Abstract:We consider the following parabolic approximation for hyperbolic system of conservation laws in 1-D with non-singular viscosity matrix $B(u)$ and $A(u)$ strictly hyperbolic,
\[u^\varepsilon_t+A(u^\varepsilon)u^\varepsilon_x=\varepsilon(B(u^\varepsilon)u^\varepsilon_x)_x.\] We prove global in time uniform $BV$ bound for solution to this parabolic system when $\varepsilon>0$ provided that the initial data is small in $BV$ and the matrix $A(u)$ and $B(u)$ commutate. Moreover, in the case where the system is conservative, we show that the sequence $(u^\varepsilon)_{\varepsilon>0}$ admits a limit $u$, which is the unique global weak solution to the limiting strictly hyperbolic system. We provide a concrete application of this result in the study of the visco-dispersive limit of the Navier-Stokes-Korteweg system.
From: Animesh Jana [view email]
[v1]
Sat, 13 Dec 2025 11:10:56 UTC (146 KB)
[v2]
Thu, 28 May 2026 04:37:31 UTC (150 KB)
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