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.
Publisher Copyright:
© 2020 American Institute of Mathematical Sciences. All rights reserved.
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
SN - 1078-0947
VL - 40
SP - 3427
EP - 3450
JO - Discrete and Continuous Dynamical Systems- Series A
JF - Discrete and Continuous Dynamical Systems- Series A
IS - 6
ER -