输运系数依赖温度的可压缩辐射流体解的整体存在性

数学物理学报 ›› 2023, Vol. 43 ›› Issue (2): 458-480.

输运系数依赖温度的可压缩辐射流体解的整体存在性

张明玉()

潍坊学院数学与信息科学学院山东潍坊261061

收稿日期:2022-04-29 修回日期:2022-10-19 出版日期:2023-04-26 发布日期:2023-04-17
作者简介:张明玉，E-mail: wfumath@126.com
基金资助:
山东省自然科学基金(ZR2021QA049)

Global Existence of the Compressible and Radiative Flux with Temperature-Dependent Transport Coefficients

Zhang Mingyu()

School of Mathematics and Information Science, Weifang University, Shandong Weifang 261061

Received:2022-04-29 Revised:2022-10-19 Online:2023-04-26 Published:2023-04-17
Supported by:
Natural Science Foundation of Shandong Province of China(ZR2021QA049)

摘要/Abstract

摘要：

该文主要研究了当粘性系数 $\lambda$ 和热传导系数 $\kappa$ 依赖于温度 $\theta$, 即 $\lambda(\theta)=\theta^\alpha$, $\kappa (\theta)=1+\theta^\beta$, 其中 $\alpha\in[0, +\infty)$, $\beta\in (2, +\infty)$ 时, 带有辐射项的可压缩 Navier-Stokes 方程解的全局存在性和非线性稳定性. 在关于参数 $\alpha$ 和初值的某些假设下, 该文得到了强解的全局存在唯一性. 此外, 在基于时间的一直估计下, 本文还证明了解的非线性指数稳定性. 需要指出的是, 如果 $\alpha$ 较小, 并且增长指数 $\beta$ 任意大, 则初值可以任意大.

关键词: 可压缩 Navier-Stokes 方程, 热辐射, 输运系数依赖温度, 大初值, 整体解

Abstract:

In this paper, we are concerned with the the global existence and non-linear stability of the compressible and radiative Navier-Stokes matrixs when the viscosity $\lambda$ and heat conductivity $\kappa$ depend on temperature $\theta$, i.e., $\lambda(\theta)=\theta^\alpha$, $\kappa(\theta)=1+\theta^\beta$ with $\alpha \in [0, +\infty)$, $\beta \in (2, +\infty)$. The global existence and uniqueness of strong solutions are obtained under the assumptions on the parameter $\alpha$ and initial data. In addition, we also proved the non-linear exponential stability of the solutions on the basis of the fundamental uniform-in-time estimates. It should be note that the initial data could be large if $\alpha$ is small and the growth exponent $\beta $ can be arbitrarily large.

Key words: Compressible Navier-Stokes matrixs, Thermal radiation, Temperature-dependent transport coefficients, Large initial data, Global solutions

中图分类号:

O175.27

张明玉. 输运系数依赖温度的可压缩辐射流体解的整体存在性[J]. 数学物理学报, 2023, 43(2): 458-480.

Zhang Mingyu. Global Existence of the Compressible and Radiative Flux with Temperature-Dependent Transport Coefficients[J]. Acta mathematica scientia,Series A, 2023, 43(2): 458-480.

参考文献 24

[1]	Antontsev S N, Kazhikhov A V, Monakhov V N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids. New York: Elsevier, 1990
[2]	Chen C Q. Global solutions to the compressible Navier-Stokes matrixs for a reacting mixture of nonlinear. SIAM J Math Anal, 1992, 23: 609-634 doi: 10.1137/0523031
[3]	Ducomet B. A model of thermal dissipation for a one-dimensional viscous reactive and radiative. Math Methods Appl Sci, 1999, 22: 1323-1349 doi: 10.1002/(ISSN)1099-1476
[4]	Jiang J, Zheng S M. Global solvability and asymptotic behavior of a free boundary problem for the one-dimensional viscous radiative and reactive gas. J Math Phy, 2012, 53
[5]	Jiang S. On initial boundary value problems for a viscous heat-conducting one-dimensional real gas. J Differential Equations, 1994, 110: 157-181 doi: 10.1006/jdeq.1994.1064
[6]	Kanel J I. A model system of matrixs for the one-dimensional motion of a gas. Differ Uravn, 1968, 4: 721-734
[7]	Kawohl B. Global existence of large solutions to initial boundary value problems for the matrixs of one- dimensional motion of viscous polytropic gases. J Differential Equations, 1985, 58: 76-103 doi: 10.1016/0022-0396(85)90023-3
[8]	Kawashima S, Nishida T. Global solutions to the initial value problem for the matrix of one-dimensional motion of viscous polytropic gases. J Math Kyoto Univ, 1981, 21: 825-837
[9]	Kazhikhov A V. To a theory of boundary value problems for matrix of one-dimensional nonstationary motion of viscous heat-conduction gases. Boundary value problems, 1981, 50: 37-62
[10]	Kazhikhov A V. Sur la solubilité global des problémes monodimensionnels aux valeurs initinales-limités pour les équations d'un gaz visqueux et calorifére. C R Acad Sci, Paris, Ser A, 1997, 284: 317-320
[11]	Kazhikhov A V, Shelukhin V V. Unique global solutions with respect to time of the initial-boundary value problems for one-dimensional matrixs of a viscous gas. J Appl Math Mech, 1977, 41: 273-282 doi: 10.1016/0021-8928(77)90011-9
[12]	Liao Y K, Zhao H J. Global solutions to one-dimensional matrixs for a self-gravitating viscous radiative and reactive gas with density-dependent viscosity. Commun Math Sci, 2017, 15: 1423-1456 doi: 10.4310/CMS.2017.v15.n5.a10
[13]	Nagasawa T. On the one-dimensional motion of the polytropic ideal gas non-fixed on the boundary. J differential Equations, 1986, 65: 49-67 doi: 10.1016/0022-0396(86)90041-0
[14]	Nagasawa T. Global asymptotics of the outer pressure problem with free boundary. Japan J Appl Math, 1988, 5: 205-224 doi: 10.1007/BF03167873
[15]	Nagasawa T. On the outer pressure problem of the one-dimensional polytropic ideal gas. Japan J Appl Math, 1988, 5: 53-85 doi: 10.1007/BF03167901
[16]	Qin Y M. Nonlinear parabolic-hyperbolic coupled systems and their attractors. Basel: Birkhäuser, 2008
[17]	Qin Y M, Hu G L, Wang T G, Huang L. Remarks on global smooth solutions to a 1D self-gravitating viscous radiative and reactive gas. J Math Anal Appl, 2013, 408: 19-26 doi: 10.1016/j.jmaa.2013.05.061
[18]	Qin Y M, Huang L. On the 1D viscous reactive and radiative gas with the first-order Arrhenius kinetics. Math Meth Appl Sci, 2019, 42: 5969-5998 doi: 10.1002/mma.v42.18
[19]	Qin Y M, Liu X, Yang X. Global existence and exponential stability for a 1D compressible and radiative MHD flow. J Differential Equations, 2012, 253: 1439-1488 doi: 10.1016/j.jde.2012.05.003
[20]	Sun Y, Zhang J W, Zhao X K. Nonlinear exponential stability for the compressible Navier-Stokes matrixs with temperature-dependent transport coefficients. J Differential Equations, 2021, 286: 676-709 doi: 10.1016/j.jde.2021.03.044
[21]	Tani A. On the first initial-boundary value problem of compressible viscous fluid motion. Publ Res Inst Math Sci, 1977, 13: 193-253 doi: 10.2977/prims/1195190106
[22]	Umehara M, Tani A. Global solution to the one-dimensional matrixs for a self-gravitating viscous radiative and reactive gas. J Differential Equations, 2007, 234: 439-463 doi: 10.1016/j.jde.2006.09.023
[23]	Wang T, Zhao H J. One-dimensional compressible heat-conducting gas with temperature-dependent viscosity. Math Models Methods Appl Sci, 2016, 26: 2237-2275 doi: 10.1142/S0218202516500524
[24]	Zhang J W, Xie F. Global solutions for a one-dimensional model problem in thermally radiative magnetohydrodynamics. J Differential Equations, 2008, 245: 1853-1882 doi: 10.1016/j.jde.2008.07.010