APDE Seminar@UPV/EHU: Sharp blow-up stability of self-similar solutions for the modified Korteweg-de Vries equation and The role of the inhomogeneity on non-radial scattering for nonlinear Schrödinger equations

1) Title: Sharp blow-up stability of self-similar solutions for the modified Korteweg-de Vries equation

Abstract

The evolution of vortex patches subject to Euler's equations can be described using the modified Korteweg-de Vries (mKdV) equation. The formation of singular geometric objects, such as corners or logarithmic spirals, corresponds to self-similar (i.e. scaling-invariant) solutions of (mKdV) at the blow-up time. Self-similar solutions present a number of critical features, from time and space decay to regularity. Moreover, at scaling-critical regularity, an instantaneous energy cascade is known to occur, blocking the derivation of a suitable well-posedness theory.

  We will discuss the blow-up stability of self-similar solutions under arbitrarily large subcritical perturbations. This is a joint work with R. Côte (U. Strasbourg).

2) Title: The role of the inhomogeneity on non-radial scattering for nonlinear Schrödinger equations

Abstract

The concentration-compactness-rigidity method, pioneered by Kenig and Merle, has become standard in the study of global well-posedness and scattering in the context of dispersive and wave equations. Albeit powerful, it requires building some heavy machinery in order to obtain the desired space-time bounds.

  In this talk, we present a simpler method, based on Tao's scattering criterion and on Dodson-Murphy's Virial/Morawetz inequalities, first proved for the 3d cubic nonlinear Schrödinger (NLS) equation. 

  Tao's criterion is, in some sense, universal, and it is expected to work in similar ways for dispersive problems. On the other hand, the Virial/Morawetz inequalities need to be established individually for each problem, as they rely on monotonicity formulae.

  This approach is versatile, as it was shown to work in the energy-subcritical setting for different nonlinearities, as well as for higher-order equations.