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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.

MSC:

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.)
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Full Text: Euclid