Holder Continuous Solutions of Active Scalar Equations

We consider active scalar equations $\partial_t θ+ \nabla \cdot (u \, θ) = 0$, where $u = T[θ]$ 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 Hölder regularity $C^{1/9-}_{t,x}$. In fact, every integral conserving scalar field can be approximated in ${\cal D}'$ 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.

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