\(L_p\)-estimates for solutions to the initial-boundary value problem for the generalized Stokes system of equations in a bounded domain (Q2773592)

Let \(\Omega\) be a bounded domain whose boundary \(S\in C^3\) consists of \(m\) connected components. The author studies the solvability of the following initial-boundary value problem: NEWLINE\[NEWLINE \begin{gathered} \frac{\partial \vec{v}}{\partial t} + A\Bigl(x,t, \frac{\partial}{\partial x} \Bigr) \vec{v} +\nabla p = \vec{f}(x,t),\;\nabla \cdot \vec{v}=0,\quad x\in \Omega\subset {\mathbb R}^n,\;t\in (0,T), \tag{1} \\ \vec{v}(x,0)=\vec{v}_0(x),\quad \vec{v}(x,t)|_{x\in S}=\vec{a}(x,t). \tag{2} \end{gathered} NEWLINE\]NEWLINE Here \(A\) is a matrix elliptic operator of the second order; i.e., NEWLINE\[NEWLINE c^{-1}|\xi|^2|\eta|^2\leq A_0(x,t,i\xi)\eta\cdot \eta \leq c |\xi|^2|\eta|^2\quad \text{~for all~} \xi, \eta \in {\mathbb R}^n, \;x\in \overline{\Omega},\;t\in [0,T], NEWLINE\]NEWLINE where \(A_0\) is the principal part of \(A\) and \(c>0\) is a constant. Let \(Q=\Omega\times (0,T)\). The symbol \(W_p^{s,s/2}(Q)\) stands for the conventional anisotropic Sobolev spaces. Under natural smoothness conditions for the data of the problem and the corresponding agreement conditions, it is proven that there exists a unique solution to problem (1), (2) such that \(\vec{v}\in W_{p}^{2,1}(Q)\) and \(\nabla p\in L_p(Q)\). The corresponding a priori bounds are also established.