Finite-time blowup of solutions (u(x, t), b(x, t)) to a generalized system of equations with applications to ideal Magnetohydrodynamics (MHD) and one-dimensional fluid convection and stretching, among other areas, is investigated. The system is parameter-dependent, our spatial domain is the unit interval or the circle, and the initial data <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(u_0(x),b_0(x))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>b</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is assumed to be smooth. Among other results, we derive precise blowup criteria for specific values of the parameters by tracking the evolution of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$u_x$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>x</mi> </msub> </math></EquationSource> </InlineEquation> along Lagrangian trajectories that originate at a point <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$x_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>x</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> at which <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$b_0(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>b</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$b_0'(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>b</mi> <mn>0</mn> <mo>′</mo> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> vanish. We employ concavity arguments, energy estimates, and ODE comparison methods. We also show that for some values of the parameters, a non-vanishing <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$b_0'(x_0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>b</mi> <mn>0</mn> <mo>′</mo> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> suppresses finite-time blowup.

On a Generalized System with Applications to Ideal Magnetohydrodynamics

期刊论文
发表日期：

2026年1月4日

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Alejandro Sarria

摘要

Finite-time blowup of solutions (u(x, t), b(x, t)) to a generalized system of equations with applications to ideal Magnetohydrodynamics (MHD) and one-dimensional fluid convection and stretching, among other areas, is investigated. The system is parameter-dependent, our spatial domain is the unit interval or the circle, and the initial data $(u_0(x),b_0(x))$ $(u_{0} (x), b_{0} (x))$ is assumed to be smooth. Among other results, we derive precise blowup criteria for specific values of the parameters by tracking the evolution of $u_x$ $u_{x}$ along Lagrangian trajectories that originate at a point $x_0$ $x_{0}$ at which $b_0(x)$ $b_{0} (x)$ and $b_0'(x)$ $b_{0}^{'} (x)$ vanish. We employ concavity arguments, energy estimates, and ODE comparison methods. We also show that for some values of the parameters, a non-vanishing $b_0'(x_0)$ $b_{0}^{'} (x_{0})$ suppresses finite-time blowup.