TY - JOUR

T1 - Hölder Continuous Solutions of Active Scalar Equations

AU - Isett, Philip

AU - Vicol, Vlad

N1 - Funding Information:
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.

PY - 2015/12/1

Y1 - 2015/12/1

N2 - 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.

AB - 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.

KW - Active scalar equations

KW - Convex integration

KW - Onsager conjecture

KW - h-principle

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U2 - 10.1007/s40818-015-0002-0

DO - 10.1007/s40818-015-0002-0

M3 - Article

AN - SCOPUS:85016249944

VL - 1

JO - Annals of PDE

JF - Annals of PDE

SN - 2524-5317

IS - 1

M1 - 2

ER -