arXiv subjects

Javier Gómez-Serrano

Publications and source records attributed to Javier Gómez-Serrano.

At least 19 recordsLinked to original sources

arXiv2026

Contractivity of Rudin's diagonal restriction operator for $H^p$ of the polydisc

We establish that Rudin's diagonal restriction operator is contractive from the Hardy space $H^p$ of the bidisc to the Bergman space $A^p$ of the disc for $2\leq p<\infty$, and characterize all cases of equality. We extend the result to polydiscs and obtain applications to polynomial point evaluation among others. The proof relies on an explicit formula, deduced from certain Green's formulas, for the defect between the $p$th powers of the two $L^p$ norms.

math.FA↗

arXiv2026

Counterexamples to Grad's conjecture

For each sufficiently large $N$, we construct smooth solutions of the magnetohydrostatic equations on embedded solid tori whose regular pressure levels are nested tori, whose magnetic field vanishes exactly on a round magnetic axis, and whose group of Euclidean symmetries, even when isometries reversing the sign of the field are admitted, is exactly the cyclic group $C_N$ of rotations through multiples of $2π/N$. For each such $N$, these equilibria form a smooth one-parameter family which is locally nontrivial in the moduli space of embedded configurations and have none of the plane-reflection, axial, or helical symmetries conjectured by Grad. A Lean 4 certification of the above results is also provided.

math.AP↗

arXiv2026

Blow-up for energy supercritical defocusing nonlinear Schrödinger equations in dimensions three, four and five

We prove finite time blow-up from smooth, radial, compactly supported initial data for the energy supercritical defocusing nonlinear Schrödinger equation $i\partial_tu+Δu-|u|^{p-1}u=0$ in $\mathbb R^d$, for $(d,p)\in\{(3,27),(4,9),(5,7)\}$. The solutions are asymptotically self-similar, their $L^\infty$ norm grows at the scaling rate $(T-t)^{-1/(p-1)}$, and their critical Sobolev norm $\dot H^{s_c}$ diverges. This closes the conjecture of Bourgain \cite{Bourgain2000} on global well-posedness and scattering for the defocusing energy supercritical equation in dimensions three and four. The proof relies on a shooting argument and is computer-assisted.

math.AP↗

arXiv2026

Double-wells in a strong magnetic field: absence of tunneling for infinitely many values of the coupling constant

We prove the fixed-potential conjecture of Fefferman, Shapiro, and Weinstein for inversion-symmetric magnetic double wells. We construct a number $L_0>0$ and one smooth, compactly supported, nonpositive, nonradial single-well potential $v$, independent of both the coupling parameter $λ$ and the well displacement, such that for every fixed $L\ge L_0$ the associated magnetic double-well Hamiltonian has infinitely many exact degeneracies of its two lowest eigenvalues as $λ\to\infty$. The magnetic hopping coefficient also vanishes along an infinite sequence tending to infinity. The potential consists of a radial core and two small, reflection-related perturbations supported on one-sided quadratic cusps. Their log-flat profiles isolate two dominant hopping contributions, which we evaluate by steepest descent after inserting the radial ground-state asymptotics. The resulting asymptotic cosine expression, together with error bounds for the even--odd splitting, yields infinitely many exact eigenvalue crossings and changes of ground-state parity.

math.AP↗

arXiv2026

An isoperimetric problem for Fourier zeros of centrally symmetric convex bodies

Given a centrally symmetric convex body, consider the zero set of the Fourier transform of its characteristic function. We study how large the distance from this set to the origin can be when the volume of the body is fixed. A 2009 conjecture of Benguria, Levitin, and Parnovski asserts that the maximum is attained by a Euclidean ball. We disprove it in all dimensions greater than one. In the plane, every regular centrally symmetric polygon with at least twelve sides outperforms the disk, albeit by a tiny margin, and the regular dodecagon is best among them. In dimensions three and higher, no maximiser exists.

math.MG↗

arXiv2026

On The Eaton-Moretó Conjecture for Principal Blocks of Finite Groups

Let $G$ be a finite group and let $p$ be a prime. If $P$ is a nonabelian Sylow $p$-subgroup of $G$ and $m(P)$ is the smallest non-linear irreducible character degree of $P$, we prove that there exists $χ\in {\rm Irr}(G)$ in the principal $p$-block of $G$ such that $1<χ(1)_p\le m(P)$, giving one inequality of the Eaton-Moretó conjecture for principal blocks. This, assuming Dade's Projective conjecture, implies the Eaton-Moretó conjecture for principal blocks.

math.RT↗

arXiv2026

The energy of fractional Allen--Cahn layers in dimension one

We study the energy $\mathcal{E}(s) := E_{s}[Φ_s]$ of the one-dimensional fractional Allen--Cahn layer solution $Φ_s$, defined as the unique odd, increasing solution of $(-Δ)^{s} Φ_s = Φ_s - Φ_s^{3}$ with $Φ_s(\pm\infty)=\pm 1$ and $Φ_s(0)=0$, for $s\in(1/2,1]$. Our main results are a sharp qualitative and quantitative description of the energy $\mathcal{E}$ on this interval. We show that the energy is continuous and strictly decreasing, and we obtain explicit asymptotic expansions at both endpoints. At the upper endpoint we prove $\mathcal{E}(s) = \frac{2\sqrt{2}}{3} + κ_1 (1-s) + o(1-s)$ with an explicit formula for $κ_1$. At the lower endpoint we prove that the energy goes to infinity as $\mathcal{E}(s)=\frac{1}{π(s-1/2)}+O(1)$. The strict decrease is proved with computer assistance. On an interior subinterval it is reduced to finitely many inequalities verified by interval-arithmetic computations. The proofs combine the Cabré--Sire construction of the layer, the minimality theorem of Palatucci--Savin--Valdinoci, an identity for the $s$ derivative of the energy, and a computer-assisted coercivity estimate at explicit approximate layers.

math.AP↗

arXiv2026

Linear instability of a Burgers--Hilbert traveling wave

We study the stability of traveling wave solutions to the Burgers--Hilbert equation on $\mathbb{T}$ in the regime of small frequency $ω$ and large wave speed $c$. For $ω= 3$ and $c \approx 1.1$, we show that the linearized operator around these solutions has an eigenvalue with negative real part, indicating spectral instability. Our approach is computer-assisted: we reduce the problem to a finite-dimensional system and solve it rigorously using interval arithmetic. The Burgers--Hilbert equation arises as a quadratic approximation of the vortex patch problem for the two-dimensional Euler equations. In this setting, our results point to the instability of threefold symmetric V-states.

math.AP↗

arXiv2026

Nested nodal loops of biharmonic functions

Given any $n\in\mathbb{N}$, we construct a real-valued biharmonic polynomial on $\mathbb{R}^2$ whose zero set contains a nest of $n$ smooth, disjoint topological loops, meaning that the $k$-th loop lies inside the domain bounded by the $(k+1)$-st loop for $k=1,\ldots,n-1$. The case $n=2$, i.e., the existence of two nested loops, is related to the failure of the Boggio-Hadamard conjecture from the early 1900s.

math.AP↗

arXiv2026

Monotonicity of the first Dirichlet eigenvalue of regular polygons

In this paper we prove that the first Dirichlet eigenvalue $λ_1^N$ of an $N$-sided regular polygon of fixed area is a monotonically decreasing function of $N$ for all $N \geq 3$, as well as the monotonicity of the quotients $\displaystyle \frac{λ_1^{N}}{λ_1^{N+1}}$. This settles a conjecture of Antunes-Freitas from 2006 [P. Antunes, P. Freitas, Experiment. Math., 15(3):333-342, 2006].

math.SP↗

arXiv2025

Homogeneous steady states for the generalized surface quasi-geostrophic equations

We consider homogeneous (stationary self-similar) solutions to the generalized surface quasi-geostrophic (gSQG) equations parametrized by the constant $0<s<1$, representing the 2D Euler equations ($s=1$), the SQG equations $(s=1/2)$, and stationary equations ($s=0$); namely, solutions whose stream function $ψ$ and advected scalar $ω$ are of the form \begin{align*} ψ=\frac{w(θ)}{r^β},\quad ω=\frac{g(θ)}{r^{β+2s}}, \end{align*} in polar coordinates $(r,θ)$ with parameter $β\in \mathbb{R}$. We classify homogeneous steady states across the full parameter space, and we identify the limiting singular regimes assuming an odd symmetric profile $(w,g)$ with Fourier modes larger than $m_0\geq 1$. Specifically, we show existence of such solutions for $-m_0-2s<β<-2s$ and $0<β<m_0+2$ ($1/2-s<β< m_0+2$ for $0<s<1/2$) and non-existence of such solutions for $-2s\leq β\leq 0$. The main result provides examples of self-similar solutions which belong to critical and supercritical regimes for the local well-posedness of the gSQG equations for $0<s<1$ and the first examples of self-similar solutions for the SQG equations and the more singular equations $0<s\leq 1/2$ in the stationary setting. We also complement our findings with a numerical illustration of the solutions.

math.AP↗

arXiv2025

Mathematical exploration and discovery at scale

AlphaEvolve (Novikov et al., 2025) is a generic evolutionary coding agent that combines the generative capabilities of LLMs with automated evaluation in an iterative evolutionary framework that proposes, tests, and refines algorithmic solutions to challenging scientific and practical problems. In this paper we showcase AlphaEvolve as a tool for autonomously discovering novel mathematical constructions and advancing our understanding of long-standing open problems. To demonstrate its breadth, we considered a list of 67 problems spanning mathematical analysis, combinatorics, geometry, and number theory. The system rediscovered the best known solutions in most of the cases and discovered improved solutions in several. In some instances, AlphaEvolve is also able to generalize results for a finite number of input values into a formula valid for all input values. Furthermore, we are able to combine this methodology with Deep Think and AlphaProof in a broader framework where the additional proof-assistants and reasoning systems provide automated proof generation and further mathematical insights. These results demonstrate that large language model-guided evolutionary search can autonomously discover mathematical constructions that complement human intuition, at times matching or even improving the best known results, highlighting the potential for significant new ways of interaction between mathematicians and AI systems. We present AlphaEvolve as a powerful new tool for mathematical discovery, capable of exploring vast search spaces to solve complex optimization problems at scale, often with significantly reduced requirements on preparation and computation time.

cs.NE↗

arXiv2025

Discovery of Unstable Singularities

Whether singularities can form in fluids remains a foundational unanswered question in mathematics. This phenomenon occurs when solutions to governing equations, such as the 3D Euler equations, develop infinite gradients from smooth initial conditions. Historically, numerical approaches have primarily identified stable singularities. However, these are not expected to exist for key open problems, such as the boundary-free Euler and Navier-Stokes cases, where unstable singularities are hypothesized to play a crucial role. Here, we present the first systematic discovery of new families of unstable singularities. A stable singularity is a robust outcome, forming even if the initial state is slightly perturbed. In contrast, unstable singularities are exceptionally elusive; they require initial conditions tuned with infinite precision, being in a state of instability whereby infinitesimal perturbations immediately divert the solution from its blow-up trajectory. In particular, we present multiple new, unstable self-similar solutions for the incompressible porous media equation and the 3D Euler equation with boundary, revealing a simple empirical asymptotic formula relating the blow-up rate to the order of instability. Our approach combines curated machine learning architectures and training schemes with a high-precision Gauss-Newton optimizer, achieving accuracies that significantly surpass previous work across all discovered solutions. For specific solutions, we reach near double-float machine precision, attaining a level of accuracy constrained only by the round-off errors of the GPU hardware. This level of precision meets the requirements for rigorous mathematical validation via computer-assisted proofs. This work provides a new playbook for exploring the complex landscape of nonlinear partial differential equations (PDEs) and tackling long-standing challenges in mathematical physics.

math.AP↗

arXiv2025

Quasiperiodic solutions of the generalized SQG equation

This monograph addresses an important problem in mathematical fluid dynamics: constructing stable, long-term solutions to certain quasilinear evolution equations. We implement an elaborate scheme for building global quasiperiodic solutions without relying on external parameters. Instead of relying on artificial external parameters, we exploit the natural structure of initial data to generate families of stable solutions. This approach offers a more robust framework for studying global solutions of quasilinear PDEs. The book combines techniques from KAM theory, a Nash-Moser iteration scheme, and pseudo-differential calculus, and provides tools that extend beyond the specific SQG context and may prove useful for other evolution equations. Specifically, we establish the existence of quasiperiodic patch solutions for the generalized Surface Quasi-Geostrophic equation (SQG) across the parameter range $α\in (1,2)$, in a neighborhood of the disk solution. These solutions exist globally in time without developing singularities, addressing an important question about the behavior of geophysical fluid models. This work provides new insights into global dynamics in a mathematically challenging regime where standard perturbative methods are insufficient. The techniques developed here offer potential applications to other evolution equations in mathematical physics, making this a valuable resource for researchers in partial differential equations, fluid dynamics, and related fields.

math.AP↗

arXiv2025

Non-radial implosion for compressible Euler and Navier-Stokes in $\mathbb{T}^3$ and $\mathbb{R}^3$

In this paper we construct smooth, non-radial solutions of the compressible Euler and Navier-Stokes equation that develop an imploding finite time singularity. Our construction is motivated by the works [Merle, Raphaël, Rodnianski, and Szeftel, Ann. of Math., 196(2):567-778, 2022, Ann. of Math., 196(2):779-889, 2022], [Buckmaster, Cao-Labora, and Gómez-Serrano, arXiv:2208.09445, 2022], but is flexible enough to handle both periodic and non-radial initial data.

math.AP↗

arXiv2025

Smooth imploding solutions for 3D compressible fluids

Building upon the pioneering work [Merle, Raphaël, Rodnianski, and Szeftel, Ann. of Math., 196(2):567-778, 2022, Ann. of Math., 196(2):779-889, 2022, Invent. Math., 227(1):247-413, 2022] we construct exact, smooth self-similar imploding solutions to the 3D isentropic compressible Euler equations for ideal gases for all adiabatic exponents $γ>1$. For the particular case $γ=\frac75$ (corresponding to a diatomic gas, e.g. oxygen, hydrogen, nitrogen), akin to the previous result, we show the existence of a sequence of smooth, self-similar imploding solutions. In addition, we provide simplified proofs of linear stability and non-linear stability, which allow us to construct asymptotically self-similar imploding solutions to the compressible Navier-Stokes equations with density independent viscosity for the case $γ=\frac75$. Moreover, the solutions constructed have density bounded away from zero and converge to a constant at infinity, representing the first example of singularity formation in such a setting.

math.AP↗

arXiv2025

A note on the existence of self-similar profiles of the hydrodynamic formulation of the focusing nonlinear Schrödinger equation

After performing the Madelung transformation, the nonlinear Schrödinger equation is transformed into a hydrodynamic equation akin to the compressible Euler equations with a certain dissipation. In this short note, we construct self-similar solutions of such system in the focusing case for any mass supercritical exponent. To the best of our knowledge these solutions are new, and may formally arise as potential blow-up profiles of the focusing NLS equation.

math.AP↗

arXiv2024

Existence of analytic non-convex V-states

V-states are uniformly rotating vortex patches of the incompressible 2D Euler equation and the only known explicit examples are circles and ellipses. In this paper, we prove the existence of non-convex V-states with analytic boundary which are far from the known examples. To prove it, we use a combination of analysis of the linearized operator at an approximate solution and computer-assisted proof techniques.

math.AP↗