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July  2001, 7(3): 449-476. doi: 10.3934/dcds.2001.7.449

A case study in vanishing viscosity

Stefano Bianchini 1, and  Alberto Bressan 2,

1. 

S.I.S.S.A., Via Beirut 4, Trieste 34014, Italy
2. 

S.I.S.S.A., Via Beirut, 2-4, 34014 Trieste

Received  September 2000 Revised  November 2000 Published  April 2001

We consider a special $2 \times 2$ viscous hyperbolic system of conservation laws of the form $u_t + A(u)u_{x} = \varepsilon u_{x x}$, where $A(u) = Df(u)$ is the Jacobian of a flux function $f$. For initialdata with smalltotalv ariation, we prove that the solutions satisfy a uniform BV bound, independent of $\varepsilon $. Letting $\varepsilon \to 0$, we show that solutions of the viscous system converge to the unique entropy weak solutions of the hyperbolic system $u_t + f(u)_{x} = 0$. 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.
Keywords: hyperbolic system of conservation laws, bounded variation, vanishing viscosity approximation.
Mathematics Subject Classification: 35K40, 35L6.
Citation: Stefano Bianchini, Alberto Bressan. A case study in vanishing viscosity. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 449-476. doi: 10.3934/dcds.2001.7.449
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