Existence, uniqueness and regularity of stationary solutions to inhomogeneous Navier-Stokes equations in $\Bbb R^n$

Farwig, R. ; Sohr, H.

Czechoslovak Mathematical Journal, Tome 59 (2009) no. 1, pp. 61-79.

Résumé

For a bounded domain $\Omega \subset \Bbb R ^n$, $n\geq 3,$ we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system $-\Delta u + u \cdot \nabla u + \nabla p=f$, $\div u = k$, $u_{|_{\partial \Omega }}=g$ with $u \in L^q$, $q \geq n$, and very general data classes for $f$, $k$, $g$ such that $u$ may have no differentiability property. For smooth data we get a large class of unique and regular solutions extending well known classical solution classes, and generalizing regularity results. Moreover, our results are closely related to those of a series of papers by Frehse Růžička, see e.g. Existence of regular solutions to the stationary Navier-Stokes equations, Math. Ann. 302 (1995), 669--717, where the existence of a weak solution which is locally regular is proved.

Détail

Classification : 35B65, 35J55, 35J65, 35Q30, 76D05, 76D07
Keywords: stationary Stokes and Navier-Stokes system; very weak solutions; existence and uniqueness in higher dimensions; regularity classes in higher dimensions