Local strong solutions of the nonhomogeneous Navier-Stokes system with control of the interval ofexistence
Local strong solutions of the nonhomogeneous Navier-Stokes system with control of the interval ofexistence
复制标题
存在区间控制的非齐次纳维-斯托克斯系统的局部强解
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
W. Varnhorn
中科院分区：
文献类型：
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作者：
R. Farwig;H. Sohr;W. Varnhorn
考虑一个有界域$varOmegasubseteq mathbb R^3$，具有光滑边界$partialvarOmega$，时间间隔$[0，T）$，$0<Tle infty$，并且在$[0，T）中 imesvarOmega$ the %完全非齐次Navier-Stokes系统$u_t - Delta u+ucdot
abla u +
abla p = f$，$u|_t=0=v_0$，$ ext
m div，u=k$，$u|_partialvarOmega = g$，具有足够平滑的数据$f，v_0，k，g $。在这种一般情况下，主要有两类已知的弱解，一类是整体弱解，类似于众所周知的情况$k=0$，$g=0$，不需要是唯一的，见citeFKS 11，一类是局部非常弱解，见citeA 02，citeA 03，citeFGS 06，这是唯一确定的，但没有可微性，不需要满足能量不等式。我们的目的是引进新的一类通常意义下的局部强解
t = 0$，$g
ot=0满足与公知的情况k=0，g=0相似的正则性和唯一性。进一步地，我们通过给定的关于存在区间$[0，T ^*）$，$0<T^* leT $的数据获得精确的信息。我们的证明基本上是基于相应的线性系统的详细分析。
Consider a bounded domain $varOmegasubseteq mathbb R^3$ with smooth boundary $partialvarOmega$, a time interval $[0,T)$, $0<Tle infty$, and in $[0,T) imesvarOmega$ the %completely nonhomogeneous Navier-Stokes system $u_t - Delta u+ucdot
abla u +
abla p = f$, $u|_t=0=v_0$, $ ext
m div,u=k$, $u|_partialvarOmega = g$, with sufficiently smooth data $f,v_0,k,g$. In this general case there are mainly known two classes of weak solutions, the class of global weak solutions, similar as in the well known case $k=0$, $g=0$ which need not be unique, see citeFKS11, and the class of local very weak solutions, see citeA02, citeA03, citeFGS06, which are uniquely determined but have no differentiability properties and need not satisfy an energy inequality. Our aim is to introduce the new class of local strong solutions in the usual sense for $k
ot= 0$, $g
ot=0$ satisfying similar regularity and uniqueness properties as in the well known case $k=0$, $g=0$. Further, we obtain precise information through the given data on the interval of existence $[0,T^*)$, $0<T^*le T$. Our proof is essentially based on a detailed analysis of the corresponding linear system.