Global solutions to the Maxwell-Navier-Stokes system in a bounded domain in 2D
Let \(\Omega \subset \mathbb{R}^2\) be a bounded and simply connected domain with smooth boundary \(\partial \Omega\). Consider the system \begin{align*} &\partial_tu+u\cdot\nabla u+ \nabla \pi -\Delta u=(E+u\times b)\times b, \\ &\partial_tE-\operatorname{rot} \ b + E+u\times b=0, \\ &\partial_tb+ \operatorname{rot} \ E=0, \\ &\operatorname{div} \, u=0, \\ &\operatorname{div} \, b=0, \ \ \mbox{in } \Omega \times (0,\infty), \\ &u\cdot n=0, \ \operatorname{rot} \ u=0, \ b\cdot n =0, \ E=0 \ \ \mbox{on } \partial \Omega \times (0, \infty), \\ &(u,E,b)(\cdot , 0)=(u_0, E_0, b_0)(\cdot ) \ \ \mbox{in } \Omega. \end{align*} Here, \(u, \pi, E\) and \(b\) denote the velocity, pressure, electric field and magnetic field, respectively. The authors state the following result: If \(u_0, E_0, b_0\in H^2(\Omega)\) satisfy \(\operatorname{div}\, u_0=\operatorname{div} \, b_0=0\) in \(\Omega\) and \(u_0\cdot n=0\), \(\operatorname{rot}\, u_0=0\), \(b_0\cdot n=0\), and \(E_0=0\) on \(\partial \Omega\), then, for any \(T>0\), there exists a unique solution of the above problem such that \(u,E,b\in L^{\infty}(0,T;H^2(\Omega))\).
scientific article; zbMATH DE number 7342394