zbMATH — the first resource for mathematics

On temporal asymptotics for the \(p\)th power viscous reactive gas. (English) Zbl 1094.76052
Summary: We investigate the long-time behaviour of solutions to the system governing a heat-conductive, viscous reactive \(p\)th power gas confined between two parallel plates. For initial-boundary value problems with the end points held at a prescribed temperature or insulated, we prove the global existence of physically relevant solutions, and establish their rate of convergence to equilibria, for generic initial data. The estimates for different boundary conditions are presented in a unified manner.

76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76V05 Reaction effects in flows
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI
[1] S.N. Antontsev, S.N. Kazhikhov, V.N. Monakhov, Boundary value problems in mechanics of nonhomogeneous fluids, Studies in Mathematics and its Applications, Vol. 22, North-Holland Publishing Co. Amsterdam, 1990. · Zbl 0696.76001
[2] Bebernes, J.; Bressan, A., Global a priori estimates for a viscous reactive gas, Proc. roy. soc. Edinburgh sect. A, 101, 321-333, (1985) · Zbl 0614.76076
[3] Besov, O.V.; Il’in, V.P.; Nikols’kii, S.M., Integral representations of functions, and embedding theorems, (1978), Winston Washington, DC
[4] Chen, G.Q.; Hoff, D.; Trivisa, K., On the navier – stokes equations for exothermically reacting compressible fluids, Acta math. appl. sinica, 18, 1-8, (2002) · Zbl 1032.76056
[5] Feireisl, E.; Petzeltova, H., Unconditional stability of stationary flows of compressible heat-conducting fluids driven by large external forces, J. math. fluid mech., 1, 168-186, (1999) · Zbl 0931.35123
[6] Guo, B.; Zhu, P., Asymptotic behaviour of the solution to the system for a viscous reactive gas, J. differential equations, 155, 177-202, (1999) · Zbl 0927.76098
[7] Hsiao, L.; Luo, T., Large-time behaviour of solutions for the outer pressure problem of a viscous heat-conductive one-dimensional real gas, Proc. roy. soc. Edinburgh sect. A, 126, 1277-1296, (1996) · Zbl 0864.35085
[8] Jiang, S., Large-time behaviour of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains, Commun. math. phys., 200, 181-193, (1999) · Zbl 0918.35021
[9] Kassoy, D.; Poland, J., The induction period of a thermal explosion in a gas between infinite parallel plates, Combust. flame, 50, 259-274, (1983)
[10] Kawashima, S.; Nishida, T., Global solutions to the initial value problem for the equations of one-dimensional motion of viscous polytropic gases, J. math. Kyoto univ., 21, 825-837, (1981) · Zbl 0478.76097
[11] Kazhikhov, A.V., Correctness “in the large” of mixed boundary value problems for a model system of equations of a viscous gas, Dinamika splošn. sredy vyp., 21, 18-47, (1975), 188 (in Russian)
[12] Kazhikhov, A.V.; Shelukhin, V.V., Unique global solution with respect to time of initial boundary value problem for one dimensional equations of a viscous gas, Prikl. mat. mekh., 41, 273-282, (1997)
[13] Lewicka, M.; Watson, S.J., Temporal asymptotics for the pth power Newtonian fluid in one space dimension, Zamp, 54, 633-651, (2003) · Zbl 1026.76039
[14] Matsumura, A.; Yanagi, S., Uniform boundedness of the solutions for a one-dimesional isentropic model system of compressible viscous gas, Comm. math. phys., 175, 259-274, (1996) · Zbl 0849.35109
[15] Mucha, P.B., Compressible navier – stokes system in 1-D, Math. methods appl. sci., 24, 607-622, (2001) · Zbl 0998.35036
[16] Nagasawa, T., Global asymptotics of the outer pressure problem with free boundary, Japan J. appl. math., 5, 205-224, (1988) · Zbl 0665.76077
[17] Watson, S.J., Unique global solvability for initial-boundary value problems in one-dimensional nonlinear thermoviscoelasticity, Arch. rational mech. anal., 153, 1-37, (2000) · Zbl 0996.74032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.