A note on a transmission problem for the Brinkman system and the generalized Darcy-Forchheimer-Brinkman system in Lipschitz domains in R³
DOI:
https://doi.org/10.24193/subbmath.2019.3.10Keywords:
Sobolev spaces, generalized Brinkman system, transmission problems, generalized Darcy-Forchheimer-Brinkman system, well-posedness result, Banach fixed point theorem.Abstract
The purpose of this paper is to treat a nonlinear transmission-type problem for a generalized version of the Darcy-Forchheimer-Brinkman system and the classical Brinkman system in complementary Lipschitz domains in R³. First of all, we define the required spaces in which we seek our solution. Next, we describe the generalized Brinkman and the generalized Darcy-Forchheimer- Brinkman systems. Further, we give important lemmas that allow us to intro- duce the trace and conormal derivative operators that appear in the formulation of our transmission problem. We invoke a result regarding the well-posedness of the (linear) transmission problem for the generalized and classical Brinkman systems in complementary Lipschitz domains in R³. The above mentioned well- posedness result in the linear case combined with Banach’s fixed point theorem will allow us to establish the main result of the paper, the well-posedness of the transmission problem for the Brinkman system and the nonlinear generalized Darcy-Forchheimer-Brinkman system in Lipschitz domains in R³.
Mathematics Subject Classification (2010): 35J25, 35Q35, 46E35.
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