arXiv subjects

Nuno J. Alves

Publications and source records attributed to Nuno J. Alves.

At least 19 recordsLinked to original sources

arXiv2026

Lorentz estimates for multilinear convolution operators

We study O'Neil-type inequalities in Lorentz spaces for two extremal geometric configurations of translation-invariant multilinear convolution operators. For the one-parameter model, we complement the existing weak-kernel theory with estimates for kernels in the Lorentz space $L^{r,s}$, obtaining a stronger target Lorentz space when $s$ is finite. For the full-dimensional model, we prove a Lorentz refinement of Oberlin's multilinear Young inequality in the interior of its admissible range. For nonnegative full-dimensional kernels, we extend Hörmander's necessary condition to the multilinear setting in Lorentz spaces and obtain an additional restriction on the secondary indices in the critical case. We also prove the failure of the full-dimensional weak $L^1$ endpoint, use additive combinatorics to construct one-parameter restricted weak-type counterexamples for $p<1$, and obtain complementary positive results for kernels of special structure.

math.FA↗

arXiv2026

From nonisothermal BGK to Euler Maxwellians via relative entropy

We study the hydrodynamic limit of the nonisothermal BGK model toward Euler Maxwellians. For a prescribed sufficiently smooth Euler solution, we use relative entropy to compare a BGK solution with the corresponding Euler Maxwellian. The result is conditional: for BGK solutions satisfying a uniform sixth velocity-moment bound and uniform $L^\infty$ bounds on their macroscopic velocity, temperature, and inverse temperature, we obtain a stability estimate uniform in time. The key new ingredient is the control of an additional velocity-cubic term in the relative entropy identity. For well-prepared initial data, this yields strong $L^1$ convergence of the BGK solution and its local Maxwellians to the target Euler Maxwellian, together with convergence of the associated macroscopic quantities.

math.AP↗

arXiv2026

An integral characterization of almost equicontinuity

We characterize the pointwise notion of almost equicontinuity for families of real-valued measurable functions on subsets of $\mathbb R^n$ of finite measure. The characterization is given by means of an integral truncated translation condition. We also provide examples showing that the finite measure assumption and the truncation are essential.

math.FA↗

arXiv2026

A truncation criterion for compactness in asymptotic $L_p$ spaces

We prove a compactness criterion for asymptotic $L_p$ spaces over arbitrary measure spaces. Total boundedness is characterized by almost equiboundedness together with total boundedness in $L_p$ of all truncations. As a consequence, we obtain a simpler proof of the Kolmogorov--Riesz compactness theorem for asymptotic $L_p$ spaces on $\mathbb R^n$.

math.FA↗

arXiv2026

A duality approach to the dense graph limit for biological transportation networks

We develop a duality-based formulation of the dense graph limit for a variational model of biological transportation networks, where edge conductivities balance pumping power against metabolic cost. In contrast to the pressure-based approach of our previous work, which required conductivities to be uniformly positive, the present formulation allows general nonnegative conductivity kernels. The kinetic energy is defined through a dual variational principle, which remains meaningful for degenerate integrable kernels and assigns infinite energy when the associated nonlocal Poisson problem is not solvable. Using this formulation, we prove $Γ$-convergence in the sense of Mosco of the semidiscrete network energies to a continuum energy on symmetric nonnegative kernels. The convergence is obtained in the natural $L^γ$ topology dictated by the metabolic term. The $Γ$-$\liminf$ inequality follows directly from the dual formulation, while $Γ$-$\limsup$ recovery sequences are constructed by positive regularization of the conductivity kernels.

math.OC↗

arXiv2026

An $L^1$-theory for $p$-Schrödinger equations with confinement in measure

We consider stationary $p$-Schrödinger equations on the whole space with integrable data and potentials that are confining in measure. We introduce asymptotic energy solutions in an asymptotic $L^p$ framework and establish existence and uniqueness in the degenerate range $p\ge2$. The proof relies on a new Rellich$\unicode{x2013}$Kondrachov-type compactness theorem of independent interest, which provides sufficient conditions for families of Sobolev functions to be precompact in asymptotic $L^p$ spaces, without any dimension-dependent restriction on the exponent. For data in the duality regime $L^1(\mathbb{R}^n)\cap L^{p'}(\mathbb{R}^n)$, asymptotic energy solutions coincide with weak energy solutions. We also show that additional compactness assumptions yield localized entropy-type solutions and, under suitable local regularity, distributional solutions.

math.AP↗

arXiv2026

Local analytic well-posedness for one-dimensional Vlasov$\unicode{x2013}$Dirac$\unicode{x2013}$Benney-type equations

We study a one-dimensional nonlinear Vlasov equation with a local self-consistent force field generated by the density, where the force is given by the spatial derivative of a real-analytic nonlinearity. For small analytic initial data, we prove local-in-time existence and uniqueness of analytic solutions. In particular, this yields a perturbative well-posedness result around the trivial equilibrium. We also give an energy-based representation of weak stationary states and discuss perturbations around spatially homogeneous stationary profiles. The proof relies on a contraction mapping argument in a complete metric space of analytic functions. As a technical byproduct, we establish quantitative composition estimates for analytic nonlinearities in the analytic norms used in the argument.

math.AP↗

arXiv2026

Strong convergence of sequences with vanishing relative entropy

We show that under natural growth conditions on the entropy function, convergence in relative entropy is equivalent to $L_p$-convergence. The main tool is the theory of Young measures, in a form that accounts for the formation of concentrations in weak limits.

math.AP↗

arXiv2026

Isometries and geometric liftings for Alexiewicz-normed $L^\infty$ spaces

We study spaces of essentially bounded functions on compact subsets of the real line, equipped with the Alexiewicz norm given by the supremum norm of the primitive. Using the associated measure projection, we classify their surjective linear isometries as weighted composition operators determined by a sign and an increasing bi-Lipschitz map between the corresponding measure intervals. We also give geometric criteria characterizing when this interval-level map lifts to a homeomorphism or to a bi-Lipschitz homeomorphism between the underlying compact sets.

math.FA↗

arXiv2026

Kolmogorov$\unicode{x2013}$Riesz compactness in asymptotic $L_p$ spaces

We extend the classical Kolmogorov-Riesz compactness theorem to the setting of asymptotic $L_p$ spaces on $\mathbb{R}^n$. These are nonlocally convex $\mathrm{F}$-spaces that contain the standard $L_p$ spaces as dense subspaces and include all measurable functions supported on sets of finite measure. In contrast with the classical $L_p$ setting, an additional almost equiboundedness condition is needed, and we prove that together with the natural tail and translation conditions it characterizes relative compactness. We conclude with illustrative examples.

math.FA↗

arXiv2026

Rigorous dense graph limit of a model for biological transportation networks

We rigorously derive the dense graph limit of a discrete model describing the formation of biological transportation networks. The discrete model, defined on undirected graphs with pressure-driven flows, incorporates a convex energy functional combining pumping and metabolic costs. It is constrained by a Kirchhoff law reflecting the local mass conservation. We first rescale and reformulate the discrete energy functional as an integral `semi-discrete' functional, where the Kirchhoff law transforms into a nonlocal elliptic integral equation. Assuming that the sequence of graphs is uniformly connected and that the limiting graphon is 0-1 valued, we prove two results: (1) rigorous Gamma-convergence of the sequence of the semi-discrete functionals to a continuum limit as the number of graph nodes and edges tends to infinity; (2) convergence of global minimizers of the discrete functionals to a global minimizer of the limiting continuum functional. Our results provide a rigorous mathematical foundation for the continuum description of biological transport structures emerging from discrete networks.

math.OC↗

arXiv2025

High-friction limit for bipolar Euler-Riesz systems

We consider a bipolar Euler-Riesz system and rigorously justify the high-friction limit of weak solutions towards a bipolar aggregation-diffusion system with Riesz interactions. The analysis is carried out via the relative entropy method in the regime where smooth solutions of the limiting equations exist. This extends previous results on the high-friction limit of bipolar Euler-Poisson systems to a more general class of interactions, and extends the one-species Euler-Riesz case to the bipolar setting.

math.AP↗

arXiv2025

Uniform boundedness of parametric bilinear fractional integrals

We provide weak-type bounds for a family of bilinear fractional integrals that arise in the study of Euler-Riesz systems. These bounds are uniform in the natural parameter that describes the family and are sharp, in the sense that they do not hold for any larger set of indices.

math.CA↗

arXiv2025

On $\mathrm{F}$-spaces of almost-Lebesgue functions

We consider the space of functions almost in $L_p$ and endow it with the topology of asymptotic $L_p$-convergence. This yields a completely metrizable topological vector space which, on finite measure spaces, coincides with the space of measurable functions equipped with the topology of (local) convergence in measure. We investigate analogs of classical results such as dominated convergence and Vitali convergence theorems. For $\mathbb{R}^d$ as the underlying measure space, we establish results on approximation by smooth functions and separability. Further aspects, including local boundedness, local convexity, and duality are examined in the $\mathbb{R}^d$ setting, revealing fundamental differences from standard $L_p$ spaces.

math.FA↗

arXiv2024

A bilinear fractional integral operator for Euler-Riesz systems

We establish a uniform estimate for a bilinear fractional integral operator via restricted weak-type endpoint estimates and Marcinkiewicz interpolation. This estimate is crucial in the integrability analysis of a tensor-valued bilinear fractional integral operator associated with Euler-Riesz systems modeling mean-field interactions induced by a singular kernel. The tensorial operator arises from a reformulation of the Euler-Riesz system that yields a gain in integrability for finite energy solutions through compensated integrability. Additionally, for smooth periodic solutions of the reformulated system, we derive a stability result.

math.AP↗

arXiv2024

Relation between asymptotic $L_p$-convergence and some classical modes of convergence

Asymptotic $L_p$-convergence, which resembles convergence in $L_p$, was introduced to address a question in diffusive relaxation. This note aims to compare asymptotic $L_p$-convergence with convergence in measure and in weak $L_p$ spaces. One of the results characterizes convergence in measure on finite measure spaces in terms of asymptotic $L_p$-convergence.

math.CA↗

arXiv2024

Weak-strong uniqueness and high-friction limit for Euler-Riesz systems

In this work we employ the relative energy method to obtain a weak-strong uniqueness principle for a Euler-Riesz system, as well as to establish its convergence in the high-friction limit towards a gradient flow equation. The main technical challenge in our analysis is addressed using a specific case of a Hardy-Littlewood-Sobolev inequality for Riesz potentials.

math.AP↗

arXiv2024

A mode of convergence arising in diffusive relaxation

In this work, a mode of convergence for measurable functions is introduced. A related notion of Cauchy sequence is given and it is proved that this notion of convergence is complete in the sense that Cauchy sequences converge. Moreover, the preservation of convergence under composition is investigated. The origin of this mode of convergence lies in the path of proving that the density of a Euler system converges almost everywhere (up to subsequences) towards the density of a non-linear diffusion system, as a consequence of the convergence in the relaxation limit.

math.CA↗