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Forward self-similar solutions to the MHD equations: existence and pointwise estimates

Yifan Yang

TL;DR

The paper addresses forward self-similar solutions to the 3D incompressible MHD equations in $\mathbb{R}^3$ with initial data homogeneous of degree $-1$. It reduces the self-similar problem to a time-independent perturbed Leray system for profile fields and employs a blow-up argument together with the Leray–Schauder fixed-point theorem on invading domains to construct a global weak solution $(V,G)$, with pressure $P$ obtained from De Rham theory. Through energy methods and kernel-based perturbation analysis using the Oseen/heat kernels, it derives decay and pointwise estimates for the perturbation $(V,G)$ and for the velocity/magnetic fields $(u,b)$, including optimal rates under higher regularity of the initial data. The results yield a smooth, forward self-similar solution $(u,b)$ that converges to the heat-flow initial data in a weak sense and provides explicit decay profiles, contributing a rigorous large-data framework for self-similar MHD behavior. Uniqueness remains open, but the methodology enables potential extensions to half-space problems and related fluid–magnetic systems, with the kernel-based approach offering sharp pointwise controls.

Abstract

In this paper, we study the forward self-similar solutions to the three-dimensional Magnetohydrodynamic equations (MHD equations) in the whole space. By employing the Leray-Schauder theorem and blow-up argument, we construct a global-time forward self-similar solutions, which is smooth in $\R^{3}\times(0,\infty)$. Furthermore, by investigating the regularity of the weak solutions to the corresponding Leray system in the weighted Sobolev space, we can derive the pointwise estimate for the forward self-similar solution.

Forward self-similar solutions to the MHD equations: existence and pointwise estimates

TL;DR

The paper addresses forward self-similar solutions to the 3D incompressible MHD equations in with initial data homogeneous of degree . It reduces the self-similar problem to a time-independent perturbed Leray system for profile fields and employs a blow-up argument together with the Leray–Schauder fixed-point theorem on invading domains to construct a global weak solution , with pressure obtained from De Rham theory. Through energy methods and kernel-based perturbation analysis using the Oseen/heat kernels, it derives decay and pointwise estimates for the perturbation and for the velocity/magnetic fields , including optimal rates under higher regularity of the initial data. The results yield a smooth, forward self-similar solution that converges to the heat-flow initial data in a weak sense and provides explicit decay profiles, contributing a rigorous large-data framework for self-similar MHD behavior. Uniqueness remains open, but the methodology enables potential extensions to half-space problems and related fluid–magnetic systems, with the kernel-based approach offering sharp pointwise controls.

Abstract

In this paper, we study the forward self-similar solutions to the three-dimensional Magnetohydrodynamic equations (MHD equations) in the whole space. By employing the Leray-Schauder theorem and blow-up argument, we construct a global-time forward self-similar solutions, which is smooth in . Furthermore, by investigating the regularity of the weak solutions to the corresponding Leray system in the weighted Sobolev space, we can derive the pointwise estimate for the forward self-similar solution.
Paper Structure (9 sections, 22 theorems, 357 equations)

This paper contains 9 sections, 22 theorems, 357 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Lorentz space
  4. The fundamental solution of the non-stationary Stokes equations
  5. Some useful lemmas
  6. Analysis of the perturbed Leray system
  7. Existence of weak solutions to the PLS in $\mathbb{R}^{3}$
  8. The decay estimate of the weak solution $(V,G)$
  9. Conclusion

Key Result

Theorem 1.1

Let $(u_{0},b_{0})\in \mathbf{L}^{\infty}_{loc}(\mathbb{R}^{3}\setminus\{0\})$ be homogeneous of degree $-1$, with $\mathrm{div} u_{0}=\mathrm{div} b_{0}=0$. Then, there exists a constant $C=C(u_{0},b_{0})>0$, such that

Theorems & Definitions (44)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.1
  • Lemma 2.1: AF,LG1,LG2,LR
  • Lemma 2.2: LG1
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 34 more