arXiv subjects

Damiano Greco

Publications and source records attributed to Damiano Greco.

10 recordsLinked to original sources

arXiv2026

Diophantine conditions in well-posedness theory for a coupled modulated Korteweg-de Vries system

We study the well-posedness theory of a coupled modulated Korteweg-de Vries (KdV) system on the circle with a time non-homogeneous modulation acting on the linear dispersion term. When the coupling parameter is equal to one, it has been recently proved that given any $s\in \mathbb{R}$, the resulting modulated KdV system is globally well-posed in $H^s(\mathbb{T})\times H^s(\mathbb{T})$, with a sufficiently irregular modulation. For couplings different from one, we use Diophantine conditions to characterize the resonances and prove that (under further restrictions on the coupling constant) for any $s\in \mathbb{R}$ the coupled modulated KdV system is globally well-posed in $H^s(\mathbb{T})\times H^s(\mathbb{T})$. This result differs from its unmodulated counterpart where it is known that global well-posedness holds for $s\ge s_*\in (5/7,1]$.

math.AP↗

arXiv2026

Phase transition for weakly interacting focusing Gibbs measures with harmonic potential

In this paper, we study the Gibbs measures on Euclidean spaces associated to the focusing nonlinear Schr\"odinger equation with harmonic potential and critical non linearity whose coupling constant tends to 0, a question initially posed by Brydges-Slade (1996) for the $\Phi^4_2$-model on $\mathbb{T}^2$. In dimension one and in the higher dimensional cases (with radial assumption), we establish a critical threshold below which the frequency-truncated measures converge to the base Gaussian measure (possibly with a renormalized $L^2$ cut-off) while, in the supercritical regime, we prove non-convergence of the frequency-truncated measures, even up to a subsequence.

math.PR↗

arXiv2026

Refined global well-posedness for the periodic modulated Korteweg-de Vries equation

We revisit the pathwise global well-posedness issue of the modulated Korteweg-de Vries equation (KdV) on the circle. In the previous work (2024), by combining the $I$-method and the sewing lemma, the second and fourth authors with C. Chouk, G. Li, and J. Li proved its global well-posedness in negative Sobolev spaces. This result was, however, restricted to the scaling subcritical regime $s > - \frac 32$ due to the use of the classical KdV scaling. In this paper, by noting that the modulated KdV enjoys additional one degree of freedom in its scaling symmetry thanks to the modulation term, we apply a non-KdV scaling to the unknown and prove that, given any $s \in \mathbb R$, the modulated KdV on the circle with a sufficiently irregular modulation is globally well-posed in $H^s(\mathbb T)$, thus going beyond the barrier of the scaling critical regularity $s = - \frac 32$.

math.AP↗

arXiv2026

Unconditional well-posedness of the stochastic Korteweg-de Vries equation on the real line

We study well-posedness issues of the stochastic Korteweg-de Vries equation (SKdV) with an additive noise, posed on the real line. By using the Fourier restriction norm method adapted to the Fourier-Lebesgue space in time, we first prove global well-posedness of SKdV in $L^2(\mathbb R)$ without assuming the homogenous Sobolev regularity, which was imposed in a work by de Bouard, Debussche, and Tsutsumi (1999). Then, by adapting the argument by Zhou (1997) to the stochastic setting, we prove optimal pathwise unconditional uniqueness for SKdV in $L^2(\mathbb R)$. In the appendix, we present a short argument for proving boundedness of the multiplication by a sharp cutoff function in the Fourier-Lebesgue and Sobolev spaces, which is of interest in its own right.

math.AP↗

arXiv2024

Critical threshold for weakly interacting log-correlated focusing Gibbs measures

We study log-correlated Gibbs measures on the $d$-dimensional torus with weakly interacting focusing quartic potentials whose coupling constants tend to $0$ as we remove regularization. In particular, we exhibit a phase transition for this model by identifying a critical threshold, separating the weakly and strongly coupling regimes; in the weakly coupling regime, we show that the frequency-truncated measures converge to the base Gaussian measure (possibly with a renormalized $L^2$-cutoff), whereas, in the strongly coupling regime, we prove non-convergence of the frequency-truncated measures, even up to a subsequence. Our result answers an open question posed by Brydges and Slade (1996).

math.PR↗

arXiv2024

Optimal decay and regularity for a Thomas--Fermi type variational problem

We study existence and qualitative properties of the minimizers for a Thomas--Fermi type energy functional defined by $$E_α(ρ):=\frac{1}{q}\int_{\mathbb{R}^d}|ρ(x)|^q dx+\frac{1}{2}\iint_{\mathbb{R}^d\times\mathbb{R}^d}\frac{ρ(x)ρ(y)}{|x-y|^{d-α}}dx dy-\int_{\mathbb{R}^d}V(x)ρ(x)dx,$$ where $d\ge 2$, $α\in (0,d)$ and $V$ is a potential. Under broad assumptions on $V$ we establish existence, uniqueness and qualitative properties such as positivity, regularity and decay at infinity of the global minimizer. The decay at infinity depends in a non--trivial way on the choice of $α$ and $q$. If $α\in (0,2)$ and $q>2$ the global minimizer is proved to be positive under mild regularity assumptions on $V$, unlike in the local case $α=2$ where the global minimizer has typically compact support. We also show that if $V$ decays sufficiently fast the global minimizer is sign--changing even if $V$ is non--negative. In such regimes we establish a relation between the positive part of the global minimizer and the support of the minimizer of the energy, constrained on the non--negative functions. Our study is motivated by recent models of charge screening in graphene, where sign--changing minimizers appear in a natural way.

math.AP↗

arXiv2024

Elementary Brezis-Browder type results and Representation formulae for s-harmonic functions

We prove Brezis--Browder type results for fractional Sobolev spaces and quantitative type estimates for $s$-harmonic functions. Furthermore, we give sufficient conditions for distributional solutions to the fractional Poisson's equation $(-\Delta)^su=T$ on $\mathbb{R}^d$ to be of the form $$u(x)=\int_{\mathbb{R}^d}\frac{T(y)}{|x-y|^{d-2s}}dy+l,\quad l\in \mathbb{R}.$$

math.AP↗

arXiv2024

Ground states of a nonlocal variational problem and Thomas-Fermi limit for the Choquard equation

We study nonnegative optimizers of a Gagliardo-Nirenberg type inequality $$\iint_{\mathbb{R}^N \times \mathbb{R}^N} \frac{|u(x)|^p\,|u(y)|^p}{|x - y|^{N-α}} dx\, dy\le C\Big(\int_{{\mathbb R}^N}|u|^2 dx\Big)^{pθ} \Big(\int_{{\mathbb R}^N}|u|^q dx\Big)^{2p(1-θ)/q},$$ that involves the nonlocal Riesz energy with $0<α \frac{N+α}{N}$, $q>\frac{2Np}{N+α}$ and $θ=\frac{(N+α)q-2Np}{Np(q-2)}$. For $p=2$, the equivalent problem has been studied in connection with the Keller-Segel diffusion-aggregation models in the past few decades. The general case $p\neq 2$ considered here appears in the study of Thomas-Fermi limit regime for the Choquard equations with local repulsion. We establish optimal ranges of parameters for the validity of the above interpolation inequality, discuss the existence and qualitative properties of the nonnegative maximizers, and in some special cases estimate the optimal constant. For $p=2$ it is known that the maximizers are Hölder continuous and compactly supported on a ball. We show that for $p<2$ the maximizers are smooth functions supported on $\mathbb{R}^N$, while for $p>2$ the maximizers consist of a characteristic function of a ball and a nonconstant nonincreasing Hölder continuous function supported on the same ball. We use these qualitative properties of the maximizers to establish the validity of the Thomas-Fermi approximations for the Choquard equations with local repulsion. The results are verified numerically with extensive examples.

math.AP↗

arXiv2023

Optimal divergence rate of the focusing Gibbs measures

We study Gibbs measures on the $d$-dimensional torus with $L^2$-(super)critical focusing interaction potentials. We establish a precise divergence rate of the partition function as we remove regularization, where the optimal constant is given by (i) (the negative of) the minimum value of the Hamiltonian given an $L^2$-constraint in the $L^2$-critical case and (ii) the optimal constant for certain Bernstein's inequality in the mass-supercritical case. In particular, our result in the $L^2$-critical case precisely quantifies the phase transition of the focusing Gibbs measure at the critical $L^2$ threshold, previously studied by Lebowitz, Rose, and Speer (1988) and Sosoe, Tolomeo, and the fourth author (2022).

math.PR↗

arXiv2023

Extension and embedding theorems for Campanato spaces on $C^{0,γ}$ domains

We consider Campanato spaces with exponents $λ, p$ on domains of class $C^{0,γ}$ in the N-dimensional Euclidean space endowed with a natural anisotropic metric depending on $γ$. We discuss several results including the appropriate Campanato's embedding theorem and we prove that functions of those spaces can be extended to the whole of the Euclidean space without deterioration of the exponents $λ, p$.

math.FA↗