Self-similar singularity formation and wellposedness theory for compressible fluids and dispersive PDE

Abstract

In this thesis, we study different problems related to singularity formation and local wellposedness of fluid equations and dispersive PDE. Regarding singularity formation, we construct radially symmetric smooth selfsimilar profiles for the compressible Euler equations which exhibit an implosion type singularity in finite time. This will be the first part of the thesis. The second part of the thesis consists on doing a non-radial stability analysis around those profiles to show singularity formation for adequate small perturbations of the profile. In particular, this stability analysis also allows to conclude existence of singularities for periodic initial data. The stability also allows to obtain singularity formation for the corresponding equation with dissipation: the compressible Navier-Stokes equations. Moreover, the self-similar profiles constructed are also intimately related to dispersive equations, and we will show how to use them to prove finite time singularity formation for some supercritical defocusing NLS equations, using its hydrodynamical formulation. The third part of the thesis consists of the study of a different dispersive equation: the Zakharov– Kuznetsov equation. The equation is a generalization of the KdV equation to higher dimensions with applications in plasma physics. We improve the deterministic local wellposedness in the cyilnder both in the deterministic and the probabilistic setting.