June  2020, 40(6): 3427-3450. doi: 10.3934/dcds.2020029

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

Dedicated to Professor Wei-Ming Ni with Respect

Received  December 2018 Revised  June 2019 Published  October 2019

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.

Citation: Zaihui Gan, Fanghua Lin, Jiajun Tong. On the viscous Camassa-Holm equations with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3427-3450. doi: 10.3934/dcds.2020029
References:
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C. Bjorland, Decay asymptotics of the viscous Camassa-Holm equations in the plane, SIAM Journal on Mathematical Analysis, 40 (2008), 516-539.  doi: 10.1137/070684070.  Google Scholar

[2]

C. Bjorland and M. E. Schonbek, On questions of decay and existence for the viscous Camassa-Holm equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 907-936.  doi: 10.1016/j.anihpc.2007.07.003.  Google Scholar

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S. ChenD. D. HolmL. G. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model, Predictability: Quantifying uncertainty in models of complex phenomena (Los Alamos, NM, 1998), Phys. D, 133 (1999), 66-83.  doi: 10.1016/S0167-2789(99)00099-8.  Google Scholar

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A. Cheskidov, Boundary layer for the Navier-Stokes-alpha model of fluid turbulence, Arch. Ration. Mech. Anal., 172 (2004), 333-362.  doi: 10.1007/s00205-004-0305-x.  Google Scholar

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M. Colombo, C. De Lellis and L. De Rosa, Ill-posedness of Leray solutions for the ipodissipative Navier-Stokes equations, Comm. Math. Phys., 362 (2018), 659-688. doi: 10.1007/s00220-018-3177-x.  Google Scholar

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P. ConstantinD. CórdobaF. GancedoL. Rodriguez-Piazza and R. M. Strain, On the Muskat problem: Global in time results in 2D and 3D, Amer. J. Math., 138 (2016), 1455-1494.  doi: 10.1353/ajm.2016.0044.  Google Scholar

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C. FoiasD. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Phys. D, 152/153 (2001), 505-519.  doi: 10.1016/S0167-2789(01)00191-9.  Google Scholar

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C. FoiasD. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, J. Dynam. Differential Equations, 14 (2002), 1-35.  doi: 10.1023/A:1012984210582.  Google Scholar

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H. Fujita and T. Kato, On the Navier-Stokes initial value problem. Ⅰ, Arch. Rational Mech. Anal., 16 (1964), 269-315.  doi: 10.1007/BF00276188.  Google Scholar

[19]

Y. Giga, Solutions for semilinear parabolic equations in ${L}^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212.  doi: 10.1016/0022-0396(86)90096-3.  Google Scholar

[20]

Y. Giga and T. Miyakawa, Solutions in ${L}^r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267-281.  doi: 10.1007/BF00276875.  Google Scholar

[21]

D. D. Holm, Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid motion, Predictability: Quantifying uncertainty in models of complex phenomena (Los Alamos, NM, 1998), Phys. D, 133 (1999), 215-269.  doi: 10.1016/S0167-2789(99)00093-7.  Google Scholar

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D. D. Holm, Karman-Howarth theorem for the Lagrangian-averaged Navier-Stokes-alpha model of turbulence, J. Fluid Mech., 467 (2002), 205-214.  doi: 10.1017/S002211200200160X.  Google Scholar

[23]

D. D. HolmJ. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721.  Google Scholar

[24]

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[25]

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[26]

F. -H. Lin and J. Tong, Solvability of the stokes immersed boundary problem in two dimensions, Comm. Pure Appl. Math., 72 (2019), 159-226.  doi: 10.1002/cpa.21764.  Google Scholar

[27]

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[28]

J. E. Marsden and S. Shkoller, Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations on bounded domains, Topological methods in the physical sciences (London, 2000), R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 359 (2001), 1449-1468.  doi: 10.1098/rsta.2001.0852.  Google Scholar

[29]

J. E. Marsden and S. Shkoller, The anisotropic Lagrangian averaged Euler and Navier-Stokes equations, Arch. Ration. Mech. Anal., 166 (2003), 27-46.  doi: 10.1007/s00205-002-0207-8.  Google Scholar

[30]

K. MohseniB. KosovićS. Shkoller and J. E. Marsden, Numerical simulations of the Lagrangian averaged Navier-Stokes equations for homogeneous isotropic turbulence, Phys. Fluids, 15 (2003), 524-544.  doi: 10.1063/1.1533069.  Google Scholar

[31]

S. Shkoller, Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid, J. Differential Geom., 55 (2000), 145-191.  doi: 10.4310/jdg/1090340568.  Google Scholar

[32]

R. Temam, Navier-Stokes Equations, Theory and numerical analysis. With an appendix by F. Thomasset. Third edition. Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984.  Google Scholar

show all references

References:
[1]

C. Bjorland, Decay asymptotics of the viscous Camassa-Holm equations in the plane, SIAM Journal on Mathematical Analysis, 40 (2008), 516-539.  doi: 10.1137/070684070.  Google Scholar

[2]

C. Bjorland and M. E. Schonbek, On questions of decay and existence for the viscous Camassa-Holm equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 907-936.  doi: 10.1016/j.anihpc.2007.07.003.  Google Scholar

[3]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903.  Google Scholar

[4]

S. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, The Camassa-Holm equations and turbulence, Predictability: Quantifying uncertainty in models of complex phenomena (Los Alamos, NM, 1998), Phys. D, 133 (1999), 49-65.  doi: 10.1016/S0167-2789(99)00098-6.  Google Scholar

[5]

S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, Camassa-Holm equations as a closure model for turbulent channel and pipe flow,, Phys. Rev. Lett., 81 (1998), 5338-5341. doi: 10.1103/PhysRevLett.81.5338.  Google Scholar

[6]

S. ChenD. D. HolmL. G. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model, Predictability: Quantifying uncertainty in models of complex phenomena (Los Alamos, NM, 1998), Phys. D, 133 (1999), 66-83.  doi: 10.1016/S0167-2789(99)00099-8.  Google Scholar

[7]

A. Cheskidov, Boundary layer for the Navier-Stokes-alpha model of fluid turbulence, Arch. Ration. Mech. Anal., 172 (2004), 333-362.  doi: 10.1007/s00205-004-0305-x.  Google Scholar

[8]

M. Colombo, C. De Lellis and L. De Rosa, Ill-posedness of Leray solutions for the ipodissipative Navier-Stokes equations, Comm. Math. Phys., 362 (2018), 659-688. doi: 10.1007/s00220-018-3177-x.  Google Scholar

[9]

P. ConstantinD. CórdobaF. GancedoL. Rodriguez-Piazza and R. M. Strain, On the Muskat problem: Global in time results in 2D and 3D, Amer. J. Math., 138 (2016), 1455-1494.  doi: 10.1353/ajm.2016.0044.  Google Scholar

[10]

P. ConstantinT. ElgindiM. Ignatova and V. Vicol, On some electroconvection models, J. Nonlinear Sci., 27 (2017), 197-211.  doi: 10.1007/s00332-016-9329-2.  Google Scholar

[11]

P. Constantin, A. J. Majda and E. Tabak, Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533. doi: 10.1088/0951-7715/7/6/001.  Google Scholar

[12]

P. Constantin and H. Q. Nguyen, Local and global strong solutions for SQG in bounded domains, Phys. D, 376/377 (2018), 195-203. doi: 10.1016/j.physd.2017.08.008.  Google Scholar

[13]

A. CórdobaD. Córdoba and M. A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math., 162 (2005), 1377-1389.  doi: 10.4007/annals.2005.162.1377.  Google Scholar

[14]

A. CórdobaD. Córdoba and F. Gancedo, Interface evolution: The Hele-Shaw and Muskat problems, Ann. of Math., 173 (2011), 477-542.  doi: 10.4007/annals.2011.173.1.10.  Google Scholar

[15]

D. CoutandJ. Peirce and S. Shkoller, Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains, Commun. Pure Appl. Anal., 1 (2002), 35-50.  doi: 10.3934/cpaa.2002.1.35.  Google Scholar

[16]

C. FoiasD. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Phys. D, 152/153 (2001), 505-519.  doi: 10.1016/S0167-2789(01)00191-9.  Google Scholar

[17]

C. FoiasD. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, J. Dynam. Differential Equations, 14 (2002), 1-35.  doi: 10.1023/A:1012984210582.  Google Scholar

[18]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem. Ⅰ, Arch. Rational Mech. Anal., 16 (1964), 269-315.  doi: 10.1007/BF00276188.  Google Scholar

[19]

Y. Giga, Solutions for semilinear parabolic equations in ${L}^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212.  doi: 10.1016/0022-0396(86)90096-3.  Google Scholar

[20]

Y. Giga and T. Miyakawa, Solutions in ${L}^r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267-281.  doi: 10.1007/BF00276875.  Google Scholar

[21]

D. D. Holm, Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid motion, Predictability: Quantifying uncertainty in models of complex phenomena (Los Alamos, NM, 1998), Phys. D, 133 (1999), 215-269.  doi: 10.1016/S0167-2789(99)00093-7.  Google Scholar

[22]

D. D. Holm, Karman-Howarth theorem for the Lagrangian-averaged Navier-Stokes-alpha model of turbulence, J. Fluid Mech., 467 (2002), 205-214.  doi: 10.1017/S002211200200160X.  Google Scholar

[23]

D. D. HolmJ. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721.  Google Scholar

[24]

D. D. Holm, J. E. Marsden and T. S. Ratiu, Euler-poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett., 80 (1998), 4173. doi: 10.1103/PhysRevLett.80.4173.  Google Scholar

[25]

A. A. Kiselev and O. A. Ladyzhenskaya, On the existence and uniqueness of the solution of the nonstationary problem for a viscous, incompressible fluid, Izv. Akad. Nauk SSSR. Ser. Mat., 21 (1957), 655-680.   Google Scholar

[26]

F. -H. Lin and J. Tong, Solvability of the stokes immersed boundary problem in two dimensions, Comm. Pure Appl. Math., 72 (2019), 159-226.  doi: 10.1002/cpa.21764.  Google Scholar

[27]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. Ⅰ. Translated from the French by P. Kenneth. Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[28]

J. E. Marsden and S. Shkoller, Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations on bounded domains, Topological methods in the physical sciences (London, 2000), R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 359 (2001), 1449-1468.  doi: 10.1098/rsta.2001.0852.  Google Scholar

[29]

J. E. Marsden and S. Shkoller, The anisotropic Lagrangian averaged Euler and Navier-Stokes equations, Arch. Ration. Mech. Anal., 166 (2003), 27-46.  doi: 10.1007/s00205-002-0207-8.  Google Scholar

[30]

K. MohseniB. KosovićS. Shkoller and J. E. Marsden, Numerical simulations of the Lagrangian averaged Navier-Stokes equations for homogeneous isotropic turbulence, Phys. Fluids, 15 (2003), 524-544.  doi: 10.1063/1.1533069.  Google Scholar

[31]

S. Shkoller, Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid, J. Differential Geom., 55 (2000), 145-191.  doi: 10.4310/jdg/1090340568.  Google Scholar

[32]

R. Temam, Navier-Stokes Equations, Theory and numerical analysis. With an appendix by F. Thomasset. Third edition. Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984.  Google Scholar

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