Well-posedness of the ideal MHD system in critical Besov spaces
From the introduction: We are concerned with the following ideal magnetohydrodynamics (MHD) system for homogeneous incompressible fluid flows and magnetic fields: \[ u_t+(u\cdot\nabla)u-(b\cdot\nabla)b-\nabla\pi=0,\tag{1.1} \] \[ b_t+(u\cdot\nabla)b-(b\cdot\nabla)u=0,\tag{1.2} \] \[ \text{div}\,u=0,\quad \text{div}\,b=0,\tag{1.3} \] with initial data \[ u(x,0)=u_0(x),\tag{1.4} \] \[ b(x,0)=b_0(x).\tag{1.5} \] Here \(u(x,t)= (u_1(x,t),u_2(x,t),\dots,u_n(x,t))\) is the velocity of the fluid flows, \(b(x,t)=\break (b_1(x,t),b_2(x,t),\dots,b_n(x,t))\) is the magnetic field, \(\pi(x,t)=p(x,t)+{1\over2}|b(x,t)|^2\) is the total pressure for \(x\in\mathbb R^n\) and \(t\geq 0\), and \(u_0(x)\) and \(b_0(x)\) are the initial velocity and initial magnetic field satisfying \(\text{div}\,u_0=0\), \(\text{div}\,b_0=0\), respectively. For simplicity, we have set the Reynolds number, magnetic Reynolds number and the corresponding coefficients to be constant 1 by a scaling transformation. In this paper, the local existence and uniqueness of the solution to the \(n\)-dimensional ideal incompressible MHD system (1.1)--(1.5) is investigated. We prove that there exists a locally unique solution in the critical Besov space \(B^{1+n/p}_{p,1}\) for \(1\leq p\leq\infty\) provided that the initial data \((u_0(x,t), b_0(x,t))\) is in the space. We obtain a priori estimates for solutions to the approximate equations by virtue of a coupling effect between the velocity \(u(x,t)\) and magnetic fields \(b(x,t)\). Our local existence and uniqueness results are as follows: Theorem 1.1. Let \(u_0(x),\;b_0(x)\in B^{1+n/p}_{p,1}(\mathbb R^n)\) be the divergence free vector for \(1\leq p\leq\infty\). There exists time \(T>0\) such that the Cauchy problem (1.1)--(1.5) has a unique solution \((u(x,t),b(x,t))\in C([0,T];B^{1+n/p}_{p,1}(\mathbb R^n))\). The plan of this paper is as follows: In Section 2 we recall succinctly the Littlewood-Paley dyadic decomposition, Besov spaces and Bony's para-product decomposition of two functions \(f(x)\) and \(g(x)\). Then we give some preliminary a priori estimates. In Section 3, we establish a priori estimates of solutions to the approximate equations, and prove that the sequence of solutions is locally bounded in the Besov space \(B^{1+n/p}_{p,1}(\mathbb R^n)\) and that there is a Cauchy sequence of solutions to the approximate equations in \(B^{n/p}_{p,1}(\mathbb R^n)\) for \(1\leq p\leq\infty\). The main theorem is also proved in Section 3.
scientific article; zbMATH DE number 5123519