }_{B_{2,2}^s}+{\parallel h(\cdot ,t)\parallel }_{B_{2,2}^s}).\hfill \end{array}$$

(27)

However, applying the results of Lemma 3.1 and Lemma 3.2 to u and h, and by the definition of $l_{\gamma }(t)$, we obtain

$$\begin{array}{rcl}{\parallel \mathrm{\nabla }u\parallel }_{L^{\mathrm{\infty }}}+{\parallel \mathrm{\nabla }h\parallel }_{L^{\mathrm{\infty }}}& \le & {\parallel D{u}^{+}\parallel }_{L^{\mathrm{\infty }}}+{\parallel D{u}^{-}\parallel }_{L^{\mathrm{\infty }}}+{\parallel D{h}^{+}\parallel }_{L^{\mathrm{\infty }}}+{\parallel D{h}^{-}\parallel }_{L^{\mathrm{\infty }}}\\ \le & C_{\gamma }({\parallel u_0\parallel }_{L^2}+{\parallel h_0\parallel }_{L^2}){(l_{\gamma }(t))}^{-\frac{5}{2}}.\end{array}$$

(28)

Therefore, we get

$$\begin{array}{c}{\parallel u(t)\parallel }_{H^s}+{\parallel h(t)\parallel }_{H^s}\hfill \\ \simeq {\parallel u(t)\parallel }_{B_{2,2}^s}+{\parallel h(t)\parallel }_{B_{2,2}^s}\hfill \\ \le ({\parallel u_0\parallel }_{B_{2,2}^s}+{\parallel h_0\parallel }_{B_{2,2}^s})exp\left\{C({\parallel u_0\parallel }_{L^2}+{\parallel h_0\parallel }_{L^2}){\int }_0^t{l}_{\gamma }{(s)}^{-\frac{5}{2}}ds\right\}\hfill \\ \simeq ({\parallel u_0\parallel }_{H^s}+{\parallel h_0\parallel }_{H^s})exp\left\{C({\parallel u_0\parallel }_{L^2}+{\parallel h_0\parallel }_{L^2}){\int }_0^t{l}_{\gamma }{(s)}^{-\frac{5}{2}}ds\right\}\hfill \end{array}$$

for $s\ge \frac{5}{2}+\gamma$. Thus we complete the proof of Theorem 2.1.