scholarly journals The Lp regularity problem for the Stokes system on Lipschitz domains

2015 ◽  
Vol 259 (4) ◽  
pp. 1275-1296 ◽  
Author(s):  
Jun Geng ◽  
Joel Kilty
Keyword(s):  
Stokes System ◽  
Lipschitz Domains ◽  
Regularity Problem ◽  
Lp Regularity

scholarly journals On transmission-type problems for the generalized Darcy-Forchheimer-Brinkman and stokes systems in complementary lipschitz domains in R3

Filomat ◽  
2019 ◽  
Vol 33 (11) ◽  
pp. 3361-3373
Author(s):  
Andrei-Florin Albişoru
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Point Theorem ◽  
Sobolev Spaces ◽  
Stokes System ◽  
Lipschitz Domains ◽  
Well Posedness ◽  
Stokes Systems

The aim of our work is to give a well-posedness result for a boundary value problem of transmission-type for the nonlinear, generalized Darcy-Forchheimer-Brinkman and Stokes systems in complementary Lipschitz domains in R3. First, we introduce the Sobolev spaces in which we seek our solution, then we define the trace operators and conormal derivative operators that are involved in the boundary conditions of our treated problem. Next, we state a result that concerns the well-posedness of the transmission problem for the generalized Brinkman and Stokes system in complementary Lipschitz domains in R3. Afterwards, we state and prove an important lemma. Finally, we obtain our desired result by employing the well-posedness of the linearized version of our problem and Banach?s fixed point theorem.

scholarly journals Lq-Estimates for stationary Stokes system with coefficients measurable in one direction

2019 ◽  
Vol 09 (01) ◽  
pp. 1950004 ◽  
Author(s):  
Hongjie Dong ◽  
Doyoon Kim
Keyword(s):  
Stokes System ◽  
A Priori ◽  
Lipschitz Constant ◽  
Variable Coefficients ◽  
Lipschitz Domains ◽  
Small Ball ◽  
Fixed Direction ◽  
Measurable Functions ◽  
Whole Space ◽  
Bounded Lipschitz Domains

We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori [Formula: see text]-estimates for any [Formula: see text] when the coefficients are merely measurable functions in one fixed direction. For the system on bounded Lipschitz domains with a small Lipschitz constant, we obtain a [Formula: see text]-estimate and prove the solvability for any [Formula: see text] when the coefficients are merely measurable functions in one direction and have locally small mean oscillations in the orthogonal directions in each small ball, where the direction is allowed to depend on the ball.

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