A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in W?1/q,q

Abstract

We develop a theory for a general class of very weak solutions to stationary Stokes and Navier-Stokes equations in a bounded domain Ω with boundary ∂Ω of class C2,1, corresponding to boundary data in the distribution space W−1/q,q(∂Ω), 1<q<∞. These solutions exist and are unique (for small data, in the nonlinear case) in their class of existence, and satisfy a correponding estimate in terms of the data. Moreover, they become regular if the data are regular. To our knowledge, the only existence result for solutions attaining such boundary data is due to Giga, [16], Proposition 2.2, for the Stokes case. However, the methods and the approach used in the present paper are different than Giga’s and cover more general issues, including the nonlinear Navier-Stokes equations and the precise way in which the boundary data are attained by the solutions. We also introduce, in the last section, a further generalization of the solution class.