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December  2013, 6(4): 883-892. doi: 10.3934/krm.2013.6.883

Energy estimate for a linear symmetric hyperbolic-parabolic system in half line

Tohru Nakamura 1, and  Shinya Nishibata 2,

1. 

Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan
2. 

Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552

Received  August 2013 Revised  September 2013 Published  November 2013

In the present paper, we study the initial boundary value problem for a linear symmetric hyperbolic-parabolic system in one-dimensional half space. We obtain a priori estimates by using an energy method developed by Matsumura--Nishida for half space problem under the assumption that a stability condition of Shizuta--Kawashima type holds. The method developed in the present paper is applicable to showing the nonlinear stability of boundary layer solutions for a system of viscous conservation laws in half space.
Keywords: asymptotic stability., Energy method, stability condition.
Mathematics Subject Classification: Primary: 35B35, 76N15; Secondary: 35B4.
Citation: Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic & Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883
References:
[1]

Y. Kagei and S. Kawashima, Local solvability of an initial boundary value problem for a quasilinear hyperbolic-parabolic system,, J. Hyperbolic Differ. Equ., 3 (2006), 195.  doi: 10.1142/S0219891606000768.  Google Scholar

[2]

T. Kato, Linear evolution equations of hyperbolic type, II,, J. Math. Soc. Japan, 25 (1973), 648.  doi: 10.2969/jmsj/02540648.  Google Scholar

[3]

S. Kawashima, Systems of A Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics,, Doctoral Thesis, (1984).   Google Scholar

[4]

S. Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications,, Proc. Roy. Soc. Edinburgh, 106 (1987), 169.  doi: 10.1017/S0308210500018308.  Google Scholar

[5]

S. Kawashima, T. Nakamura, S. Nishibata and P. Zhu, Stationary waves to viscous heat-conductive gases in half-space: existence, stability and convergence rate,, Math. Models Methods Appl. Sci., 20 (2010), 2201.  doi: 10.1142/S0218202510004908.  Google Scholar

[6]

S. Kawashima, S. Nishibata and P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space,, Comm. Math. Phys., 240 (2003), 483.   Google Scholar

[7]

B. Kwon, M. Suzuki and M. Takayama, Large-time behavior of solutions to an outflow problem for a shallow water model,, J. Differential Equations, 255 (2013), 1883.  doi: 10.1016/j.jde.2013.05.025.  Google Scholar

[8]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases,, J. Math. Kyoto Univ., 20 (1980), 67.   Google Scholar

[9]

A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids,, Comm. Math. Phys., 89 (1983), 445.  doi: 10.1007/BF01214738.  Google Scholar

[10]

A. Matsumura and K. Nishihara, Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas,, Comm. Math. Phys., 222 (2001), 449.  doi: 10.1007/s002200100517.  Google Scholar

[11]

T. Nakamura and S. Nishibata, Stationary waves for symmetric hyperbolic-parabolic systems in half line and application to fluid dynamics,, preprint., ().   Google Scholar

[12]

T. Nakamura and S. Nishibata, Convergence rate toward planar stationary waves for compressible viscous fluid in multi-dimensional half space,, SIAM J. Math. Anal., 41 (2009), 1757.  doi: 10.1137/090755357.  Google Scholar

[13]

T. Nakamura and S. Nishibata, Stationary wave associated with an inflow problem in the half line for viscous heat-conductive gas,, J. Hyperbolic Differ. Equ., 8 (2011), 651.  doi: 10.1142/S0219891611002524.  Google Scholar

[14]

T. Nakamura, S. Nishibata and T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line,, J. Differential Equations, 241 (2007), 94.  doi: 10.1016/j.jde.2007.06.016.  Google Scholar

[15]

Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation,, Hokkaido Math. J., 14 (1985), 249.   Google Scholar

[16]

T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electromagnetofluid dynamics,, Japan J. Appl. Math., 1 (1984), 435.  doi: 10.1007/BF03167068.  Google Scholar

show all references

References:
[1]

Y. Kagei and S. Kawashima, Local solvability of an initial boundary value problem for a quasilinear hyperbolic-parabolic system,, J. Hyperbolic Differ. Equ., 3 (2006), 195.  doi: 10.1142/S0219891606000768.  Google Scholar

[2]

T. Kato, Linear evolution equations of hyperbolic type, II,, J. Math. Soc. Japan, 25 (1973), 648.  doi: 10.2969/jmsj/02540648.  Google Scholar

[3]

S. Kawashima, Systems of A Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics,, Doctoral Thesis, (1984).   Google Scholar

[4]

S. Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications,, Proc. Roy. Soc. Edinburgh, 106 (1987), 169.  doi: 10.1017/S0308210500018308.  Google Scholar

[5]

S. Kawashima, T. Nakamura, S. Nishibata and P. Zhu, Stationary waves to viscous heat-conductive gases in half-space: existence, stability and convergence rate,, Math. Models Methods Appl. Sci., 20 (2010), 2201.  doi: 10.1142/S0218202510004908.  Google Scholar

[6]

S. Kawashima, S. Nishibata and P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space,, Comm. Math. Phys., 240 (2003), 483.   Google Scholar

[7]

B. Kwon, M. Suzuki and M. Takayama, Large-time behavior of solutions to an outflow problem for a shallow water model,, J. Differential Equations, 255 (2013), 1883.  doi: 10.1016/j.jde.2013.05.025.  Google Scholar

[8]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases,, J. Math. Kyoto Univ., 20 (1980), 67.   Google Scholar

[9]

A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids,, Comm. Math. Phys., 89 (1983), 445.  doi: 10.1007/BF01214738.  Google Scholar

[10]

A. Matsumura and K. Nishihara, Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas,, Comm. Math. Phys., 222 (2001), 449.  doi: 10.1007/s002200100517.  Google Scholar

[11]

T. Nakamura and S. Nishibata, Stationary waves for symmetric hyperbolic-parabolic systems in half line and application to fluid dynamics,, preprint., ().   Google Scholar

[12]

T. Nakamura and S. Nishibata, Convergence rate toward planar stationary waves for compressible viscous fluid in multi-dimensional half space,, SIAM J. Math. Anal., 41 (2009), 1757.  doi: 10.1137/090755357.  Google Scholar

[13]

T. Nakamura and S. Nishibata, Stationary wave associated with an inflow problem in the half line for viscous heat-conductive gas,, J. Hyperbolic Differ. Equ., 8 (2011), 651.  doi: 10.1142/S0219891611002524.  Google Scholar

[14]

T. Nakamura, S. Nishibata and T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line,, J. Differential Equations, 241 (2007), 94.  doi: 10.1016/j.jde.2007.06.016.  Google Scholar

[15]

Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation,, Hokkaido Math. J., 14 (1985), 249.   Google Scholar

[16]

T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electromagnetofluid dynamics,, Japan J. Appl. Math., 1 (1984), 435.  doi: 10.1007/BF03167068.  Google Scholar

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