Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.26547

Global strong solutions to the aligned 2.5D non-resistive full MHD system with high temperature and a strong magnetic field

Yachun Li·Zixuan Qu·Zhaoyang Shang
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the Cauchy problem for an aligned vertical-field reduction of the full compressible non-resistive MHD equations in two space variables. The shear and bulk viscosities are proportional to $θ^α$, and the heat conductivity is proportional to $θ^β$. For $2<α<β<α+2$, we prove global existence and uniqueness of strong solutions when the background temperature and the comparable magnetic background are sufficiently large, while the positive density background is fixed. The velocity and temperature perturbations belong initially to $H^2$, and the transported magnetic perturbation belongs to $H^1\cap D^{1,4}$; their absolute norms may be bounded by an arbitrary fixed constant. Alignment is essential and implies a small density perturbation relative to the magnetic background. Our proof combines relative energy estimates, uniform in the background temperature and magnetic field, with coupled estimates for the material derivatives of the velocity and temperature. These bounds control the source terms in the damped magnetic-gradient equation, allowing us to close the global a priori estimates without a separate large-time decay assumption. We construct the global strong solution by smooth approximation and Sobolev energy estimates for the approximating solutions, and prove uniqueness by an energy estimate for the difference of two solutions.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yachun Li, Zixuan Qu, Zhaoyang Shang. 2026-10-19. Global strong solutions to the aligned 2.5D non-resistive full MHD system with high temperature and a strong magnetic field. https://arxiv.org/abs/2610.26547

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Convergence of solutions to the $p$-Laplace evolution equation as $p$ goes to 1

We prove that the set of solutions to the parabolic singular $p$-Laplace equation with Dirichlet boundary conditions on a bounded Lipschitz domain $Ω$ for all space dimensions is continuous in the parameter $p\in [1,+\infty)$ and the initial data. The highly singular limit case $p=1$ is included. In particular, we show that the solutions $u_p$ converge strongly in $L^2(Ω)$, uniformly in time, to the solution $u_1$ of the parabolic $1$-Laplace equation as $p\to 1$.

math.AP↗
arXiv2026

Complex Monge-Ampère equation for positive $(p,p)$-forms on compact Kähler manifolds

A complex Monge-Ampère equation for differential $(p,p)$-forms is introduced on compact Kähler manifolds. For any $1 \leq p < n$, we show the existence of smooth solutions unique up to adding constants. For $p=1$, this corresponds to the Calabi-Yau theorem proved by S. T. Yau, and for $p=n-1$, this gives the Monge-Ampère equation for $(n-1)$ plurisubharmonic functions studied by Tosatti-Weinkove. For other $p$ values, this defines a non-linear PDE that falls outside of the general framework of Caffarelli-Nirenberg-Spruck. Further, we define a geometric flow for higher-order forms that extends the Kähler-Ricci flow naturally to $(p,p)$-forms. As a consequence of our main theorem, we show that this flow exists in a maximal time interval and can be shown to converge under some assumptions. A modified flow is introduced and the convergence of the associated normalized flow is shown.

math.AP↗
arXiv2026

Existence and Smoothing for a Nondivergence-Form Degenerate Diffusion from Plasma-Wave Theory

We prove existence and positive-time smoothing of weak solutions to the degenerate parabolic Cauchy--Dirichlet problem $\partial_t u = ρ_λ(x)u\,\partial_x^2 u + ρ_λ(x)g(x)u$ on the half-line, where $ρ_λ$ vanishes at the boundary and grows at infinity. This scalar problem arises from a formal reduction of the one-dimensional quasilinear system for plasma waves, which couples an electron distribution to a wave-energy spectrum: we develop a solution theory for the scalar problem that lifts to this system. For the scalar problem, we construct nonnegative weak solutions under weighted integrability assumptions on the initial data and boundedness of the reaction coefficient $ρ_λg$, admitting unbounded and discontinuous data pertinent to the physical model. We establish a one-sided bound on $\partial_t u$ which yields positive-time smoothing: solutions become locally Hölder continuous in both space and time and locally Lipschitz continuous in space. Explicit examples show that this spatial regularity is sharp. Under natural assumptions on the initial data, our scalar solutions lift to weak solutions of the particle--wave system that preserve nonnegativity, conserve particle mass, smooth initial particle measures to locally bounded densities at positive times, and exhibit quantitative decay of the wave spectrum at large wavenumber.

math.AP↗