Hölder Continuous Solutions of Active Scalar Equations

Philip Isett; Vlad Vicol

doi:10.1007/s40818-015-0002-0

Hölder Continuous Solutions of Active Scalar Equations

Philip Isett, Vlad Vicol

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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 language	English (US)
Article number	2
Journal	Annals of PDE
Volume	1
Issue number	1
DOIs	https://doi.org/10.1007/s40818-015-0002-0
State	Published - Dec 1 2015

Keywords

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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10.1007/s40818-015-0002-0

Cite this

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title = "H{\"o}lder Continuous Solutions of Active Scalar Equations",

abstract = "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 H{\"o}lder 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.",

keywords = "Active scalar equations, Convex integration, Onsager conjecture, h-principle",

author = "Philip Isett and Vlad Vicol",

note = "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. 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. Publisher Copyright: {\textcopyright} 2015, Springer International Publishing AG.",

year = "2015",

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doi = "10.1007/s40818-015-0002-0",

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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. 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. Publisher Copyright: © 2015, Springer International Publishing AG.

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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JO - Annals of PDE

JF - Annals of PDE

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M1 - 2

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Hölder Continuous Solutions of Active Scalar Equations

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Keywords

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