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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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.

Article information

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

Dates
First available in Project Euclid: 18 August 2015

Permanent link to this document
https://projecteuclid.org/euclid.ade/1439901072

Mathematical Reviews number (MathSciNet)
MR3388894

Zentralblatt MATH identifier
1328.35170

Subjects
Primary: 35Q10

Citation

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. https://projecteuclid.org/euclid.ade/1439901072.


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