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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.

MSC:

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
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References:

[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
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