Abstract
We study the nonhomogeneous Dirichlet problem for the stationary Stokes equations on exterior smooth domains $\Omega$ in $\mathbb R^n , n \ge 3$. 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 $n/(n-2) < p < \infty , 1 \le q \le \infty$. 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.
Citation
Dugyu Kim. Hyunseok Kim. Sungyong Park. "Very weak solutions of the stationary Stokes equations on exterior domains." Adv. Differential Equations 20 (11/12) 1119 - 1164, November/December 2015. https://doi.org/10.57262/ade/1439901072
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