Random data wave equations
Abstract
Nowadays we have many methods allowing to exploit the regularising properties of the linear part of a nonlinear dispersive equation (such as the KdV equation, the nonlinear wave or the nonlinear Schroedinger equations) in order to prove wellposedness in low regularity Sobolev spaces. By wellposedness in low regularity Sobolev spaces we mean that less regularity than the one imposed by the energy methods is required (the energy methods do not exploit the dispersive properties of the linear part of the equation). In many cases these methods to prove wellposedness in low regularity Sobolev spaces lead to optimal results in terms of the regularity of the initial data. By optimal we mean that if one requires slightly less regularity then the corresponding Cauchy problem becomes illposed in the Hadamard sense. We call the Sobolev spaces in which these illposedness results hold spaces of supercritical regularity. More recently, methods to prove probabilistic wellposedness in Sobolev spaces of supercritical regularity were developed. More precisely, by probabilistic wellposedness we mean that one endows the corresponding Sobolev space of supercritical regularity with a non degenerate probability measure and then one shows that almost surely with respect to this measure one can define a (unique) global flow. However, in most of the cases when the methods to prove probabilistic wellposedness apply, there is no information about the measure transported by the flow. Very recently, a method to prove that the transported measure is absolutely continuous with respect to the initial measure was developed. In such a situation, we have a measure which is quasiinvariant under the corresponding flow. The aim of these lectures is to present all of the above described developments in the context of the nonlinear wave equation.
 Publication:

arXiv eprints
 Pub Date:
 April 2017
 arXiv:
 arXiv:1704.01191
 Bibcode:
 2017arXiv170401191T
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Probability
 EPrint:
 Lecture notes based on a course given at a CIME summer school in August 2016