Local existence and blow-up criterion for the Euler equations in Besov spaces of weak type
The author introduce Besov type function spaces, based on the weak \(L^p\) spaces instead of the standard \(L^p\) spaces. Some important properties of those space are proved. In particular, embedding theorems and Fourier multiplier theorem are presented. The Cauchy problem to the Euler equations is studied in Besov type function spaces. \[ \begin{aligned} &\frac{\partial v}{\partial t}+(v\cdot\nabla)v+\nabla p=0,\quad \text{div}\,v=0, \quad x\in \mathbb R^n,\;t\in (0,\infty),\\ &v(x,0)=v_0(x), \quad x\in \mathbb R^n. \end{aligned} \] Here \(v(x,t)=(v_1,\dots,v_n)\) is the velocity field, \(p\,(x,t)\) is the pressure, \(v_0(x)\) is the given initial velocity satisfying div\,\(v_0=0\), \(n\geq 3\). A local-in-time unique existence and blow-up criterion of solutions to the problem are proved in Besov spaces of weak type.
scientific article; zbMATH DE number 5562658