zbMATH — the first resource for mathematics

Local existence of the solution to the initial-boundary value problem in nonlinear thermodiffusion in micropolar medium. (English) Zbl 0955.35003
Summary: We prove a theorem about local existence (in time) of the solution to the first initial-boundary value problem for a nonlinear hyperbolic-parabolic system of eight coupled partial differential equations of second-order describing the process of thermodiffusion in a three-dimensional micropolar medium. At first, we prove existence, uniqueness and regularity of the solution to this problem for the associated linearized system by using the Faedo-Galerkin method and semigroup theory. Next, we prove (based on this theorem) an energy estimate for the solution to the linearized system by applying the method of Sobolev spaces. Finally, we prove by the Banach fixed point theorem that the solution of our nonlinear problem exists and is unique.

35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
74F05 Thermal effects in solid mechanics
47D06 One-parameter semigroups and linear evolution equations
80A20 Heat and mass transfer, heat flow (MSC2010)
Full Text: DOI EuDML
[1] Adams, R. A.: Sobolev Spaces. New York: Academic Press 1973. · Zbl 0262.46036
[2] Agmon, I. A.: Lectures on Elliptic Boundary-Value Problems. Van Nostrand: Princeton 1965. · Zbl 0142.37401
[3] Dafermos, C. M. and W. J. Hrusa: Energy methods for quasilinear hyperbolic initial boundary-value problems. Applications to elektrodynamics. Arch. Rat. Mech Anl. 87 (1985), 267 - 292. · Zbl 0586.35065
[4] Gawinecki, J. and K. Sierpiński: Existence, uniqueness and regularity of the solution of the first boundary initial value problem for the equations if thermodiffusion in a solid body. Bull. Acad. Polon. Sci. Ser. Techn. 30 (1982), 163 - 171. · Zbl 0553.73001
[5] Gawinecki, J. and P. Kacprzyk: Existence, uniqueness and regularity of the solution to the first boundary-initial value problem of linear thermodiffusion in micropolar medium. Bull. Acad. Polon. Sci., Ser. Techn. 42 (1994), 341 - 359. · Zbl 0821.73004
[6] Gawinecki, J., Kacprzyk, P. and J. J\?edrzejewski: On energy estimate for some coupled parabolic systems of partial differential equations. Biul. WAT XLV 12 (1996), 7 - 13.
[7] Kato, T.: Abstract Differential Equations and Nonlinear Mixed Problems Center for Pure and Appl. Math. Report Univ. of California, Berkely Published in Fermi Lectures Scuola Normale Sup., Pisa 1985.
[8] Kawashima, S.: Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics. Thesis. Kyoto: University 1983.
[9] Kawashima, S. and M. Matsumura: Mixed problems for symmetric hyperbolic-parabolic systems. Manuscript 1989.
[10] Kawashima, S., Yanagisawa, T. and Y. Shizuta: Mixed problem for quasilinear symmetric hyperbolic systems. Proc. Japan Acad. Sci. (Ser. A) 63 (1987), 243 - 241. · Zbl 0673.35069
[11] Ladyzhenskaya, O. A., Solonnikov, V. A. and N. N. Uralceva: Linear and Quasi-Linear Equations of Parabolic Types (Trans. Math. Monographs: Vol. 23). Providence (R.I.): Amer. Math. Soc. 1968. 451 · Zbl 0174.15403
[12] Li, D.: The nonlinear initial-boundary value problem and the existence of multi-dimensio- nal, shock wave for quasilinear hyperbolic-parabolic coupled systems. Chin. Ann. Math. 8B(2) (1987), 252 - 280. · Zbl 0678.35071
[13] Milani, A. J.: A regularity result for strongly elliptic systems. Bull. Uni. Math. Itali. (Ser. 2B) 6 (1983), 641 - 651. · Zbl 0524.35035
[14] Naerlović-Vejlković, N. and M. Plav\check sić: Thermodiffusion in elastic solids with microstruc- tura. Bull. Acad. Polon. Sci., Ser. Techn. 22 (1974), 623 - 632.
[15] Nowacki, W.: Certains problems of thermodiffusion in solids. Arch. Mech. Stos. 23 (1971), 731 - 754. · Zbl 0264.73006
[16] Nowacki, W.: Dynamic problems of thermodiffusion in solids. Parts I - III. Bull. Acad. Polon. Sci., Ser. Techn. 23 (1974), 55 - 64, 205 - 211 and 257 - 266.
[17] Nowacki, W.: Theory of Asymmetric Elasticity. Warszawa: Pol. Sci. Publ. 1986. · Zbl 0604.73020
[18] Plav\check sić, M. and N. Naerlović-Vejlković: Field equations for thermodiffusion in elastic solids with microstructure. Bull. Acad. Polon. Sci., Ser. Techn. 23 (1975), 483 - 492. · Zbl 0326.73002
[19] Shibata, Y.: On a local existence theorem for some quasilinear hyperbolic-parabolic coupled system with Neumann type boundary condition. Manuscript. · Zbl 0736.35004
[20] Shibata, Y. and Y. Tsutsumi: Local existence of solution for the initial-boundary value problem of fully nonlinear equation. Nonlin. Arch. Theory Math. Appl. 11 (1987), 335 -365. · Zbl 0651.35053
[21] Zaj\?aczkowski, W.: Mixed problems for nonlinear symmetric hyperbolic system. Math. Meth. Appl. Sci. 11 (1989), 139 - 168. · Zbl 0684.35070
[22] Zheng, S.: Initial boundary value problems for quasilinear hyperbolic-parabolic coupled systems in higher dimensonal spaces. Chin. Ann. Math. 4B (1983), 443 - 462. · Zbl 0509.35056
[23] Zheng, S. and W. Shen: Global solutions to the Cauchy problem of quasilinear hyperbolic parabolic coupled systems. Sci. Sinica Ser. 30 (1987), 1133 - 1149. · Zbl 0649.35013
[24] Volpert, A. J. and S. T. Hudajev: On the Cauchy problem for composite systems of nonlinear differential equations. Math. USSR Sbornik 16 (1972), 517 - 544. · Zbl 0251.35064
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.