Global existence and optimal decay rates of solutions to a reduced gravity two and a half layer model

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May 2015, 14(3): 981-1000. doi: 10.3934/cpaa.2015.14.981

Global existence and optimal decay rates of solutions to a reduced gravity two and a half layer model

Haibo Cui ^1, , Lei Yao ^2, and Zheng-An Yao ^3,

1.	School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, China
2.	School of Mathematics and Center for Nonlinear Studies, Northwest University, Xi'an 710127
3.	Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275

Received September 2014 Revised January 2015 Published March 2015

Abstract
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In this paper, we consider global existence and optimal time decay rates of global smooth solutions to three-dimensional reduced gravity two and a half layer model. Indeed we show that the upper and middle layer thicknesses and horizontal velocities converge to their equilibrium state at the $L^2$-rate $(1+t)^{-\frac{3}{4}}$ or $L^\infty$-rate $(1+t)^{-\frac{3}{2}}$, respectively. These convergence rates are also shown to be optimal. The proof is based on the detailed analysis of the Green's function to the linearized system and elaborate energy estimates to the nonlinear system.

Keywords: global existence, Cauchy problem, decay estimates., Two and a half layer model.

Mathematics Subject Classification: Primary: 76T99, 76N1.

Citation: Haibo Cui, Lei Yao, Zheng-An Yao. Global existence and optimal decay rates of solutions to a reduced gravity two and a half layer model. Communications on Pure & Applied Analysis, 2015, 14 (3) : 981-1000. doi: 10.3934/cpaa.2015.14.981

References:

[1]	R. Duan and C. H. Zhou, On the compactness of the reduced-gravity two-and-a-half layer equations,, \emph{J. Differential Equations}, 252 (2012), 3506. doi: 10.1016/j.jde.2011.12.012.
[2]	R. J. Duan, Green's function and large time behavior of the Navier-Stokes-Maxwell system,, \emph{Anal. Appl.}, 10 (2012), 133. doi: 10.1142/S0219530512500078.
[3]	R. J. Duan, Global smooth flows for the compressible Euler-Maxwell system The relaxation case,, \emph{J. Hyperbolic Differential Equations}, 8 (2011), 375. doi: 10.1142/S0219891611002421.
[4]	R. J. Duan and H. F. Ma, Global existence and convergence rates for the 3-D compressible Navier-Stokes equations without heat conductivity,, \emph{Indiana Univ. Math. J.}, 57 (2008), 2299. doi: 10.1512/iumj.2008.57.3326.
[5]	Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces,, \emph{Comm. Partial Differential Equations}, 37 (2012), 2165. doi: 10.1080/03605302.2012.696296.
[6]	Z. H. Guo, Z. L. Li and L. Yao, Existence of global weak solution for a reduced gravity two and half layer model,, \emph{J. Math. Phys.}, 54 (2013), 1. doi: 10.1063/1.4836775.
[7]	D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow,, \emph{Indiana Univ. Math. J.}, 44 (1995), 603. doi: 10.1512/iumj.1995.44.2003.
[8]	D. L. Li, The Greens function of the Navier-Stokes equations for gas dynamics in $R^3$,, \emph{Comm. Math. Phys.}, 257 (2005), 579. doi: 10.1007/s00220-005-1351-4.
[9]	H. L. Li, A. Matsumura and G. J. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $R^3$,, \emph{Arch. Ration. Mech. Anal.}, 196 (2010), 681. doi: 10.1007/s00205-009-0255-4.
[10]	H. L. Li and T. Zhang, Large time behavior of isentropic compressible Navier-Stokes system in $R^3$,, \emph{Math. Meth. Appl. Sci.}, 34 (2011), 670. doi: 10.1002/mma.1391.
[11]	H. L. Li and T. Zhang, Large time behavior of solutions to 3D compressible Navier-Stokes-Poisson system,, \emph{Sci. China Math.}, 55 (2012), 159. doi: 10.1007/s11425-011-4280-z.
[12]	T. P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions,, \emph{Comm. Math. Phys.}, 196 (1998), 145. doi: 10.1007/s002200050418.
[13]	A. Matsumura and T. Nishida, The initial value problem for the equation of motion of viscous and heat-conductive gases,, \emph{J. Math. Kyoto Univ.}, 20 (1980), 67.
[14]	W. K. Wang and X. F. Yang, The pointwise estimates of solutions to the isentropic Navier-Stokes equations in even space-dimensions,, \emph{J. Hyperbolic Differential Equations}, 2 (2005), 673. doi: 10.1142/S0219891605000580.
[15]	G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation,, Cambridge University Press, (2006).
[16]	J. D. Zabsonre and G. Narbona-Reina, Existence of a global weak solution for a 2D viscous bi-layer Shallow Water model,, \emph{Nonlinear Anal. Real World Appl.}, 10 (2009), 2971. doi: 10.1016/j.nonrwa.2008.09.004.
[17]	G. J. Zhang, H. L. Li and C. J. Zhu, Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in $R^3$,, \emph{J. Differential Equations}, 250 (2011), 866. doi: 10.1016/j.jde.2010.07.035.

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References:

[1]	R. Duan and C. H. Zhou, On the compactness of the reduced-gravity two-and-a-half layer equations,, \emph{J. Differential Equations}, 252 (2012), 3506. doi: 10.1016/j.jde.2011.12.012.
[2]	R. J. Duan, Green's function and large time behavior of the Navier-Stokes-Maxwell system,, \emph{Anal. Appl.}, 10 (2012), 133. doi: 10.1142/S0219530512500078.
[3]	R. J. Duan, Global smooth flows for the compressible Euler-Maxwell system The relaxation case,, \emph{J. Hyperbolic Differential Equations}, 8 (2011), 375. doi: 10.1142/S0219891611002421.
[4]	R. J. Duan and H. F. Ma, Global existence and convergence rates for the 3-D compressible Navier-Stokes equations without heat conductivity,, \emph{Indiana Univ. Math. J.}, 57 (2008), 2299. doi: 10.1512/iumj.2008.57.3326.
[5]	Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces,, \emph{Comm. Partial Differential Equations}, 37 (2012), 2165. doi: 10.1080/03605302.2012.696296.
[6]	Z. H. Guo, Z. L. Li and L. Yao, Existence of global weak solution for a reduced gravity two and half layer model,, \emph{J. Math. Phys.}, 54 (2013), 1. doi: 10.1063/1.4836775.
[7]	D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow,, \emph{Indiana Univ. Math. J.}, 44 (1995), 603. doi: 10.1512/iumj.1995.44.2003.
[8]	D. L. Li, The Greens function of the Navier-Stokes equations for gas dynamics in $R^3$,, \emph{Comm. Math. Phys.}, 257 (2005), 579. doi: 10.1007/s00220-005-1351-4.
[9]	H. L. Li, A. Matsumura and G. J. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $R^3$,, \emph{Arch. Ration. Mech. Anal.}, 196 (2010), 681. doi: 10.1007/s00205-009-0255-4.
[10]	H. L. Li and T. Zhang, Large time behavior of isentropic compressible Navier-Stokes system in $R^3$,, \emph{Math. Meth. Appl. Sci.}, 34 (2011), 670. doi: 10.1002/mma.1391.
[11]	H. L. Li and T. Zhang, Large time behavior of solutions to 3D compressible Navier-Stokes-Poisson system,, \emph{Sci. China Math.}, 55 (2012), 159. doi: 10.1007/s11425-011-4280-z.
[12]	T. P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions,, \emph{Comm. Math. Phys.}, 196 (1998), 145. doi: 10.1007/s002200050418.
[13]	A. Matsumura and T. Nishida, The initial value problem for the equation of motion of viscous and heat-conductive gases,, \emph{J. Math. Kyoto Univ.}, 20 (1980), 67.
[14]	W. K. Wang and X. F. Yang, The pointwise estimates of solutions to the isentropic Navier-Stokes equations in even space-dimensions,, \emph{J. Hyperbolic Differential Equations}, 2 (2005), 673. doi: 10.1142/S0219891605000580.
[15]	G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation,, Cambridge University Press, (2006).
[16]	J. D. Zabsonre and G. Narbona-Reina, Existence of a global weak solution for a 2D viscous bi-layer Shallow Water model,, \emph{Nonlinear Anal. Real World Appl.}, 10 (2009), 2971. doi: 10.1016/j.nonrwa.2008.09.004.
[17]	G. J. Zhang, H. L. Li and C. J. Zhu, Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in $R^3$,, \emph{J. Differential Equations}, 250 (2011), 866. doi: 10.1016/j.jde.2010.07.035.

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