We consider a special 2 x 2 viscous hyperbolic system of conservation laws of the form ut + A(u)ux = εuxx, where A(u) = Df(u) is the Jacobian of a flux function f. For initial data with small total variation, we prove that the solutions satisfy a uniform BV bound, independent of ε. Letting ε → 0, we show that solutions of the viscous system converge to the unique entropy weak solutions of the hyperbolic system ut + f(u)x = 0. Within the proof, we introduce two new Lyapunov functionals which control the interaction of viscous waves of the same family. This provides a first example where uniform BV bounds and convergence of vanishing viscosity solutions are obtained, for a system with a genuinely nonlinear field where shock and rarefaction curves do not coincide.

A case study in vanishing viscosity / Bianchini, S.; Bressan, A.. - In: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. - ISSN 1078-0947. - 7:3(2001), pp. 449-476. [10.3934/dcds.2001.7.449]

A case study in vanishing viscosity

Bianchini, S.;
2001-01-01

Abstract

We consider a special 2 x 2 viscous hyperbolic system of conservation laws of the form ut + A(u)ux = εuxx, where A(u) = Df(u) is the Jacobian of a flux function f. For initial data with small total variation, we prove that the solutions satisfy a uniform BV bound, independent of ε. Letting ε → 0, we show that solutions of the viscous system converge to the unique entropy weak solutions of the hyperbolic system ut + f(u)x = 0. Within the proof, we introduce two new Lyapunov functionals which control the interaction of viscous waves of the same family. This provides a first example where uniform BV bounds and convergence of vanishing viscosity solutions are obtained, for a system with a genuinely nonlinear field where shock and rarefaction curves do not coincide.
2001
7
3
449
476
Bianchini, S.; Bressan, A.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/16398
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