arXiv subjects

Antonio J. Fernández

Publications and source records attributed to Antonio J. Fernández.

At least 19 recordsLinked to original sources

arXiv2026

Desingularization of vortex sheets for the 2D Euler equations

We show how to regularize vortex sheets by means of smooth, compactly supported vorticities that asymptotically evolve according to the Birkhoff-Rott vortex sheet dynamics. More precisely, consider a vortex sheet initial datum $ω^0_{\mathrm{sing}}$, which is a signed Radon measure supported on a closed curve. We construct a family of initial vorticities $ω^0_\varepsilon \in C^\infty_c(\mathbb{R}^2)$ converging to $ω^0_{\mathrm{sing}}$ distributionally as $\varepsilon \to 0^+$, and show that the corresponding solutions $ω_\varepsilon(x,t)$ to the 2D incompressible Euler equations converge to the measure defined by the Birkhoff-Rott system with initial datum $ω^0_{\mathrm{sing}}$. The regularization relies on a layer construction designed to exploit the key observation that the Kelvin-Helmholtz instability has a strongly anisotropic effect: while vorticities must be analytic in the "tangential" direction, the way layers can be arranged in the "normal" direction is essentially arbitrary.

math.AP↗

arXiv2026

Hölder estimates up to the singular time for the Córdoba-Córdoba-Fontelos equation

We prove that the solution to the Córdoba-Córdoba-Fontelos equation $θ_t-uθ_x=0$, $u=Hθ$, on the real line with analytic initial datum $θ_0(x)=(1+x^2)^{-1}$ develops a finite-time singularity while remaining uniformly bounded in $C^{2/5}$. We also prove that, as the solution approaches the singular time, the $C^{2/3}$ seminorms of both $θ$ and $u$ diverge. The main ingredient is an integral representation in terms of a probability measure that is preserved by the nonlinear evolution. From this representation we derive a pointwise quadratic estimate for the key nonlinear term in the equation for $u$, using a comparison of positive kernels. Together with evolution identities for the moments of the measure, this estimate yields both the uniform Hölder bound and the divergence of the higher seminorms.

math.AP↗

arXiv2026

Smooth nonradial stationary solutions to SQG via the half-Yamabe equation

We prove the existence of infinitely many smooth nonradial stationary solutions to the surface quasi-geostrophic (SQG) equation with finite kinetic energy. Our construction is based on a family of nonradial sign-changing solutions to the two-dimensional half-Yamabe equation, obtained via a Lyapunov--Schmidt reduction and concentrated at the vertices of a regular polygon. As the number of vertices tends to infinity, the associated stationary SQG solutions converge to a radial stationary profile centered at the origin, together with a lower-order vortex sheet correction.

math.AP↗

arXiv2026

Vortex-sheet desingularization for three-dimensional ideal fluids

We prove a desingularization theorem for analytic vortex sheets of the 3D incompressible Euler equations. Starting from an analytic solution of the corresponding Birkhoff-Rott system, we construct, for every sufficiently small thickness parameter $\varepsilon>0 $, an exact Euler vorticity supported on a tubular neighborhood of width $ O(\varepsilon) $ around the sheet, and defined on a time interval that does not shrink to 0 as $\varepsilon \to 0$. We show that, as $\varepsilon \to 0$, these vorticities converge, in the sense of distributions, to the prescribed vortex sheet. In particular, we conclude that analytic 3D vortex sheet motions arise as limits of exact Euler flows with lifespan bounded from below independently of $ \varepsilon $. The proof hinges on the study of vorticities defined in terms of a time-dependent foliation by almost parallel surfaces and of divergence-free vector fields tangent to these surfaces.

math.AP↗

arXiv2025

Uniformly rotating Euler flows with compactly supported velocity

For any positive integer $k$, we prove the existence of nontrivial $C^k$-smooth uniformly rotating solutions to the 2D incompressible Euler equations with compact spatial support. These solutions, which can be chosen to be small perturbations of radial flows, are the first example of smooth rotating flows with finite energy which are not locally radial. We also prove new rigidity results for rotating solutions which show that the geometric structure of these flows is severely constrained.

math.AP↗

arXiv2025

Blowing-up solutions to competitive critical systems in dimension 3

We study the critical system of $m\geq 2$ equations \begin{equation*} -Δu_i = u_i^5 + \sum_{j = 1,\,j\neq i}^m β_{ij} u_i^2 u_j^3\,, \quad u_i \gneqq 0 \quad \mbox{in } \mathbb{R}^3\,, \quad i \in \{1, \ldots, m\}\,, \end{equation*} where $β_{κ\ell} =α\in\mathbb{R}$ if $κ\neq\ell$, and $β_{\ell m}=β_{m κ} =β<0$, for $ κ, \ell \in \{1,\ldots, m-1\}$. We construct solutions to this system in the case where $β\to-\infty$ by means of a Ljapunov-Schmidt reduction argument. This allows us to identify the explicit form of the solution at main order: $u_1$ will look like a perturbation of the standard radial positive solution to the Yamabe equation, while $u_2$ will blow-up at the $k$ vertices of a regular planar polygon. The solutions to the other equations will replicate the blowing-up structure under an appropriate rotation that ensures $u_i\neq u_j$ for $i\neq j$. The result provides the first almost-explicit example of non-synchronized solutions to competitive critical systems in dimension 3.

math.AP↗

arXiv2025

A contractible Schiffer counterexample on the half-sphere

We show the existence of a family of nontrivial smooth contractible domains on the sphere that admit Neumann eigenfunctions of the Laplacian which are constant on the boundary. These domains are contained on the half-sphere, in stark contrast with the rigidity literature for Serrin-type problems. The proof relies on a local bifurcation argument around the family of geodesic disks centered at the north pole. We combine the use of anisotropic Hölder spaces for the functional setting with computer-assisted techniques to check the bifurcation conditions.

math.AP↗

arXiv2024

A Schiffer-type problem for annuli with applications to stationary planar Euler flows

If on a smooth bounded domain $Ω\subset\mathbb{R}^2$ there is a nonconstant Neumann eigenfunction $u$ that is locally constant on the boundary, must $Ω$ be a disk or an annulus? This question can be understood as a weaker analog of the well known Schiffer conjecture, in that the function $u$ is allowed to take a different constant value on each connected component of $\partial Ω$ yet many of the known rigidity properties of the original problem are essentially preserved. Our main result provides a negative answer by constructing a family of nontrivial doubly connected domains $Ω$ with the above property. As a consequence, a certain linear combination of the indicator functions of the domains $Ω$ and of the bounded component of the complement $\mathbb{R}^2\backslash\overlineΩ$ fails to have the Pompeiu property. Furthermore, our construction implies the existence of continuous, compactly supported stationary weak solutions to the 2D incompressible Euler equations which are not locally radial.

math.AP↗

arXiv2024

Smooth nonradial stationary Euler flows on the plane with compact support

We prove the existence of nonradial classical solutions to the 2D incompressible Euler equations with compact support. More precisely, for any positive integer $k$, we construct compactly supported stationary Euler flows of class $C^k(\mathbb{R}^2)$ which are not locally radial. The proof uses a degree-theory-based bifurcation argument which hinges on three key ingredients: a novel approach to stationary Euler flows through elliptic equations with non-autonomous nonlinearities; a set of sharp regularity estimates for the linearized operator, which involves a potential that blows up as the inverse square of the distance to the boundary of the support; and overcoming a serious problem of loss of derivatives by the introduction of anisotropic weighted functional spaces between which the linearized operator is Fredholm.

math.AP↗

arXiv2024

Small spheres with prescribed nonconstant mean curvature in Riemannian manifolds

Given a function $f$ on a smooth Riemannian manifold without boundary, we prove that if $p \in M$ is a non-degenerate critical point of $f$, then a neighborhood of $p$ contains a foliation by spheres with mean curvature proportional to $f$. This foliation is essentially unique. The nondegeneracy assumption can be substantially relaxed, at the expense of losing the property that the family of spheres with prescribed mean curvature defines a foliation.

math.DG↗

arXiv2023

Global fractional Calderón-Zygmund type regularity

We obtain a global fractional Calderón-Zygmund regularity theory for the fractional Poisson problem. More precisely, for $Ω\subset \mathbb{R}^N$, $N \geq 2$, a bounded domain with boundary $\partial Ω$ of class $C^2$, $s \in (0,1)$ and $f \in L^m(Ω)$ for some $m \geq 1$, we consider the problem $$ \left. \begin{aligned} (-Δ)^s u = f \quad \mbox{in } Ω, \qquad\ u = 0 \quad \mbox{in } \mathbb{R}^N \setminus Ω, \end{aligned} \right. $$ and, according to $m$, we find the values of $s \leq t < \min\{1,2s\}$ and of $1 < p < +\infty$ such that $u \in L^{t,p}(\mathbb{R}^N)$ and such that $u \in W^{t,p}(\mathbb{R}^N)$.

math.AP↗

arXiv2023

Non-uniqueness for the nonlocal Liouville equation in $\mathbb{R}$ and applications

We construct multiple solutions to the nonlocal Liouville equation \begin{equation} \label{eqk} \tag{L} (-Δ)^{\frac{1}{2}} u = K(x) e^u \quad \mbox{ in } \mathbb{R}. \end{equation} More precisely, for $K$ of the form $K(x) = 1+\varepsilon κ(x)$ with $\varepsilon \in (0,1)$ small and $κ\in C^{1,α}(\mathbb{R}) \cap L^{\infty}(\mathbb{R})$ for some $α> 0$, we prove existence of multiple solutions to \eqref{eqk} bifurcating from the bubbles. These solutions provide examples of flat metrics in the half-plane with prescribed geodesic curvature $K(x)$ on its boundary. Furthermore, they imply the existence of multiple ground state soliton solutions for the Calogero-Moser derivative NLS.

math.AP↗

arXiv2022

Deterministic KPZ-type equations with nonlocal "gradient terms"

The main goal of this paper is to prove existence and non-existence results for deterministic Kardar-Parisi-Zhang type equations involving non-local "gradient terms". More precisely, let $Ω\subset \mathbb{R}^N$, $N \geq 2$, be a bounded domain with boundary $\partial Ω$ of class $C^2$. For $s \in (0,1)$, we consider problems of the form \[ \tag{KPZ} \left\{ \begin{aligned} (-Δ)^s u & = μ(x) |\mathbb{D}(u)|^q + λf(x), \quad && \mbox{ in } Ω,\\ u & = 0, && \mbox{ in } \mathbb{R}^N \setminus Ω, \end{aligned} \right. \] where $q > 1$ and $λ> 0$ are real parameters, $f$ belongs to a suitable Lebesgue space, $μ$ belongs to $L^{\infty}(Ω)$ and $\mathbb{D}$ represents a nonlocal "gradient term". Depending on the size of $λ> 0$, we derive existence and non-existence results. In particular, we solve several open problems posed in [4, Section 6] and [2, Section 7].

math.AP↗

arXiv2022

Some non-homogeneous Gagliardo-Nirenberg inequalities and application to a biharmonic non-linear Schrödinger equation

We study the standing waves for a fourth-order Schrödinger equation with mixed dispersion that minimize the associated energy when the $L^2-$norm (the \textit{mass}) } is kept fixed. We need some non-homogeneous Gagliardo-Nirenberg-type inequalities and we develop a method to prove such estimates that should be useful elsewhere. We prove optimal results on the existence of minimizers in the {\it mass-subcritical } and {\it mass-critical } cases. In the { \it mass supercritical} case we show that global minimizers do not exist, and we investigate the existence of local minimizers. If the mass does not exceed some threshold $ μ_0 \in (0,+\infty)$, our results on "best" local minimizers are also optimal.

math.AP↗

arXiv2022

Blowing-up solutions for a nonlocal Liouville type equation

We consider the nonlocal Liouville type equation $$ (-Δ)^{\frac{1}{2}} u = \varepsilon κ(x) e^u, \quad u > 0, \quad \mbox{in } I, \qquad u = 0, \quad \mbox{in } \mathbb{R} \setminus I, $$ where $I$ is a union of $d \geq 2$ disjoint bounded intervals, $κ$ is a smooth bounded function with positive infimum and $\varepsilon > 0$ is a small parameter. For any integer $1 \leq m \leq d$, we construct a family of solutions $(u_\varepsilon)_{\varepsilon}$ which blow up at $m$ interior distinct points of $I$ and for which $\varepsilon \int_I κe^{u_\varepsilon} \, \rightarrow 2 m π$, as $\varepsilon \to 0$. Moreover, we show that, when $d = 2$ and $m$ is suitably large, no such construction is possible.

math.AP↗

arXiv2020

Some remarks on a minimization problem associated to a fourth order nonlinear Schrödinger equation

Let $γ> 0\,$, $β> 0\,$, $α> 0$ and $0 < σN < 4$. In the present paper, we study, for $c > 0$ given, the constrained minimization problem \begin{equation*} \label{MinL2fixed} m(c):=\inf_{u\in S (c) }E(u), \end{equation*} where \begin{equation*} E (u):=\fracγ{2}\int_{\mathbb{R}^N}|Δu|^2\, dx -\fracβ{2}\int_{\mathbb{R}^N}|\nabla u|^2\, dx-\fracα{2σ+2}\int_{\mathbb{R}^N}|u|^{2σ+2}\, dx, \end{equation*} and \begin{equation*} S(c):=\left\{u\in H^2(\mathbb{R}^N):\int_{\mathbb{R}^N}|u|^{2}\, dx=c\right\}. \end{equation*} The aim of our study is twofold. On one hand, this minimization problem is related to the existence and orbital stability of standing waves for the mixed dispersion nonlinear biharmonic Schrödinger equation \begin{equation*} i \partial_t ψ-γΔ^2 ψ- βΔψ+ α|ψ|^{2σ} ψ=0, \quad ψ(0, x)=ψ_0 (x),\quad (t, x) \in \mathbb{R} \times \mathbb{R}^N. \end{equation*} On the other hand, in most of the applications of the Concentration-Compactness principle of P.-L. Lions, the difficult part is to deal with the possible dichotomy of the minimizing sequences. The problem under consideration provides an example for which, to rule out the dichotomy is rather standard while, to rule out the vanishing, here for $c > 0$ small, is challenging. We also provide, in the limit $c \to 0$, a precise description of the behaviour of the minima. Finally, some extensions and open problems are proposed.

math.AP↗

arXiv2020

The nonlinear Schrödinger equation in the half-space

The present paper is concerned with the half-space Dirichlet problem \begin{equation} \tag{$P_c$} \label{problem-abstract} -Δv + v = |v|^{p-1}v,\ \mbox{ in } \mathbb{R}^N_{+}, \qquad v = c,\ \mbox{ on } \partial \mathbb{R}^N_{+},\ \qquad \lim_{x_N \to \infty} v(x',x_N) = 0 \mbox{ uniformly in }x' \in \mathbb{R}^{N-1}, \end{equation} where $\mathbb{R}^N_{+} := \{\,x \in \mathbb{R}^N: x_N > 0\, \}$ for some $N \geq 1$ and $p > 1$, $c > 0$ are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to \eqref{problem-abstract}. We prove that the existence and multiplicity of bounded positive solutions to \eqref{problem-abstract} depend in a striking way on the value of $c > 0$ and also on the dimension $N$. We find an explicit number $c_p \in (1,\sqrt{e})$, depending only on $p$, which determines the threshold between existence and non-existence. In particular, in dimensions $N \geq 2$, we prove that, for $0 < c < c_p$, problem \eqref{problem-abstract} admits infinitely many bounded positive solutions, whereas, for $c > c_p$, there are no bounded positive solutions to \eqref{problem-abstract}.

math.AP↗

arXiv2019

Existence and multiplicity for an elliptic problem with critical growth in the gradient and sign-changing coefficients

Let $Ω\subset \mathbb{R}^N$, $N \geq 2$, be a smooth bounded domain. We consider the boundary value problem \begin{equation} \label{Plambda-Abstract-ch3} \tag{$P_λ$} -Δu = c_λ(x) u + μ|\nabla u|^2 + h(x)\,, \quad u \in H_0^1(Ω) \cap L^{\infty}(Ω)\,, \end{equation} where $c_λ$ and $h$ belong to $L^q(Ω)$ for some $q > N/2$, $μ$ belongs to $\mathbb{R} \setminus \{0\}$ and we write $c_λ$ under the form $c_λ:= λc_{+} - c_{-}$ with $c_{+} \gneqq 0$, $c_{-} \geq 0$, $c_{+} c_{-} \equiv 0$ and $λ\in \mathbb{R}$. Here $c_λ$ and $h$ are both allowed to change sign. As a first main result we give a necessary and sufficient condition which guarantees the existence of a unique solution to \eqref{Plambda-Abstract-ch3} when $λ\leq 0$. Then, assuming that $(P_0)$ has a solution, we prove existence and multiplicity results for $λ> 0$. Our proofs rely on a suitable change of variable of type $v = F(u)$ and the combination of variational methods with lower and upper solution techniques.

math.AP↗