Time-dependent invariant regions for parabolic systems related to one- dimensional nonlinear elasticity. (English) Zbl 0709.73013

Summary: A parabolic system arising as a viscosity regularization of the quasilinear one-dimensional telegraph equation is considered. The existence of \(L_{\infty}\)- a priori estimates, independent of viscosity, is shown. The results are achieved by means of generalized invariant regions.


74B20 Nonlinear elasticity
35B65 Smoothness and regularity of solutions to PDEs
35B45 A priori estimates in context of PDEs
35K45 Initial value problems for second-order parabolic systems
Full Text: EuDML


[1] K. N. Chueh C. C. Conley J. A. Smoller: Positively invariant regions for systems of non linear diffusion equations. Indiana Univ. Math. J. 26 (1977), 372-7411. · Zbl 0368.35040
[2] C. M. Dafermos: Estimates for conservation laws with little viscosity. SIAM J. Math. Anal. 18 (1987), 409-421. · Zbl 0655.35055
[3] R. J. DiPerna: Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82 (1983), 27-70. · Zbl 0519.35054
[4] M. Rascle: Un résultat de ,,compacité par compensation à coefficients variables”. Application à l’élasticité nonlinéaire. Compt. Rend. Acad. Sci. Paris, Série I, 302 (1986), 311 - 314. · Zbl 0606.35054
[5] D. Serre: Domaines invariants pour les systèmes hyperboliques de lois de conservation. J. Differential Equations 69 (1987), 46-62. · Zbl 0626.35061
[6] D. Serre: La compacité par compensation pour les systèmes hyperboliques non linéaires de deux équations a une dimension d’espace. J. Math. pures et appl. 65 (1986), 423 - 468. · Zbl 0601.35070
[7] T. D. Venttseľ: Estimates of solutions of the one-dimensional system of equations of gas dynamics with ”viscosity” nondepending on ”viscosity”. Soviet Math. J., 31 (1985), 3148- –3153. · Zbl 0575.76076
[8] E. Feireisl: Compensated compactness and time-periodic solutions to non-autonomous quasilinear telegraph equations. Apl. mat. 35 (1990), 192-208. · Zbl 0737.35040
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.