On blow-up criteria of smooth solutions to the 3-D Euler equations in a bounded domain.
The Euler equations for ideal incompressible fluids in a bounded domain \(\Omega\subset\mathbb{R}^3\) with smooth boundary \(\partial\Omega\) \[ \begin{cases} \partial_t u+ u\cdot\nabla u+\nabla p= 0,\quad \text{div\,}u= 0,\;t\geq 0,\quad & x\in\Omega,\\ u\cdot n= 0,\quad t\geq 0,\;x\in\partial\Omega,\;u(x,0)= a(x),\quad & x\in\Omega,\end{cases}\tag{1} \] are considered. Here \(u= (u^1(x, t), u^2(x, t), u^3(x, t))\) denotes the unknown velocity vector field and \(p= p(x,t)\) the pressure scalar of the fluid the point \(x\in\Omega\) and at time \(t\in (0,\infty)\); \(a= (a^1(x), a^2(x), a^3(x))\) is a given initial velocity and \(n= n(x)= (n^1(x), n^2(x), n^3(x))\) is the unit outward normal at \(x\in\partial\Omega\). \textit{T. Kato} and \textit{C. Y. Lai} [J. Funct. Anal. 56, 15--28 (1984; Zbl 0545.76007)] proved that for every \(a\in H^m(\Omega)\) with \(\text{div\,}a= 0\) where \(m\geq 3\) is an integer, there exist \(T> 0\) and a unique solution \(u\) of (1) on \([0,T)\) in the class \(C([0, T); H^m(\Omega))\) where \(T\) depends only on \(\| a\|_{H^3}\). \textit{J. T. Beale, T. Kato} and \textit{A. Majda} [Commun. Math. Phys. 94, 61--66 (1984; Zbl 0573.76029)] showed that the \(L^\infty\)-norm of the vorticity \(\text{rot\,}u\) controls the breakdown of smooth solutions to (1), i.e.: \[ \limsup_{t\nearrow T}\,\| u(t)\|_{H^m(\Omega)}= \infty \] if and only if \[ \int^t_0\|\text{rot\,}u(\tau)\|_{L^\infty(\Omega)} d\tau\nearrow \infty\quad\text{as }t\nearrow T.\tag{2} \] They discovered the special estimate \[ \|\nabla u\|_{L^\infty}\leq C(1+\| \text{rot\,}u\|_{L^\infty}(1+ \log^+\| u\|_{H^3})+ \|\text{rot\,}u\|_{L^2})\quad\text{if }u= 0,\tag{3} \] which plays a crucial role to establish the blow-up criterion (2). In this paper, the estimate (3) is improved in order to establish new blow-up criteria by means of BMO-norm (BMO is strictly wider than \(L^\infty\) and includes \(\log 1/| X|\)).
scientific article; zbMATH DE number 1915652
scientific article