Depicting instabilities in linear dispersive PDEs: a dynamical systems approach

In the present thesis, we study instabilities in different classes of linear dispersive PDEs, in the form of weak turbulence formation and energy cascades. We use the growth in time of positive Sobolev norms of solutions as an indicator of instability in terms of transfer of energy towards higher frequencies. We identify a resonant mechanism responsible for instability formation in different models. More precisely, we base our procedure on a combination of resonant normal forms, pseudodifferential techniques and positive commutator arguments, and we reduce the instability problem to the study of the underlying classical dynamics. We obtain three different results. The first concerns time dependently perturbed quantum harmonic oscillators in $\R^2$, where the perturbation is a selfadjoint pseudodifferential operator of degree zero, $2\pi$-periodic in time. We identify sufficient conditions on the principal symbol of the potential that ensure existence of solutions exhibiting unbounded growth in time of their positive Sobolev norms and we show that the class of symbols satisfying such conditions is generic in the Fréchet space of classical $2\pi$-time periodic symbols of order zero. To prove our result we apply the abstract Theorem of \cite{Mas22}: the main difficulty is to find a conjugate operator $A$ for the resonant average of the perturbation. We construct explicitly the symbol of the conjugate operator $A$, called escape function, combining techniques from microlocal analysis, dynamical systems and contact topology. In the second result, we study linear and time-dependent perturbations of periodic transport equations on the two-dimensional torus. For generic perturbations, we prove the existence of a large class of initial data whose Sobolev norms diverge exponentially fast. In higher dimensions, the same conclusion holds under a Morse-Smale assumption on the resonant part of the perturbation. In both cases, the proof is based on a normal form procedure and on the study of Sobolev instabilities for time-dependent perturbations of Morse-Smale transport equations. The latter are analyzed on general compact manifolds using techniques from microlocal analysis and hyperbolic dynamics. Finally, we consider the one-dimensional instance of the previous model: we study linear, time-dependent and skewadjoint perturbations of periodic transport equations on the one-dimensional torus. In this low-dimensional setting, we are able to describe the long-time behavior of solutions for all non-degenerate perturbations in resonant regime, proving that either there exist solutions whose Sobolev norms grow exponentially fast, provoking an energy transfer towards higher frequencies, or all solutions remain stable over arbitrarily long time scales. In this case as well, the proof combines pseudodifferential tools with dynamical systems techniques: a resonant normal form procedure reduces the problem to the analysis of the classical dynamics associated with the resonant equation. The main difficulty lies in the instability mechanism, for which we explicitly construct an escape function adapted to the dynamics and use the associated operator for a positive commutator estimate.