Very weak solutions of the stationary Stokes equations on exterior domains. (English) Zbl 1328.35170

Summary: We study the nonhomogeneous Dirichlet problem for the stationary Stokes equations on exterior smooth domains \(\Omega\) in \(\mathbb{R}^n\), \(n\geq 3\). Our main result is the existence and uniqueness of very weak solutions in the Lorentz space \(L^{p,q}(\Omega)^n\), where \((p,q)\) satisfies either \((p,q)=(n/(n-2),\infty)\) or \(n/(n-2)<p<\infty\), \(1\leq q\leq \infty\). This is deduced by a duality argument from our new solvability results on strong solutions in homogeneous Sobolev-Lorentz spaces. Homogeneous Sobolev-Lorentz spaces are also studied in quite details: particularly, we establish basic interpolation and density results, which are not only essential to our results for the Stokes equations but also themselves of independent interest.


35Q35 PDEs in connection with fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
35D30 Weak solutions to PDEs
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Full Text: Euclid