January  2018, 38(1): 311-328. doi: 10.3934/dcds.2018015

Vanishing viscosity limit of the rotating shallow water equations with far field vacuum

School of Mathematics and Statistics, Fuyang Normal College, Fuyang 236037, China

* Corresponding author: Zhigang Wang

Received  April 2017 Revised  July 2017 Published  January 2018

Fund Project: Zhigang Wang is supported by Chinese National Natural Science Foundation under grant 11401104 and China Postdoctoral Science Foundation under grant 2015M581579.

In this paper, we consider the Cauchy problem of the rotating shallow water equations, which has height-dependent viscosities, arbitrarily large initial data and far field vacuum. Firstly, we establish the existence of the unique local regular solution, whose life span is uniformly positive as the viscosity coefficients vanish. Secondly, we prove the well-posedness of the regular solution for the inviscid flow. Finally, we show the convergence rate of the regular solution from the viscous flow to the inviscid flow in $L^{\infty}([0, T]; H^{s'})$ for any $s'\in [2, 3)$ with a rate of $\epsilon^{1-\frac{s'}{3}}$.

Citation: Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015
References:
[1]

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

[2]

D. Bresch and B. Desjardins, On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models, J. Math. Pures Appl., 86 (2006), 362-368.  doi: 10.1016/j.matpur.2006.06.005.  Google Scholar

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D. BreschB. Desjardins and C. K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868.  doi: 10.1081/PDE-120020499.  Google Scholar

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Q. ChenC. Miao and Z. Zhang, Well-posedness for the viscous shallow water equations in critical spaces, SIAM J. Math. Anal., 40 (2008), 443-474.  doi: 10.1137/060660552.  Google Scholar

[5]

Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manu. Math., 120 (2006), 91-129.  doi: 10.1007/s00229-006-0637-y.  Google Scholar

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M. Ding and S. Zhu, Vanishing viscosity Limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow with far field vacuum, J. Math. Pures Appl., 107 (2017), 288-314.  doi: 10.1016/j.matpur.2016.07.001.  Google Scholar

[7]

B. DuanY. Zheng and Z. Luo, Local existence of classical solutions to Shallow water equations with Cauchy data containing vacuum, SIAM J. Math. Anal., 44 (2012), 541-567.  doi: 10.1137/100817887.  Google Scholar

[8]

Z. GuoQ. Jiu and Z. Xin, Radially symmetric isentropic compressible flows with density-dependent viscosity coefficients, SIAM J. Math. Anal., 39 (2008), 1402-1427.  doi: 10.1137/070680333.  Google Scholar

[9]

C. HaoL. Hsiao and H. Li, Cauchy problem for viscous rotating shallow water equations, J. Differential Equations, 247 (2009), 3234-3257.  doi: 10.1016/j.jde.2009.09.008.  Google Scholar

[10]

P. E. 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

[11]

H. LiJ. Li and Z. Xin, Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations, Comm. Math. Phys., 281 (2008), 401-444.  doi: 10.1007/s00220-008-0495-4.  Google Scholar

[12]

Y. LiR. Pan and S. Zhu, On Classical Solutions to 2D Shallow Water Equations with Degenerate Viscosities, J. Math. Fluid Mech., 19 (2017), 151-190.  doi: 10.1007/s00021-016-0276-3.  Google Scholar

[13]

Y. Li, R. Pan and S. Zhu, On classical solutions for viscous polytropic fluids with degenerate viscosities and vacuum, submitted, preprint, arXiv: 1503. 05644. Google Scholar

[14]

Y. LiR. Pan and S. Zhu, Recent progress on classical solutions for compressible isentropic Navier-Stokes equations with degenerate viscosities and vacuum, Bull. Braz. Math. Soc. (N.S.), 47 (2016), 507-519.  doi: 10.1007/s00574-016-0165-7.  Google Scholar

[15]

Y. Li and S. Zhu, Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum, J. Differential Equations, 256 (2014), 3943-3980.  doi: 10.1016/j.jde.2014.03.007.  Google Scholar

[16]

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

[17]

L. Sundbye, Global existence for the Cauchy problem for the viscous shallow water equations, Rocky Mountain J. Math., 28 (1998), 1135-1152.  doi: 10.1216/rmjm/1181071760.  Google Scholar

[18]

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

[19]

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

[20]

S. Zhu, Existence results for viscous polytropic fluids with degenerate viscosity coefficients and vacuum, J. Differential Equations, 259 (2015), 84-119.  doi: 10.1016/j.jde.2015.01.048.  Google Scholar

[21]

S. Zhu, Well-Posedness and Singularity Formation of the Compressible Isentropic Navier-Stokes Equations, Ph. D Thesis, Shanghai Jiao Tong University, 2015. Google Scholar

show all references

References:
[1]

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

[2]

D. Bresch and B. Desjardins, On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models, J. Math. Pures Appl., 86 (2006), 362-368.  doi: 10.1016/j.matpur.2006.06.005.  Google Scholar

[3]

D. BreschB. Desjardins and C. K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868.  doi: 10.1081/PDE-120020499.  Google Scholar

[4]

Q. ChenC. Miao and Z. Zhang, Well-posedness for the viscous shallow water equations in critical spaces, SIAM J. Math. Anal., 40 (2008), 443-474.  doi: 10.1137/060660552.  Google Scholar

[5]

Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manu. Math., 120 (2006), 91-129.  doi: 10.1007/s00229-006-0637-y.  Google Scholar

[6]

M. Ding and S. Zhu, Vanishing viscosity Limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow with far field vacuum, J. Math. Pures Appl., 107 (2017), 288-314.  doi: 10.1016/j.matpur.2016.07.001.  Google Scholar

[7]

B. DuanY. Zheng and Z. Luo, Local existence of classical solutions to Shallow water equations with Cauchy data containing vacuum, SIAM J. Math. Anal., 44 (2012), 541-567.  doi: 10.1137/100817887.  Google Scholar

[8]

Z. GuoQ. Jiu and Z. Xin, Radially symmetric isentropic compressible flows with density-dependent viscosity coefficients, SIAM J. Math. Anal., 39 (2008), 1402-1427.  doi: 10.1137/070680333.  Google Scholar

[9]

C. HaoL. Hsiao and H. Li, Cauchy problem for viscous rotating shallow water equations, J. Differential Equations, 247 (2009), 3234-3257.  doi: 10.1016/j.jde.2009.09.008.  Google Scholar

[10]

P. E. 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

[11]

H. LiJ. Li and Z. Xin, Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations, Comm. Math. Phys., 281 (2008), 401-444.  doi: 10.1007/s00220-008-0495-4.  Google Scholar

[12]

Y. LiR. Pan and S. Zhu, On Classical Solutions to 2D Shallow Water Equations with Degenerate Viscosities, J. Math. Fluid Mech., 19 (2017), 151-190.  doi: 10.1007/s00021-016-0276-3.  Google Scholar

[13]

Y. Li, R. Pan and S. Zhu, On classical solutions for viscous polytropic fluids with degenerate viscosities and vacuum, submitted, preprint, arXiv: 1503. 05644. Google Scholar

[14]

Y. LiR. Pan and S. Zhu, Recent progress on classical solutions for compressible isentropic Navier-Stokes equations with degenerate viscosities and vacuum, Bull. Braz. Math. Soc. (N.S.), 47 (2016), 507-519.  doi: 10.1007/s00574-016-0165-7.  Google Scholar

[15]

Y. Li and S. Zhu, Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum, J. Differential Equations, 256 (2014), 3943-3980.  doi: 10.1016/j.jde.2014.03.007.  Google Scholar

[16]

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

[17]

L. Sundbye, Global existence for the Cauchy problem for the viscous shallow water equations, Rocky Mountain J. Math., 28 (1998), 1135-1152.  doi: 10.1216/rmjm/1181071760.  Google Scholar

[18]

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

[19]

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

[20]

S. Zhu, Existence results for viscous polytropic fluids with degenerate viscosity coefficients and vacuum, J. Differential Equations, 259 (2015), 84-119.  doi: 10.1016/j.jde.2015.01.048.  Google Scholar

[21]

S. Zhu, Well-Posedness and Singularity Formation of the Compressible Isentropic Navier-Stokes Equations, Ph. D Thesis, Shanghai Jiao Tong University, 2015. Google Scholar

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