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Existence, uniqueness and regularity of stationary solutions to inhomogeneous Navier-Stokes equations in \(\mathbb R^n\). (English) Zbl 1224.76034
Summary: For a bounded domain \(\Omega \subset \mathbb R ^n\), \(n\geq 3\), we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system \(-\Delta u + u \cdot \nabla u + \nabla p=f\), \(\operatorname {div} u = k\), \(u| _{\partial \Omega }=g\) with \(u \in L^{q}\), \(q \geq n\) and very general data classes for \(f\), \(k\) and \(g\) such that \(u\) may be non differentiable. 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 J. Frehse and M. Růžička [see, e.g., Math. Ann. 302, No. 4, 699–717 (1995; Zbl 0861.35074)], where the existence of a weak solution which is locally regular was proved.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
35J65 Nonlinear boundary value problems for linear elliptic equations
35Q30 Navier-Stokes equations
76D07 Stokes and related (Oseen, etc.) flows
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