Decay of large solutions around shocks to multi-D viscous conservation law with strictly convex flux
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| Format: | Preprint |
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2025
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| _version_ | 1866913654396420096 |
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| author | Kang, Moon-Jin Oh, HyeonSeop |
| author_facet | Kang, Moon-Jin Oh, HyeonSeop |
| contents | We consider a planar viscous shock for a scalar viscous conservation law with a strictly convex flux in multi-dimensional setting, where the transversal direction is periodic. We first show the contraction property for any solutions evolving from a large bounded initial perturbation in $L^2$ of the viscous shock. The contraction holds up to a dynamical shift, and it is measured by a weighted relative entropy. This result for the contraction extends the existing result in 1D \cite{Kang19} to the multi-dimensional case. As a consequence, if the large bounded initial $L^2$-perturbation is also in $L^1$, then the large perturbation decays of rate $t^{-1/4}$ in $L^2$, up to a dynamical shift that is uniformly bounded in time. This is the first result for the quantitative estimate converging to a planar shock under large perturbations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_04311 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decay of large solutions around shocks to multi-D viscous conservation law with strictly convex flux Kang, Moon-Jin Oh, HyeonSeop Analysis of PDEs We consider a planar viscous shock for a scalar viscous conservation law with a strictly convex flux in multi-dimensional setting, where the transversal direction is periodic. We first show the contraction property for any solutions evolving from a large bounded initial perturbation in $L^2$ of the viscous shock. The contraction holds up to a dynamical shift, and it is measured by a weighted relative entropy. This result for the contraction extends the existing result in 1D \cite{Kang19} to the multi-dimensional case. As a consequence, if the large bounded initial $L^2$-perturbation is also in $L^1$, then the large perturbation decays of rate $t^{-1/4}$ in $L^2$, up to a dynamical shift that is uniformly bounded in time. This is the first result for the quantitative estimate converging to a planar shock under large perturbations. |
| title | Decay of large solutions around shocks to multi-D viscous conservation law with strictly convex flux |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.04311 |