PDE in Barcelona

Multi-peak solutions for the non local Schrödinger equation with Dirichlet datum

We will analyze the existence and non existence of multi-spike solutions for a non local Schrödinger type equation in a bounded domain, under Dirichlet boundary conditions. We will pay special attention to the location of the peaks, and the interaction among them.

Multifractality in the evolution of vortex filaments

Vortex filaments that evolve according to the binormal flow are expected to exhibit turbulent properties. Aiming to quantify this, I will discuss the multifractal properties of the family of functions $$ R_{x_0}(t) = \sum_{n \neq 0} \frac{e^{2\pi i ( n^2 t + n x_0 ) } }{n^2}, \quad x_0 \in [0,1], $$ that approximate the trajectories of regular polygonal vortex filaments. These functions are a generalization of the classical Riemann's non-differentiable function, which we recover when $x_0 = 0$. I will highlight how the analysis seems to critically depend on $x_0$, and I will discuss the important role played by Gauss sums, a restricted version of Diophantine approximation, the Duffin-Schaeffer theorem, and the mass transference principle.

This talk is based on the article doi.org/10.1007/s00208-024-02971-0 in collaboration with Valeria Banica (Sorbonne Université), Andrea R. Nahmod (University of Massachusetts) and Luis Vega (BCAM, UPV/EHU).

Dark-bright solitons to one-dimensional defocusing Schrödinger systems

Solitons are localized structures that propagate at constant speed without changing shape. A visual analogy would be a traveling water wave in a narrow channel. These structures typically arise in physical systems governed by coupled nonlinear Schrödinger equations, such as mixtures of Bose-Einstein condensates (BECs) and multimode nonlinear optics. Solitons are classified as dark or bright, depending on whether the density is a positive constant or zero at infinity, and emerge under repulsive (defocusing) or attractive (focusing) interactions, respectively. Experimental and numerical studies confirm the existence of stable dark-bright solitons, even when self-defocusing interactions are present in the bright component. In this talk, we will present recent analytical results that rigorously validate this phenomenon. Specifically, we will show that symmetric and radially monotone (in modulus) dark-bright solitons can be derived as energy minimizers subject to a momentum-mass constraint.

Yau's conjecture for (non)local minimal surfaces

We will explain a recent result on the existence of infinitely many nonlocal minimal surfaces, obtained as min-max critical points of a canonical definition of nonlocal perimeter, on any closed Riemannian manifold. We will moreover describe further directions regarding the convergence of nonlocal minimal surfaces to classical minimal surfaces, as well as a Weyl-type Law for these objects, which combined lead to a new approach to Yau's conjecture on the existence of classical minimal surfaces in three-dimensional closed manifolds.

The talk is based on work with Michele Caselli and Joaquim Serra.

Sharp Lipschitz Estimates for Solutions to the Poisson Equation Involving Surface Measures

The Poisson equation with measure data has been extensively studied since the 20th century, and its general theory is now well understood. For finite measures, Littman, Stampacchia, and Weinberger (1963) showed in their seminal work that $L^1$-weak solutions belong to the Sobolev space $W^{1,p}_{\text{loc}}$, for all $p \in [1, \tfrac{n}{n-1})$, via the duality method. A central question in regularity theory is to determine conditions on the measure under which weak solutions are in $W^{1,\infty}_{\text{loc}}$, that is, locally Lipschitz. Classical potential theory shows that such regularity holds when the localized Riesz potential of the measure is bounded.

In this talk, we will focus on surface measures, that is, measures supported on $(n-1)$-dimensional manifolds, which represent a borderline case for this Lipschitz criterion. Our main goal is to identify minimal regularity conditions on the surface that ensure the Lipschitz continuity of weak solutions. We will see that Dini spaces play a central role in this theory, and we will present sharp counterexamples that demonstrate the minimality of our assumptions.

This is joint work with Iñigo U. Erneta (Rutgers University).

Concentration inequality on holomorphic polynomials

In the space of one-dimensional complex polynomials of bounded degree, the reproducing kernels are the unique optimizers of both the local concentration in domains and of a global version using a Wehrl-type entropy. The latter result means that the Wehrl entropy of quantum Bloch states is minimum when it is a coherent state.

In this talk, we will focus on the stability of the previous extremal properties. Namely, if the local concentration is close to the maximal one among all sets of a given measure, we quantify how close the set and the polynomial are to a disc and to a reproducing kernel, respectively.

This is a joint work with Joaquim Ortega-Cerdà (UB-CRM).

An improved quantitative fractional isoperimetric inequality

We present an improved version of the quantitative fractional isoperimetric inequality, in which a stronger notion of asymmetry appears. In particular, we show that the square root of the isoperimetric deficit controls, not only the Fraenkel asymmetry, but also a sort of "oscillation of the boundary". In the classical local setting the analogue result was obtained by Fusco and Julin. This is a joint project with E. M. Merlino and B. Ruffini.

Instability, non-uniqueness and global smooth solutions for SQG

In this talk, we prove a sharp non-uniqueness result for the forced $\alpha$-SQG equation, encompassing the 2D Euler equation ($\alpha = 0$), the Surface Quasi-Geostrophic (SQG) equation ($\alpha = 1$), and the intermediate cases. A key step in the analysis is the construction of non-linearly unstable vortices that are smooth and compactly supported. As a by-product, we also show the existence of global smooth solutions to the unforced $\alpha$-SQG equation that are neither rotating nor traveling.

Renormalizing singular SPDEs via spectral gap inequalities

In singular stochastic PDEs, approximations by mollification fail to converge due to the low regularity of the noise, which creates divergencies in some nonlinear terms. “Renormalizing” is removing such divergencies, adding divergent counterterms to the equation, in a way that convergence is restorted. Within the theory of regularity structures (Hairer 2014), we introduce a systematic renormalization method assuming that the noise satisfies a spectral gap inequality. This inequality is well-suited for our purpose for two reasons: On the one hand, the expectation of the noise functionals determines the counterterms, thus fitting a BPHZ-type choice of renormalization; on the other, the "noise" derivative of the functional controls the variance in a more regular space of modelled distributions. Based on joint work with Felix Otto, Markus Tempelmayr and Pavlos Tsatsoulis.

Vortex caps on the rotating unit sphere

In this talk, we will analytically study the existence of periodic vortex cap solutions for the homogeneous and incompressible Euler equations on the rotating unit 2-sphere, which was numerically conjectured by Dritschel-Polvani and Kim-Sakajo-Sohn. Such solutions are piecewise constant vorticity distributions, subject to the Gauss constraint and rotating uniformly around the vertical axis. The proof is based on the bifurcation from zonal solutions given by spherical caps. This is a collaboration with Z. Hassainia and E. Roulley.

A Schiffer-type problem for annular domains

The so-called Schiffer conjecture can be stated as follows:

If a nonconstant Neumann eigenfunction of the Laplacian on a smooth bounded domain is constant on the boundary, then the domain is a ball.

In this talk we will consider a version of such question for domains with disconnected boundary. Specifically, we consider Neumann eigenfunctions that are locally constant on the boundary, and we wonder if the domain has to be necessarily a disk or an annulus. We will point out that this question shares some rigidity features inherent to the original Schiffer conjecture.

However, we will show that the answer is negative: there are nonradial Neumann eigenfunctions which are locally constant on the boundary of the domain. The proof uses a local bifurcation argument in a suitable framework that avoids a problem of loss of derivatives. This is joint work with A. Enciso, A. J. Fernández and P. Sicbaldi.

A PDE construction of singular Hamiltonian-stationary surfaces

I will introduce the subject of Lagrangian surfaces in $\mathbb{C}^2$ which are stationary for the area with respect to variations by Hamiltonian vector fields. As for classical minimal submanifolds, these surfaces satisfy a monotonicity formula (due to Schoen and Wolfson) and their singularities are modeled on cones, but these may be non-flat already in two dimensions and exhibit an interesting geometric behavior. In a joint work with F. Gaia and T. Rivière, we present a new PDE characterization of these surfaces and use it to construct families of Hamiltonian-stationary Lagrangian surfaces gluing arbitrarily many such conical singularities. The construction is based on a variational method relying on an unexpected connection with the sharp $L^\infty$ Wente inequality.

Streamline Geometries of Steady Euler Flows

Steady states of the two-dimensional Euler equations generally come in infinite-dimensional families and play an important role in the long-time dynamics of generic initial data. In this talk we will review several classes of steady states, we will discuss recent results on their structures and we will provide a geometric characterization of them in the periodic channel and annulus.

Local smoothing estimates for the wave equation

The local smoothing phenomenon for the wave equation consists in obtaining a regularity gain over the known fixed-time $L^p$ estimates for the half-wave propagator if one takes an $L^p$-integration in time over the interval $[1,2]$. This phenomenon was first observed by Sogge, who conjectured the sharp regularity gain depending on the Lebesgue exponent. Obtaining the conjectured bounds has become a central problem in Harmonic Analysis. It continues open in dimensions 3 and higher, but there has been lots of partial progress in recent years.

In this talk we present a fractal version of that conjecture, in which the integration is now taken over an arbitrary subset $E$ of $[1,2]$. This new family of conjectured inequalities "interpolates" between the fixed-time estimates and the standard local smoothing estimates. The conjectured regularity gain for each Lebesgue exponent depends on a quantity involving the Assouad spectrum of $E$ and the Legendre transform. We provide positive evidence for this conjecture, verifying it for radial functions and any Lebesgue exponent, and for arbitrary functions in the off-diagonal Strichartz regime. This is joint work with Joris Roos, Alex Rutar and Andreas Seeger.

Mathematical modelling in adsorption processes

In this talk, we will present recent advances in the mathematical modeling of adsorption processes, which are widely used to remove contaminants from liquids and gases. This field has traditionally been dominated by experimental and numerical approaches, whose applicability is often limited to the laboratory scale.

We begin by introducing the system of partial differential equations that arises from coupling fluid dynamics with surface reactions. Many parameters in these models cannot be directly measured and are typically estimated by fitting experimental data to theoretical predictions. One of the key challenges is developing models that remain valid across both laboratory and industrial scales. A common simplification in modeling is to assume that adsorbent materials are homogeneous, even though real materials often exhibit significant heterogeneity. In this talk, we will present results that quantify the impact of such heterogeneities on adsorption performance, using asymptotic homogenisation techniques.

On $C^1$ regularity for degenerate elliptic equations in the plane

I will present joint work with Thibault Lacombe, where we show that Lipschitz solutions $u$ of $\mathrm{div}\, G(\nabla u)=0$ in a planar domain are $C^1$, for strictly monotone vector fields $G\in C^0(\mathbb R^2;\mathbb R^2)$ satisfying a very mild ellipticity condition. When the vector field $G$ is the gradient of a strictly convex function, this extends results by De Silva and Savin (Duke Math. J. 2010). When $G$ is not a gradient, the ellipticity assumption needs to be interpreted in a specific way and we provide an example highlighting the nontrivial effect of the antisymmetric part of $\nabla G$.

Normalizing flows and factorizations of transport maps

Sampling from complex distributions is a fundamental challenge in unsupervised learning. A recent line of work models this task via normalizing flows, which are invertible transformations pushing forward a simple distribution (e.g., Gaussian) onto the target one.

Neural ODEs offer a natural way to model continuous transformations while preserving invertibility and differentiability, making them well-suited for approximating transport maps. In this talk, I will present a constructive approach to approximately match probability densities via continuity equations whose velocity fields are defined at each time instance by a shallow (two-layer) neural network.

Existence results to some fully nonlinear problems with Wiener data

We will see some fourth-order problems arising in Physics which model different processes, such as the growth of crystal surfaces and wetting-dewetting processes. Mathematically speaking, we will focus on global existence and regularity results for problems as $$ u_t=F(t,x,\nabla u,\ldots,\Delta^2 u) \text{ in } [0,T]\times\mathbb{T}^N,$$ $$ u(0,x)=u_0(x) \text{ in }\mathbb{T}^N, $$ where $\mathbb{T}^N=[-\pi,\pi]^N$ is the $N$-dimensional torus and the initial data $u_0$ belong to Wiener spaces. The particular choices of $F$ will describe the model in object.

These results are contained in doi.org/10.1016/j.nonrwa.2024.104137 and in a joint work with R. Granero Belinchón doi.org/10.3934/dcds.2019088.

The Muskat problem in critical spaces

The Muskat equation describes the movement of two immiscible and incompressible fluids propagating in a porous medium in the presence of gravity. We will discuss the small-data critical-regularity theory for this quasilinear equation and, in particular, we will focus on initial interfaces with corners.

Nonlocal interaction kernels inference in nonlinear gradient flow equations

When applying nonlinear aggregation-diffusion equations to model real life phenomenon, a major challenge lies on the choice of the interaction potential. Previous numerical and theoretical studies typically required predetermination of terms and the goal is often to reproduce the observed dynamics qualitatively, not quantitatively. In this talk, we address the inverse problem of identifying nonlocal interaction potentials in nonlinear aggregation-diffusion equations from noisy discrete trajectory data. Our approach involves formulating and solving a regularised variational problem, which requires minimising a quadratic error functional across a set of hypothesis functions. A key theoretical contribution is our novel stability estimate for the PDE, validating the error functional ability in controlling the 2-Wasserstein distance between solutions generated using the true and estimated interaction potentials. We demonstrate the effectiveness of the methods through various 1D and 2D examples showcasing collective behaviours

Plateau’s problem and surfaces of minimal capacity

We will discuss a novel family of free boundary problems that arise in the study of variational problems with topological constraints in their level sets. Our two main examples are the classical problem surfaces of minimal capacity spanning a given wire frame and the Allen-Cahn approximation to solutions of the Plateau problem. The heart of the matter in this reformulation of classical variational problems is the notion of spanning, which corresponds to a measure theoretical version of the so-called homotopic spanning condition introduced by Harrison and Pugh. Based on joint works with Anna Skoborogatova, Michael Novack, and Francesco Maggi.

Smooth nonradial stationary 2d Euler flows with compact support

In this talk we will show how to construct nonradial classical solutions to the 2d incompressible Euler equations. More precisely, for any positive integer $k$, we will see how to construct compactly supported stationary Euler flows of class $C^k(\mathbb{R}^2)$ which are not locally radial. The talk is based on a joint work with Alberto Enciso (Madrid) and David Ruiz (Granada).

Computer-Assisted Techniques as Tools for Stability Problems

We will focus on the use of computer-assisted techniques in the stability study of self-similar blowup solutions, especially in the case of stability up to finite codimension. We will present different techniques both for exact solutions and for approximate solutions with very small residual error. The techniques are general enough to be applied to various types of PDEs (compressible Euler equations, focusing/defocusing NLS, relativistic Euler equations, Nonlinear Wave Equation, CCF...). If time permits, we will also briefly discuss other possible applications (such as self-similar non-uniqueness in the Jia-Sverak scenario, or stability of solitons).