% Submitted to Jesus Ildefonso Diaz on July 30, 2017. cauchy02@naver.com % \documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 103, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/103\hfil 3D incompressible MHD equations with Hall term] {Regularity criteria for weak solutions to 3D incompressible MHD equations with Hall term} \author[J.-M. Kim \hfil EJDE-2018/103\hfilneg] {Jae-Myoung Kim} \address{Jae-Myoung Kim \newline Department of Mathematical Sciences, Seoul National University, Seoul, Korea} \email{cauchy02@naver.com} \dedicatory{Communicated by Jesus Ildefonso Diaz} \thanks{Submitted July 30, 2017. Published May 7, 2018.} \subjclass[2010]{35B65, 35Q35, 76W05} \keywords{Magnetohydrodynamics equation; weak solution; \hfill\break\indent regularity condition} \begin{abstract} We study the regularity conditions for a weak solution to the incompressible 3D magnetohydrodynamic equations with Hall term in the whole space $\mathbb{R}^3$. In particular, we show the regularity criteria in view of gradient vectors in various spaces. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} We consider the incompressible 3D magneto hydro dynamic (MHD) equations with Hall term \begin{gather} \partial_tu -\Delta u+u\cdot\nabla u+\nabla\pi =b\cdot\nabla b,\label{HMHD-1}\\ \partial_tb-\Delta b+u\cdot\nabla b-b\cdot\nabla u+\nabla \times ((\nabla \times b)\times b)=0,\label{HMHD-2}\\ \operatorname{div} u=\operatorname{div} b=0,\label{HMHD-3} \end{gather} Here $u:Q_T:=\mathbb{R}^3\times [0,T)\to\mathbb{R}^3$ is the flow velocity vector, $b:Q_T\to\mathbb{R}^3$ is the magnetic vector, $\pi=p+ \frac{|b|^2}{2}:Q_T\to\mathbb{R}$ is the total pressure. We consider the initial value problem of \eqref{HMHD-1}--\eqref{HMHD-3}, which requires initial conditions \begin{equation}\label{HMHD-4} u(x,0)=u_0(x) \quad \text{and} \quad b(x,0)=b_0(x) \quad x\in\mathbb{R}^3 \end{equation} The initial conditions satisfy the compatibility condition, i.e. \[ \operatorname{div} u_0(x)=0, \quad \text{and} \quad \operatorname{div} b_0(x)=0. \] \begin{definition} \rm A weak solution pair $(u,b)$ of the incompressible 3D MHD equations with the Hall term \eqref{HMHD-1}--\eqref{HMHD-4} is regular in $Q_T$ provided that $\|u\|_{L^{\infty}(Q_T)}+\|b\|_{L^{\infty}(Q_T)}<\infty$. \end{definition} For a long time, the effects of Hall current on fluids has been a subject of great interest to researchers. A current induced in a direction normal to the electric and magnetic fields is commonly called Hall current \cite{Sato}. In particular, the effects of Hall current are very important if the strong magnetic field is applied The mathematical derivations of the incompressible 3D MHD equations with the Hall term could be given in \cite{ADFL} from either two-fluids or kinetic models. It is well-known that the global existence of weak solutions, local existence and uniqueness of smooth solutions to the system \eqref{HMHD-1}--\eqref{HMHD-4} were established in \cite{CJ2,CJ14}. Recently, various results for this equation were proved in view of partial regularity, temporary decay and regularity or blow-up conditions (see \cite{CJ2,CJ14,CS13,CJ215,CJ216,FLN14,FFNZ15,RY15,Weng16,Zhang15} and references therein.) We list only some results relevant to our concerns. In view of the regularity conditions in Lorentz space, He and Wang \cite{HW07} proved that a weak solution pair $(u, b)$ becomes regular in the presence of a certain type of the integral conditions, typically referred to as Serrin's condition, namely, \[ u \in L^{q,\infty}(0,\, T; L^{p,\infty}({\mathbb{R}}^3))\quad \text{with } 3/p+2/q \leq 1, \; 3
0} \frac{1}{B(x,R)}\int_{B(x,R)} | f (y)- f_{B_R} (y)|dy<\infty. $$ Here, $f_{B_R}$ is the average of $f$ over all ball $B_R(x)$ in $\mathbb{R}^3$. It will be convenient to define BMO in terms of its dual space, $\mathcal{H}^1$. On the other hand, following \cite{KT01} let $w$ be the solution to the heat equation $w_t-\Delta w=0$ with initial data $v$. Then $$ \|v\|^2_{BMO}= \sup_{x\in \mathbb{R}^3} \sup_{R>0}\frac{1}{B(x,R)}\int_{B(x,R)}\int_{0}^{R^2} | w|^2\,dt\,dy. $$ and define the $BMO^{-1}$-norm by $$ \|v\|^2_{BMO^{-1}}= \sup_{x\in \mathbb{R}^3} \sup_{R>0}\frac{1}{B(x,R)}\int_{B(x,R)}\int_{0}^{R^2} | \nabla w|^2\,dt\,dy. $$ We note that if $u$ is a tempered distribution. Then $u \in BMO^{-1}$ if and only if there exist $f^i \in BMO$ with $u=\sum \partial_i f^i$ in \cite[Theorem 1]{KT01}. Let $m(\varphi,t) $ be the Lebesgue measure of the set $\{x\in \mathbb{R}^3:|\varphi(x)|> t\}$, i.e. $$ m(\varphi,t):=m\{x\in \mathbb{R}^3:|\varphi(x)|> t\}. $$ We denote by the Lorentz space $L^{p,q}(\mathbb{R}^3)$ with $1\leq p$, $q\leq \infty $ with the norm \cite{Tr} \begin{equation}\label{poiseuille} \|\varphi\|_{L^{p,q}(\mathbb{R}^3)} =\begin{cases} \Big(\int_0^{\infty}t^q(m(\varphi,t))^{q/p} \frac{dt}{t} \Big)^{1/q}<\infty, & \text{for } 1\leq q ,\\ \sup_{t\geq 0}\{t(m(\varphi,t))^{1/p}\} , &\text{for } q=\infty\,. \end{cases} \end{equation} Followed in \cite{Tr}, Lorentz space $L^{p,q}(\mathbb{R}^3)$ may be defined by real interpolation methods \begin{equation}\label{interpolation-lorentz} L^{p,q}(\mathbb{R}^3) =(L^{p_{1}}(\mathbb{R}^3),\,L^{p_{2}}(\mathbb{R}^3))_{\alpha,q}, \end{equation} with $ \frac{1}{p}=\frac{1-\alpha}{p_{1}}+\frac{\alpha}{p_{2}}$, $1\leq p_{1}