arXiv subjects

Ryan L. Acosta Babb

Publications and source records attributed to Ryan L. Acosta Babb.

4 recordsLinked to original sources

arXiv2025

The well-posedness and convergence of higher-order Hartree equations in critical Sobolev spaces on $\mathbb{T}^3$

In this article, we consider Hartree equations generalised to $2p+1$ order nonlinearities. These equations arise in the study of the mean-field limits of Bose gases with $p$-body interactions. We study their well-posedness properties in $H^{s_c}(\mathbb{T}^3)$, where $\mathbb{T}^3$ is the three dimensional torus and $s_c = 3/2 - 1/p$ is the scaling-critical regularity. The convergence of solutions of the Hartree equation to solutions of the nonlinear Schrödinger equation is proved. We also consider the case of mixed nonlinearities, proving local well-posedness in $s_c$ by considering the problem as a perturbation of the higher-order Hartree equation. In the particular case of the (defocusing) quintic-cubic Hartree equation, we also prove global well-posedness for all initial conditions in $H^1(\mathbb{T}^3)$. This is done by viewing it as a perturbation of the local quintic NLS.

math.AP↗

arXiv2024

A partial converse to the Riemann--Lebesgue lemma for Bessel--Fourier series of order zero

It is known that the Bessel--Fourier coefficients $f_m$ of a function $f$ such that $\sqrt{x}f(x)$ is integrable over $[0,1]$ satisfy $f_m/\sqrt{m}\to 0$. We show a partial converse, namely that for $0\leq α<1/2$ and any non-negative $a_m\to 0$, there is a function $f$ such that $x^{α+1}f(x)$ is integrable and its Bessel--Fourier coefficients $f_m$ satisfy $m^{-α}f_m\geq a_m$ and $m^{-α}f_m\to 0$. We conjecture that the same should be true when $α=\frac{1}{2}$, and discuss some consequences of this conjecture.

math.CA↗

arXiv2024

$L^p$ continuity of eigenprojections for 2-d Dirichlet Laplacians under perturbations of the domain

We generalise results by Lamberti and Lanza de Cristoforis (2005) concerning the continuity of projections onto eigenspaces of self-adjoint differential operators with compact inverses as the (spatial) domain of the functions is perturbed in $\mathbb{R}^2$. Our main case of interest is the Dirichlet Laplacian. We extend these results from bounds from $H_0^1$ to $H_0^1$ to bounds from $L^p$ to $L^p$, under the assumption that $(-Δ^{-1}-z)^{-1}$ is $L^p$ bounded when $z$ lies outside of the spectrum of $-Δ^{-1}$. We show that this assumption is met if the initial domain is a square or a rectangle.

math.AP↗

arXiv2022

Remarks on the $L^p$ convergence of Bessel--Fourier series on the disc

The $L^p$ convergence of eigenfunction expansions for the Laplacian on planar domains is largely unknown for $p\neq 2$. After discussing the classical Fourier series on the 2-torus, we move onto the disc, whose eigenfunctions are explicitly computable as products of trigonometric and Bessel functions. We summarise a result of Balodis and Córdoba (1999) regarding the $L^p$ convergence of the Bessel--Fourier series in the mixed norm space $L^p_{\mathrm{rad}}(L^2_{\mathrm{ang}})$ on the disk for the range $\tfrac{4}{3}<p<4$. We then describe how to modify their result to obtain $L^p(\mathbb{D}, r\,\mathrm{d}r\,\mathrm{d}t)$ norm convergence in the subspace $L^p_{\mathrm{rad}}(L^q_{\mathrm{ang}})$($\tfrac{1}{p}+\tfrac{1}{q}=1$) for the restricted range ${2\leq p < 4}$.

math.CA↗