# American Institue of Mathematical Sciences

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 - A, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015
##### References:
 [1] D. Bresch, 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. [2] D. Bresch, 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. [3] D. Bresch, B. Desjardins, 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. [4] Q. Chen, C. Miao, 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. [5] Y. Cho, 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. [6] M. Ding, 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. [7] B. Duan, Y. Zheng, 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. [8] Z. Guo, Q. Jiu, Z. Xin, Radially symmetric isentropic compressible flows with density-dependent viscosity coefficients, SIAM J. Math. Anal., 39 (2008), 1402-1427. doi: 10.1137/070680333. [9] C. Hao, L. Hsiao, 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. [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. [11] H. Li, J. Li, 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. [12] Y. Li, R. Pan, 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. [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. [14] Y. Li, R. Pan, 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. [15] Y. Li, 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. [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. [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. [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. [19] W. Wang, C. Xu, The Cauchy problem for viscous shallow water equations, Rev. Mat. Iberoamericana, 21 (2005), 1-24. doi: 10.4171/RMI/412. [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. [21] S. Zhu, Well-Posedness and Singularity Formation of the Compressible Isentropic Navier-Stokes Equations, Ph. D Thesis, Shanghai Jiao Tong University, 2015.

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##### References:
 [1] D. Bresch, 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. [2] D. Bresch, 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. [3] D. Bresch, B. Desjardins, 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. [4] Q. Chen, C. Miao, 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. [5] Y. Cho, 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. [6] M. Ding, 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. [7] B. Duan, Y. Zheng, 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. [8] Z. Guo, Q. Jiu, Z. Xin, Radially symmetric isentropic compressible flows with density-dependent viscosity coefficients, SIAM J. Math. Anal., 39 (2008), 1402-1427. doi: 10.1137/070680333. [9] C. Hao, L. Hsiao, 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. [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. [11] H. Li, J. Li, 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. [12] Y. Li, R. Pan, 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. [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. [14] Y. Li, R. Pan, 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. [15] Y. Li, 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. [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. [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. [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. [19] W. Wang, C. Xu, The Cauchy problem for viscous shallow water equations, Rev. Mat. Iberoamericana, 21 (2005), 1-24. doi: 10.4171/RMI/412. [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. [21] S. Zhu, Well-Posedness and Singularity Formation of the Compressible Isentropic Navier-Stokes Equations, Ph. D Thesis, Shanghai Jiao Tong University, 2015.
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