Seminario de Ecuaciones Diferenciales

Resumen: En mecánica, las restricciones en las configuraciones de un sistema se denominan "holónomas". Un ejemplo sencillo es la longitud constante del péndulo. Sistemas mecánicos con restricciones en las velocidades que no pueden reducirse a restricciones en las posiciones se llaman "no-holónomas". Un ejemplo clásico es una esfera que rueda sin resbalar en una mesa.

El reto en el estudio de los sistemas mecánicos no-holónomos aparece debido a que las ecuaciones de movimiento no poseen una estructura Hamiltoniana. En su lugar, la dinámica es descrita en términos de un corchete de funciones que no satisface la identidad de Jacobi. Hablamos entonces de una "corchete casi-Poisson".

La pérdida de la identidad de Jacobi da lugar a fenómenos que no son posibles en los sistemas Hamiltonianos clásicos. Algunas preguntas abiertas en el área de mecánica no-holónoma incluyen determinar condiciones para la existencia de una medida conservada y de existencia de equilibrios asintóticos, relación entre simetrías y leyes de conservación, reducción e integrabilidad.

En la primera parte de la charla presentaré una introducción básica a los sistemas no-holónomos rica en ejemplos y después me concentraré en el problema de existencia de medidas invariantes suaves para estos sistemas.

Resumen: In this talk I will discuss some aspects on the leapfrogging phenomenon in the vortex dynamics for Euler equations in the plane subject to a linear shear flow. We show the existence of non rigid time periodic solutions in the local frame which is translating uniformly. We use some techniques borrowed from KAM theory. This is a joint work with Zineb Hassainia and Nader Masmoudi.

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Resumen: In this talk we will discuss a class of alignment models with self-propulsion and Rayleigh-type friction forces, which describes the collective behavior of agents with individual characteristic parameters. We describe the long time dynamics via a method which allows to reduce analysis from the multidimensional system to a simpler family of two-dimensional systems parametrized by a proper Grassmannian. With this method we demonstrate exponential alignment for a large (and sharp) class of initial velocity configurations confined to a sector of opening less than $\pi$. In the case when characteristic parameters remain frozen, the system governs dynamics of opinions for a set of players with constant convictions. Viewed as a dynamical non-cooperative game, the system is shown to possess a unique stable Nash equilibrium, which represents a settlement of opinions most agreeable to all agents. Such an agreement is furthermore shown to be a global attractor for any set of initial opinions.

Resumen: In this talk I will speak about periodic solutions with a fixed period of the Lorentz force equation with the Kepler electric potential. The set of T-periodic solutions is the set of Szulkin critical point of the action functional associated to the Poincare lagrangian. Using the Ekeland variational principle and the Lusternik - Schnirelman category we prove that the action functional has infinitely many critical points, so the Lorentz force equation has infinitely many T-periodic solutions with a fixed period T.

Resumen: The theory of viscosity solutions was developed to deal with Hamilton-Jacobi type problems in the late 80s and 90s by Crandall, Lions, and others. The surname "viscosity" comes from their construction through the vanishing viscosity method. In many scenearios have well-posedness and select the physical solution in many settings. Furthermore, their stability properties makes them suitable to study different approximations: finite differences and asymptotic limits as $t \to \infty$. The aim of this talk is to introduce this notion of solution and show its usefulness to study the time limit of Aggregation-Diffusion equations. The talk presents joint work with: J.A. Carrillo, A. Fernández-Jiménez, and J.L. Vázquez.

Resumen: We will discuss several recent results for aggregation-diffusion equations related to partial concentration of the density of particles. Nonlinear diffusions with homogeneous kernels will be reviewed quickly in the case of degenerate diffusions to have a full picture of the problem. Most of the talk will be devoted to discuss the less explored case of fast diffusion with homogeneous kernels with positive powers. We will first concentrate in the case of stationary solutions by looking at minimisers of the associated free energy showing that the minimiser must consist of a regular smooth solution with singularity at the origin plus possibly a partial concentration of the mass at the origin. We will give necessary conditions for this partial mass concentration to and not to happen. We will then look at the related evolution problem and show that for a given confinement potential this concentration happens in infinite time under certain conditions. We will briefly discuss the latest developments when we introduce the aggregation term. This talk is based on a series of works in collaboration with M. Delgadino, J. Dolbeault, A. Fernández, R. Frank, D. Gómez-Castro, F. Hoffmann, M. Lewin, and J. L. Vázquez.

Resumen: We consider the inhomogeneous incompressible Euler equations describing two fluids with different constant densities under the influence of gravity. Initially the fluids are supposed to be at rest and separated by a flat horizontal interface with the heavier fluid being on top of the lighter one. Due to gravity this configuration is unstable, the two fluids begin to mix in a more and more turbulent way. This is called the Rayleigh-Taylor instability. In the talk we will see how admissible solutions to the Euler equations reflecting a turbulent mixing of the two fluids in a quadratically growing zone can be constructed. Furthermore, if time allows, we will discuss an arising selection problem for the averaged motion of solutions. This is based on joint works with József Kolumbán, László Székelyhidi, Jonas Hirsch.

Resumen: We construct multiple solutions to the Liouville type equation  $$ (-\Delta)^{\frac12} u = k(x) e^u, \quad \textup{ in } \mathbb{\mathbb{R}} $$ More precisely, for $k$ of the form $k(x) = 1+\epsilon\kappa(x)$ with $\epsilon \in (0,1)$ small and $\kappa \in C^{1,\alpha}(\R) \cap L^{\infty}(\mathbb{\R})$ for some $\alpha > 0$, we prove the existence of multiple solutions to the above equation bifurcating from the so-called Aubin-Talenti bubbles. These solutions provide examples of flat metrics in the half-plane with prescribed geodesic curvature $k(x)$ on its boundary. Moreover, they imply the existence of multiple ground state soliton solutions for the Calogero-Moser derivative NLS. The talk is based on joint works with L. Battaglia (Roma), M. Cozzi (Milano) and A. Pistoia (Roma).

Resumen: Para grafos con peso hay su ciente literatura sobre el corte de Cheeger y el problema de Cheeger, pero para los grafos métricos hay pocos resultados sobre estos problemas. Nuestro objetivo es estudiar el corte Cheeger y el problema de Cheeger en grafos métricos. Para eso, necesitamos introducir los conceptos de variación total y perímetro en grafos métricos, de tal forma que tengan en cuenta los saltos en los vértices de las funciones de la variación acotada. También necesitamos una fórmula de integración por partes. Además, estudiamos el problema del valores propios para operador 1-laplacio en grafos métricos, mediante el cual damos un método para resolver el problema de corte óptimo de Cheeger.

Resumen: Our aim is to study a nonlocal Cahn-Hilliard model (CHE) in the framework of random walk spaces, which includes as particular cases, the CHE on locally finite weighted connected graphs, the CHE determined by finite Markov chains or the Cahn-Hilliard Equations driven by convolution integrable kernels. We consider different transitions for the phase and the chemical potential, and a large class of potentials including obstacle ones. We prove existence and uniqueness of solutions in L1 of the Cahn-Hilliard Equation. We also show that the Cahn-Hilliard equation is the gradient flow of the Ginzburg-Landau free energy functional on an appropriate Hilbert space. We finally study the asymptotic behaviour of the solutions. Joint work with J. Toledo

Resumen: Nos proponemos estudiar algunas relaciones entre el grado de Brouwer de un campo vectorial y la dinámica del flujo inducido. Estamos interesados, en particular, en las propiedades dinámicas y topológicas de los conjuntos invariantes aislados y de sus variedades inestables. Estudiamos también relaciones análogas para el índice de un campo vectorial y obtenemos de este modo nuevas formas del teorema de Poincaré-Hopf. También obtenemos algunas consecuencias relativas a los teoremas antipodales de Borsuk y de Hirsch. Como aplicación, calculamos el grado de Brouwer y el índice de campos vectoriales en algunas situaciones topologicamente relevantes, obtenemos criterios para la detección de orbitas conectantes en las descomposiciones atractor-repulsor de los conjuntos invariantes aislados y calculamos el grado de Brouwer del campo vectorial de las ecuaciones de Lorenz en bloques aislantes del conjunto extraño. Estos resultados han sido obtenidos en colaboración con Héctor Barge.

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Resumen: In this preliminary work we will concern about the "Finite Morse Index" scenario (instead of the usual stability one) of the following long-standing conjecture: "Let u be a (compactly supported) weak stable solution of the general non-linear Poisson equation and assume that the non-linearity is positive, non-decreasing, convex, and superlinear at +∞, and let n<10. Then u is bounded." Recently, Cabré, Figalli, Ros-Oton and Serra end up a complete proof in the classical stability setting: W^{1,2}-stable solutions are universally bounded for n<10 (and therefore smooth by classical elliptic regularity theory); namely they are bounded in terms only of their L^1 norm, with a constant that is independent of the non-linearity. This conjecture is in a sense equivalent to another problem stated before by Brezis: Is it possible to prove that the extremal solution of the so-called Gelfand problem is smooth at least in low dimensions?. We will take this as the starting point to reproduce our new results recovering the existing results by S. Villegas in the (semi-)stable case. Using some estimates on the "size" of the local stability behavior of finite Morse Index solutions we provide a uniform a priori bound and some pointwise estimates of that solutions that partially answer positively the long standing conjecture in this more general setting.

Resumen: We propose a numerical method for a Vlasov-Poisson-Fokker-Planck model and prove quantitative results ensuring that it is Asymptotic-Preserving for the linearized model in both the macroscopic and the long time regime. We illustrate these results with various numerical experiments in which we observe, among others, transition phase between macroscopic and long time behavior as well as oscillations and instability phenomena.

Resumen: We consider the Gauss equation that governs the minimal immersion of a closed surface of genus greater than two in a hyperbolic three manifold for which the second fundamental form is prescribed. The PDE admits two solutions and we will analyze the behavior of such solutions when the norm of the second fundamental form is small.

Resumen: We provide a qualitative explanation of the co-orbital motion of two small moons orbiting a planet. The two small bodies revolve about the planet in nearly circular orbits with almost equal radii. The system is modelled as a planar three-body problem whose Hamiltonian is expanded as a perturbation of two uncoupled Kepler problems. A combination of averaging, normal form, symplectic scaling and Hamiltonian reduction theories and the application of a KAM theorem for high-order degenerate systems allows us to establish the existence of quasi-periodic motions and KAM 4-tori related to the co-orbital motion of the moons. By conveniently selecting a suitable region of the reduced phase space (which is the Cartesian product of a two-dimensional sphere and one sheet of a two-sheet hyperboloid of revolution), we are able to establish the existence of these quasi-periodic motions  that are valid for any value of an action variable, related to the angular momenta of the two moons. This is a joint work with Josep M. Cors and Patricia Yanguas.

Resumen: We will present the derivation of a nucleation boundary condition to the Lifshitz-Slyozov equation, the well-posedness of the Cauchy problem as well as the long-time behaviour of the solutions.

Resumen: We present results on a stochastic version of a well-known kinetic nucleation model for phase transition phenomena. In the Becker-Döring model, aggregates grow or shrink by addition or removal of one-by-one particle at a time. Under certain conditions, very large aggregates emerge and are interpreted as a phase transition. We study stationary and quasi-stationary properties of the stochastic Becker-Döring model in the limit of infinite total number of particles, and compare with results from the deterministic nucleation theory. Our findings are largely inspired from recent results from stochastic chemical reaction network theory.

Resumen: In this talk, we present the existence of nontrivial simply connected domains of the sphere such that the overdetermined elliptic problem admits a positive solution by a local bifurcation argument. This shows in particular that Serrin's theorem for overdetermined problems in the Euclidean space cannot be generalized to the sphere even for simply connected domains.

Resumen: The existence of a local curve of corotating vortex pairs was proven by Hmidi and Mateu via a desingularization of a pair of point vortices. In this talk, we will present a global continuation of these local curves. That is, we consider solutions which are more than a mere perturbation of trivial solution. Indeed, while the local analysis relies on the study of the linear equation at the trivial solution, the global analysis requires on a deeper understanding of topological properties of the nonlinear problem. For our proof, we adapt the powerful analytic global bifurcation theorem due to Buffoni and Toland, to allow for the singularity at the bifurcation point. This is a collaboration with Susanna V. Haziot.

Resumen: Motivated by a popular kinetic model by Saintillan and Shelley for the dynamics of suspensions of active elongated particles, we study phase-mixing and enhanced dissipation on the sphere. In particular, we show that, up to log errors, the phase mixing estimate persists until  the enhanced dissipation takes over. This is proved by combining an optimized hypocoercive approach with the vector field method. Joint work with Michele Coti Zelati and David Gérard-Varet (https://arxiv.org/abs/2207.08431).

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