American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits
  • KRM Home
  • This Issue
  • Next Article
    Nonlinear stability of Broadwell model with Maxwell diffuse boundary condition
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

[1]

Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108
[2]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450
[3]

Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1207-1223. doi: 10.3934/dcds.2020315
[4]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316
[5]

Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021017
[6]

Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021006
[7]

Xiaoxiao Li, Yingjing Shi, Rui Li, Shida Cao. Energy management method for an unpowered landing. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020180
[8]

Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021015
[9]

Md. Masum Murshed, Kouta Futai, Masato Kimura, Hirofumi Notsu. Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1063-1078. doi: 10.3934/dcdss.2020230
[10]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366
[11]

Meihua Dong, Keonhee Lee, Carlos Morales. Gromov-Hausdorff stability for group actions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1347-1357. doi: 10.3934/dcds.2020320
[12]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432
[13]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
[14]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292
[15]

Hongguang Ma, Xiang Li. Multi-period hazardous waste collection planning with consideration of risk stability. Journal of Industrial & Management Optimization, 2021, 17 (1) : 393-408. doi: 10.3934/jimo.2019117
[16]

Laure Cardoulis, Michel Cristofol, Morgan Morancey. A stability result for the diffusion coefficient of the heat operator defined on an unbounded guide. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020054
[17]

Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362
[18]

Yi Guan, Michal Fečkan, Jinrong Wang. Periodic solutions and Hyers-Ulam stability of atmospheric Ekman flows. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1157-1176. doi: 10.3934/dcds.2020313
[19]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003
[20]

Do Lan. Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021002

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences