论文标题
关于Navier-Stokes方程的局部压力扩展
On the local pressure expansion for the Navier-Stokes equations
论文作者
论文摘要
我们表明，与整个空间上Navier-Stokes方程的分布解决方案相关的压力可以满足局部扩展为分布的局部扩展，并且仅当解决方案是轻度时。这给出了Lemarié-Rieusset的“等效定理”的新观点。在这里，在不使用Littlewood-Paley分解的情况下定义了由梯度组成的Leray投影操作员。对局部扩张的足够条件假定的空间衰减或梯度上的估计值，这意味着所考虑的解决方案是轻度的。一个重要的工具是对泊松方程的有界平均振荡解决方案的明确描述，我们会详细检查。作为应用，我们包括作者在莫雷空间中的唯一性标准的改进，并重新审视了由于Grujić和Xu而动态限制的本地莫雷空间中规则性标准的证明。
We show that the pressure associated with a distributional solution of the Navier-Stokes equations on the whole space satisfies a local expansion defined as a distribution if and only if the solution is mild. This gives a new perspective on Lemarié-Rieusset's "equivalence theorem." Here, the Leray projection operator composed with a gradient is defined without using the Littlewood-Paley decomposition. Prior sufficient conditions for the local expansion assumed spatial decay or estimates on the gradient and imply the considered solution is mild. An important tool is an explicit description of the bounded mean oscillation solution to a Poisson equation, which we examine in detail. As applications we include an improvement of a uniqueness criteria by the authors in Morrey spaces and revisit a proof of a regularity criteria in dynamically restricted local Morrey spaces due to Grujić and Xu.
