Mathematics Colloquia and Seminars

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 with symbol $m$. Motivated by questions arising in the Kolmogorov/Kraichnan/Onsager theory of turbulence, we consider weak solutions that do not conserve energy. We prove that when $m$ is not an odd function of frequency, there are nontrivial, compactly supported solutions weak solutions, with H\"older 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 the multiplier m is odd, weak limits of solutions are solutions, so that the h-principle for odd active scalars may not be expected. This is joint work with Philip Isett.