Hölder Continuous Solutions of Active Scalar Equations

Philip Isett, Vlad Vicol

Research output: Contribution to journalArticlepeer-review


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 Ct,x1/9-. 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.

Original languageEnglish (US)
Article number2
JournalAnnals of PDE
Issue number1
StatePublished - Dec 1 2015


  • Active scalar equations
  • Convex integration
  • Onsager conjecture
  • h-principle

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Geometry and Topology
  • Mathematical Physics
  • General Physics and Astronomy


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