arXiv subjects

Monica Musso

Publications and source records attributed to Monica Musso.

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 $\Omega\subset{\mathbb R}^3$ for the energy functional $$ \mathcal E (\Omega) = {\rm Per}\,(\Omega) + \frac 12 \iint_{\Omega\times \Omega } \frac{dx\,dy}{|x-y|} $$ under the volume constraint $|\Omega|=V$. We look for smooth, embedded, compact surfaces $\partial\Omega$ 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 $V^{\frac 43}$.

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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.

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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 \[ u_t=\Delta u+u^3, \qquad n\geq 5. \] 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 \[ \xi_r(t)\sim \sqrt{2(n-4)(T-t)}. \] At the same time, concentration takes place transversely to the sphere at the much smaller scale \[ \lambda(t)\sim \kappa_* \frac{T-t}{|\log(T-t)|^{\frac n{n-2}}}, \] for some $\kappa_*>0$. More precisely, in cylindrical coordinates $r=|x'|$, $z\in\mathbb R^3$, the leading profile is \[ u(x,t) \sim \frac{1}{\lambda(t)} U\left( \frac{r-\xi_r(t)}{\lambda(t)}, \frac{z}{\lambda(t)} \right), \] 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 $\lambda(t)$. 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.

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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}-\Delta v=|\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.

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arXiv2026

Uncountably many non-rotationally symmetric type II ancient Yamabe flows on the sphere

For every $n \ge 3$, we construct uncountably many families of type II ancient solutions to the Yamabe flow on the unit round $n$-sphere $\Ss^n$. These families are pairwise distinct up to conformal equivalence, and no member is conformally equivalent to a rotationally symmetric solution. At every negative time, the Ricci curvature tensor of each solution is indefinite at some point. Moreover, the associated backward limit space is a wedge sum of finitely many isometric copies of $\Ss^n$. These examples show that the collection of ancient Yamabe flows on $\Ss^n$ has a much richer structure than suggested by two natural comparison problems: the compact ancient Ricci flows on $\Ss^2$, all of which are known to be rotationally symmetric, and the elliptic Yamabe equation on $\R^n$, whose positive entire solutions are only the standard bubbles. The construction uses a non-radial inner--outer gluing scheme. After stereographic projection, we reformulate the flow as a conformally invariant parabolic problem on $\R^n$. By exploiting Kelvin invariance and switching between the Euclidean and spherical formulations as needed, we control the non-radial modes directly without reducing the problem to one space dimension. Weighted H\"older estimates provide the pointwise control needed to establish the Type II behavior, the Ricci-sign property, conformal inequivalence, and the description of the backward limits in a straightforward manner.

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arXiv2026

Bifurcations in Isoperimetric Problems with Nonlocal Interactions

We study isoperimetric problems modeled on the liquid drop model, with nonlocal interactions under a volume constraint. While balls are natural critical points, we show that, for an unbounded sequence of radii, non-spherical solutions bifurcate from the family of balls. These new solutions lie arbitrarily close to balls and can have arbitrarily large volume. Conversely, at radii outside this sequence, no bifurcation occurs, and nearby solutions are trivial, arising only from rigid motions.

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arXiv2026

Clustered vortex helices with compactly supported cross-sectional vorticity in the 3D Euler equations

We consider the three-dimensional incompressible Euler equations for helical flows without swirl. By adapting gluing techniques, we construct the first smooth multi-vortex solution in the whole space $\mathbb{R}^3$ exhibiting a cluster of collapsing helical filaments, with the associated cross-sectional vorticity remaining compactly supported in $\mathbb{R}^2$ for all times. Our result generalises previous collapsing configurations in $\mathbb{R}^3$ with rapidly decaying vorticity cores, and extends related variational solutions obtained in infinite cylindrical domains.

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arXiv2025

Helical vortex filaments with compactly supported cross-sectional vorticity for the incompressible Euler equations in $\mathbb{R}^3$

We revisit the vortex filament conjecture for three-dimensional inviscid and incompressible Euler flows with helical symmetry and no swirl. Using gluing arguments, we provide the first construction of a smooth helical vortex filament in the whole space $\mathbb{R}^3$ whose cross-sectional vorticity is compactly supported in $\mathbb{R}^2$ for all times. The construction extends to a multi-vortex solution comprising several helical filaments arranged along a regular polygon. Our approach yields fine asymptotics for the vorticity cores, thus improving related variational results for smooth solutions in bounded helical domains and infinite pipes, as well as non-smooth vortex patches in the whole space.

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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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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 $\epsilon>0$, and show that the dynamics is governed at leading order as $\epsilon\to0$ 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.

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arXiv2025

Nearly parallel helical vortex filaments in the three dimensional Euler equations

Klein, Majda, and Damodaran have previously developed a formalized asymptotic motion law describing the evolution of nearly parallel vortex filaments within the framework of the three-dimensional Euler equations for incompressible fluids. In this study, we rigorously justify this model for two configurations: the central configuration consisting of regular polygons of $N$ helical-filaments rotating with constant speed, and the central configurations of $N+1$ vortex filaments, where an $N$-polygonal central configuration surrounds a central straight filament.

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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 $\omega_{\varepsilon}(x,t)$ concentrated around points $\xi_{j}(t)$ 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 $\xi_{j}(t)$ follow an expanding self similar trajectory of spirals, with the support of the vorticities contained in balls of radius $3\varepsilon$ around each $\xi_{j}$.

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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 $W(x)=w(r)e^{i\theta}$. Using explicit representation formulae for the Fourier modes in $\theta$, 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.

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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 $\Sigma =\partial \Omega$ in $\mathbb{R}^3$ that is critical for the energy $$ E(\Omega) = {\rm Per\,} (\Omega ) + \frac 12 \int_\Omega\int_\Omega \frac {dxdy}{|x-y|} $$ under the volume constraint $|\Omega| = m$. The term ${\rm Per\,} (\Omega ) $ corresponds to the surface area of $\Sigma$. The associated Euler-Lagrange equation is $$ H_\Sigma (x) + \int_{\Omega } \frac {dy}{|x-y|} = \lambda \quad \hbox{ for all } x\in \Sigma, \quad $$ where $H_\Sigma$ stands for the mean curvature of the surface, and where $\lambda\in\mathbb{R}$ is the Lagrange multiplier associated to the constraint $|\Omega|=m$. 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 $H_\Sigma = \lambda$, the only compact embedded solutions are spheres, as stated by the classical Alexandrov result.

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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, H\'elein, 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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arXiv2023

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

Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system $\begin{equation} \begin{cases} u_{t} =\Delta u - \nabla \cdot(u \nabla v) \ \ \ \text{in } \mathbb{R}^2\times(0,T),\\[5pt] v = (-\Delta_{\mathbb{R}^2})^{-1} u := \displaystyle\frac {1}{2\pi} \displaystyle\int_{\mathbb{R}^2} \log \frac {1}{|x-z|}u(z,t) dz, \ \ \ \ \ \ \ \ \ (\star)\\[5pt] u(\cdot ,0) = u_{0}^{\star} \ge 0 \ \ \ \text{in } \mathbb{R}^2. \end{cases} \end{equation}$ We show that there exists $\varepsilon>0$ such that for any $m$ satisfying $8\pi<m\le 8\pi+\varepsilon$ 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 $$u(x,t)=\sum_{j=1}^{k}\frac{1}{\lambda_{j}^{2}(t)}U\left(\frac{x-\xi_{j}(t)}{\lambda_{j}(t)}\right)(1+o(1)), U(y)=\frac{8}{(1+|y|^{2})^{2}},$$ with $\lambda_{j}(t) \approx 2e^{-\frac{\gamma+2}{2}}\sqrt{T-t}e^{-\sqrt{\frac{|\ln(T-t)|}{2}}} $ where $\gamma=0.57721...$ is the Euler-Mascheroni constant, $\xi_{j}(t)\to q_{j}\in \mathbb{R}^{2}$ 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}.

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arXiv2023

Nodal cluster solutions for the Brezis-Nirenberg problem in dimensions $N\geq 7$

We show that the classical Brezis-Nirenberg problem $$\Delta u + |u|^{4 \over N-2} u + \varepsilon u = 0 ,\quad {\mbox {in}} \quad \Omega, \quad u= 0 , \quad {\mbox {on}} \quad \partial \Omega$$ admits nodal solutions clustering around a point on the boundary of $\Omega$ as $\varepsilon \to 0$, for smooth bounded domains $\Omega \subset \mathbb{R}^N $ in dimensions $N\geq 7$.

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arXiv2023

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 $$ \omega_t + \nabla^\perp \Psi\cdot \nabla \omega = 0 , \quad \Psi = (-\Delta)^{-1} \omega, \quad \hbox{in } \mathbb{R}^2 \times (0,\infty)$$ is a travelling wave solution of the form $\omega(x,t) = W(x_1-ct,x_2 )$ 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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