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IMI one-day workshop on PDEs

  • Fechas:

    Del 18/01/19 al 18/01/19

  • Lugar:

    Facultad de Ciencias Matemáticas, UCM, Madrid, España (mapa)

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Workshop on PDEs held at Universidad Complutense de Madrid

Organized by Instituto de Matemática Interdisciplinar

Organizing Comittee:

J.I. Díaz (UCM)

J.L. Vázquez (UAM, UCM)

 

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18
Ene 2019
  • 09:15 - 09:30
    Welcome to participants
  • 09:30 - 10:30
    A. Rodríguez-Bernal: Nonlinear-like behavior of solutions of the linear heat equation in R^N

    By introducing an (optimal) class of initial data which are large at infinity, we show that certain solutions of the linear heat equation in R^N exhibit a nonlinear-like behavior: spatially localized blow-up in finite time or unbounded wild oscillations. Concerning blow-up we even show that some solutions blow-up in finite time only at an arbitrarily prescribed convex set. These phenomena are  due to a mechanism of “mass coming from infinity”. This is a joint work with J. Robinson (U. Warwick). 

  • 10:30 - 11:30
    J.L. Rodrigo: Results motivated by the study of the evolution of isolated vortex lines for 3D Euler

    In the study of an isolated vortex line for 3D Euler one is trying to make sense of the evolution of a curve, where the vorticity (a distribution in this case) is supported, and tangential to the curve. This idealised vorticity generates a velocity field that is too singular (like the inverse of the distance to the curve and therefore not in $L^2$) and making rigorous sense of the evolution of the curve remains a fundamental problem.

    In the talk I will present examples of simple globally divergence-free velocity fields for which an initial delta function in one point (in 2D, with analogous results in 3D) becomes a delta supported on a set of Hausdorff dimension 2.  In this examples the velocity does not correspond to an active scalar equation.

    I will also present a construction of an active scalar equation in 2D, with a milder singularity than that present in Euler for which there exists an an initial data given by a point delta becomes a one dimensional set. These results are joint with C. Fefferman and B. Pooley.

    These are examples in which we have non-uniqueness for the evolution of a singular "vorticity". For the Surface Quasi-Geostrophic equation, an equation with great similarities with 3D Euler, the evolution of a sharp front is the analogous scenario to a vortex line for 3D Euler. I will describe a geometric construction using "almost-sharp" fronts than ensure the evolution according to the equation derived heuristically. This part is joint work with C. Fefferman

  • 11:30 - 11:45
    Coffee Break
  • 11:45 - 12:45
    N. Abatangelo: Between the Laplacian and the bilaplacian: the case of higher-order fractional Laplacians

    We will review some results concerning higher-order fractional Laplacians, which are nonlocal operators of non-integer order greater than 2. These pertain the validity (or rather the failure) of maximum principles, the extension of Boggio’s formula, the structure and the boundary regularity of corresponding harmonics, the formulation of suitable natural boundary conditions, and the pointwise evaluation of the operators. This was the outcome of a collaboration with Sven Jarohs (Frankfurt) and Alberto Saldaña (Karlsruhe).

  • 12:45 - 13:45
    D. Gómez-Castro: The fractional Schrödinger equation with singular potential

    In this talk I will present new results obtained in collaboration with J.I. Díaz and J.L. Vázquez on existence and uniqueness of solution of the fractional Schrödinger equation when a singular non-negative potential V is considered.

    We will study two case: first, the case of singularity on the boundary for the so-called Restricted Fractional Laplacian and, secondly, singularity at interior points for general fractional Laplacians.

    We construct our solutions as limit of the "good" case of bounded data f and potential V.

    We will also pay some attention to the singular data: functions and measures. We extend our results to the optimal class for the fractional Laplacians.

  • 13:45 - 16:00
    Lunch
  • 16:00 - 17:00
    F. del Teso: On numerical approximations of the spectral fractional Laplacian via the method of semigroups

    As it is well-known, there are several (non-equivalent) ways of defining fractional operators in bounded domains. In this talk we will focus on the so-called spectral fractional Laplacian. Following the heat semi-group formula we consider a family of operators which are boundary conditions dependent and discuss a suitable approach for their numerical discretizations. We will also discuss the numerical treatment of the associated homogeneous boundary value problems. In the end we will talk about possible extensions that can be treated with our approach such as non-homogeneous boundary conditions and discretizations of fractional operators in $\mathbb{R}^N$.

    Joint works with: Nicole Cusimano (BCAM), Luca Gerardo-Giorda (BCAM), Gianni Pagnini (BCAM), Jörgen Endal (NTNU) and Espen R. Jakobsen (NTNU).

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