Time-asymptotic stability of composite weak planar waves for a general $n\times n$ multi-D viscous system
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| Format: | Preprint |
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2025
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| _version_ | 1866912202632462336 |
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| author | Meng, Jiayun |
| author_facet | Meng, Jiayun |
| contents | We prove the time-asymptotic stability of the superposition of a weak planar viscous 1-shock and either a weak planar n-rarefaction or a weak planar viscous n-shock for a general $n\times n$ multi-D viscous system. In 2023, Kang-Vasseur-Wang [11] showed the stability of the superposition of a viscous shock and a rarefaction for 1-D compressible barotropic Navier-Stokes equations and solved a long-standing open problem officially introduced by Matsumura-Nishihara [23] in 1992. Our work is an extension of [11], where a general $n\times n$ multi-D viscous system is studied. Same as in [11], we apply the $a$-contraction method with shifts, an energy based method invented by Kang and Vasseur in [9], for both viscous shock and rarefaction at the level of the solution. In such a way, we can work with general perturbations and compositions of waves. Finally, a technique to classify and control higher-order terms is developed to work in multi-D. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_14188 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Time-asymptotic stability of composite weak planar waves for a general $n\times n$ multi-D viscous system Meng, Jiayun Analysis of PDEs We prove the time-asymptotic stability of the superposition of a weak planar viscous 1-shock and either a weak planar n-rarefaction or a weak planar viscous n-shock for a general $n\times n$ multi-D viscous system. In 2023, Kang-Vasseur-Wang [11] showed the stability of the superposition of a viscous shock and a rarefaction for 1-D compressible barotropic Navier-Stokes equations and solved a long-standing open problem officially introduced by Matsumura-Nishihara [23] in 1992. Our work is an extension of [11], where a general $n\times n$ multi-D viscous system is studied. Same as in [11], we apply the $a$-contraction method with shifts, an energy based method invented by Kang and Vasseur in [9], for both viscous shock and rarefaction at the level of the solution. In such a way, we can work with general perturbations and compositions of waves. Finally, a technique to classify and control higher-order terms is developed to work in multi-D. |
| title | Time-asymptotic stability of composite weak planar waves for a general $n\times n$ multi-D viscous system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.14188 |