Noticias en español
Reversal in the Stationary Prandtl Equations.
Abstract: We investigate reversal and recirculation for the stationary Prandtl equations. Reversal describes the solution after the Goldstein singularity, and is characterized by spatio-temporal regions in which $u > 0$ and $u < 0$. The classical point of view of regarding the Prandtl equations as an evolution $x$ completely breaks down. Instead, we view the problem as a quasilinear, mixed-type, free-boundary problem. Joint work with Nader Masmoudi.
Read MoreEnergy decay for classes of nonlocal dispersive equations.
Abstract:We consider the long-time dynamics of large solutions to a special class of evolution equations. Using virial techniques, we describe regions of space where every solution in a suitable Sobolev space must decay to zero along sequences of times. Moreover, in the case of interior regions, we prove decay for a sequence of times. The classes of nonlocal dispersive equations which we will treat are as follows: {∂t u + Lαu + u∂x u = 0, x, t ∈ R,u(x, 0) = u0(x), where α > 0, and the operator Lα is the Fourier multiplier operator by a real-valued odd function belonging to (C 1(R) ∩ C...
Read MoreSobre las propiedades de continuación única discreta y sus aplicaciones al control y problemas inversos.
Abstract: Es bien conocido que las Propiedades de Continuación Única juegan un importante papel en el estudio de los problemas de controlabilidad y problemas inversos. También es natural preguntarse que sucede con las discretizaciones de las EDP y sus propiedades. En esta charla mostraremos que al considerar la discretización, en diferencias finitas, de operadores diferenciales, este tipo de propiedades pierden su validez, por lo que es válido preguntarse si resultados de controlabialidad, identificabilidad y estabilidad en los problemas inversos siguen siendo válidos en el caso discreto....
Read MoreNew mathematical model for Tsunamis with precise time arrival predictions.
Abstract: We propose in this work a new system of equations modeling Tsunamis. It is a coupled system accounting for both water compressibility and viscoelasticity of the earth. Adding these latter physical effects is responsible for the closest-to-reality time arrival predictions (among existing models), capturing the negative peak before the main wave hump, and the exhibition of the negative dispersion phenomena. This comes in remarkable agreement with previous experiments and studies on the topic. The system is also delivered in a relatively simple mathematical structure of equations that...
Read MoreOn the collapse of the local Rayleigh condition for the hydrostatic Euler equations.
Abstract: The hydrostatic Euler equations are derived from the incompressible Euler equations by means of the hydrostatic approximation. Among the different stability criteria that arise in the study of linear stability for the incompressible Euler equations, we can mention Rayeligh’s stability criterion, which gives rise to the local Rayleigh condition. Linear and nonlinear instability of the hydrostatic Euler equations around certain shear flows is well-known, as well as the finite time blow-up of certain solutions that do not satisfy the local Rayleigh condition. On the other hand,...
Read MoreCordes-nirenberg type results for nonlocal equations with deforming kernels.
Abstract. We derive Cordes-Nirenberg type results for nonlocal elliptic equations with deforming kernels using a compactness method.Under a natural integrability assumption for the Monge-Ampere solution, we are able to prove a stability lemma that allows the ellipticity class to vary. As a consequence, we get that the limit equation, up to a rotation, behaves like rough fractional Laplacians where the known regularity theory for this class of equation can be applied.
Read More