On the mixed problem for the semilinear Darcy-Forchheimer-Brinkman PDE system in Besov spaces on creased Lipschitz domains

摘要

The purpose of this paper is to study the mixed Dirichlet-Neumann boundaryvalue problem for the semilinear Darcy-Forchheimer-Brinkman system in$L_p$-based Besov spaces on a bounded Lipschitz domain in ${\mathbb R}^3$, with$p$ in a neighborhood of $2$. This system is obtained by adding the semilinearterm $|{\bf u}|{\bf u}$ to the linear Brinkman equation. {First, we providesome results about} equivalence between the Gagliardo and non-tangentialtraces, as well as between the weak canonical conormal derivatives and thenon-tangential conormal derivatives. Various mapping and invertibilityproperties of some integral operators of potential theory for the linearBrinkman system, and well posedness results for the Dirichlet and Neumannproblems in $L_p$-based Besov spaces on bounded Lipschitz domains in ${\mathbbR}^n$ ($n\geq 3$) are also presented. Then, employing integral potentialoperators, we show the well-posedness in $L_2$-based Sobolev spaces for themixed problem of Dirichlet-Neumann type for the linear Brinkman system on abounded Lipschitz domain in ${\mathbb R}^n$ $(n\geq 3)$. Further, by using somestability results of Fredholm and invertibility properties and exploringinvertibility of the associated Neumann-to-Dirichlet operator, we extend thewell-posedness property to some $L_p$-based Sobolev spaces. Next we use thewell-posedness result in the linear case combined with a fixed point theorem inorder to show the existence and uniqueness for a mixed boundary value problemof Dirichlet and Neumann type for the semilinear Darcy-Forchheimer-Brinkmansystem in $L_p$-based Besov spaces, with $p\in (2-\varepsilon ,2+\varepsilon)$and some parameter $\varepsilon >0$.