Mathematics Colloquia and Seminars

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We present recent results on spatial decay and properties of non-uniqueness for the 3D Navier-Stokes equations. We show decay rates for the ‘non-linear’ part of scaling invariant flows with data in subcritical classes, and improved decay for data in Holder classes. Motivated by recent work on non-uniqueness and numerical evidence supporting conjectured non-uniqueness in the weak Leray class, we investigate how non-uniqueness would evolve in the local energy class. By extending our subcritical asymptotics to approximations by Picard iterates, we may bound the rate at which two solutions, evolving from the same data, may separate pointwise. This application also holds for Lorentz solutions with no scaling assumption, and locally subcritical data. Here, we all but recover the separation rate achieved for self-similar solutions, locally.