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. |