半空间中满足 Cattaneo 定律的双曲系统的稳定性

Abstract

Abstract:

This paper is concerned with the time-asymptotically nonlinear stability of stationary solutions to the initial boundary value problem of hyperbolic equations with Cattaneo's law in one-dimensional half space. We construct the stationary solutions to such an initial boundary value problem and show their regularities. Moreover, by introducing a correction function, the asymptotic stability of the above stationary solutions under small initial perturbations is proved by using the $ L^2 $-energy method and Poincaré-type inequalities in half space.

Key words: Cattaneo's Law, initial-boundary value problem, stationary solutions, asymptotic stability, Poincaré-type inequalities

CLC Number:

O175.2

Junyuan Deng, Junhao Zhang. Stability to a Hyperbolic System with Cattaneo's Law in the Half Space[J].Acta mathematica scientia,Series A, 2025, 45(5): 1444-1462.

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References

[1]	Cattaneo C. Sulla conduzione del calore. (Italian) Atti Sem Mat Fis Univ Modena, 1949, 3: 83—101
[2]	Vernotte P. Les paradoxes de la theorie continue de l'equation de la chaleur. C R Acad Sci Paris, 1958, 246: 3154—3155
[3]	Morse P M, Feshbach H. Methods of Theoretical Physics. New York: McGraw-Hill, 1953
[4]	H. Fernández Sare, Racke R, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law. Arch Ration Mech Anal, 2009, 194(1): 221—251
[5]	Nakamura T, Kawashima S. Viscous shock profile and singular limit for hyperbolic systems with Cattaneo's law. Kinet Relat Models, 2018, 11(4): 795—819
[6]	Nakamura K, Nakamura T, Kawashima S. Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws. Kinet Relat Models, 2019, 12(4): 923—944
[7]	Racke R. Thermoelasticity with second sound-exponential stability in linear and nonlinear 1-d. Math Methods Appl Sci, 2002, 25(5): 409—441
[8]	Liu T P, Nishihara K. Asymptotic behavior for scalar viscous conservation laws with boundary effect. J Differential Equations, 1997, 133: 296—320
[9]	Liu T P, Yu S H. Propagation of a stationary shock layer in the presence of a boundary. Arch Ration Mech Anal, 1997, 139(1): 57—82
[10]	Liu T P, Matsumura A, Nishihara K. Behaviors of solutions for the burgers equation with boundary corresponding to rarefaction waves. SIAM J Math Anal, 1998, 29: 293—308
[11]	Nishihara K. Boundary effect on a stationary viscous shock wave for scalar viscous conservation laws. J Math Anal Appl, 2001, 255: 535—550
[12]	Fan L L, Liu H X, Wang T, Zhao H J. Inflow problem for the one-dimensional compressible navier-stokes equations under large initial perturbation. J Differential Equations, 2014, 257(10): 3521—3553
[13]	Hong H, Wang T. Large-time behavior of solutions to the inflow problem of full compressible navier-stokes equations with large perturbation. Nonlinearity, 2017, 30: 3010—3039
[14]	Huang F M, Matsumura A, Shi X D. Viscous shock wave and boundary layer solution to an inflow problem for compressible viscous gas. Comm Math Phys, 2003, 239: 261—285
[15]	Huang F M, Matsumura A. Stability of a composite wave of two viscous shock waves for the full compressible Navier-Stokes equation. Comm Math Phys, 2009, 289: 841—861
[16]	Kawashima S, Nishibata S, Zhu P C. Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space. Comm Math Phys, 2003, 240(3): 483—500
[17]	Kawashima S, Zhu P C. Asymptotic stability of rarefaction wave for the Navier-Stokes equations for a compressible fluid in the half space. Arch Ration Mech Anal, 2009, 194(1): 105—132
[18]	Matsumura A. Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas. Methods Appl Anal, 2001, 8: 645—666
[19]	Matsumura A, Mei M. Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary. Arch Ration Mech Anal, 1999, 146: 1—22
[20]	Matsumura A, Nishihara K. 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, 2001, 222: 449—474
[21]	Qin X H, Wang Y. Stability of wave patterns to the inflow problem of full compressible Navier-Stokes equations. SIAM J Math Anal, 2009, 41: 2057—2087
[22]	Qin X H, Wang Y. Large-time behavior of solutions to the inflow problem of full compressible Navier-Stokes equations. SIAM J Math Anal, 2011, 43: 341—366
[23]	Wan L, Wang T, Zou Q Y. Stability of stationary solutions to the outflow problem for full compressible Navier-Stokes equations with large initial perturbation. Nonlinearity, 2016, 29(4): 1329—1354
[24]	Qin X H. Boundary layer to a system of viscous hyperbolic conservation laws. Acta Math Appl Sin Engl Ser, 2008, 24(3): 523—528
[25]	Nakamura T, Nishibata S. Existence and asymptotic stability of stationary waves for symmetric hyperbolic-parabolic systems in half-line. Math Models Methods Appl Sci, 2017, 27(11): 2071—2110
[26]	Kawashima S, Nishibata S, Nishikawa M. $ L^p $ energy method for multi-dimensional viscous conservation laws and application to the stability of planar waves. J Hyperbolic Differ Equ, 2004, 1(3): 581—603
[27]	Bai Y S, He L, Zhao H J. Nonlinear stability of rarefaction waves for a hyperbolic system with Cattaneo's law. Commun Pure Appl Anal, 2021, 20(7/8): 2441—2474
[28]	Bai Y S, Fan L L, Zhao H J. Nonlinear stability of viscous shock profiles for a hyperbolic system with Cattaneo's law in a one-dimensional half space. Math Meth Appl Sci, 2024, 47(6): 5207—5242
[29]	Friedrichs K. Symmetric hyperbolic linear differential equations. Comm Pure Appl Math, 1954, 7: 345—392
[30]	Friedrichs K. Symmetric positive linear differential equations. Comm Pure Appl Math, 1958, 11: 333—418
[31]	Gu C H. Differentiable solutions of symmetric positive partial differential equations. Chinese Math, 1964, 5: 541—555
[32]	Lax P, Phillips R. Local boundary conditions for dissipative symmetric linear differential operators. Comm Pure Appl Math, 1960, 13: 427—455
[33]	Kawashima S, Yanagisawa T, Shizuta Y. Mixed problems for quasi-linear symmetric hyperbolic systems. Proc Japan Acad Ser A Math Sci, 1987, 63(7): 243–246

Stability to a Hyperbolic System with Cattaneo's Law in the Half Space

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