zbMATH — the first resource for mathematics

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.

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
PDF BibTeX Cite