# Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {L}}^p$$\end{document}-solutions of deterministic and stochastic convective Brinkman–Forchheimer equations

Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym}... 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {L}}^p$$\end{document}-solutions of deterministic and stochastic convective Brinkman–Forchheimer equations

, Volume 11 (4) – Dec 1, 2021
33 pages      /lp/springer-journals/lp-documentclass-12pt-minimal-usepackage-amsmath-usepackage-wasysym-Rc5T9Z1BuM
Publisher
Springer Journals
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-021-00595-0
Publisher site
See Article on Publisher Site

### 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.

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Dec 1, 2021

Keywords: Convective Brinkman–Forchheimer equations; Lévy noise; Fractional Brownian motion; Mild solution; Primary 76D06; Secondary 35Q30; 76D03; 47D03