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

Maroua Ltifi

TL;DR

This work analyzes the 3D incompressible MHD equations with damping to model porous-media effects, introducing two damping schemes: a power-law term $\alpha|u|^{\beta-1}u$ with $\beta>3$ and a generalized damping $\alpha f(|u|^{2})|u|^{2}u$ where $f$ belongs to a class that includes logarithmic forms. Under small initial data in $H^{1}(\mathbb{R}^{3})$, the authors prove global existence and uniqueness of strong solutions for the $MHD_D$ system (and discuss the critical $\beta=3$ regime, which they treat via the auxiliary $MHD_f$ formulation). The analysis combines Friedrichs regularization, sharp energy estimates in $L^{2}$ and $H^{1}$, and interpolation/semigroup tools to control nonlinear terms, yielding uniform-in-time bounds and an exponential-in-time growth bound for the $H^{1}$-norm. These results advance understanding of damped 3D MHD models in porous-media contexts and provide a framework for handling logarithmic or near-logarithmic damping nonlinearities.

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 $β>3$ and $α>0$. However, when $β=3$, the problem is limited to the case $0<α<\frac{1}{2}$ as the above inequality is unsolvable for these values of $α$ 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 $\log(\log(\log((e^{e})^{e}+z^{2})))$.

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

TL;DR

This work analyzes the 3D incompressible MHD equations with damping to model porous-media effects, introducing two damping schemes: a power-law term with and a generalized damping where belongs to a class that includes logarithmic forms. Under small initial data in , the authors prove global existence and uniqueness of strong solutions for the system (and discuss the critical regime, which they treat via the auxiliary formulation). The analysis combines Friedrichs regularization, sharp energy estimates in and , and interpolation/semigroup tools to control nonlinear terms, yielding uniform-in-time bounds and an exponential-in-time growth bound for the -norm. These results advance understanding of damped 3D MHD models in porous-media contexts and provide a framework for handling logarithmic or near-logarithmic damping nonlinearities.

Abstract

This study delves into a comprehensive examination of the three-dimensional incompressible magneto-hydrodynamic equations in . 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 and . However, when , the problem is limited to the case as the above inequality is unsolvable for these values of using our method. To support our statement, we will add a "slight disturbance" of the function f of the type or or even .
Paper Structure (7 sections, 7 theorems, 69 equations)

This paper contains 7 sections, 7 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminary results
  3. Notations
  4. Preliminary results
  5. Existence and uniqueness of strong solution
  6. Proof of Theorem $\ref{['the1']}$
  7. Proof of Theorem $\ref{['the2']}$

Key Result

Theorem 1.1

For $\alpha>0$ and $\beta>3$, consider divergence-free vector fields $u^{0}$ and $b^{0}$$\in H^{1}(\mathbb{R}^{3})$ such that $\|(u^{0},b^{0})\|_{H^{1}}<<\epsilon_{0},$ with $\epsilon_{0}$ is small enough. There exists a global solution $w=(u,b)$ of the magnetohydrodynamics equation $(MHD_{D})$ sati and where, $c_{\alpha,\beta}=\frac{1}{2}\frac{\beta-3}{\beta-1}.(\frac{\alpha(\beta-1)}{2})^{-\fra

Theorems & Definitions (10)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Remark 3.1