$\mathbb{L}^p$-solutions of deterministic and stochastic convective Brinkman-Forchheimer equations

TL;DR

This paper proves existence and uniqueness of solutions for both deterministic and stochastic convective Brinkman-Forchheimer equations, including cases with Lévy and fractional Brownian noise, using fixed point and contraction mapping methods.

Contribution

It introduces new existence and uniqueness results for local solutions of both deterministic and stochastic CBF equations with various noise types.

Findings

Established local mild solution existence for deterministic CBF equations.

Proved pathwise mild solution existence for stochastic CBF equations with Lévy noise.

Extended results to stochastic CBF equations with fractional Brownian noise.

Abstract

In the first part of this work, we establish the existence and uniqueness of a local mild solution to the deterministic convective Brinkman-Forchheimer (CBF) equations defined on the whole space, by using properties of the heat semigroup and fixed point arguments based on an iterative technique. The second part is devoted for establishing the existence and uniqueness of a pathwise mild solution upto a random time to the stochastic CBF equations perturbed by L\'evy noise by exploiting the contraction mapping principle. We also discuss the local solvability of the stochastic CBF equations subjected to fractional Brownian noise.