$L^2$-contraction and asymptotic stability of large shock for scalar viscous conservation laws
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| Format: | Preprint |
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2025
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| _version_ | 1866908517154160640 |
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| author | Vasseur, Alexis F. Wang, Yi Zhang, Jian |
| author_facet | Vasseur, Alexis F. Wang, Yi Zhang, Jian |
| contents | We investigate $L^2$-contraction and time-asymptotic stability of large shock for scalar viscous conservation laws with polynomial flux. For the strictly convex flux $f(u)=u^p $ with $2\leq p \leq 4$, we can prove $L^2$-contraction and time-asymptotic stability of arbitrarily large viscous shock profile in $H^1$-framework by using $a$-contraction method with time-dependent shift and suitable weight function. Additionally, if the initial perturbation belongs to $L^1$, then $L^2$ time-asymptotic decay rate $t^{-\frac{1}{4}}$ can be obtained. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_02965 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $L^2$-contraction and asymptotic stability of large shock for scalar viscous conservation laws Vasseur, Alexis F. Wang, Yi Zhang, Jian Analysis of PDEs Mathematical Physics We investigate $L^2$-contraction and time-asymptotic stability of large shock for scalar viscous conservation laws with polynomial flux. For the strictly convex flux $f(u)=u^p $ with $2\leq p \leq 4$, we can prove $L^2$-contraction and time-asymptotic stability of arbitrarily large viscous shock profile in $H^1$-framework by using $a$-contraction method with time-dependent shift and suitable weight function. Additionally, if the initial perturbation belongs to $L^1$, then $L^2$ time-asymptotic decay rate $t^{-\frac{1}{4}}$ can be obtained. |
| title | $L^2$-contraction and asymptotic stability of large shock for scalar viscous conservation laws |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2509.02965 |