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Time-dependent singularities in the heat equation
Global existence and optimal decay rates of solutions to a reduced gravity two and a half layer model
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 |
References:
[1] |
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. |
[2] |
R. J. Duan, Green's function and large time behavior of the Navier-Stokes-Maxwell system, Anal. Appl., 10 (2012), 133-197.
doi: 10.1142/S0219530512500078. |
[3] |
R. J. Duan, Global smooth flows for the compressible Euler-Maxwell system The relaxation case, J. Hyperbolic Differential Equations, 8 (2011), 375-413.
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, Indiana Univ. Math. J., 57 (2008), 2299-2319.
doi: 10.1512/iumj.2008.57.3326. |
[5] |
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. |
[6] |
Z. H. Guo, Z. L. Li and L. Yao, Existence of global weak solution for a reduced gravity two and half layer model, J. Math. Phys., 54 (2013), 1-19.
doi: 10.1063/1.4836775. |
[7] |
D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676.
doi: 10.1512/iumj.1995.44.2003. |
[8] |
D. L. Li, The Greens function of the Navier-Stokes equations for gas dynamics in $R^3$, Comm. Math. Phys., 257 (2005), 579-619.
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 $R^3$, Arch. Ration. Mech. Anal., 196 (2010), 681-713.
doi: 10.1007/s00205-009-0255-4. |
[10] |
H. L. Li and T. Zhang, Large time behavior of isentropic compressible Navier-Stokes system in $R^3$, Math. Meth. Appl. Sci., 34 (2011), 670-682.
doi: 10.1002/mma.1391. |
[11] |
H. L. Li and T. Zhang, Large time behavior of solutions to 3D compressible Navier-Stokes-Poisson system, Sci. China Math., 55 (2012), 159-177.
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, Comm. Math. Phys., 196 (1998), 145-173.
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, J. Math. Kyoto Univ., 20 (1980), 67-104. |
[14] |
W. K. Wang and X. F. Yang, The pointwise estimates of solutions to the isentropic Navier-Stokes equations in even space-dimensions, J. Hyperbolic Differential Equations, 2 (2005), 673-695.
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, Nonlinear Anal. Real World Appl., 10 (2009), 2971-2984.
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 $R^3$, J. Differential Equations, 250 (2011), 866-891.
doi: 10.1016/j.jde.2010.07.035. |
show all references
References:
[1] |
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. |
[2] |
R. J. Duan, Green's function and large time behavior of the Navier-Stokes-Maxwell system, Anal. Appl., 10 (2012), 133-197.
doi: 10.1142/S0219530512500078. |
[3] |
R. J. Duan, Global smooth flows for the compressible Euler-Maxwell system The relaxation case, J. Hyperbolic Differential Equations, 8 (2011), 375-413.
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, Indiana Univ. Math. J., 57 (2008), 2299-2319.
doi: 10.1512/iumj.2008.57.3326. |
[5] |
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. |
[6] |
Z. H. Guo, Z. L. Li and L. Yao, Existence of global weak solution for a reduced gravity two and half layer model, J. Math. Phys., 54 (2013), 1-19.
doi: 10.1063/1.4836775. |
[7] |
D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676.
doi: 10.1512/iumj.1995.44.2003. |
[8] |
D. L. Li, The Greens function of the Navier-Stokes equations for gas dynamics in $R^3$, Comm. Math. Phys., 257 (2005), 579-619.
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 $R^3$, Arch. Ration. Mech. Anal., 196 (2010), 681-713.
doi: 10.1007/s00205-009-0255-4. |
[10] |
H. L. Li and T. Zhang, Large time behavior of isentropic compressible Navier-Stokes system in $R^3$, Math. Meth. Appl. Sci., 34 (2011), 670-682.
doi: 10.1002/mma.1391. |
[11] |
H. L. Li and T. Zhang, Large time behavior of solutions to 3D compressible Navier-Stokes-Poisson system, Sci. China Math., 55 (2012), 159-177.
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, Comm. Math. Phys., 196 (1998), 145-173.
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, J. Math. Kyoto Univ., 20 (1980), 67-104. |
[14] |
W. K. Wang and X. F. Yang, The pointwise estimates of solutions to the isentropic Navier-Stokes equations in even space-dimensions, J. Hyperbolic Differential Equations, 2 (2005), 673-695.
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, Nonlinear Anal. Real World Appl., 10 (2009), 2971-2984.
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 $R^3$, J. Differential Equations, 250 (2011), 866-891.
doi: 10.1016/j.jde.2010.07.035. |
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