EVENT DETAILS AND ABSTRACT

Title: Holder continuous solutions of active scalar equations
Speaker: Vlad Vicol
Speaker Info: Princeton University
Brief Description:
Special Note:
Abstract:

We consider active scalar equations \(\partial_t \theta + \nabla \cdot (u\, \theta) = 0\), where \(u = T[\theta]\) is a divergence-free velocity field, and \(T\) is a Fourier multiplier operator. We prove that when \(T\) is not an odd multiplier, there are nontrivial, compactly supported solutions weak solutions, with Holder regularity \(C^{1/9-}_{t,x}\). In fact, every integral conserving scalar field can be approximated in \(D'\) by such solutions, and these weak solutions may be obtained from arbitrary initial data. We also show that when \(T\) is odd, weak limits of solutions are solutions, so that the h-principle for odd active scalars may not be expected. This is a joint work with Phillip Isett (MIT).