On the very weak solution for the Oseen and Navier-Stokes equations
The main objective of the paper consists in the development of a unified theory of very weak solutions of the Dirichlet problem for Stokes, Oseen and Navier-Stokes equations (and also for the Laplace equation). An important question in this context is the rigorous definition of traces of the vector functions that are belonging to subspaces of \({\mathbf L}^{p}(\Omega)\). The authors prove existence and regularity of very weak solutions \((\mathbf u, q) \in {\mathbf L}^{p}(\Omega) \times W^{-1,p}(\Omega)\) of Stokes and Oseen equations for any \(1 < p < \infty\) with arbitrary large data belonging to some Sobolev spaces with negative order. In the case of the Navier-Stokes equations the existence of very weak solutions is proved for arbitrary large external forces. Here a smallness condition is necessary for both the compressibility \(h\) and the boundary conditions \(\mathbf g\). If the data are small enough then the uniqueness of the very weak solution can also been proved.
scientific article; zbMATH DE number 5760005