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 $m$. We prove that when $m$ is not an odd function of frequency, 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 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.
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