4:00 pm Thursday, October 9, 2014

Vlad Vicol (Princeton)

We consider active scalar equations ∂_{t}θ + ∇ ⋅ (*u*θ) = 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 *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.