<p>We study the ideal magnetohydrodynamics (MHD) equations in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {R}^{3}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> under the asymptotic condition <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(u,B)\rightarrow (u_{\infty }, B_{\infty }) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mi>∞</mi> </msub> <mo>,</mo> <msub> <mi>B</mi> <mi>∞</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$|x|\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> for constant vectors <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$u_{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$B_{\infty }\in \mathbb {R}^{3}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mi>∞</mi> </msub> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. We prove that the explicit traveling wave solution given by <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$u=u_{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>=</mo> <msub> <mi>u</mi> <mi>∞</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$B=U_{C}(x-u_{\infty }t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo>=</mo> <msub> <mi>U</mi> <mi>C</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <msub> <mi>u</mi> <mi>∞</mi> </msub> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$U_{C}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>U</mi> <mi>C</mi> </msub> </math></EquationSource> </InlineEquation> is an axisymmetric nonlinear force-free field with swirl, is orbitally stable in weak ideal limits of axisymmetric Leray–Hopf solutions to the viscous and resistive MHD equations.</p>

Stability of Force-Free Fields in Weak Ideal Limits of Leray–Hopf Solutions II: Nonlinear Force-Free Fields

期刊论文
发表日期：

2026年4月4日

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Ken Abe

摘要

We study the ideal magnetohydrodynamics (MHD) equations in $\mathbb {R}^{3}$ $R^{3}$ under the asymptotic condition $(u,B)\rightarrow (u_{\infty }, B_{\infty }) $ $(u, B) \to (u_{\infty}, B_{\infty})$ as $|x|\rightarrow \infty $ $| x | \to \infty$ for constant vectors $u_{\infty }$ $u_{\infty}$ , $B_{\infty }\in \mathbb {R}^{3}$ $B_{\infty} \in R^{3}$ . We prove that the explicit traveling wave solution given by $u=u_{\infty }$ $u = u_{\infty}$ and $B=U_{C}(x-u_{\infty }t)$ $B = U_{C} (x - u_{\infty} t)$ , where $U_{C}$ $U_{C}$ is an axisymmetric nonlinear force-free field with swirl, is orbitally stable in weak ideal limits of axisymmetric Leray–Hopf solutions to the viscous and resistive MHD equations.