Ergodicity of the viscous scalar conservation laws with a degenerate noise
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909746053775360 |
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| author | Peng, Xuhui Su, Houqi |
| author_facet | Peng, Xuhui Su, Houqi |
| contents | This paper establishes the ergodicity in $H^\nn,\nn=\lfloor\frac{d}{2}+1\rfloor$ of the viscous scalar conservation laws on torus $\mT^d$ with general polynomial flux and a degenerate noise. The noise could appear in as few as several directions. We introduce a localized framework that restricts attention to trajectories with controlled energy growth, circumventing the limitations of traditional contraction-based approaches. This localized method allows for a demonstration of e-property and consequently proves the uniqueness of invariant measure under a H{ö}rmander-type condition. Furthermore, we characterize the absolute continuity of the invariant measure's projections onto any finite-dimensional subspaces under requirement on a new algebraically non-degenerate condition for the flux. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_15828 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ergodicity of the viscous scalar conservation laws with a degenerate noise Peng, Xuhui Su, Houqi Probability 35R60, 60H15, 37A50, 35L65 This paper establishes the ergodicity in $H^\nn,\nn=\lfloor\frac{d}{2}+1\rfloor$ of the viscous scalar conservation laws on torus $\mT^d$ with general polynomial flux and a degenerate noise. The noise could appear in as few as several directions. We introduce a localized framework that restricts attention to trajectories with controlled energy growth, circumventing the limitations of traditional contraction-based approaches. This localized method allows for a demonstration of e-property and consequently proves the uniqueness of invariant measure under a H{ö}rmander-type condition. Furthermore, we characterize the absolute continuity of the invariant measure's projections onto any finite-dimensional subspaces under requirement on a new algebraically non-degenerate condition for the flux. |
| title | Ergodicity of the viscous scalar conservation laws with a degenerate noise |
| topic | Probability 35R60, 60H15, 37A50, 35L65 |
| url | https://arxiv.org/abs/2503.15828 |