Invariant Gibbs measures and global strong solutions for periodic 2D nonlinear Schrödinger equations.
Invariant Gibbs measures and global strong solutions for periodic 2D nonlinear Schrödinger equations.
In this talkÌýwe willÌýfirst give a quick background overview of Bourgain's approach to prove the invariance of the GibbsÌýmeasure forÌýthe periodic cubicÌýnonlinear Schrodinger equation in 2DÌýand of the para-controlled calculusÌýofÌýGubinelli-Imkeller and PerkowskiÌýin the context ofÌýparabolic stochastic equations.ÌýWe will then presentÌýour resolution ofÌýthe long-standing problem ofÌýproving almost sure global well-posednessÌý(i.e. existenceÌýwith uniqueness) for the periodic nonlinear Schrödinger equation (NLS) inÌý2D on the supportÌýof the Gibbs measure,Ìýfor any (defocusing and renormalized) odd power nonlinearity.ÌýConsequently we get theÌý¾±²Ô±¹²¹°ù¾±²¹²Ô³¦±ðÌýof the Gibbs measure.ÌýThis is achieved by a new methodÌýwe callÌýrandom averaging operatorsÌýwhichÌý±è°ù±ð³¦¾±²õ±ð±ô²âÌý³¦²¹±è³Ù³Ü°ù±ð²õÌýthe intrinsic randomnessÌýstructure of the problematic high-lowÌýfrequencyÌýinteractions at the heart of this problem.ÌýThis is joint work with Yu Deng (USC) and Haitian YueÌý(USC).