Hölder Continuous Solutions of Active Scalar Equations

• Published in: Annals of PDE Vol. 1; no. 1; p. 2
• Main Authors: Isett, Philip, Vicol, Vlad
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2015; Springer Nature B.V
• Subjects: Divergence; Energy; Mathematical analysis; Mathematical Methods in Physics; Multipliers; Operators (mathematics); Partial Differential Equations; Physics; Physics and Astronomy; Scalars; Velocity; Velocity distribution; Convex integration; principle; Onsager conjecture; Active scalar equations
• ISSN: 2199-2576; 2524-5317; 2199-2576
• DOI: 10.1007/s40818-015-0002-0

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 . We prove that when m is not an odd function of frequency, there are nontrivial, compactly supported solutions weak solutions, with Hölder regularity C t , x 1 / 9 - . 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.