Hölder Continuous Solutions of Active Scalar Equations
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.
© 2015 Springer International Publishing AG. Received: 1 June 2015; Accepted: 3 November 2015; Published online: 14 November 2015. The work of P.I. was in part supported by the NSF Postdoctoral Fellowship DMS-1402370, while the work of V.V. was in part supported by the NSF grant DMS-1348193 and an Alfred P. Sloan Fellowship.
Submitted - 1405.7656.pdf