LONG TIME BEHAVIOR OF LINEAR AND NONLINEAR WAVES:LARGE LARGE AMPLITUDE TRAVELING QUASI-PERIODIC SOLUTIONS IN FLUID MECHANICS AND DYNAMICS OF LINEAR WAVE EQUATIONS

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, large amplitude solutions are constructed for the 2D non-resistive MHD system with periodic boundary conditions. For the $\beta$-plane equation the invertibility of the linearized operator requires completely different strategy since the equation is dispersive and the dispersion relation is sub-linear. The main difficulty is due to the fact that the dispersion relation is highly degenerate - indicating an infinite-dimensional kernel for the linear principal operator. This degeneracy challenge is overcome by implementing a novel normal form methods based on the sub-linear growth of the unperturbed frequencies, the traveling-wave structure and the conservation of momentum which allows to cancel out some resonant terms. The third result proved in this thesis regards instead a model which finds applications in General Relativity. We consider the linear Klein-Gordon (KG) equation $\psi_{tt}-\psi_{xx}+\mathtt{m}\psi+\mathcal{Q}(\omega t)\psi=0 $ on $\mathbb{T}$ where $ \mathcal{Q} (\omega t)$ is a quasi-periodic differential operator of {\it order $ 2 $}. Under suitable condition on this perturbation we prove that all the solutions are {\it almost periodic} in time and uniformly bounded for all times. This result is obtained by reducing the KG equation to a diagonal constant coefficient system with purely imaginary eigenvalues. The main difficulty is the presence in the perturbation $ \mathcal{Q} (\omega t) $ of the maximal order. This requires a normalization procedure of the highest order which is particularly delicate due to the linear dispersion relation which implies a strong interaction between space derivative and time derivative. This is delicate from a quantitative point of view.