doi: 10.3934/dcdsb.2018267

Global solution and decay rate for a reduced gravity two and a half layer model

1. 

School of Mathematics, Northwest University, Xi'an 710127, China

2. 

School of Mathematics and Center for Nonlinear Studies, Northwest University, Xi'an 710127, China

* Corresponding author: Lei Yao

Received  December 2017 Revised  April 2018 Published  October 2018

Fund Project: Liu and Yao were supported by the National Natural Science Foundation of China #11571280, 11331005, FANEDD #201315

In this paper we investigate the reduced gravity two and a half model in oceanic fluid dynamics. In a finite domain (for the initial-boundary value problem), we obtain time-independent estimates, which allow us to show the existence and uniqueness of regular solutions as well as the decay rate estimates. A collection of the decay rate estimates for $h_i-\widetilde{h}_i$ (with $\widetilde{h}_i$ being the stationary layer thickness) and $u_i(i = 1,2)$ in $L^2(Ω)$-norm as well as $H^1(Ω)$-norm as time $t \to \infty $ are established.

Citation: Yongming Liu, Lei Yao. Global solution and decay rate for a reduced gravity two and a half layer model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018267
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.

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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.

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H. B. CuiL. Yao and Z. A. Yao, Global existence and optimal decay rates of solutions to a reduced gravity two and a half layer model, Commun. Pure Appl. Anal., 14 (2015), 981-1000. doi: 10.3934/cpaa.2015.14.981.

[6]

R. Duan and C. H. Zhou, On the compactness of the reduced-gravity two-and-a-half layer equations, J. Differential Equations, 252 (2012), 3506-3519. doi: 10.1016/j.jde.2011.12.012.

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R. J. Duan and H. F. Ma, Global existence and convergence rates for the 3-D compressible Navier-Stokes equations without heat conductivity, Indiana Univ. Math. J., 57 (2008), 2299-2319. doi: 10.1512/iumj.2008.57.3326.

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S. EvjeH. Y. Wen and L. Yao, Global solutions to a one-dimensional non-conservative two-phase model, Discrete Contin. Dyn. Syst., 36 (2016), 1927-1955. doi: 10.3934/dcds.2016.36.1927.

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Z. H. Guo, Z. L. Li and L. Yao, Existence of global weak solution for a reduced gravity two and a half layer model, J. Math. Phys., 54 (2013), 121503, 19 pp. doi: 10.1063/1.4836775.

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J. Li and Z. P. Xin, Global existence of weak solution to the barotropic compressible Navier-Stokes flows with degenerate viscosities, arXiv: 1504.06826.

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A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equations, Comm. Partial Differential Equations, 32 (2007), 431-452. doi: 10.1080/03605300600857079.

[23]

I. Straškraba and A. Zlotnik, On a decay rate for 1D-viscous compressible barotropic fluid equations, J. Evol. Equ., 2 (2002), 69-96. doi: 10.1007/s00028-002-8080-3.

[24]

I. Straškraba and A. Zlotnik, Global properties of solutions to 1D-viscous compressible barotropic fluid equations with density dependent viscosity, Z. Angew. Math. Phys., 54 (2003), 593-607. doi: 10.1007/s00033-003-1009-z.

[25]

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[26]

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A. F. Vasseur and C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations, Invent. Math., 206 (2016), 935-974. doi: 10.1007/s00222-016-0666-4.

[28]

S. W. VongT. Yang and C. J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum (Ⅱ), J. Differential Equations, 192 (2003), 475-501. doi: 10.1016/S0022-0396(03)00060-3.

[29]

W. K. Wang and X. F. Yang, The pointwise estimates of solutions to the isentropic Navier-Stokes equations in even space-dimensions, J. Hyperbolic Differ. Equ., 2 (2005), 673-695. doi: 10.1142/S0219891605000580.

[30]

L. YaoZ. L. Li and W. J. Wang, Existence of spherically symmetric solutions for a reduced gravity two-and-a-half layer system, J. Differential Equations, 261 (2016), 1637-1668. doi: 10.1016/j.jde.2016.04.012.

[31]

T. YangZ. A. Yao and C. J. Zhu, Compressible Navier-Stokes equations with density-dependent viscosity and vacuum, Comm. Partial Differential Equations, 26 (2001), 965-981. doi: 10.1081/PDE-100002385.

[32]

T. Yang and H. J. Zhao, A vacuum problem for the one-dimensional Compressible Navier-Stokes equations with density-dependent viscosity, J. Differential Equations, 184 (2002), 163-184. doi: 10.1006/jdeq.2001.4140.

[33]

T. Yang and C. J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum, Comm. Math. Phys., 230 (2002), 329-363. doi: 10.1007/s00220-002-0703-6.

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.

[2]

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.

[3]

D. Bresch and B. Desjardins, Stabilité de solutions faibles globales pour les équations de Navier-Stokes compressible avec température, C. R. Math. Acad. Sci., 343 (2006), 219-224. doi: 10.1016/j.crma.2006.05.016.

[4]

Y. ChoH. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004.

[5]

H. B. CuiL. Yao and Z. A. Yao, Global existence and optimal decay rates of solutions to a reduced gravity two and a half layer model, Commun. Pure Appl. Anal., 14 (2015), 981-1000. doi: 10.3934/cpaa.2015.14.981.

[6]

R. Duan and C. H. Zhou, On the compactness of the reduced-gravity two-and-a-half layer equations, J. Differential Equations, 252 (2012), 3506-3519. doi: 10.1016/j.jde.2011.12.012.

[7]

R. J. Duan and H. F. Ma, Global existence and convergence rates for the 3-D compressible Navier-Stokes equations without heat conductivity, Indiana Univ. Math. J., 57 (2008), 2299-2319. doi: 10.1512/iumj.2008.57.3326.

[8]

S. EvjeH. Y. Wen and L. Yao, Global solutions to a one-dimensional non-conservative two-phase model, Discrete Contin. Dyn. Syst., 36 (2016), 1927-1955. doi: 10.3934/dcds.2016.36.1927.

[9]

D. Y. Fang and T. Zhang, Compressible Navier-Stokes equations with vacuum state in one dimension, Commun. Pure Appl. Anal., 3 (2004), 675-694. doi: 10.3934/cpaa.2004.3.675.

[10]

Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208. doi: 10.1080/03605302.2012.696296.

[11]

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

[12]

Z. H. GuoH. L. Li and Z. P. Xin, Lagrange structure and dynamics for solutions to the spherically symmetric compressible Navier-Stokes equations, Comm. Math. Phys., 309 (2012), 371-412. doi: 10.1007/s00220-011-1334-6.

[13]

Z. H. Guo, Z. L. Li and L. Yao, Existence of global weak solution for a reduced gravity two and a half layer model, J. Math. Phys., 54 (2013), 121503, 19 pp. doi: 10.1063/1.4836775.

[14]

X. D. Huang and J. Li, Existence and blowup behavior of global strong solutions to the two-dimensional barotrpic compressible Navier-Stokes system with vacuum and large initial data, J. Math. Pures Appl., 106 (2016), 123-154. doi: 10.1016/j.matpur.2016.02.003.

[15]

S. JiangZ. P. Xin and P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity, Methods Appl. Anal., 12 (2005), 239-251. doi: 10.4310/MAA.2005.v12.n3.a2.

[16]

Q. S. JiuY. Wang and Z. P. Xin, Global well-posedness of the Cauchy problem of two-dimensional compressible Navier-Stokes equations in weighted spaces, J. Differential Equations, 255 (2013), 351-404. doi: 10.1016/j.jde.2013.04.014.

[17]

Q. S. JiuY. Wang and Z. P. Xin, Global well-posedness of 2D compressible Navier-Stokes equations with large data and vacuum, J. Math. Fluid Mech., 16 (2014), 483-521. doi: 10.1007/s00021-014-0171-8.

[18]

D. L. Li, The Green's function of the Navier-Stokes equations for gas dynamics in R3, Comm. Math. Phys., 257 (2005), 579-619. doi: 10.1007/s00220-005-1351-4.

[19]

H. L. Li and T. Zhang, Large time behavior of isentropic compressible Navier-Stokes system in R3, Math. Methods Appl. Sci., 34 (2011), 670-682. doi: 10.1002/mma.1391.

[20]

J. Li and Z. P. Xin, Global existence of weak solution to the barotropic compressible Navier-Stokes flows with degenerate viscosities, arXiv: 1504.06826.

[21]

T. P. LiuZ. P. Xin and T. Yang, Vacuum states for compressible flow, Discrete Contin. Dyn. Syst., 4 (1998), 1-32. doi: 10.3934/dcds.1996.2.1.

[22]

A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equations, Comm. Partial Differential Equations, 32 (2007), 431-452. doi: 10.1080/03605300600857079.

[23]

I. Straškraba and A. Zlotnik, On a decay rate for 1D-viscous compressible barotropic fluid equations, J. Evol. Equ., 2 (2002), 69-96. doi: 10.1007/s00028-002-8080-3.

[24]

I. Straškraba and A. Zlotnik, Global properties of solutions to 1D-viscous compressible barotropic fluid equations with density dependent viscosity, Z. Angew. Math. Phys., 54 (2003), 593-607. doi: 10.1007/s00033-003-1009-z.

[25]

V. A. Vaigant and A. V. Kazhikhov, On existence of global solutions to the twodimensional Navier-Stokes equations for a compressible viscous fluid, Sib. Math. J., 36 (1995), 1283-1316. doi: 10.1007/BF02106835.

[26]

G. K. Vallis, Atmospheric and oceanic fluid dynamics: Fundamentals and large-scale circulation, Cambridge University Press, 2006.

[27]

A. F. Vasseur and C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations, Invent. Math., 206 (2016), 935-974. doi: 10.1007/s00222-016-0666-4.

[28]

S. W. VongT. Yang and C. J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum (Ⅱ), J. Differential Equations, 192 (2003), 475-501. doi: 10.1016/S0022-0396(03)00060-3.

[29]

W. K. Wang and X. F. Yang, The pointwise estimates of solutions to the isentropic Navier-Stokes equations in even space-dimensions, J. Hyperbolic Differ. Equ., 2 (2005), 673-695. doi: 10.1142/S0219891605000580.

[30]

L. YaoZ. L. Li and W. J. Wang, Existence of spherically symmetric solutions for a reduced gravity two-and-a-half layer system, J. Differential Equations, 261 (2016), 1637-1668. doi: 10.1016/j.jde.2016.04.012.

[31]

T. YangZ. A. Yao and C. J. Zhu, Compressible Navier-Stokes equations with density-dependent viscosity and vacuum, Comm. Partial Differential Equations, 26 (2001), 965-981. doi: 10.1081/PDE-100002385.

[32]

T. Yang and H. J. Zhao, A vacuum problem for the one-dimensional Compressible Navier-Stokes equations with density-dependent viscosity, J. Differential Equations, 184 (2002), 163-184. doi: 10.1006/jdeq.2001.4140.

[33]

T. Yang and C. J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum, Comm. Math. Phys., 230 (2002), 329-363. doi: 10.1007/s00220-002-0703-6.

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