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New estimates for the steady-state Stokes problem in exterior domains with applications to the Navier-Stokes problem. (English) Zbl 0823.35142
The steady-state Stokes problem is investigated: $\Delta v= \nabla p+f,\;\;\nabla\cdot v=0 \quad \text{in } \Omega, \qquad v=0 \quad \text{on }\partial \Omega,$ in an exterior domain $$\Omega$$ of $$\mathbb{R}^ n$$, $$n\geq 3$$ (it means $$\Omega$$ is the complement of a compact set in $$\mathbb{R}^ n$$ of class $$C^ 2$$). The existence, uniqueness and some estimates of solutions $v\in D_ 0^{1,q} (\Omega), \quad p\in L^ q (\Omega) \quad \text{for } q> {\textstyle {n\over \alpha}} \quad \text{and} \quad n\geq 3,$ ($$D_ 0^{1,q} (\Omega)$$ is the completion with respect to the $$L^ q$$-norm of the gradient of the space of solenoidal, smooth functions of compact support in $$\Omega$$) are established under the assumption that $$f$$ is of type $$\nabla \cdot F$$, where $$F$$ is a second order tensor field, such that $$\sup_{x\in \Omega} (1+| x|^ \alpha) | F(x) |< \infty$$, $$\alpha$$ is either 2 or $$n- 1$$.
These estimates are used to show the existence of a unique solution for the nonlinear Navier-Stokes problem corresponding to zero velocity at infinity and in the case when the body force $$f$$ is the divergence of some tensor $$F$$ with $$\sup_{x\in \Omega} (1+| x|^ 2) | F(x) |< \infty$$.
The authors give the proof in the case $$n=3$$ and state the generalization to $$n\geq 4$$ as a remark.

##### MSC:
 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids
##### Keywords:
Stokes problem; existence of solution; exterior domain