LONG TIME BEHAVIOR OF LINEAR AND NONLINEAR WAVES:LARGE LARGE AMPLITUDE TRAVELING QUASI-PERIODIC SOLUTIONS IN FLUID MECHANICS AND DYNAMICS OF LINEAR WAVE EQUATIONS
type: CreativeWork
description: In this thesis we study quasilinear PDEs on tori and we prove existence results of quasi-periodic large amplitude solutions for fluid-dynamic models in more than one space-dimension and a stability result for the linear Klein-Gordon Equation with a small quasi-periodic perturbation. In the first part of the thesis, where we consider two different models on $\mathbb{T}^2$: the non-resistive Magnetohydrodynamics (MHD) system and the 2D-Euler equation for rotating fluids ($\beta$-plane equation), both in presence of a large amplitude traveling wave external force with large velocity speed $\lambda \omega$, where $\lambda \gg 1$ is a large parameter. For both equations we prove the existence of large amplitude traveling quasi-periodic solutions. In particular, for the MHD system we construct solutions which are bi-periodic and stationary in a moving frame, while for the case of the $\beta$-plane equation we also prove the stability of the solutions, provided that the external force fulfills some additional algebraic property. Due to the presence of small divisors, the proofs rely on a nonlinear Nash-Moser scheme tailored to construct nonlinear waves of large size. The key point in this strategy is to study the invertibility of the linearized operator at a quasi-periodic approximate solution, giving ``good'' tame estimates. In both cases the linearized operator are unbounded diagonal operators plus a large amplitude perturbation and therefore for the invertibility of the linearized operator new ideas are required. Another key obstruction is the fact that the problems are high dimensional. For the MHD system the principal symbol of the linearized operator has super-linear growth. The invertibility procedure is then performed by using tools from micro-local analysis and normal forms together with a sharp analysis of high and low frequency regimes w.r.t. the large parameter $\lambda$. Moreover, to the best of our knowledge, this is the first result in which global in time, …
abstract: In this thesis we study quasilinear PDEs on tori and we prove existence results of quasi-periodic large amplitude solutions for fluid-dynamic models in more than one space-dimension and a stability result for the linear Klein-Gordon Equation with a small quasi-periodic perturbation. In the first part of the thesis, where we consider two different models on $\mathbb{T}^2$: the non-resistive Magnetohydrodynamics (MHD) system and the 2D-Euler equation for rotating fluids ($\beta$-plane equation), both in presence of a large amplitude traveling wave external force with large velocity speed $\lambda \omega$, where $\lambda \gg 1$ is a large parameter. For both equations we prove the existence of large amplitude traveling quasi-periodic solutions. In particular, for the MHD system we construct solutions which are bi-periodic and stationary in a moving frame, while for the case of the $\beta$-plane equation we also prove the stability of the solutions, provided that the external force fulfills some additional algebraic property. Due to the presence of small divisors, the proofs rely on a nonlinear Nash-Moser scheme tailored to construct nonlinear waves of large size. The key point in this strategy is to study the invertibility of the linearized operator at a quasi-periodic approximate solution, giving ``good'' tame estimates. In both cases the linearized operator are unbounded diagonal operators plus a large amplitude perturbation and therefore for the invertibility of the linearized operator new ideas are required. Another key obstruction is the fact that the problems are high dimensional. For the MHD system the principal symbol of the linearized operator has super-linear growth. The invertibility procedure is then performed by using tools from micro-local analysis and normal forms together with a sharp analysis of high and low frequency regimes w.r.t. the large parameter $\lambda$. Moreover, to the best of our knowledge, this is the first result in which global in time, …
author: TERRACINA, SHULAMIT
language: eng
keywords: Fluid Mechanic, Magnetohydrodynamic, Traveling wave, KAM for PDE, Klein-Gordon Equation, Quasi-periodic solutions, Settore MATH-03/A - Analisi matematica, Settore MATH-04/A - Fisica matematica
date published: 2025-02-21
date modified: 2026-07-27T20:18:58Z
provider: Università degli Studi di Milano
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: Università degli Studi di Milano

PropertyValue
property ID: Handle
value: https://hdl.handle.net/2434/1138896

PropertyValue
property ID: DOI
value: 10.13130/terracina-shulamit_phd2025-02-21