Vanishing viscosity solutions of nonlinear hyperbolic systems

• Autores: Stefano Bianchini, Alberto Bressan
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 161, Nº 1, 2005, págs. 223-342
• Idioma: inglés
• DOI: 10.4007/annals.2005.161.223
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• Resumen

  • We consider the Cauchy problem for a strictly hyperbolic, n ¿~ n system in one-space dimension: ut + A(u)ux = 0, assuming that the initial data have small total variation. We show that the solutions of the viscous approximations ut + A(u)ux = ¿Ãuxx are defined globally in time and satisfy uniform BV estimates, independent of ¿Ã. Moreover, they depend continuously on the initial data in the L1 distance, with a Lipschitz constant independent of t, ¿Ã. Letting ¿Ã ¿¿ 0, these viscous solutions converge to a unique limit, depending Lipschitz continuously on the initial data. In the conservative case where A = Df is the Jacobian of some flux function f : Rn ¿¿ Rn, the vanishing viscosity limits are precisely the unique entropy weak solutions to the system of conservation laws ut + f(u)x = 0.