TY - JOUR

T1 - On the viscous Camassa-Holm equations with fractional diffusion

AU - Gan, Zaihui

AU - Lin, Fanghua

AU - Tong, Jiajun

N1 - Funding Information:
Foundation of China under grants 11571254. Fanghua Lin and Jiajun Tong are partially supported by National Science Foundation under Award Number DMS-1501000. The research was initiated while the first author was visiting the Courant Institute in the Fall of 2015.
Funding Information:
Acknowledgments. Zaihui Gan is partially supported by the National Science
Funding Information:
Zaihui Gan is partially supported by the National Science Foundation of China under grants 11571254. Fanghua Lin and Jiajun Tong are partially supported by National Science Foundation under Award Number DMS-1501000. The research was initiated while the first author was visiting the Courant Institute in the Fall of 2015.

PY - 2020

Y1 - 2020

N2 - 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 ∈ [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,+∞)(D(A)) ∩ L2 [0,+∞),loc(D(A1+s/2)) whenever the initial data u0 ∈ 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.

AB - 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 ∈ [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,+∞)(D(A)) ∩ L2 [0,+∞),loc(D(A1+s/2)) whenever the initial data u0 ∈ 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.

KW - Fractional diffusion

KW - Global well-posedness

KW - Improved regularity

KW - Lagrangian averaged Navier-Stokes equations

KW - Viscous Camassa-Holm equations

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U2 - 10.3934/dcds.2020029

DO - 10.3934/dcds.2020029

M3 - Article

AN - SCOPUS:85082511432

VL - 40

SP - 3427

EP - 3450

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 6

ER -