Intermediate Domains for Scalar Conservation Laws
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910413025705984 |
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| author | Ancona, Fabio Bressan, Alberto Marconi, Elio Talamini, Luca |
| author_facet | Ancona, Fabio Bressan, Alberto Marconi, Elio Talamini, Luca |
| contents | For a scalar conservation law with strictly convex flux, by Oleinik's estimates the total variation of a solution with initial data $\overline{u}\in \bf{L}^\infty(\mathbb R)$ decays like $t^{-1}$. This paper introduces a class of intermediate domains $\mathcal P_α$, $0<α<1$, such that for $\overline u\in \mathcal P_α$ a faster decay rate is achieved: $\mathrm{Tot.Var.}\bigl\{ u(t,\cdot)\bigr\}\sim t^{α-1}$. A key ingredient of the analysis is a ``Fourier-type" decomposition of $\overline u$ into components which oscillate more and more rapidly. The results aim at extending the theory of fractional domains for analytic semigroups to an entirely nonlinear setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_10905 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Intermediate Domains for Scalar Conservation Laws Ancona, Fabio Bressan, Alberto Marconi, Elio Talamini, Luca Analysis of PDEs 35L65 For a scalar conservation law with strictly convex flux, by Oleinik's estimates the total variation of a solution with initial data $\overline{u}\in \bf{L}^\infty(\mathbb R)$ decays like $t^{-1}$. This paper introduces a class of intermediate domains $\mathcal P_α$, $0<α<1$, such that for $\overline u\in \mathcal P_α$ a faster decay rate is achieved: $\mathrm{Tot.Var.}\bigl\{ u(t,\cdot)\bigr\}\sim t^{α-1}$. A key ingredient of the analysis is a ``Fourier-type" decomposition of $\overline u$ into components which oscillate more and more rapidly. The results aim at extending the theory of fractional domains for analytic semigroups to an entirely nonlinear setting. |
| title | Intermediate Domains for Scalar Conservation Laws |
| topic | Analysis of PDEs 35L65 |
| url | https://arxiv.org/abs/2404.10905 |