American Institute of Mathematical Sciences

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

On the viscous Camassa-Holm equations with fractional diffusion

Zaihui Gan 1, , Fanghua Lin 2, and  Jiajun Tong 2,,

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.

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.
Citation: Zaihui Gan, Fanghua Lin, Jiajun Tong. On the viscous Camassa-Holm equations with fractional diffusion. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3427-3450. doi: 10.3934/dcds.2020029
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. Chen, C. Foias, D. D. Holm, E. Olson, E. 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. Chen, D. D. Holm, L. 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. Constantin, D. Córdoba, F. Gancedo, L. 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. Constantin, T. Elgindi, M. 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. Córdoba, D. 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. Córdoba, D. 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. Coutand, J. 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. Foias, D. 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. Foias, D. 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. Holm, J. 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. Mohseni, B. Kosović, S. 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. Chen, C. Foias, D. D. Holm, E. Olson, E. 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. Chen, D. D. Holm, L. 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. Constantin, D. Córdoba, F. Gancedo, L. 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. Constantin, T. Elgindi, M. 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. Córdoba, D. 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. Córdoba, D. 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. Coutand, J. 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. Foias, D. 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. Foias, D. 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. Holm, J. 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. Mohseni, B. Kosović, S. 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

[1]

Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35
[2]

Joel Avrin. Global existence and regularity for the Lagrangian averaged Navier-Stokes equations with initial data in $H^{1//2}$. Communications on Pure & Applied Analysis, 2004, 3 (3) : 353-366. doi: 10.3934/cpaa.2004.3.353
[3]

Ying Fu, Changzheng Qu, Yichen Ma. Well-posedness and blow-up phenomena for the interacting system of the Camassa-Holm and Degasperis-Procesi equations. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1025-1035. doi: 10.3934/dcds.2010.27.1025
[4]

Quanrong Li, Shijin Ding. Global well-posedness of the Navier-Stokes equations with Navier-slip boundary conditions in a strip domain. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3545-3565. doi: 10.3934/cpaa.2021121
[5]

Daoyuan Fang, Ruizhao Zi. On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3517-3541. doi: 10.3934/dcds.2013.33.3517
[6]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations I: Local well-posedness. Evolution Equations & Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195
[7]

Matthias Hieber, Sylvie Monniaux. Well-posedness results for the Navier-Stokes equations in the rotational framework. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5143-5151. doi: 10.3934/dcds.2013.33.5143
[8]

Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101
[9]

Weimin Peng, Yi Zhou. Global well-posedness of axisymmetric Navier-Stokes equations with one slow variable. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3845-3856. doi: 10.3934/dcds.2016.36.3845
[10]

Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1
[11]

Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139
[12]

Yingdan Ji, Wen Tan. Global well-posedness of a 3D Stokes-Magneto equations with fractional magnetic diffusion. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3271-3278. doi: 10.3934/dcdsb.2020227
[13]

Lei Zhang, Bin Liu. Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2655-2685. doi: 10.3934/dcds.2018112
[14]

H. A. Erbay, S. Erbay, A. Erkip. On the decoupling of the improved Boussinesq equation into two uncoupled Camassa-Holm equations. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3111-3122. doi: 10.3934/dcds.2017133
[15]

Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809
[16]

Jishan Fan, Yasuhide Fukumoto, Yong Zhou. Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations. Kinetic & Related Models, 2013, 6 (3) : 545-556. doi: 10.3934/krm.2013.6.545
[17]

Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148
[18]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142
[19]

Keyan Wang, Yao Xiao. Local well-posedness for Navier-Stokes equations with a class of ill-prepared initial data. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2987-3011. doi: 10.3934/dcds.2020158
[20]

Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (268)
  • HTML views (366)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences