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

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.

Name thesis:degree_name

Doctoral

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2024

Author dc:creator

• Cao Labora, Gonzalo

Advisor dc:contributor.advisor

• Staffilani, Gigliola

Statement dc:rights

• In Copyright - Educational Use Permitted
• Copyright retained by author(s)

Licence dc:rights.uri

• https://rightsstatements.org/page/InC-EDU/1.0/

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/157346

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/157346