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November  2020, 25(11): 4277-4293. doi: 10.3934/dcdsb.2020097

On global large energy solutions to the viscous shallow water equations

Xiaoping Zhai 1, and  Hailong Ye 1,2,,

1. 

School of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China
2. 

Shenzhen Key Laboratory of Advanced Machine Learing and Applications, Shenzhen University, Shenzhen, 518060, China

* Corresponding author: Hailong Ye

Received  July 2011 Published  November 2020 Early access  April 2020

By exploring the smooth effect of the heat flows and the weighted-Chemin-Lerner technique, we obtain the global solutions of large energy to the viscous shallow water equations with initial data in the critical Besov spaces, which improves the previous small energy type arguments [5], [13]. Moreover, the method used here is quiet different from [5], [13].

Keywords: Shallow water equations, Global solutions, Large energy, Besov space, Littlewood-Paley theory.
Mathematics Subject Classification: Primary: 35Q35, 76N10; Secondary: 35B40.
Citation: Xiaoping Zhai, Hailong Ye. On global large energy solutions to the viscous shallow water equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4277-4293. doi: 10.3934/dcdsb.2020097
References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, Vol. 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[2]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223.  doi: 10.1007/s00220-003-0859-8.

[3]

D. Bresch, B. Desjardins and G. Métivier, Recent mathematical results and open problem about shallow water equations, in Analysis and Simulation of Fluid Dynamics, Birkhäuser, Basel, 2006, 15–31. doi: 10.1007/978-3-7643-7742-7_2.

[4]

J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes (French), J. Differential Equations, 121 (1995), 314-328.  doi: 10.1006/jdeq.1995.1131.

[5]

Q. Chen, C. Miao and Z. Zhang, On the well-posedness for the viscous shallow water equations, SIAM J. Math. Anal., 40 (2008), 443-474.  doi: 10.1137/060660552.

[6]

R. Danchin and J. Xu, Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical $L^p$ framework, Arch. Ration. Mech. Anal., 224 (2017), 53-90.  doi: 10.1007/s00205-016-1067-y.

[7]

P. Kloeden, Global existence of classical solutions in the dissipative shallow water equations, SIAM J. Math. Anal., 16 (1985), 301-315.  doi: 10.1137/0516022.

[8]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322.

[9] C. Miao, J. Wu and Z. Zhang, Littlewood-Paley Theory and its Applications: Hydrodynamic Equations (Chinese Edition), Scientific Press, 2012. 
[10]

T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter Series in Nonlinear Analysis Applications, Vol. 3, Walter de Gruyter & Co., Berlin, 1996. doi: 10.1515/9783110812411.

[11]

L. Sundbye, Global existence for the Dirichlet problem for the viscous shallow water equations, J. Math. Anal. Appl., 202 (1996), 236-258.  doi: 10.1006/jmaa.1996.0315.

[12]

B. A. Ton, Existence and uniqueness of a classical solution of an initial boundary value problem of the theory of shallow waters, SIAM J. Math. Anal., 12 (1981), 229-241.  doi: 10.1137/0512022.

[13]

W. Wang and C.-J. Xu, The Cauchy problem for viscous shallow water equations, Rev. Mat. Iberoamericana, 21 (2005), 1-24.  doi: 10.4171/RMI/412.

show all references

References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, Vol. 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[2]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223.  doi: 10.1007/s00220-003-0859-8.

[3]

D. Bresch, B. Desjardins and G. Métivier, Recent mathematical results and open problem about shallow water equations, in Analysis and Simulation of Fluid Dynamics, Birkhäuser, Basel, 2006, 15–31. doi: 10.1007/978-3-7643-7742-7_2.

[4]

J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes (French), J. Differential Equations, 121 (1995), 314-328.  doi: 10.1006/jdeq.1995.1131.

[5]

Q. Chen, C. Miao and Z. Zhang, On the well-posedness for the viscous shallow water equations, SIAM J. Math. Anal., 40 (2008), 443-474.  doi: 10.1137/060660552.

[6]

R. Danchin and J. Xu, Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical $L^p$ framework, Arch. Ration. Mech. Anal., 224 (2017), 53-90.  doi: 10.1007/s00205-016-1067-y.

[7]

P. Kloeden, Global existence of classical solutions in the dissipative shallow water equations, SIAM J. Math. Anal., 16 (1985), 301-315.  doi: 10.1137/0516022.

[8]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322.

[9] C. Miao, J. Wu and Z. Zhang, Littlewood-Paley Theory and its Applications: Hydrodynamic Equations (Chinese Edition), Scientific Press, 2012. 
[10]

T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter Series in Nonlinear Analysis Applications, Vol. 3, Walter de Gruyter & Co., Berlin, 1996. doi: 10.1515/9783110812411.

[11]

L. Sundbye, Global existence for the Dirichlet problem for the viscous shallow water equations, J. Math. Anal. Appl., 202 (1996), 236-258.  doi: 10.1006/jmaa.1996.0315.

[12]

B. A. Ton, Existence and uniqueness of a classical solution of an initial boundary value problem of the theory of shallow waters, SIAM J. Math. Anal., 12 (1981), 229-241.  doi: 10.1137/0512022.

[13]

W. Wang and C.-J. Xu, The Cauchy problem for viscous shallow water equations, Rev. Mat. Iberoamericana, 21 (2005), 1-24.  doi: 10.4171/RMI/412.

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