Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2503.12901

On Mañé's critical value for the two-component Hunter-Saxton system and a infnite dimensional magnetic Hopf-Rinow theorem

Levin Maier
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we introduce a nonlinear system of partial differential equations, the magnetic two-component Hunter-Saxton system (M2HS). This system is formulated as a magnetic geodesic equation on an infinite-dimensional Lie group equipped with a right-invariant metric, the $\dot{H}^1$ -metric, which is closely related to the infinite-dimensional Fisher-Rao metric, and the derivative of an infinite-dimensional contact-type form as the magnetic field. We define Mañé's critical value for exact magnetic systems on Hilbert manifolds in full generality and compute it explicitly for the (M2HS). Moreover, we establish an infinite-dimensional Hopf-Rinow theorem for this magnetic system, where Mañé's critical value serves as the threshold beyond which the Hopf-Rinow theorem no longer holds. This geometric framework enables us to thoroughly analyze the blow-up behavior of solutions to the (M2HS). Using this insight, we extend solutions beyond blow-up by introducing and proving the existence of global conservative weak solutions. This extension is facilitated by extending the Madelung transform from an isometry into a magnetomorphism, embedding the magnetic system into a magnetic system on an infinite-dimensional sphere equipped with the derivative of the standard contact form as the magnetic field. Crucially, this setup can always be reduced, via a dynamical reduction theorem, to a totally magnetic three-sphere, providing a deeper understanding of the underlying dynamics.

Explore related subjects

math.APmath.DGmath.DSmath.SG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Levin Maier. 2026-04-19. On Mañé's critical value for the two-component Hunter-Saxton system and a infnite dimensional magnetic Hopf-Rinow theorem. https://arxiv.org/abs/2503.12901

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

KEEP EXPLORING

Related papers

arXiv2026

Renormalized Lambert-W Cascade and Finite-Time Amplification and Blowup for reconstructed b Dynamics on T^3 for the 3D Navier Stokes Equations

This article extracts and consolidates the renormalized Lambert-$W$ branch-point cascade, its distinguished phase reduction, the exact characteristic invariant and finite-time amplification mechanism, and the extended reconstructed $b_i$ equation on $\mathbb T^3$. Repeated historical derivations are removed while the principal proofs and terminal reconstruction estimates are retained. The presentation separates exact finite-depth statements from coupled-depth asymptotics and records the hypotheses required for the extended PDE reconstruction. This paper further supports a recent paper \cite {moschandreou2026exploration} published by the corresponding author which claims that the Navier Stokes equations lose smoothness in finite time from initial smooth data.

math.AP↗
arXiv2026

Unconditional uniqueness for the derivative nonlinear Schrödinger equation by normal form approach

We prove uniqueness of solutions to the Cauchy problem for the derivative nonlinear Schrödinger equation in $L^\infty_tH^{1/2}_x$. Our proof is based on the method of normal form reduction (NFR), which has been employed to obtain the uniqueness in $C_tH^s_x$, $s>1/2$. To overcome logarithmic divergences at the $H^{1/2}$ regularity, we exploit the $B^{0+}_{\infty,1}$ control of solutions provided by a refined Strichartz estimate. Our NFR argument consists of two stages: we first use NFR finitely many times to derive an intermediate equation in which the main cubic nonlinearity is restricted to a certain type of frequency interaction; we then apply the infinite NFR scheme to the intermediate equation. Moreover, we modify the usual NFR argument relying on continuity in time of solutions so that the uniqueness in the class $L^\infty_tH^{1/2}_x$ can be obtained directly.

math.AP↗
arXiv2026

Equivalence between solvability of the Dirichlet and Regularity problem under an $L^1$ Carleson condition on $\partial_t A$

We study an elliptic operator $L:=-\mathrm{div}(A\nabla \cdot)$ on the upper half space. It is known that solvability of the Regularity problem in $\dot{W}^{1,p}$ implies solvability of the adjoint Dirichlet problem in $L^{p'}$. Previously, Shen (2007) established a partial reverse result. In our work, we show that if we assume a mixed \(L^1-L^\infty\) condition on only \(|\partial_t A|\), the full reverse direction holds. As a result, we obtain equivalence between solvability of the Dirichlet problem $(D)^*_{p'}$ and the Regularity problem $(R)_p$ under this condition. As a further consequence, we can extend the class of operators for which the $L^p$ Regularity problem is solvable by operators satisfying the mixed $L^1-L^\infty$ condition. Additionally in the case of the upper half plane, this class includes operators satisfying this this mixed \(L^1-L^\infty\) condition.

math.AP↗