Very weak solutions of the stationary Navier-Stokes equations for an incompressible fluid past obstacles

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• 作者：Dugyu Kim ; ^{dugyu@sogang.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Hyunseok Kim ^{kimh@sogang.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35Q10
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：147
• 期：Complete
• 页码：145-168
• 全文大小：775 K

文摘

We consider the stationary motion of an incompressible Navier–Stokes fluid past obstacles in R³${\mathbb{R}}^3$, subject to the given boundary velocity v_b$v_b$, external force width="60" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0362546X16301894-si21.gif"> and nonzero constant vector ke₁$k{e}_1$ at infinity. Our main result is the existence of at least one very weak solution v$v$ in ke₁+L³(Ω)$k{e}_1+L^3\left(\Omega \right)$ for arbitrary large F∈L^3/2(Ω)+L^12/7(Ω)$F\in L^{3/2}\left(\Omega \right)+L^{12/7}\left(\Omega \right)$ provided that the flux of v_b−ke₁$v_b-k{e}_1$ on the boundary of each body is sufficiently small with respect to the viscosity ν$\nu$. The uniqueness of very weak solutions is proved by assuming that F$F$ and v_b−ke₁$v_b-k{e}_1$ are suitably small. Moreover, we establish weak and strong regularity results for very weak solutions. In particular, our existence and regularity results enable us to prove the existence of a weak solution v$v$ satisfying ∇v∈L^3/2(Ω)$\nabla v\in L^{3/2}\left(\Omega \right)$.