Patterns in Partial Differential Equations Arising from Fluid Mechanics

Abstract

This dissertation is centered around the existence of time–periodic solutions for Hamiltonian models that arise from Fluid Mechanics. In the first part, we explore relative equilibria taking the form of rigid motion (pure rotations or translations) in the plane with uniform and non uniform distributions for standard models like the incompressible Euler equations or the generalized quasi–geostrophic equation. In the second part, we focus on a similar study for the 3D quasi–geostrophic system. The study of this model shows a remarkable diversity compared to the 2D models due to the existence of a large set of stationary solutions or the variety of the associated spectral problems. In the last part, we show some works in progress of this dissertation, and also some conclusions and perspectives. In what follows, we briefly explain the contents of this thesis and the works contained in it. • Chapter 1 deals with a general introduction to the above mentioned models, the contribution of this dissertation and related literature. It is divided in two sections: two– dimensional Euler equations and three–dimensional quasi–geostrophic system. • Chapter 2 is devoted to the work [67], which is a collaboration with my thesis advisors T. HMIDI and J. SOLER. This work is currently accepted for publication in Archive for Rational Mechanics and Analysis. There, we focus on the existence of non uniform rotating solutions for the 2D Euler equations, which are compactly supported in bounded domains. The main idea is the bifurcation from stationary radial solutions. The system reduces to two coupled nonlinear equations for the shape of the support and the density inside it. We will deeply analyze the bifurcation diagram around a quadratic profile, i.e. (A|x| 2 + B)1D, by using Crandall–Rabinowitz theorem and also refined properties of hypergeometric functions. • Chapter 3 refers to the work [65], which is published in Nonlinearity. This chapter aims to provide a robust model for the well–known phenomenon of Karm´ an Vortex Street arising ´ in nonlinear transport equations. The first theoretical attempt to model this pattern was given by VON KARM ´ AN´ [89, 90] using a system of point vortices. The author considered two parallel staggered rows of Dirac masses, with opposite strength, that translate at the same speed. Following the numerical simulations of SAFFMAN and SCHATZMAN [134], we propose to study this phenomenon in a more realistic way using two infinite arrows of vortex patches. Hence, by desingularizating the system of point vortices, we are able to rigorously prove these numerical observations • Chapter 4 is the content of [66], which is a collaboration with my thesis advisor T. HMIDI and with J. MATEU, and is currently submitted for publication. It aims to study time periodic solutions for the 3D inviscid quasi–geostrophic model. We show the existence of non trivial rotating patches by suitable perturbation of stationary solutions given by generic revolution shapes around the vertical axis. The construction of those special solutions is achieved through bifurcation theory. In general, the spectral problem is very delicate and strongly depends on the shape of the initial stationary solutions. Restricting ourselves to a particular class of revolution shapes and exploiting the particular structure of our model, we are able to implement the bifurcation at the largest eigenvalue of a family of 1D Fredholm type operators. • Chapter 5 is devoted to explain some works in progress of this dissertation. Some conclusions of the above mentioned works together with some new perspective and future works are also given at the end of this chapter. • Finally, Appendices A, B and C collect some necessary results about bifurcation theory, potential theory and special functions.