Strong solution of the three-dimensional $(3D)$ incompressible magneto-hydrodynamic $(MHD)$ equationss with a modified damping

TL;DR

This paper proves the global existence and uniqueness of solutions for a modified 3D incompressible MHD equations with small initial data, incorporating a nonlinear damping term relevant to porous media flows.

Contribution

It establishes the first rigorous results on global solutions for the modified 3D MHD equations with nonlinear damping under specific parameter conditions.

Findings

Global existence of solutions for $eta > 3$ and $eta=3$ with $ ext{0}< ext{alpha}<rac{1}{2}$

Uniqueness of solutions under small initial data conditions

Applicability to porous media flows with nonlinear damping

Abstract

This study delves into a comprehensive examination of the three-dimensional $(3D)$ incompressible magneto-hydrodynamic $(MHD)$ equations in $H^{1}(\R^{3})$. The modification involves incorporating a power term in the nonlinear convection component, a particularly relevant adjustment in porous media scenarios, especially when the fluid adheres to the Darcy-Forchheimer law instead of the conventional Darcy law. Our main contributions include establishing global existence over time and demonstrating the uniqueness of solutions. It is important to note that these achievements are obtained with smallness conditions on the initial data, but under the condition that $\beta >3$ and $\alpha>0$. However, when $\beta=3$, the problem is limited to the case $0<\alpha<\frac{1}{2}$ as the above inequality is unsolvable for these values of $\alpha$ using our method. To support our statement, we will add a "slight disturbance" of the function f of the type $f(z)=log(e+z^{2})$ or $\log(\log(e^{e}+z^{2}))$ or even $\log(\log(\log((e^{e})^{e}+z^{2})))$.