American Institute of Mathematical Sciences

Advanced
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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Yao Nie, Jia Yuan. The Littlewood-Paley $ pth $-order moments in three-dimensional MHD turbulence. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020397
[2]

Md. Masum Murshed, Kouta Futai, Masato Kimura, Hirofumi Notsu. Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1063-1078. doi: 10.3934/dcdss.2020230
[3]

Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021002
[4]

Bilal Al Taki, Khawla Msheik, Jacques Sainte-Marie. On the rigid-lid approximation of shallow water Bingham. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 875-905. doi: 10.3934/dcdsb.2020146
[5]

Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322
[6]

Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021002
[7]

Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229
[8]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268
[9]

Tianwen Luo, Tao Tao, Liqun Zhang. Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3737-3765. doi: 10.3934/dcds.2019230
[10]

Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021003
[11]

Liam Burrows, Weihong Guo, Ke Chen, Francesco Torella. Reproducible kernel Hilbert space based global and local image segmentation. Inverse Problems & Imaging, 2021, 15 (1) : 1-25. doi: 10.3934/ipi.2020048
[12]

Marcello D'Abbicco, Giovanni Girardi, Giséle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Equipartition of energy for nonautonomous damped wave equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 597-613. doi: 10.3934/dcdss.2020364
[13]

Kuo-Chih Hung, Shin-Hwa Wang. Classification and evolution of bifurcation curves for a porous-medium combustion problem with large activation energy. Communications on Pure & Applied Analysis, 2021, 20 (2) : 559-582. doi: 10.3934/cpaa.2020281
[14]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[15]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037
[16]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163
[17]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458
[18]

Petr Čoupek, María J. Garrido-Atienza. Bilinear equations in Hilbert space driven by paths of low regularity. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 121-154. doi: 10.3934/dcdsb.2020230
[19]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361
[20]

Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (102)
  • HTML views (230)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences