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Holder Continuous Solutions of Active Scalar Equations

4:00 pm Thursday, October 9, 2014
Vlad Vicol (Princeton)

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. Motivated by questions arising in the Kolmogorov/Kraichnan/Onsager theory of turbulence, we consider weak solutions that do not conserve energy. We prove that when m is not an odd function of frequency, there are nontrivial, compactly supported solutions weak solutions, with Holder regularity C1/9−.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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