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

TL;DR

This paper establishes the well-posedness of a mixed boundary value problem for a semilinear PDE system in Besov spaces on Lipschitz domains, extending linear results to nonlinear cases using potential theory and fixed point methods.

Contribution

It introduces new well-posedness results for the semilinear Darcy-Forchheimer-Brinkman system in Besov spaces on Lipschitz domains, extending linear theory with nonlinear analysis techniques.

Findings

Proved equivalence of trace and conormal derivative definitions.

Established invertibility of integral operators for the linear system.

Extended well-posedness from linear to nonlinear systems using fixed point theorem.

Abstract

The purpose of this paper is to study the mixed Dirichlet-Neumann boundary value 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 semilinear term $|{\bf u}|{\bf u}$ to the linear Brinkman equation. {First, we provide some results about} equivalence between the Gagliardo and non-tangential traces, as well as between the weak canonical conormal derivatives and the non-tangential conormal derivatives. Various mapping and invertibility properties of some integral operators of potential theory for the linear Brinkman system, and well posedness results for the Dirichlet and Neumann problems in $L_p$-based Besov spaces on bounded Lipschitz domains in ${\mathbb R}^n$ ($n\geq 3$) are also presented. Then, employing integral potential operators, we show the well-posedness in $L_2$-based Sobolev spaces for the mixed problem of Dirichlet-Neumann type for the linear Brinkman system on a bounded Lipschitz domain in ${\mathbb R}^n$ $(n\geq 3)$. Further, by using some stability results of Fredholm and invertibility properties and exploring invertibility of the associated Neumann-to-Dirichlet operator, we extend the well-posedness property to some $L_p$-based Sobolev spaces. Next we use the well-posedness result in the linear case combined with a fixed point theorem in order to show the existence and uniqueness for a mixed boundary value problem of Dirichlet and Neumann type for the semilinear Darcy-Forchheimer-Brinkman system in $L_p$-based Besov spaces, with $p\in (2-\varepsilon ,2+\varepsilon)$ and some parameter $\varepsilon >0$.