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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
##### Keywords:
Stokes system; a priori bounds; uniqueness and existence