arXiv subjects

Matteo Giardi

Publications and source records attributed to Matteo Giardi.

2 recordsLinked to original sources

arXiv2026

Ideal MHD below the classical well-posedness threshold

We establish local existence and uniqueness of solutions for the ideal incompressible magnetohydrodynamics system posed on $[0,T]\times\mathbb{R}^n$, $n\ge2$, with a nonzero constant initial magnetic field $\mathbf{B}_0$ and arbitrary divergence-free velocity data $v_0\in H^s$, in the range $(n+1)/2<s\le n/2+1$. The proof uses a Lagrangian wave--Hodge reformulation and exploits an Alfv\'en null--structure hidden in the pressure forcing. In particular, the constructed Eulerian solutions are induced by a bi-Lipschitz measure-preserving flow map. Zhang first identified this null-structure in \cite{Zhang2024}; the present work provides a self-contained bridge from that Lagrangian theory to the Eulerian Cauchy problem.

math.AP↗

arXiv2026

$C^{1/5^{-}}$ Convex Integration Solutions of Ideal MHD

For any $0\leq \gamma < 1/5$, we construct weak solutions $(v, B, p )$ of the Ideal MHD Equations which do not conserve the total kinetic energy, the cross-helicity and lie in $C^\gamma(\mathbb{T}^3\times\mathbb{R})$. In the spirit of Arnold's formulation of ideal hydrodynamics, a solution is thought of as a path of volume-preserving diffeomorphisms; the proof is then based on the interplay between classical convex integration techniques and geometric constructions at the level of the Lie algebra of this Lie group. Our work substantially extends the recent work of and building on the recent work of Enciso, Pe\~nafiel-Tom\'as and Peralta-Salas.

math.AP↗