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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2512.15620 |
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| _version_ | 1866910268089434112 |
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| author | Haspot, Boris Jana, Animesh |
| author_facet | Haspot, Boris Jana, Animesh |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15620 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vanishing viscosity limit for $n\times n$ hyperbolic system of conservation laws in 1-d with nonlinear viscosity: Part-I Uniform BV estimates Haspot, Boris Jana, Animesh Analysis of PDEs 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. |
| title | Vanishing viscosity limit for $n\times n$ hyperbolic system of conservation laws in 1-d with nonlinear viscosity: Part-I Uniform BV estimates |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.15620 |