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A new class of weak solutions of the Navier-Stokes equations with nonhomogeneous data. (English) Zbl 1104.35032
Summary: We investigate a class of weak solutions, the so-called very weak solutions, to stationary and nonstationary Navier-Stokes equations in a bounded domain $$\Omega \subseteq {\mathbb R}^{3}$$. This notion was introduced by H. Amann [in: Nonlinear problems in mathematical physics and related topics II, New York: Kluwer Academic Publishers. Int. Math. Ser., N.Y. 2, 1–28 (2002; Zbl 1201.76038)] for the nonstationary case with nonhomogeneous boundary data leading to a very large solution class of low regularity. Here we are mainly interested in the investigation of the “largest possible” class of solutions $$u$$ for the more general problem with arbitrary divergence $$k = \operatorname{div} u$$, boundary data $$g = u |_{\partial \Omega}$$ and an external force $$f$$, as weak as possible, but maintaining uniqueness. In principle, we follow Amann’s approach.

##### MSC:
 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids 35J25 Boundary value problems for second-order elliptic equations 35J65 Nonlinear boundary value problems for linear elliptic equations 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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