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Estimates of the solution of model evolution generalized Stokes problem in weighted Hölder spaces. (English) Zbl 1127.35051
Zap. Nauchn. Semin. POMI 336, 211-238 (2006) and J. Math. Sci., New York 143, No. 2, 2969-2986 (2007).
The author considers the Cauchy problem and the half-space initial boundary value problem to the generalized nonstationary Stokes equations
\[ \begin{aligned} &\frac{\partial v}{\partial t}+A_0\left(\frac{\partial}{\partial x}\right) v+\nabla p=f(x,t), \quad \operatorname{div} v=0, \quad x\in\mathbb R^n,\;t\in (0,T),\\ &v(x,0)=v_0(x).\end{aligned}\tag{1} \]
\[ \begin{aligned} &\frac{\partial v}{\partial t}+A_0\left(\frac{\partial}{\partial x}\right) v+\nabla p=f(x,t),\quad \operatorname{div} v=0, \quad x\in\mathbb R^n_+,\quad t\in (0,T)\\ &v(x,0)=0,\quad v(x,t)| _{x_n=0}=b(x',t),\quad x'=(x_1,\dots,x_{n-1})\in\mathbb R^{n-1}.\end{aligned}\tag{2} \] Here \(A_0(\frac{\partial}{\partial x})\) is the matrix strongly elliptic differential operator containing only the second derivatives, \(\mathbb R^n_+=\{x\in\mathbb R^n:x_n>0\}\). The Stokes system corresponds to \(A_0=-\nu I\nabla^2\), \(\nu= \text{const}>0\).
Coercive estimates in anisotropic weighted Hölder spaces are proved to solve both the problems (1) and (2).

MSC:
35Q35 PDEs in connection with fluid mechanics
35B45 A priori estimates in context of PDEs
76D07 Stokes and related (Oseen, etc.) flows
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