arXiv subjects

Juan Dávila

Publications and source records attributed to Juan Dávila.

17 recordsLinked to original sources

arXiv2026

Global in Time Vortex Configurations for the $2$D Euler Equations

We consider the problem of finding a solution to the incompressible Euler equations $$ ω_t + v\cdot \nabla ω= 0 \quad \hbox{ in } \mathbb{R}^2 \times (0,\infty), \quad v(x,t) = \frac 1{2π} \int_{{\mathbb R}^2} \frac {(y-x)^\perp}{|y-x|^2} ω(y,t)\, dy $$ that is close to a superposition of traveling vortices as $t\to \infty$. We employ a constructive approach by gluing classical traveling waves: two vortex-antivortex pairs traveling at main order with constant speed in opposite directions. More precisely, we find an initial condition that leads to a 4-vortex solution of the form $$ ω(x,t) = ω_0(x-ct\, e ) - ω_0 ( x+ ct \, e) + o(1) \ \hbox{ as } t\to\infty $$ where $$ ω_0( x ) = \frac 1{\varepsilon^{2}} W \left ( \frac {x-q} \varepsilon \right ) - \frac 1{\varepsilon^{2}}W \left ( \frac {x+q} \varepsilon \right ) + o(1) \ \hbox{ as } \varepsilon \to 0 $$ and $W(y)$ is a certain fixed smooth profile, radially symmetric, positive in the unit disc zero outside.

math.AP↗

arXiv2025

Finite-time blow-up for the three dimensional axially symmetric Keller-Segel system

We construct axially symmetric finite-time blow-up solutions to the three-dimensional Keller-Segel system. By adapting gluing techniques, we derive a precise asymptotic expansion for Type II singularities that generalizes the recent work of Hou, Nguyen, and Song. In our construction the mass concentrates along multiple rings and we obtain a refined expansion for the blow-up rate.

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arXiv2024

An Expanding Self-Similar Vortex Configuration for the 2D Euler Equations

This paper addresses the long-time dynamics of solutions to the 2D incompressible Euler equations. We construct solutions with continuous vorticity $ω_{\varepsilon}(x,t)$ concentrated around points $ξ_{j}(t)$ that converge to a sum of Dirac delta masses as $\varepsilon\to0$. These solutions are associated with the Kirchhoff-Routh point-vortex system, and the points $ξ_{j}(t)$ follow an expanding self similar trajectory of spirals, with the support of the vorticities contained in balls of radius $3\varepsilon$ around each $ξ_{j}$.

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arXiv2024

Overhanging solitary water waves

We provide the first construction of overhanging gravity water waves having the approximate form of a disk joined to a strip by a thin neck. The waves are solitary with constant vorticity, and exist when an appropriate dimensionless gravitational constant $g>0$ is sufficiently small. Our construction involves combining three explicit solutions to related problems: a disk of fluid in rigid rotation, a linear shear flow in a strip, and a rescaled version of an exceptional domain discovered by Hauswirth, Hélein, and Pacard \cite{hauswirth-helein-pacard}. The method developed here is related to the construction of constant mean curvature surfaces through gluing.

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arXiv2024

Asymptotic properties of vortex-pair solutions for incompressible Euler equations in $\mathbb{R}^2$

A {\em vortex pair} solution of the incompressible $2d$ Euler equation in vorticity form $$ ω_t + \nabla^\perp Ψ\cdot \nabla ω= 0 , \quad Ψ= (-Δ)^{-1} ω, \quad \hbox{in } \mathbb{R}^2 \times (0,\infty)$$ is a travelling wave solution of the form $ω(x,t) = W(x_1-ct,x_2 )$ where $W(x)$ is compactly supported and odd in $x_2$. We revisit the problem of constructing solutions which are highly $\varepsilon$-concentrated around points $ (0, \pm q)$, more precisely with approximately radially symmetric, compactly supported bumps with radius $\varepsilon$ and masses $\pm m$. Fine asymptotic expressions are obtained, and the smooth dependence on the parameters $q$ and $\varepsilon$ for the solution and its propagation speed $c$ are established. These results improve constructions through variational methods in [14] and in [5] for the case of a bounded domain.

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arXiv2021

Interacting helical traveling waves for the Gross-Pitaevskii equation

We consider the 3D Gross-Pitaevskii equation \begin{equation}\nonumber i\partial_t ψ+Δψ+(1-|ψ|^2)ψ=0 \text{ for } ψ:\mathbb{R}\times \mathbb{R}^3 \rightarrow \mathbb{C} \end{equation} and construct traveling waves solutions to this equation. These are solutions of the form $ψ(t,x)=u(x_1,x_2,x_3-Ct)$ with a velocity $C$ of order $\varepsilon|\log\varepsilon|$ for a small parameter $\varepsilon>0$. We build two different types of solutions. For the first type, the functions $u$ have a zero-set (vortex set) close to an union of $n$ helices for $n\geq 2$ and near these helices $u$ has degree 1. For the second type, the functions $u$ have a vortex filament of degree $-1$ near the vertical axis $e_3$ and $n\geq 4$ vortex filaments of degree $+1$ near helices whose axis is $e_3$. In both cases the helices are at a distance of order $1/(\varepsilon\sqrt{|\log \varepsilon|)}$ from the axis and are solutions to the Klein-Majda-Damodaran system, supposed to describe the evolution of nearly parallel vortex filaments in ideal fluids. Analogous solutions have been constructed recently by the authors for the stationary Gross-Pitaevskii equation, namely the Ginzburg-Landau equation. To prove the existence of these solutions we use the Lyapunov-Schmidt method and a subtle separation between even and odd Fourier modes of the error of a suitable approximation.

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arXiv2020

Travelling helices and the vortex filament conjecture in the incompressible Euler equations

We consider the Euler equations in ${\mathbb R}^3$ expressed in vorticity form. A classical question that goes back to Helmholtz is to describe the evolution of solutions with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called binormal curvature flow. Existence of true solutions concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings. We provide what appears to be the first rigorous construction of {\em helical filaments}, associated to a translating-rotating helix. The solution is defined at all times and does not change form with time. The result generalizes to multiple similar helical filaments travelling and rotating together.

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arXiv2019

Interacting helical vortex filaments in the 3-dimensional Ginzburg-Landau equation

For each given $n\geq 2$, we construct a family of entire solutions $u_\varepsilon (z,t)$, $\varepsilon>0$, with helical symmetry to the 3-dimensional complex-valued Ginzburg-Landau equation \begin{equation*}\nonumber Δu+(1-|u|^2)u=0, \quad (z,t) \in \mathbb{R}^2\times \mathbb{R} \simeq \mathbb{R}^3. \end{equation*} These solutions are $2π/\varepsilon$-periodic in $t$ and have $n$ helix-vortex curves, with asymptotic behavior as $\varepsilon\to 0$ $$ u_\varepsilon (z,t) \approx \prod_{j=1}^n W\left( z- \varepsilon^{-1} f_j(\varepsilon t) \right), $$ where $W(z) =w(r) e^{iθ} $, $z= re^{iθ},$ is the standard degree $+1$ vortex solution of the planar Ginzburg-Landau equation $ ΔW+(1-|W|^2)W=0 \text{ in } \mathbb{R}^2 $ and $$ f_j(t) = \frac { \sqrt{n-1} e^{it}e^{2 i (j-1)π/ n }}{ \sqrt{|\log\varepsilon|}}, \quad j=1,\ldots, n. $$ Existence of these solutions was previously conjectured, being ${\bf f}(t) = (f_1(t),\ldots, f_n(t))$ a rotating equilibrium point for the renormalized energy of vortex filaments there derived, $$ \mathcal W_\varepsilon ( {\bf f} ) :=π\int_0^{2π} \Big ( \, \frac{|\log \varepsilon|} 2 \sum_{k=1}^n|f'_k(t)|^2-\sum_{j\neq k}\log |f_j(t)-f_k(t)| \, \Big ) \mathrm{d} t, $$ corresponding to that of a planar logarithmic $n$-body problem. These solutions satisfy $$ \lim_{|z| \to +\infty } |u_\varepsilon (z,t)| = 1 \quad \hbox{uniformly in $t$} $$ and have nontrivial dependence on $t$, thus negatively answering the Ginzburg-Landau analogue of the Gibbons conjecture for the Allen-Cahn equation, a question originally formulated by H. Brezis.

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arXiv2019

Long-time asymptotics for evolutionary crystal dislocation models

We consider a family of evolution equations that generalize the Peierls-Nabarro model for crystal dislocations. They can be seen as semilinear parabolic reaction-diffusion equations in which the diffusion is regulated by a fractional Laplace operator of order $2 s \in (0, 2)$ acting in one space dimension and the reaction is determined by a $1$-periodic multi-well potential. We construct solutions of these equations that represent the typical propagation of $N \ge 2$ equally oriented dislocations of size $1$. For large times, the dislocations occur around points that evolve according to a repulsive dynamical system. When $s \in (1/2, 1)$, these solutions are shown to be asymptotically stable with respect to odd perturbations.

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arXiv2016

Theory of light-matter interaction in nematic liquid crystals and the second Painlevé equation

We study global minimizers of an energy functional arising as a thin sample limit in the theory of light-matter interaction in nematic liquid crystals. We show that depending on the parameters various defects are predicted by the model. In particular we show existence of a new type of topological defect which we call the {\it shadow kink}. Its local profile is described by the second Painlevé equation. As part of our analysis we find new solutions to this equation thus generalizing the well known result of Hastings and McLeod.

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arXiv2015

Nonlocal Delaunay surfaces

We construct codimension 1 surfaces of any dimension that minimize a periodic nonlocal perimeter functional among surfaces that are periodic, cylindrically symmetric and decreasing. These surfaces may be seen as a nonlocal analogue of the classical Delaunay surfaces (onduloids). For small volume, most of their mass tends to be concentrated in a periodic array and the surfaces are close to a periodic array of balls (in fact, we give explicit quantitative bounds on these facts).

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arXiv2015

Finite topology self-translating surfaces for the mean curvature flow in $\mathbb R^3$

Finite topology self translating surfaces to mean curvature flow of surfaces constitute a key element for the analysis of Type II singularities from a compact surface, since they arise in a limit after suitable blow-up scalings around the singularity. We find in $\mathbb R^3$ a surface $M$ orientable, embedded and complete with finite topology (and large genus) with three ends asymptotically paraboloidal, such that the moving surface $Σ(t) = M + te_z$ evolves by mean curvature flow. This amounts to the equation $H_M = ν\cdot e_z$ where $H_M$ denotes mean curvature, $ν$ is a choice of unit normal to $M$, and $e_z$ is a unit vector along the $z$-axis. The surface $M$ is in correspondence with the classical 3-end Costa-Hoffmann-Meeks minimal surface with large genus, which has two asymptotically catenoidal ends and one planar end, and a long array of small tunnels in the intersection region resembling a periodic Scherk surface. This example is the first non-trivial one of its kind, and it suggests a strong connection between this problem and the theory of embedded, complete minimal surfaces with finite total curvature.

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arXiv2014

Bubbling solutions for nonlocal elliptic problems

We investigate bubbling solutions for the nonlocal equation \[ A_Ω^s u =u^p,\ u >0 \quad \mbox{in } Ω, \] under homogeneous Dirichlet conditions, where $Ω$ is a bounded and smooth domain. The operator $A_Ω^s$ stands for two types of nonlocal operators that we treat in a unified way: either the spectral fractional Laplacian or the restricted fractional Laplacian. In both cases $s \in (0,1)$ and the Dirichlet conditions are different: for the spectral fractional Laplacian, we prescribe $u=0$ on $\partial Ω$ and for the restricted fractional Laplacian, we prescribe $u=0$ on $\mathbb R^n \setminus Ω$. We construct solutions when the exponent $p = (n+2s)/(n-2s) \pm \varepsilon$ is close to the critical one, concentrating as $\varepsilon \to 0$ near critical points of a reduced function involving the Green and Robin functions of the domain

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arXiv2014

Nonlocal $s$-minimal surfaces and Lawson cones

The nonlocal $s$-fractional minimal surface equation for $Σ= \partial E$ where $E$ is an open set in $R^N$ is given by $$ H_Σ^ s (p) := \int_{R^N} \frac {χ_E(x) - χ_{E^c}(x)} {|x-p|^{N+s}}\, dx \ =\ 0 \quad \text{for all } p\in Σ. $$ Here $0<s<1$, $χ$ designates characteristic function, and the integral is understood in the principal value sense. The classical notion of minimal surface is recovered by letting $s\to 1$. In this paper we exhibit the first concrete examples (beyond the plane) of nonlocal $s-$minimal surfaces. When $s$ is close to $1$, we first construct a connected embedded $s$-minimal surface of revolution in $R^3$, the {\bf nonlocal catenoid}, an analog of the standard catenoid $|x_3| = \log (r + \sqrt{r^2 -1})$. Rather than eventual logarithmic growth, this surface becomes asymptotic to the cone $|x_3|= r\sqrt{1-s}$. We also find a two-sheet embedded $s$-minimal surface asymptotic to the same cone, an analog to the simple union of two parallel planes. On the other hand, for any $0<s<1$, $n,m\ge 1$, $s-$minimal Lawson cones $|v|=α|u|$, $(u,v)\in R^n\times R^m$, are found to exist. In sharp contrast with the classical case, we prove their stability for small $s$ and $n+m=7$, which suggests that unlike the classical theory (or the case $s$ close to 1), the regularity of $s$-area minimizing surfaces may not hold true in dimension $7$.

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arXiv2013

Concentrating standing waves for the fractional nonlinear Schrödinger equation

We consider the semilinear equation $$ ε^{2s} (-Δ)^s u + V(x)u - u^p = 0, \quad u>0, \quad u\in H^{2s}(\R^N) $$ where $0 0$, and $ε>0$ is a small number. Letting $w_λ$ be the radial ground state of $(-Δ)^s w_λ+ λw_λ- w_λ^p=0$ in $H^{2s}(\R^N)$, we build solutions of the form $$ u_ε(x) \sim \sum_{i=1}^k w_{λ_i} ((x-ξ_i^ε)/ε),$$ where $λ_i = V(ξ_i^ε)$ and the $ξ_i^ε$ approach suitable critical points of $V$. Via a Lyapunov Schmidt variational reduction, we recover various existence results already known for the case $s=1$. In particular such a solution exists around $k$ nondegenerate critical points of $V$. For $s=1$ this corresponds to the classical results by Floer-Weinstein and Oh.

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arXiv2013

Regular solutions to a supercritical elliptic problem in exterior domains

We consider the supercritical elliptic problem -Δu = λe^u, λ> 0, in an exterior domain $Ω= \mathbb{R}^N \setminus D$ under zero Dirichlet condition, where D is smooth and bounded in \mathbb{R}^N, N greater or equal than 3. We prove that, for λsmall, this problem admits infinitely many regular solutions.

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arXiv2010

Regularity of radial extremal solutions for some non local semilinear equations

We investigate stable solutions of elliptic equations of the type \begin{equation*} \left \{ \begin{aligned} (-Δ)^s u&=λf(u) \qquad {\mbox{ in $B_1 \subset \R^{n}$}} \\ u&= 0 \qquad{\mbox{ on $\partial B_1$,}}\end{aligned}\right . \end{equation*} where $n\ge2$, $s \in (0,1)$, $λ\geq 0$ and $f$ is any smooth positive superlinear function. The operator $(-Δ)^s$ stands for the fractional Laplacian, a pseudo-differential operator of order $2s$. According to the value of $λ$, we study the existence and regularity of weak solutions $u$.

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