Long time behaviour for nonlinear evolution equations

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Partial Differential Equations (PDEs) are used to describe a wide variety of phenomena in the physical world, from the motion of fluids and plasma, to the description of elementary particles, to the motion of stars. In this proposal we are interested in nonlinear PDEs describing the complex nature of the nonlinear interactions of waves, particles, fields... In particular, we will investigate the dynamics of various physical systems, study their evolution over large periods of time, and their asymptotic states. ******The proposed research activities can be group into two main categories, corresponding to two fundamental issues in the analysis of PDEs: the long-time dynamics around special solutions, and the behavior of quasilinear evolution equations in periodic settings.******Several nonlinear equations possess special solutions such as stationary solutions (that do not change with time) and traveling waves and solitons (solutions that keep their shape as they move). These coherent structures, emerging from the balance between dispersive effects and the focusing mechanisms of nonlinear interactions, can represent various physical phenomena (e.g., traveling waves on the surface of the sea, particle-like objects in field theories, or black-holes) and play a major role in the global dynamics of the system. Studying their stability is then a crucial task from both a mathematical and physical perspective. We propose projects related to the global behavior around (topological) solitons for some models in mathematical physics and fluid dynamics, such as the phi-4 model, long waves in shallow water, and equations from multidimensional field theories.******Many of the fundamental equations that model inviscid fluids, plasmas, gravitational waves, are quasilinear, which means that the highest order derivatives of the unknowns appear nonlinearly in the equations. Building on the pioneering works of Shatah, Klainerman, and Christodoulou-Klainerman in the `80s, several powerful techniques have emerged in the past 5-10 years in the study of quasilinear equations. These led to important achievements, such as the first global existence results for several water waves systems and plasma models.***One of the major open questions in the field is what happens in non-purely-dispersive settings, such as a bounded domain (typical for a water wave), or a periodic box (a suitable scenario for plasma experiments). The second part of our project concentrates on this type of questions. Our aim is to investigate the long-time existence of periodic solutions of quasilinear systems, with particular emphasis on the water waves problem, which describes the motion of an irrotational and incompressible fluid with a free moving boundary, such as waves on the surface of the ocean. Related problems for the nonlinear Schrodinger equations, and connections to the general theory of Hamiltonian system will also be sought.