On the Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain
On the Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain
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DOI：
10.1515/crelle.2008.013
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发表时间：
2008-02
期刊：
影响因子：
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通讯作者：
Y. Shibata;Senjo Shimizu
中科院分区：
文献类型：
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作者：
Y. Shibata;Senjo Shimizu
**摘要**：在本文中，我们证明了在有界区域中具有非齐次边界条件和散度条件的斯托克斯方程的诺伊曼问题解的\(L^p - L^q\)最大正则性。该结果首先由索洛尼科夫（V. A. Solonnikov）在《关于粘性不可压缩流体孤立体积的瞬态运动》（Math. USSR Izvest. 31 (1988), 381–405）中给出，但他假设\(p = q > 3\)且仅考虑了有限时间区间的情况。在本文中，我们不仅考虑\(1 < p,q < \infty\)的情况，还考虑了无限时间区间的情况。特别地，我们得到了在无限时间区间上具有指数稳定性的\(L^p - L^q\)最大正则性定理。我们的方法可应用于具有合适边界条件（该边界条件生成一个解析半群）的抛物型方程的任何初边值问题，例如具有无滑移、滑移或罗宾边界条件的斯托克斯方程。
Abstract In this paper, we prove the Lp-Lq maximal regularity of solutions to the Neumann problem for the Stokes equations with non-homogeneous boundary condition and divergence condition in a bounded domain. The result was first stated by Solonnikov V. A. Solonnikov, On the transient motion of an isolated volume of viscous incompressible fluid, Math. USSR Izvest. 31 (1988), 381–405., but he assumed that p = q > 3 and considered only the finite time interval case. In this paper, we consider not only the case: 1 < p, q < ∞ but also the infinite time interval case. Especially, we obtain the Lp-Lq maximal regularity theorem with exponential stability on the infinite time interval. Our method can be applied to any initial boundary value problem for the equation of parabolic type with suitable boundary condition which generates an analytic semigroup, for example the Stokes equation with non-slip, slip or Robin boundary conditions.