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

Abstract

In the first part of this work, we establish the existence and uniqueness of a local mild solution to 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. Moreover, we prove that the solution exists globally. 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 Lévy noise by exploiting the contraction mapping principle. Then by using stopping time arguments, we show that the pathwise mild solution exists globally. We also discuss the local and global solvability of the stochastic CBF equations forced by fractional Brownian noise.