Advances in Differential Equations

Very weak solutions of the stationary Stokes equations on exterior domains

Dugyu Kim, Hyunseok Kim, and Sungyong Park

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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.

Article information

Adv. Differential Equations, Volume 20, Number 11/12 (2015), 1119-1164.

First available in Project Euclid: 18 August 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q10


Kim, Dugyu; Kim, Hyunseok; Park, Sungyong. Very weak solutions of the stationary Stokes equations on exterior domains. Adv. Differential Equations 20 (2015), no. 11/12, 1119--1164.

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