PhD defense of Haochen YANG

Haochen YANG

Existence, instability and energy partition for linear or nonlinear evolution equations issued from fluid mechanics

Abstract

This thesis contains three themes: microlocal partition of energy, the virial theorem, and the well-posedness of jet flows. The first theme explores how the energy of solutions to dispersive equations is equally distributed inside and outside the light cone. This phenomenon has been frequently used to study the long-term behavior of (non)linear wave equations. We will extend it to a large class of linear dispersive equations, replacing the light cone by a hypersurface in phase space and characterizing the energy partition via pseudo-differential operators. The second theme examines the equipartition of kinetic and potential energies for two-phase water-wave systems. This study leads to a quantitative description of the instability of the system when the lower fluid is no denser than the upper one, which is known as the Rayleigh-Taylor instability and the Kelvin-Helmoltz instability. The third theme addresses the Cauchy problem of the evolution of cylindrical fluids with free boundary, namely jets. We establish the local well-posedness theory for this evolution equation in Sobolev spaces without any symmetric assumptions. This result constitutes a fundamental step towards understanding the long-term behaviour of jet flows.

Keywords

Microlocal analysis,Dispersive equation,Partition of energy,Virial theorem,Rayleigh-Taylor instability,Jets

Supervision

• Thomas ALAZARD
• Jean-Marc DELORT

Jury 

• Anne-Laure DALIBARD, Professeure, Sorbonne Université,  Examinateur
• François ALOUGES, Professeur, École normale supérieure Paris-Saclay, Examinateur
• Mihaela IFRIM, Professeure, University of Wisconsin–Madison, Examinateur
• Daniel TATARU, Professeur, University of California, Berkeley, Examinateur