Hölder Continuous Solutions of Active Scalar Equations

Abstract

We consider active scalar equations ∂_tθ + ∇ ⋅ (uθ)=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 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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