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Asymptotic behavior of solution of the non-resistive 2D MHD equations on the half space

Jiakun Jin, Yoshiyuki Kagei, Xiaoxia Ren, Lei Wang, Cuili Zhai

Abstract

In this paper, we obtain the global well-posedness and the asymptotic behavior of solution of non-resistive 2D MHD problem on the half space. We overcome the difficulty of zero spectrum gap by building the relationship between half space and the whole space, and get the resolvent estimate for the weak diffusion system. We use the two-tier energy method that couples the boundedness of high-order $(H^3)$ energy to the decay of low-order energy, the latter of which is necessary to control the growth of the highest energy.

Asymptotic behavior of solution of the non-resistive 2D MHD equations on the half space

Abstract

In this paper, we obtain the global well-posedness and the asymptotic behavior of solution of non-resistive 2D MHD problem on the half space. We overcome the difficulty of zero spectrum gap by building the relationship between half space and the whole space, and get the resolvent estimate for the weak diffusion system. We use the two-tier energy method that couples the boundedness of high-order energy to the decay of low-order energy, the latter of which is necessary to control the growth of the highest energy.
Paper Structure (9 sections, 7 theorems, 317 equations)

This paper contains 9 sections, 7 theorems, 317 equations.

Table of Contents

  1. Introduction
  2. The resolvent estimate of linearized problem
  3. Solution formula of the linearized problem
  4. The contour integration of the linear part
  5. The resolvent estimate of the linear part
  6. The decay rate of nonlinear system
  7. Local well-posedness
  8. High order energy estimate
  9. acknowledgement

Key Result

Theorem 1.1

Assume that the initial data $(u_0, b_0)$ satisfies $(u_0, b_0)\in W^{2, 1}(\Omega)\cap H^3(\Omega)$, $u_0\in H_0^1(\Omega)$, $b_{1, 0}=\partial_2 b_{1, 0}=0$ on $\partial\Omega$, $\mathcal{P}(\Delta u_0 -u_0 \cdot \nabla u_0 +b_0 \cdot \nabla b_0) \in H_0^1(\Omega)$ (Here $\mathcal{P}$ is the Helmh with $\varepsilon$ is a small positive constants. Then the MHD system (eq:MHDT) has a unique global

Theorems & Definitions (15)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 5 more