论文标题
在有限域中流体力压力的双Hölder规律性上
On Double Hölder Regularity of the Hydrodynamic Pressure in Bounded Domains
论文作者
论文摘要
我们证明，与速度$ u \相关的流体动力压力$ p $ in c^θ（ω）$，$θ\ in（0,1）$，是一个不可压缩的液体，在一个有界和简单连接的域$ω\ subset \ subset \ subbb r^d $ c^$ c^$ c^$ c^$ c^{2+sounderies c^{2+sounderies c^{2+sounderies c^{ $θ\ leq \ frac12 $和$ p \ in C^{1,2θ-1}（ω）$ for $θ> \ frac12 $。此外，当$ \ partialω\ in c^{3+} $中时，我们证明了几乎doublehölder规律性$ p \ in c^{2θ-}（ω）$即使在$θ<\ frac12 $中也容纳。这扩展并改善了在平面案例中获得的Bardos和Titi的最新结果到每个维度$ d \ ge2 $，并且也使压力规律性增加了一倍。与Bardos和Titi不同，我们没有引入压力的新边界条件，而是与天然的边界条件一起使用。在$ d $维圆环的无边界情况下，我们表明，在较弱的假设中，速度的差异足够规律，因此不一定是零的，因此压力的双重规律性实际上可以实现。
We prove that the hydrodynamic pressure $p$ associated to the velocity $u\in C^θ(Ω)$, $θ\in(0,1)$, of an inviscid incompressible fluid in a bounded and simply connected domain $Ω\subset \mathbb R^d$ with $C^{2+}$ boundary satisfies $p\in C^θ(Ω)$ for $θ\leq \frac12$ and $p\in C^{1,2θ-1}(Ω)$ for $θ>\frac12$. Moreover, when $\partial Ω\in C^{3+}$, we prove that an almost double Hölder regularity $p\in C^{2θ-}(Ω)$ holds even for $θ<\frac12$. This extends and improves the recent result of Bardos and Titi obtained in the planar case to every dimension $d\ge2$ and it also doubles the pressure regularity. Differently from Bardos and Titi, we do not introduce a new boundary condition for the pressure, but instead work with the natural one. In the boundary-free case of the $d$-dimensional torus, we show that the double regularity of the pressure can be actually achieved under the weaker assumption that the divergence of the velocity is sufficiently regular, thus not necessarily zero.
