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Viscous approximation of triangular system in 1-d with nonlinear viscosity
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abstract
We study the vanishing viscosity limit for $2\times2$ triangular system of hyperbolic conservation laws when the viscosity coefficients are non linear. In this article, we assume that the viscosity matrix $B(u)$ is commutating with the convective part $A(u)$. We show the existence of global smooth solution to the parabolic equation satisfying uniform total variation bound in $\varepsilon$ provided that the initial data is small in $BV$. This extends the previous result of Bianchini and Bressan [Commun. Pure Appl. Anal. (2002)] which was considering the case $B(u)=I$.
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Cited by 1 Pith paper
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Vanishing viscosity limit for $n\times n$ hyperbolic system of conservation laws in 1-d with nonlinear viscosity: Part-I Uniform BV estimates
Small-BV data for u_t + A(u)u_x = ε(B(u)u_x)_x with commuting A, B yield a uniform-in-ε total-variation bound, and the conservative case has a unique Liu-admissible vanishing-viscosity limit.
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