On the viscous Camassa-Holm equations with fractional diffusion

Zaihui Gan; Fanghua Lin; Jiajun Tong

doi:10.3934/dcds.2020029

Article Contents

2020, Volume 40, Issue 6: 3427-3450. Doi: 10.3934/dcds.2020029 open access

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On the viscous Camassa-Holm equations with fractional diffusion

1.
Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
2.
Courant Institute, New York University, 251 Mercer Street, New York, NY 10012, USA

^* Corresponding author: Jiajun Tong
^* Corresponding author: Jiajun Tong

Dedicated to Professor Wei-Ming Ni with Respect

Received: November 30, 2018

Revised: May 31, 2019

Published: March 2020

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Abstract

We study Cauchy problem of a class of viscous Camassa-Holm equations (or Lagrangian averaged Navier-Stokes equations) with fractional diffusion in both smooth bounded domains and in the whole space in two and three dimensions. Order of the fractional diffusion is assumed to be $ 2s $ with $ s\in [n/4,1) $, which seems to be sharp for the validity of the main results of the paper; here $ n = 2,3 $ is the dimension of space. We prove global well-posedness in $ C_{[0,+\infty)}(D(A))\cap L^2_{[0,+\infty),loc}(D(A^{1+s/2})) $ whenever the initial data $ u_0\in D(A) $, where $ A $ is the Stokes operator. We also prove that such global solutions gain regularity instantaneously after the initial time. A bound on a higher-order spatial norm is also obtained.

Keywords:

Viscous Camassa-Holm equations,
Lagrangian averaged Navier-Stokes equations,
fractional diffusion,
global well-posedness,
improved regularity.

Mathematics Subject Classification: Primary: 35A01, 35Q35; Secondary: 35G25, 35K30.

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References

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