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Decay of positive waves in nonlinear systems of conservation laws. (English) Zbl 0906.35059
The paper is concerned with BV solutions to a strictly hyperbolic system of conservation laws in one space dimension. The aim of the paper is to extend classical decay estimates for entropic solutions due to Oleinik and derive certain a priori bounds on the positive waves of genuinely nonlinear families. The authors use a technique of wave-front tracking approximations to derive generalized a priori estimates on the distributional space derivative of the solution.

MSC:
35L65 Hyperbolic conservation laws
35B45 A priori estimates in context of PDEs
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References:
[1] P. Baiti - A. Bressan , Lower Semicontinuity of Weighted Path Length in BV , In ” Geometrical Optics and Related Topics ”, F. Colombini and N. Lerner (eds.), Birkäuser , 1997 . MR 2033490 | Zbl 0891.35089 · Zbl 0891.35089
[2] A. Bressan , Global solutions to systems of conservation laws by wave-front tracking , J. Math. Anal. Appl. 170 ( 1992 ), 414 - 432 . MR 1188562 | Zbl 0779.35067 · Zbl 0779.35067
[3] A. Bressan , ” Lecture Notes on Systems of Conservation Laws ”, S.I.S.S.A. , Trieste , 1995 . · Zbl 0867.35060
[4] A. Bressan , The unique limit of the Glimm scheme , Arch. Rational Mech. Anal. 130 ( 1995 ), 205 - 230 . MR 1337114 | Zbl 0835.35088 · Zbl 0835.35088
[5] A. Bressan , The semigroup approach to systems of conservation laws , Mathematica Contemporanea 10 ( 1996 ), 21 - 74 . MR 1425453 | Zbl 0866.35064 · Zbl 0866.35064
[6] A. Bressan - R.M. Colombo , The semigroup generated by 2 x 2 conservation laws , Arch. Rational Mech. Anal. 133 ( 1995 ), 1 - 75 . MR 1367356 | Zbl 0849.35068 · Zbl 0849.35068
[7] A. Bressan - R.M. Colombo , Unique solutions of 2 x 2 conservation laws with large data , Indiana Univ. Math. J. 44 ( 1995 ), 677 - 725 . MR 1375345 | Zbl 0852.35092 · Zbl 0852.35092
[8] A. Bressan - G. Crasta - B. Piccoli , Well posedness of the Cauchy Problem for n x n systems of conservation laws , preprint S.I.S.S.A. , Trieste , 1996 . · Zbl 0958.35001
[9] C.M. Dafermos , Generalized characteristics in hyperbolic systems of conservation laws , Arch. Rational Mech. Anal. 107 ( 1989 ), 127 - 155 . Zbl 0714.35046 · Zbl 0714.35046
[10] A.F. Filippov , ” Differential Equations with Discontinuous Righthand Sides ”, Kluwer Academic Publisher , Dordrecht , 1988 . Zbl 0664.34001 · Zbl 0664.34001
[11] J. Glimm , Solutions in the large for nonlinear hyperbolic systems of equations , Comm. Pure Appl. Math. 18 ( 1965 ), 697 - 715 . Zbl 0141.28902 · Zbl 0141.28902
[12] P. Lax , Hyperbolic systems of conservation laws II , Comm. Pure Appl. Math. 10 ( 1957 ), 537 - 566 . Zbl 0081.08803 · Zbl 0081.08803
[13] O. Oleinik , Discontinuous solutions of nonlinear differential equations , Usp. Mat. Nauk. 12 ( 1957 ), 3 - 73 ; English translation: Amer. Math. Soc. Transl. 26 ( 1963 ), 95 - 172 . Zbl 0131.31803 · Zbl 0131.31803
[14] M. Schatzman , Continuous Glimm functionals and uniquess of solutions of the Riemann problem , Indiana Univ. Math. J. 34 ( 1985 ), 533 - 589 . Zbl 0579.35053 · Zbl 0579.35053
[15] J. Smoller , ” Shock Waves and Reaction-Diffusion Equations ”, Springer-Verlag , New York , 1983 . Zbl 0508.35002 · Zbl 0508.35002
[16] B. Temple , Systems of conservation laws with invariant submanifolds , Trans. Amer.Math. Soc. 280 ( 1983 ), 781 - 795 . Zbl 0559.35046 · Zbl 0559.35046
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