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

Abstract

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 R 3 . 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 R 3 . 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.

A note on a transmission problem for the Brinkman system and the generalized Darcy-Forchheimer-Brinkman system in Lipschitz domains in R3

“…where Z + is a right inverse of the trace operator T r + : 3 . The operator t + P,ν is linear, bounded and does not depend on the choice of the right inverse Z + of the trace operator T r + .…”

“…where α > 0 is a given constant. The main result of the paper reads as follows (see e.g., [3,Theorem 3.2], and [8, Theorem 5.2] in the case P = αI, where α, k, β > 0 are constants). Theorem 3.2.…”

“…Let us write the problem (3.5) in the equivalent form: 9) where J k,β,Ω+ (w + ) ∈ H −1 (Ω + ) 3 is given by Lemma 2.8. Now, we fix w + ∈ H 1 div (Ω + ) 3 and consider the following linear transmission problem for the generalized and classical Brinkman systems with the unknowns (w 0…”

“…for all w + ∈ H 1 div (Ω + ) 3 . Indeed, such an inequality is provided by Lemma 2.8 and c 0 > 0 is the constant involved in Lemma 2.8.…”

“…If we are able to show this property, then the fixed point w + ∈ H 1 div (Ω + ) 3 , together with the fields w − = W − (w + ) and with p ± = P ± (w + ) will give a solution of our nonlinear problem (3.9). Now, we use similar ideas to those in the proof of Theorem 5.2 in [8], to show that W + maps a closed ball B λ to the same closed ball in H 1 div (Ω + ) 3 and that W + is a contraction on that ball.…”