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On estimation for solutions of the Stokes system in exterior domains. (Russian. English summary) Zbl 0719.35068

Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 180, 105-120 (1990).
[For the entire collection see Zbl 0698.00040.]
The authors study the following boundary value problem: \[ -\Delta v(x)+\nabla q(x)=f(x),\quad\nabla \cdot v(x)=0,\quad x\in \Omega; \]
\[ v=0,\quad x\in \Gamma;\quad v\to 0,\quad | x| \to \infty\quad (n>2); \]
\[ v\to \text{const},\quad | x| \to \infty\quad (n=2). \]
Here \(\Omega\subset\mathbb R^ n\) is an unbounded exterior domain with a smooth compact boundary \(\Gamma\). One of the main results is the following a priori estimate: \[ \sum_{i,j}\| \partial^ 2v/\partial x_ i\partial x_ j\|_ p+\| \nabla q\|_ p\leq c\cdot \| f\|_ p \] where \(\| \cdot \|_ p\) is the norm in \(L_ p(\Omega)\), \(1<p<n/2\). The estimate is valid for \(p\geq n/2\) if and only if the exterior force \(f\) satisfies some additional conditions.

MSC:

35Q30 Navier-Stokes equations
35B45 A priori estimates in context of PDEs
76D07 Stokes and related (Oseen, etc.) flows

Citations:

Zbl 0698.00040
Full Text: EuDML