Attractors in inhomogeneous conservation laws and parabolic regularizations
Authors:
Hai Tao Fan and Jack K. Hale
Journal:
Trans. Amer. Math. Soc. 347 (1995), 12391254
MSC:
Primary 35L65; Secondary 35B25, 58F39
MathSciNet review:
1270661
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Abstract 
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Abstract: The asymptotic behavior of inhomogeneous conservation laws is considered. The attractor of the equation is characterized. The relationship between attractors of the equation and that of its parabolic regularization is studied.
 [A]
V. Babin and M. I. Vishik (1989), Attractors of evolutionary equations, "Nauka", Moscow; English transl., NorthHolland, Amsterdam, 1992.
 [C]
M. Dafermos (1977), Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J. 26, 10971119.
 [L]
Evans (1990), Weak convergence methods for nonlinear partial differential equations, CBMS Regional Conf. Ser. in Math., vol. 74, Amer. Math. Soc., Providence, RI.
 [H]
Fan and J. K. Hale (1992), Large time behavior in inhomogeneous conservation laws, Arch. Rat. Mech. Anal. (to appear).
 [B]
Fiedler and J. MalletParet (1989), A PoincaréBendixson theorem for scalar reaction diffusion equations, Arch. Rational. Mech. Anal. 107, 325345.
 [A]
F. Filippov (1960), Differential equations with discontinuous right hand side, Mat. Sb. (N.S.) 42, 99128; English transl., Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, RI.
 [J]
K. Hale (1988), Asymptotic behavior of dissipative systems, Amer. Math. Soc., Providence, RI.
 [S]
N. Kruzkov (1970), First order quasilinear equations in several independent variables, Mat. Sb. (N.S.) 81, 228255; English transl., Math. USSRSb. 10, 217243.
 [A]
N. Lyberopoulos (1992), Large time structure of solutions of scalar conservation laws with a nonlinear source field, preprint.
 [F]
Murat (1978), Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5, 485507.
 [L]
Tartar (1979), Compensated compactness and applications to partial differential equations, HeriotWatt Sympos. in Vol. IV, Pitman, New York.
 [R]
Temam (1988), Infinite dimensional dynamical systems in mechanics and physics, Appl. Math. Sci., no. 68, SpringerVerlag, Berlin and New York.
 [A]
I. Vol'pert (1967), The space and quasilinear equations, Mat. Sb. (N.S.) 73; English transl., Math. USSRSb. 2, 225267.
 [A]
 V. Babin and M. I. Vishik (1989), Attractors of evolutionary equations, "Nauka", Moscow; English transl., NorthHolland, Amsterdam, 1992.
 [C]
 M. Dafermos (1977), Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J. 26, 10971119.
 [L]
 Evans (1990), Weak convergence methods for nonlinear partial differential equations, CBMS Regional Conf. Ser. in Math., vol. 74, Amer. Math. Soc., Providence, RI.
 [H]
 Fan and J. K. Hale (1992), Large time behavior in inhomogeneous conservation laws, Arch. Rat. Mech. Anal. (to appear).
 [B]
 Fiedler and J. MalletParet (1989), A PoincaréBendixson theorem for scalar reaction diffusion equations, Arch. Rational. Mech. Anal. 107, 325345.
 [A]
 F. Filippov (1960), Differential equations with discontinuous right hand side, Mat. Sb. (N.S.) 42, 99128; English transl., Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, RI.
 [J]
 K. Hale (1988), Asymptotic behavior of dissipative systems, Amer. Math. Soc., Providence, RI.
 [S]
 N. Kruzkov (1970), First order quasilinear equations in several independent variables, Mat. Sb. (N.S.) 81, 228255; English transl., Math. USSRSb. 10, 217243.
 [A]
 N. Lyberopoulos (1992), Large time structure of solutions of scalar conservation laws with a nonlinear source field, preprint.
 [F]
 Murat (1978), Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5, 485507.
 [L]
 Tartar (1979), Compensated compactness and applications to partial differential equations, HeriotWatt Sympos. in Vol. IV, Pitman, New York.
 [R]
 Temam (1988), Infinite dimensional dynamical systems in mechanics and physics, Appl. Math. Sci., no. 68, SpringerVerlag, Berlin and New York.
 [A]
 I. Vol'pert (1967), The space and quasilinear equations, Mat. Sb. (N.S.) 73; English transl., Math. USSRSb. 2, 225267.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199512706619
PII:
S 00029947(1995)12706619
Keywords:
Attractors,
conservation laws,
dynamical systems
Article copyright:
© Copyright 1995
American Mathematical Society
