arXiv 1109.53282011-09math.AP

具有真空和大初值的完全可压缩Navier-斯托克斯方程的整体对称经典解和强解

Global symmetric classical and strong solutions of the full compressible Navier-Stokes equations with vacuum and large initial data

Huanyao Wen, Changjiang Zhu

备注：: arXiv admin note: text overlap with arXiv:1103.1421

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摘要：

首先，我们得到了三维空间中完全可压缩Navier-斯托克斯方程的经典解和强解的整体存在性，其中初值是大的，并且是球对称或柱对称的。允许出现真空。特别地，如果初始数据是球对称的，则空间维度可以取为不小于小于2。该分析基于一些精细的{ \it先验}估计，这些估计在时间上全局地依赖于假设$κ=O(1+θ^q)$，其中$q>r$（$r$可以为零），这放宽了[14，29，42]中的条件$q\ge2+2r$。这可以被视为[18]的扩展工作，其中方程在密度为正的集合中的分布意义上成立，初始数据是大的，不连续的，并且在三维空间中是球对称或圆柱对称的。最后，在真空可能出现和解不一定对称的假设下，我们建立了强解的$\|ρ\|_{L^\infty_tL_x^\infty}$和$\|ρθ\|_{L^4_tL^(12/5)_x}$爆破准则.

英文摘要：

First of all, we get the global existence of classical and strong solutions of the full compressible Navier-Stokes equations in three space dimensions with initial data which is large and spherically or cylindrically symmetric. The appearance of vacuum is allowed. In particular, if the initial data is spherically symmetric, the space dimension can be taken not less than two. The analysis is based on some delicate {\it a priori} estimates globally in time which depend on the assumption $κ=O(1+θ^q)$ where $q>r$ ($r$ can be zero), which relaxes the condition $q\ge2+2r$ in [14,29,42]. This could be viewed as an extensive work of [18] where the equations hold in the sense of distributions in the set where the density is positive with initial data which is large, discontinuous, and spherically or cylindrically symmetric in three space dimension. Finally, with the assumptions that vacuum may appear and that the solutions are not necessarily symmetric, we establish a blow-up criterion in terms of $\|ρ\|_{L^\infty_tL_x^\infty}$ and $\|ρθ\|_{L^4_tL^(12/5)_x}$ for strong solutions.

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