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\(L_p\)-estimates for a solution to the nonstationary Stokes equations. (English. Russian original) Zbl 0987.35121
J. Math. Sci., New York 106, No. 3, 3073-3077 (2001); translation from Probl. Mat. Anal. 22, 240-245 (2001).
Under consideration is the following initial-boundary value problem for the Stokes system: \[ \begin{gathered} \frac{\partial \vec{v}}{\partial t} - \Delta \vec{v} +\nabla p = \nabla \cdot{\mathcal F},\quad \nabla \cdot \vec{v}=0,\quad x\in \Omega\subset {\mathbb R}^n,\;t\in (0,T), \tag{1} \\ \vec{v}(x,0)=0,\quad\vec{v}(x,t)|_{x\in S}=\vec{a}(x,t),\quad S=\partial \Omega, \tag{2} \end{gathered} \] where \({\mathcal F}= \{F_{i,k}\}\) is an \(n\times n\)-matrix and \(\nabla \cdot{\mathcal F}= (\sum_{i=1}^n \frac{\partial F_{i,k}}{\partial x_i})_{k=1,\dots, n}\). Let \(Q=\Omega\times (0,T)\) (\(\Omega\) is a bounded convex domain) and let \(\Sigma=S\times (0,T)\). By the symbols \(W_p^s(Q)\) and \(W_p^{s,s/2}(Q)\) the author means the conventional Sobolev spaces (anisotropic in the latter case). The symbols \(W_{p,0}^s(Q)\) and \(W_{p,0}^{s,s/2}(Q)\) denote the subspaces of \(W_p^s(Q)\) and \(W_p^{s,s/2}(Q)\) comprising functions \(f(x,t)\) whose zero extensions to \(\Omega\times (-\infty,T)\) belong to the same Sobolev class; by definition, the norm of a function \(f(x,t)\) in \(W_{p,0}^s(Q)\) (\(W_{p,0}^{s,s/2}(Q)\)) coincides with the norm of the corresponding extension \(f_0(x,t)\) equal to \(f\) on \((0,T)\) and zero on \((-\infty,0]\) in the space \(W_p^{s}(\Omega\times (-\infty,T))\) (\(W_{p}^{s,s/2}(\Omega\times (-\infty,T))\)). The vector \(\vec{a}\) with the property \(\int_{S}\vec{a}\cdot \vec{n} dS=0 \) is assumed to be representable as \(\vec{a}\cdot \vec{n}=\text{div}_S \vec{{\mathcal A}}\), where the symbol \(\vec{n}\) stands for the unit normal to \(S\). The main result of the article can be stated as follows:
Theorem. Let \(S\in C^2\). Then a solution \(\vec{v}\) to problem (1), (2) meets the inequality \[ \|\vec{v}\|_{W_{p,0}^{1,1/2}(Q)}\leq c(T)( \|{\mathcal F}\|_{L_p(Q)} + \|\vec{a}\|_{W_{p,0}^{1-1/p,1/2-1/(2p)}(\Sigma)}+ \|\vec{{\mathcal A}}\|_{W_{p,0}^{1/2}(0,T;W_p^{1-1/p}(S))})\equiv c N, \tag{3} \] the pressure \(p(x,t)\) is representable as \(p=p_1 + \frac{\partial P}{\partial t}\) with \(P\) a harmonic function, and \[ \|p_1\|_{L_p(Q)}+\|\nabla P\|_{W_{p,0}^{1,1/2}(Q)}\leq c N. \tag{4} \] The constants \(c\) in (3) and (4) are nondecreasing functions of \(T\).
If \(\vec{{\mathcal A}}\in W_{p,0}^{1-1/(2p)-r/2}(0,T;W_p^{r}(S))\) (\(r\in (0,1-1/p]\)) then the estimate (4) may be refined: \[ \|P\|_{W_{p,0}^{1-1/(2p)-r/2}(0,T;W_p^{1/p+r}(\Omega))}\leq c(T)(N+ \|\vec{{\mathcal A}}\|_{W_{p,0}^{1-1/(2p)-r/2}(0,T;W_p^{r}(S))}). \]

MSC:
35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D07 Stokes and related (Oseen, etc.) flows
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