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Large time behavior of solutions to the 2D damped wave-type magnetohydrodynamic equations

Yaowei Xie, Huan Yu

Abstract

In this paper, we are concerned with the 2D damped wave-type magnetohydrodynamic system (abbreviated as MHD-wave system). The purpose of this paper is to study the large time behavior of solutions to the MHD-wave system, espesically to investigate the influence of the bad term $γ\partial_{tt}b$ on the large time behavior. Rates of decay are obtained for both the solutions and higher derivatives in different Sobolev spaces with explicit rates of $γ$, which shows that the decay rates closely align with that of the MHD system under the same norm, for any fixed $γ>0$. In this sense, these decay rates are optimal.

Large time behavior of solutions to the 2D damped wave-type magnetohydrodynamic equations

Abstract

In this paper, we are concerned with the 2D damped wave-type magnetohydrodynamic system (abbreviated as MHD-wave system). The purpose of this paper is to study the large time behavior of solutions to the MHD-wave system, espesically to investigate the influence of the bad term on the large time behavior. Rates of decay are obtained for both the solutions and higher derivatives in different Sobolev spaces with explicit rates of , which shows that the decay rates closely align with that of the MHD system under the same norm, for any fixed . In this sense, these decay rates are optimal.

Paper Structure

This paper contains 9 sections, 11 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. Challenges and methodology
  5. Preliminaries
  6. Global well-posedness and $L^q~(q\geq 2)$ decay
  7. Global existence and uniqueness
  8. The decay estimates of $(u,b)$ in $L^q\times L^q$, for $q\geq 2$
  9. The Decay of $(u, b)$ in $\dot{H}^\beta\times \dot{H}^\varrho$, for $0\leq \beta\leq m, 0\leq\varrho<m+1$

Key Result

Theorem 1.1

Let $m>0$ and $\gamma>0$. Assume that the initial data $(u_0,b_0,a_0)\in X^m(\mathbb{R}^2)$ satisfying $\nabla\cdot u_0=\nabla \cdot b_0=\nabla\cdot a_0=0.$ If there exists a suitable constant $\varepsilon> 0$, such that then there exists a unique global solution $(u,b)$ to system mhdwave satisfying, for any $t>0$, Furthermore, the global solution obeys the following $L^q$ decay estimates: for

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2: gag-nirenberg
  • Lemma 2.3: fourier-2009-nonlinear-ponce-hausdroffyoung
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • ...and 8 more