On the stability of the standard Riemann semigroup

Bianchini, Stefano; Colombo, Rinaldo

doi:10.1090/S0002-9939-02-06568-1

On the stability of the standard Riemann semigroup
HTML articles powered by AMS MathViewer

by Stefano Bianchini and Rinaldo M. Colombo PDF

Proc. Amer. Math. Soc. 130 (2002), 1961-1973 Request permission

Abstract:

We consider the dependence of the entropic solution of a hyperbolic system of conservation laws \[ \left \{ \begin {array}{c} u_t + f(u)_x = 0, u(0,\cdot ) = u_0 \end {array} \right . \] on the flux function $f$. We prove that the solution is Lipschitz continuous w.r.t. the $C^0$ norm of the derivative of the perturbation of $f$. We apply this result to prove the convergence of the solution of the relativistic Euler equation to the classical limit.

References

F. Ancona, A. Marson, Well posedness for general $2\times 2$ systems of conservation laws, preprint.
Paolo Baiti and Alberto Bressan, The semigroup generated by a Temple class system with large data, Differential Integral Equations 10 (1997), no. 3, 401–418. MR 1744853
Stefano Bianchini, On the shift differentiability of the flow generated by a hyperbolic system of conservation laws, Discrete Contin. Dynam. Systems 6 (2000), no. 2, 329–350. MR 1739381, DOI 10.3934/dcds.2000.6.329
Alberto Bressan, The unique limit of the Glimm scheme, Arch. Rational Mech. Anal. 130 (1995), no. 3, 205–230. MR 1337114, DOI 10.1007/BF00392027
A. Bressan, On the Cauchy problem for nonlinear hyperbolic systems, preprint S.I.S.S.A., 1998.
A. Bressan, “Hyperbolic Systems of Conservation Laws: the One Dimensional Cauchy Problem”, Oxford Univ. Press (2000).
Alberto Bressan and Rinaldo M. Colombo, The semigroup generated by $2\times 2$ conservation laws, Arch. Rational Mech. Anal. 133 (1995), no. 1, 1–75. MR 1367356, DOI 10.1007/BF00375350
Alberto Bressan, Graziano Crasta, and Benedetto Piccoli, Well-posedness of the Cauchy problem for $n\times n$ systems of conservation laws, Mem. Amer. Math. Soc. 146 (2000), no. 694, viii+134. MR 1686652, DOI 10.1090/memo/0694
F. Bouchut and B. Perthame, Kružkov’s estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc. 350 (1998), no. 7, 2847–2870 (English, with English and French summaries). MR 1475677, DOI 10.1090/S0002-9947-98-02204-1
Rinaldo M. Colombo and Nils H. Risebro, Continuous dependence in the large for some equations of gas dynamics, Comm. Partial Differential Equations 23 (1998), no. 9-10, 1693–1718. MR 1641709, DOI 10.1080/03605309808821397
C. M. Dafermos, Generalized characteristics in hyperbolic systems of conservation laws, Arch. Rational Mech. Anal. 107 (1989), no. 2, 127–155. MR 996908, DOI 10.1007/BF00286497
P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI 10.1002/cpa.3160100406
Tai Ping Liu, The Riemann problem for general systems of conservation laws, J. Differential Equations 18 (1975), 218–234. MR 369939, DOI 10.1016/0022-0396(75)90091-1
Lu Min and Seiji Ukai, Non-relativistic global limits of weak solutions of the relativistic Euler equation, J. Math. Kyoto Univ. 38 (1998), no. 3, 525–537. MR 1661165, DOI 10.1215/kjm/1250518065
A. I. Panasyuk, Quasidifferential equations in a metric space, Differentsial′nye Uravneniya 21 (1985), no. 8, 1344–1353, 1468 (Russian). MR 806264
Joel Smoller and Blake Temple, Global solutions of the relativistic Euler equations, Comm. Math. Phys. 156 (1993), no. 1, 67–99. MR 1234105
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336