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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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