On the energy decay of a weak solution of the MHD. equations in a three- dimensional exterior domain
Let O be a bounded domain in \(R^ 3\) with smooth boundary \(\partial O\). Set \(\Omega =R^ 3-O\), and assume that \(\Omega\) is simply connected. In \(Q:=\Omega \times (0,\infty)\) consider the following magnetohydrodynamic problem: \[ \partial_ tu-\Delta u+(u,\nabla)u+B\times rot B+\nabla \pi =f, \] \[ \partial_ tB-\Delta B+(u,\nabla)B-(B,\nabla)u=0,\quad div u=0,\quad div B=0, \] in Q, with \(u=B\cdot v=rot B\times v=0\) on \(\partial \Omega \times (0,\infty)\) and \(u|_{t=0}=u_ 0\), \(B|_{t=0}=B_ 0\). The author constructs a weak solution on (0,\(\infty)\) of this exterior problem with the property that \[ E(t):=\int_{\Omega}(| u(x,t)|^ 2+| B(x,t)|^ 2)dt \] tends to zero as \(t\to \infty\).
scientific article; zbMATH DE number 4035482