A new class of weak solutions of the Navier-Stokes equations with nonhomogeneous data. (English) Zbl 1104.35032

Farwig, Reinhard; Galdi, Giovanni P.; Sohr, Hermann

J. Math. Fluid Mech. 8, No. 3, 423-444 (2006).

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

Keywords:

Stokes and Navier-Stokes equations; very weak solutions; nonhomogeneous data; Serrin’s class