On singularity formation in nonlinear evolution equations

Pierre Raphael, University of Cambridge

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Linear waves propagate by spreading a fixed initial quantum of energy over the whole space. Some nonlinear waves behave very differently: they can concentrate part of their initial energy, and in some dramatic situations this can lead to the formation of a singularity. Whether or not a given nonlinear partial differential equation may produce such singularities, and what is the nature of these, is a classical problem in mathematical physics, and is in particular at the heart of the 6th Millennium problem on incompressible fluid dynamics. Over the last twenty years, spectacular progress has been made on simplified canonical models with a so called focusing structure, and two essential concepts have emerged: the notion of super criticality, and the key role played by special non linear waves (solitons) to sustain the energy concentration mechanism. The concept of being a focusing problem remains however very unclear, in particular in the realm of fluid mechanics. In a recent joint work with Merle (IHES), Rodnianski (Princeton) and Szeftel (Paris Sorbonne), we produce the first example of singularity formation for a canonical defocusing model (the energy super critical non linear Schrodinger equation). The construction goes hand in hand with the first explicit description of singularity formation for the three dimensional compressible Navier Stokes equations (blow up by implosion) which opens up a new route for the study of singularity formation in fluid mechanics.