arXiv subjects

Manuel del Pino

Publications and source records attributed to Manuel del Pino.

At least 19 recordsLinked to original sources

arXiv2026

The double sphere solution in the liquid drop model

We consider the problem of finding critical domains $Ω\subset{\mathbb R}^3$ for the energy functional $$ \mathcal E (Ω) = {\rm Per}\,(Ω) + \frac 12 \iint_{Ω\times Ω} \frac{dx\,dy}{|x-y|} $$ under the volume constraint $|Ω|=V$. We look for smooth, embedded, compact surfaces $\partialΩ$ that solve this problem. We construct an axially symmetric, non-minimizing solution that, for a sufficiently small $V>0$, resembles the union of two balls with volume $V/2$ connected by a tiny, approximately catenoidal neck with a width of the order $V^{\frac 43}$.

math.AP↗

arXiv2026

A Supercritical Problem in Dimension Two

Let $Ω\subset\mathbb R^2$ be a smooth bounded domain containing the origin and invariant under reflection across the coordinate axes, and let $0<λ<λ_1(Ω)$, where $Λ_1 (Ω)$ is the first eigenvalue for $-Δ$ on $Ω$ under Dirichlet boundary conditions. For every fixed integer $k\ge1$ and all sufficiently small $\varepsilon>0$, we construct a positive solution of \[ -Δu=λu e^{u^{2+\varepsilon}} \quad\hbox{in }Ω, \qquad u=0\quad\hbox{on }\partialΩ, \] which blows up at the origin as a tower of $k$ Liouville bubbles. The concentration scales are strongly separated. The proof is based on a Lyapunov--Schmidt reduction adapted to these different scales.

math.AP↗

arXiv2026

On the vortex filament conjecture for the Gross-Pitaevskii equation

We establish one form of the vortex filament conjecture for the three-dimensional Gross-Pitaevskii equation. Given any smooth closed embedded binormal flow of curves on a compact time interval, we construct, in the small-core limit $\varepsilon\to0$, a family of exact solutions whose degree-one vortex filaments converge uniformly to the prescribed flow. Near the filament the solutions have the standard planar vortex profile, while away from it their phase gradients converge to the associated Biot-Savart field. We also derive a refined modulation law for the vortex filament.

math.AP↗

arXiv2026

A Bernstein problem for translating solutions to the mean curvature flow

We study entire graphical translating solutions of the mean curvature flow, \[ \operatorname{div}\!\left(\frac{\nabla G}{\sqrt{1+|\nabla G|^2}}\right) =\frac{1}{\sqrt{1+|\nabla G|^2}} \qquad\text{in }\mathbb R^N. \] Every such graph is mean-convex, since its mean curvature is the vertical component of its unit normal. In dimension two, mean-convex translating solitons are convex, and an entire graphical translator is therefore the rotationally symmetric bowl soliton. In higher dimensions Wang constructed non-rotational entire convex translating graphs. We prove that a further loss of rigidity occurs at the Bernstein dimension: for every $N\ge 8$ there exists a one-parameter family of entire graphical translators that are mean-convex but not convex. The construction starts from the Bombieri--De Giorgi--Giusti (BDG) entire minimal graph in $\mathbb R^8$ and develops a singular perturbation theory for the translator equation around it. The main new feature is a transition layer near Simons' cone: the translating term breaks the odd symmetry of the minimal graph, and after a suitable recentering the matching problem is governed by a parabolic inner equation. A detailed analysis of this layer, together with weighted Jacobi theory on the BDG graph and global barriers, yields the desired entire solutions.

math.DG↗

arXiv2026

Nondegeneracy and Morse Index of Ginzburg--Landau Vortices

We prove that the standard degree-two and degree-three vortex solutions of the Ginzburg-Landau equation are nondegenerate. Their Morse indices are also computed. The proof relies on new explicit upper and lower bounds of the modulus of these solutions and a comparison argument. It is expected that our method can be generalized to study higher degree solutions.

math.AP↗

arXiv2026

A two-dimensional Allen-Cahn theory for interfaces with boundary

We develop a two-dimensional Allen-Cahn theory for interfaces with boundary in the line-bundle framework introduced by Fröhlich and Struwe. The central new object is a model solution on the punctured plane whose nodal set is a half-line. Near the boundary of an interface, this solution plays the role of the heteroclinic profile in the classical interior theory. However, unlike the interior setting, where the analysis in directions normal to the interface reduces to the ODE satisfied by the heteroclinic, the boundary model has non-flat level sets and must be studied as a genuinely two-dimensional elliptic solution. In the first part of this paper, we construct this model solution and develop its stability and invertibility theory. In the second part of this paper, we present the main application: we construct Allen--Cahn sections whose nodal sets concentrate on any prescribed finite collection of disjoint line segments in the plane. The construction uses a Lyapunov-Schmidt reduction, but the boundary introduces a new difficulty: it creates large error terms along the interior of the interface. Controlling these terms requires a refined ansatz and new gluing arguments beyond the standard interior Allen-Cahn theory.

math.AP↗

arXiv2026

Collapsing-tube type II blow-up for the energy-supercritical heat equation

We construct a new type II finite-time blow-up mechanism for the energy-supercritical heat equation \[ u_t=Δu+u^3, \qquad n\geq 5. \] The solution is positive and blows up only at the origin, but in a highly anisotropic fashion. As $t\nearrow T$, the solution concentrates in a thin tubular region around an $(n-4)$-dimensional sphere whose radius shrinks to zero at the self-similar scale \[ ξ_r(t)\sim \sqrt{2(n-4)(T-t)}. \] At the same time, concentration takes place transversely to the sphere at the much smaller scale \[ λ(t)\sim κ_* \frac{T-t}{|\log(T-t)|^{\frac n{n-2}}}, \] for some $κ_*>0$. More precisely, in cylindrical coordinates $r=|x'|$, $z\in\mathbb R^3$, the leading profile is \[ u(x,t) \sim \frac{1}{λ(t)} U\left( \frac{r-ξ_r(t)}{λ(t)}, \frac{z}{λ(t)} \right), \] where $U$ is the Aubin--Talenti bubble in $\mathbb R^4$. The construction reveals a two-scale singularity mechanism in which a critical transverse bubble concentrates around a geometric set that itself collapses. The concentration tube evolves at the parabolic scale $\sqrt{T-t}$, whereas its transverse thickness is governed by the much smaller type II scale $λ(t)$. The logarithmic blow-up law is determined by a nonlocal modulation equation arising from the interaction between the four-dimensional critical bubble and the axisymmetric heat kernel. To our knowledge, this seems to be the first Type II blowup that quantifies the effect of a self-similar collapsing tube. The exponent $p=3$ is energy-supercritical in dimensions $n\geq5$, but lies below the Joseph--Lundgren exponent for $5\leq n\leq 12$, in a regime where positive radial type II blow-up is ruled out. The present result provides the first example of a positive type II, single-point blow-up through a collapsing thin-tube geometry.

math.AP↗

arXiv2026

Stable blow-up on a sphere for a quadratic-derivative nonlinear wave equation

We study finite-time blow-up for the nonlinear wave equation \begin{equation*} v_{tt}-Δv=|\nabla_x v|^2 \end{equation*} in dimensions $n\geq2$, under radial symmetry. For every prescribed radius $r_0>0$, we construct solutions which blow up in finite time $T>0$ on the sphere $\{|x|=r_0\}$ with logarithmic Type-I rate. The leading singular dynamics are governed by ``generalised self-similar'' profiles of the associated one-dimensional equation, while the radial geometry generates a curvature correction of size $\mathcal{O}((\frac{T}{r_0})^2)$. A key simplification in our approach is a logarithmic radial correction which removes the first-order radial drift and reduces the geometry to a decaying inverse-square forcing. We further prove asymptotic stability of the resulting family under radial perturbations. A new feature compared with the one-dimensional theory is that the stable blow-up family is not fully explicit. To overcome this, we develop spectral and semigroup estimates on an extended light cone, together with Lipschitz dependence on the modulation parameters for the spectral projections, the stable flow, and the non-explicit correction.

math.AP↗

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.

math.AP↗

arXiv2025

Vortex dynamics for the Gross-Pitaevskii equation

We rigorously establish the formal asymptotics of Neu for Gross-Pitaevskii vortex dynamics in the plane. Given any integer $n\geq2$, we construct a family of $n$-vortex solutions with vortices of degree $\pm1$, and describe precisely the solution profile and associated vortex dynamics on an arbitrarily large, finite time interval. We compute an asymptotic expansion of the vortex positions in terms of the vortex core size $ε>0$, and show that the dynamics is governed at leading order as $ε\to0$ by the classical Helmholtz-Kirchhoff system. Moreover, we show that the first correction to the leading order dynamics is determined by the solution of a linear wave equation, justifying a formal expansion found by Ovchinnikov and Sigal.

math.AP↗

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 $ξ_{j}$.

math.AP↗

arXiv2024

Solvability for the Ginzburg-Landau equation linearized at the degree-one vortex

We consider the Ginzburg-Landau equation in the plane linearized around the standard degree-one vortex solution $W(x)=w(r)e^{iθ}$. Using explicit representation formulae for the Fourier modes in $θ$, we obtain sharp estimates for the inverse of the linearized operator which hold for a large class of right-hand sides. This theory can be applied, for example, to estimate the inverse after dropping the usual orthogonality conditions.

math.AP↗

arXiv2024

Delaunay-like compact equilibria in the liquid drop model

The liquid drop model was introduced by Gamow in 1928 and Bohr-Wheeler in 1938 to model atomic nuclei. The model describes the competition between the surface tension, which keeps the nuclei together, and the Coulomb force, corresponding to repulsion among protons. More precisely, the problem consists of finding a surface $Σ=\partial Ω$ in $\mathbb{R}^3$ that is critical for the energy $$ E(Ω) = {\rm Per\,} (Ω) + \frac 12 \int_Ω\int_Ω\frac {dxdy}{|x-y|} $$ under the volume constraint $|Ω| = m$. The term ${\rm Per\,} (Ω) $ corresponds to the surface area of $Σ$. The associated Euler-Lagrange equation is $$ H_Σ(x) + \int_{Ω} \frac {dy}{|x-y|} = λ\quad \hbox{ for all } x\in Σ, \quad $$ where $H_Σ$ stands for the mean curvature of the surface, and where $λ\in\mathbb{R}$ is the Lagrange multiplier associated to the constraint $|Ω|=m$. Round spheres enclosing balls of volume $m$ are always solutions. They are minimizers for sufficiently small $m$. Since the two terms in the energy compete, finding non-minimizing solutions can be challenging. We find a new class of compact, embedded solutions with large volumes, whose geometry resembles a "pearl necklace" with an axis located on a large circle, with a shape close to a Delaunay's unduloid surface of constant mean curvature. The existence of such equilibria is not at all obvious, since for the closely related constant mean curvature problem $H_Σ= λ$, the only compact embedded solutions are spheres, as stated by the classical Alexandrov result.

math.AP↗

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.

math.AP↗

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.

math.AP↗

arXiv2024

Infinite time blow-up for the three dimensional energy critical heat equation in bounded domains

We consider the Dirichlet problem for the energy-critical heat equation \begin{equation*} \begin{cases} u_t=Δu+u^5,~&\mbox{ in } Ω\times \mathbb{R}^+,\\ u(x,t)=0,~&\mbox{ on } \partial Ω\times \mathbb{R}^+,\\ u(x,0)=u_0(x),~&\mbox{ in } Ω, \end{cases} \end{equation*} where $Ω$ is a bounded smooth domain in $\mathbb{R}^3$. Let $H_γ(x,y)$ be the regular part of the Green function of $-Δ-γ$ in $Ω$, where $γ\in (0,λ_1)$ and $λ_1$ is the first Dirichlet eigenvalue of $-Δ$. Then, given a point $q\in Ω$ such that $3γ(q)<λ_1$, where $$ γ(q)=\sup\{ γ>0: H_γ(q,q)>0 \}, $$ we prove the existence of a non-radial global positive and smooth solution $u(x,t)$ which blows up in infinite time with spike in $q$. The solution has the asymptotic profile $$ u(x,t)\sim 3^{\frac{1}{4}} \bigg(\frac{μ(t)}{μ(t)^2+|x-ξ(t)|^2}\bigg)^{\frac{1}{2}} \quad \text{as}\quad t \to \infty, $$ where $$ -\ln μ(t)= 2γ(q) t(1+o(1)),\quad ξ(t)=q+O\big(μ(t)\big) \quad \text{as}\quad t \to \infty. $$

math.AP↗

arXiv2023

Existence of finite time blow-up in Keller-Segel system

Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system $\begin{equation} \begin{cases} u_{t} =Δu - \nabla \cdot(u \nabla v) \ \ \ \text{in } \mathbb{R}^2\times(0,T),\\[5pt] v = (-Δ_{\mathbb{R}^2})^{-1} u := \displaystyle\frac {1}{2π} \displaystyle\int_{\mathbb{R}^2} \log \frac {1}{|x-z|}u(z,t) dz, \ \ \ \ \ \ \ \ \ (\star)\\[5pt] u(\cdot ,0) = u_{0}^{\star} \ge 0 \ \ \ \text{in } \mathbb{R}^2. \end{cases} \end{equation}$ We show that there exists $\varepsilon>0$ such that for any $m$ satisfying $8π<m\le 8π+\varepsilon$ and any $k$ given points $q_{1},...,q_{k}$ in $\mathbb{R}^{2}$ there is an initial data $u_0^*$ of $(\star)$ for which the solution $u(x,t)$ blows-up in finite time as $t\to T$ with the approximate profile $$u(x,t)=\sum_{j=1}^{k}\frac{1}{λ_{j}^{2}(t)}U\left(\frac{x-ξ_{j}(t)}{λ_{j}(t)}\right)(1+o(1)), U(y)=\frac{8}{(1+|y|^{2})^{2}},$$ with $λ_{j}(t) \approx 2e^{-\frac{γ+2}{2}}\sqrt{T-t}e^{-\sqrt{\frac{|\ln(T-t)|}{2}}} $ where $γ=0.57721...$ is the Euler-Mascheroni constant, $ξ_{j}(t)\to q_{j}\in \mathbb{R}^{2}$ and such that $\int_{\mathbb{R}^2}u(x,t)dx=km.$ This construction generalizes the existence result of the stable blow-up dynamics recently proved in \cite{CGMN1,CGMN2}.

math.AP↗