Saturating spaces for the squared-gradient nonlinearity on compact Riemannian manifolds via Nash embeddings

Buffe Rémi; Duca Alessandro; Parada Hugo

Home/Archive/arXiv

arXiv · 2610.29952

Saturating spaces for the squared-gradient nonlinearity on compact Riemannian manifolds via Nash embeddings

Buffe Rémi·Duca Alessandro·Parada Hugo

View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the saturation property on smooth compact Riemannian manifolds without boundary. Starting from a finite-dimensional space of smooth functions, one considers successive iterations of the nonlinear map $f\mapsto|\nabla_g f|_g^2$ and convex combinations. We extend to the general setting of compact manifolds existing results on particular geometries, namely the torus, spheres and Lie groups. Our approach relies on the existence of an isometric embedding of the manifold into $\mathbb{R}^n$ due to Nash. As an application, we obtain global approximate controllability in arbitrarily small time for the bilinear Schrödinger equation on any connected compact Riemannian manifold, with finitely many potentials.

Explore related subjects

math.AP math.OC

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Buffe Rémi, Duca Alessandro, Parada Hugo. 2026-10-21. Saturating spaces for the squared-gradient nonlinearity on compact Riemannian manifolds via Nash embeddings. https://arxiv.org/abs/2610.29952

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↗