On the well posedness of vanishing viscosity limits
Journées équations aux dérivées partielles (2002), article no. 4, 10 p.

This paper provides a survey of recent results concerning the stability and convergence of viscous approximations, for a strictly hyperbolic system of conservation laws in one space dimension. In the case of initial data with small total variation, the vanishing viscosity limit is well defined. It yields the unique entropy weak solution to the corresponding hyperbolic system.

@article{JEDP_2002____A4_0,
     author = {Bressan, Alberto},
     title = {On the well posedness of vanishing viscosity limits},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {4},
     publisher = {Universit\'e de Nantes},
     year = {2002},
     doi = {10.5802/jedp.602},
     mrnumber = {1968200},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.602/}
}
Bressan, Alberto. On the well posedness of vanishing viscosity limits. Journées équations aux dérivées partielles (2002), article  no. 4, 10 p. doi : 10.5802/jedp.602. http://www.numdam.org/articles/10.5802/jedp.602/

[BB] S. Bianchini and A. Bressan, Vanishing viscosity solutions to nonlinear hyperbolic systems, Preprint S.I.S.S.A., Trieste 2001.

[B1] A. Bressan, Unique solutions for a class of discontinuous differential equations, Proc. Amer. Math. Soc. 104 (1988), 772-778. | MR 964856 | Zbl 0692.34004

[B2] A. Bressan, The unique limit of the Glimm scheme, Arch. Rational Mech. Anal. 130 (1995), 205-230. IV-9 | MR 1337114 | Zbl 0835.35088

[B3] A. Bressan, Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem. Oxford University Press, 2000. | MR 1816648 | Zbl 0997.35002

[BG] A. Bressan and P. Goatin, Oleinik type estimates and uniqueness for n x n conservation laws, J. Diff. Equat. 156 (1999), 26-49. | MR 1701818 | Zbl 0990.35095

[BLY] A. Bressan, T. P. Liu and T. Yang, L 1 stability estimates for n Ã— n conservation laws, Arch. Rational Mech. Anal. 149 (1999), 1-22. | MR 1723032 | Zbl 0938.35093

[BS] A. Bressan and W. Shen, Uniqueness for discontinuous O.D.E. and conservation laws, Nonlinear Analysis, T. M. A. 34 (1998), 637-652. | MR 1634652 | Zbl 0948.34006

[DP] R. Diperna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), 27-70. | MR 684413 | Zbl 0519.35054

[G] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697-715. | MR 194770 | Zbl 0141.28902

[GX] J. Goodman and Z. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Rational Mech. Anal. 121 (1992), 235- 265. | MR 1188982 | Zbl 0792.35115

[K] S. Kruzhkov, First order quasilinear equations with several space variables, Math. USSR Sbornik 10 (1970), 217-243. | Zbl 0215.16203

[J] H. K. Jenssen, Blowup for systems of conservation laws, SIAM J. Math. Anal. 31 (2000), 894-908. | MR 1752421 | Zbl 0969.35091

[L] T. P. Liu, Admissible solutions of hyperbolic conservation laws, Amer. Math. Soc. Memoir 240 (1981). | MR 603391 | Zbl 0446.76058

[O] O. Oleinik, Discontinuous solutions of nonlinear differential equations (1957), Amer. Math. Soc. Translations 26, 95-172. | MR 151737 | Zbl 0131.31803

[V] A. Vanderbauwhede, Centre manifolds, normal forms and elementary bifurcations, Dynamics Reported, Vol. 2 (1989 | MR 1000977 | Zbl 0677.58001