A one-phase hyperbolic Stefan problem in multi-dimensional space

Li, De Ning

doi:10.1090/S0002-9947-1990-1005936-7

A one-phase hyperbolic Stefan problem in multi-dimensional space
HTML articles powered by AMS MathViewer

by De Ning Li PDF

Trans. Amer. Math. Soc. 318 (1990), 401-415 Request permission

Abstract:

The hyperbolic heat transfer model is obtained by replacing the classical Fourier’s law with the relaxation relation $\tau \vec qt + \vec q = - k\nabla T$. The sufficient and necessary conditions are derived for the local existence and uniqueness of classical solutions for multi-${\text {D}}$ Stefan problem of hyperbolic heat transfer model where phase change is accompanied with delay of latent heat storage.

References

An Ton Bui and De Ning Li, Double shock fronts for hyperbolic systems of conservation laws in multidimensional space, Trans. Amer. Math. Soc. 316 (1989), no. 1, 233–250. MR 935939, DOI 10.1090/S0002-9947-1989-0935939-1
Avner Friedman and Bei Hu, The Stefan problem for a hyperbolic heat equation, J. Math. Anal. Appl. 138 (1989), no. 1, 249–279. MR 988334, DOI 10.1016/0022-247X(89)90334-X
Lars Gȧrding, Le problème de la dérivée oblique pour l’équation des ondes, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 12, A773–A775 (French, with English summary). MR 458511
J. M. Greenberg, A hyperbolic heat transfer problem with phase changes, IMA J. Appl. Math. 38 (1987), no. 1, 1–21. MR 983526, DOI 10.1093/imamat/38.1.1
Mitsuru Ikawa, A mixed problem for hyperbolic equations of second order with a first order derivative boundary condition, Publ. Res. Inst. Math. Sci. 5 (1969), 119–147. MR 0277890, DOI 10.2977/prims/1195194627
D. D. Joseph and Luigi Preziosi, Heat waves, Rev. Modern Phys. 61 (1989), no. 1, 41–73. MR 977943, DOI 10.1103/RevModPhys.61.41
Tosio Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal. 58 (1975), no. 3, 181–205. MR 390516, DOI 10.1007/BF00280740
De Ning Li, The nonlinear initial-boundary value problem and the existence of multidimensional shock wave for quasilinear hyperbolic-parabolic coupled systems, Chinese Ann. Math. Ser. B 8 (1987), no. 2, 252–280. A Chinese summary appears in Chinese Ann. Math. Ser. A 8 (1987), no. 2, 271. MR 901389
De Ning Li, The well-posedness of a hyperbolic Stefan problem, Quart. Appl. Math. 47 (1989), no. 2, 221–231. MR 998097, DOI 10.1090/S0033-569X-1989-0998097-9
Andrew Majda, The existence of multidimensional shock fronts, Mem. Amer. Math. Soc. 43 (1983), no. 281, v+93. MR 699241, DOI 10.1090/memo/0281
Andrew Majda and Enrique Thomann, Multidimensional shock fronts for second order wave equations, Comm. Partial Differential Equations 12 (1987), no. 7, 777–828. MR 890631, DOI 10.1080/03605308708820509
Sadao Miyatake, Mixed problem for hyperbolic equation of second order, J. Math. Kyoto Univ. 13 (1973), 435–487. MR 333467, DOI 10.1215/kjm/1250523319
Jeffrey B. Rauch and Frank J. Massey III, Differentiability of solutions to hyperbolic initial-boundary value problems, Trans. Amer. Math. Soc. 189 (1974), 303–318. MR 340832, DOI 10.1090/S0002-9947-1974-0340832-0
R. E. Showalter and N. J. Walkington, A hyperbolic Stefan problem, Quart. Appl. Math. 45 (1987), no. 4, 769–781. MR 917025, DOI 10.1090/S0033-569X-1987-0917025-3
A. D. Solomon, V. Alexiades, D. G. Wilson, and J. Drake, On the formulation of hyperbolic Stefan problems, Quart. Appl. Math. 43 (1985), no. 3, 295–304. MR 814228, DOI 10.1090/S0033-569X-1985-0814228-9

A hyperbolic Stefan problem with discontinuous temperature