THE REGULARITY OF THE BOUNDARY OF VORTEX PATCHES FOR SOME NONLINEAR TRANSPORT EQUATIONS

Juan Carlos Cantero; Joan Mateu; Joan Orobitg; Joan Verdera

doi:10.2140/apde.2023.16.1621

THE REGULARITY OF THE BOUNDARY OF VORTEX PATCHES FOR SOME NONLINEAR TRANSPORT EQUATIONS

Juan Carlos Cantero, Joan Mateu, Joan Orobitg, Joan Verdera

Department of Mathematics

Universitat Autònoma de Barcelona (UAB)

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

2 Citations (Scopus)

Abstract

We prove the persistence of boundary smoothness of vortex patches for a nonlinear transport equation in Rⁿ. with velocity field given by convolution of the density with an odd kernel, homogeneous of degree −(n−1) and of class C²(Rⁿ \{0},Rⁿ). This allows the velocity field to have nontrivial divergence. The quasigeostrophic equation in R³ and the Cauchy transport equation in the plane are examples.

Original language	English
Pages (from-to)	1621-1650
Number of pages	30
Journal	Analysis and PDE
Volume	16
Issue number	7
DOIs	https://doi.org/10.2140/apde.2023.16.1621 https://doi.org/10.2140/apde.2023.16.1621
Publication status	Published - 2023

Keywords

potential theory
transport equation
vortex patch

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title = "THE REGULARITY OF THE BOUNDARY OF VORTEX PATCHES FOR SOME NONLINEAR TRANSPORT EQUATIONS",

abstract = "We prove the persistence of boundary smoothness of vortex patches for a nonlinear transport equation in Rn. with velocity field given by convolution of the density with an odd kernel, homogeneous of degree −(n−1) and of class C2(Rn \{0},Rn). This allows the velocity field to have nontrivial divergence. The quasigeostrophic equation in R3 and the Cauchy transport equation in the plane are examples.",

keywords = "potential theory, transport equation, vortex patch",

author = "Cantero, {Juan Carlos} and Joan Mateu and Joan Orobitg and Joan Verdera",

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AU - Orobitg, Joan

AU - Verdera, Joan

N1 - Publisher Copyright: © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). Open Access made possible by subscribing institutions via Subscribe to Open.

PY - 2023

Y1 - 2023

N2 - We prove the persistence of boundary smoothness of vortex patches for a nonlinear transport equation in Rn. with velocity field given by convolution of the density with an odd kernel, homogeneous of degree −(n−1) and of class C2(Rn \{0},Rn). This allows the velocity field to have nontrivial divergence. The quasigeostrophic equation in R3 and the Cauchy transport equation in the plane are examples.

AB - We prove the persistence of boundary smoothness of vortex patches for a nonlinear transport equation in Rn. with velocity field given by convolution of the density with an odd kernel, homogeneous of degree −(n−1) and of class C2(Rn \{0},Rn). This allows the velocity field to have nontrivial divergence. The quasigeostrophic equation in R3 and the Cauchy transport equation in the plane are examples.

KW - potential theory

KW - transport equation

KW - vortex patch

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DO - 10.2140/apde.2023.16.1621

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VL - 16

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JO - Analysis and PDE

JF - Analysis and PDE

IS - 7

ER -

THE REGULARITY OF THE BOUNDARY OF VORTEX PATCHES FOR SOME NONLINEAR TRANSPORT EQUATIONS

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Keywords

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