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Bruno Vergara

Publications and source records attributed to Bruno Vergara.

6 recordsLinked to original sources
arXiv2024

Pointwise decay for radial solutions of the Schrödinger equation with a repulsive Coulomb potential

We study the long-time behavior of solutions to the Schrödinger equation with a repulsive Coulomb potential on $\mathbb{R}^3$ for spherically symmetric initial data. Our approach involves computing the distorted Fourier transform of the action of the associated Hamiltonian $H=-Δ+\frac{q}{|x|}$ on radial data $f$, which allows us to explicitly write the evolution $e^{itH}f$. A comprehensive analysis of the kernel is then used to establish that, for large times, $\|e^{i t H}f\|_{L^{\infty}} \leq C t^{-\frac{3}{2}}\|f\|_{L^1}$. Our analysis of the distorted Fourier transform is expected to have applications to other long-range repulsive problems.

math.AP↗
arXiv2023

Approximate boundary controllability for parabolic equations with inverse square infinite potential wells

We consider heat operators on a bounded domain $\Omega \subseteq \mathbb{R}^n$, with a critically singular potential diverging as the inverse square of the distance to $\partial \Omega$. While null boundary controllability for such operators was recently proved in all dimensions in arXiv:2112.04457, it crucially assumed (i) $\Omega$ was convex, (ii) the control must be prescribed along all of $\partial \Omega$, and (iii) the strength of the singular potential must be restricted to a particular subrange. In this article, we prove instead a definitive approximate boundary control result for these operators, in that we (i) do not assume convexity of $\Omega$, (ii) allow for the control to be localized near any $x_0 \in \partial \Omega$, and (iii) treat the full range of strength parameters for the singular potential. Morever, we lower the regularity required for $\partial \Omega$ and the lower-order coefficients. The key novelty is a local Carleman estimate near $x_0$, with a carefully chosen weight that takes into account both the appropriate boundary conditions and the local geometry of $\partial \Omega$.

math.AP↗
arXiv2021

Controllability of parabolic equations with inverse square infinite potential wells via global Carleman estimates

We consider heat operators on a convex domain $\Omega$, with a critically singular potential that diverges as the inverse square of the distance to the boundary of $\Omega$. We establish a general boundary controllability result for such operators in all dimensions, in particular providing the first such result in more than one spatial dimension. The key step in the proof is a novel global Carleman estimate that captures both the appropriate boundary conditions and the $H^1$-energy for this problem. The estimate is derived by combining two intermediate Carleman inequalities with distinct and carefully constructed weights involving non-smooth powers of the boundary distance.

math.AP↗
arXiv2020

Carleman estimates with sharp weights and boundary observability for wave operators with critically singular potentials

We establish a new family of Carleman inequalities for wave operators on cylindrical spacetime domains containing a potential that is critically singular, diverging as an inverse square on all the boundary of the domain. These estimates are sharp in the sense that they capture both the natural boundary conditions and the natural $H^1$-energy. The proof is based around three key ingredients: the choice of a novel Carleman weight with rather singular derivatives on the boundary, a generalization of the classical Morawetz inequality that allows for inverse-square singularities, and the systematic use of derivative operations adapted to the potential. As an application of these estimates, we prove a boundary observability property for the associated wave equations.

math.AP↗
arXiv2018

Convexity of Whitham's highest cusped wave

We prove the existence of a periodic traveling wave of extreme form of the Whitham equation that has a convex profile between consecutive stagnation points, at which it is known to feature a cusp of exactly $C^{1/2}$ regularity. The convexity of Whitham's highest cusped wave had been conjectured by Ehrnström and Wahlén.

math.AP↗