Asymptotic stability of composite waves of shock profile and rarefaction for the Navier-Stokes-Poisson system
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866908489375285248 |
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| author | Shim, Wanyong |
| author_facet | Shim, Wanyong |
| contents | We study the stability of composite waves consisting of a shock profile and a rarefaction wave for the one-dimensional isothermal Navier--Stokes--Poisson (NSP) system, which describes the ion dynamics in a collision-dominated plasma. More precisely, we prove that if the initial data are sufficiently close in the $H^2$ norm to the Riemann data corresponding to a solution consisting of a shock and a rarefaction wave of the associated quasi-neutral Euler system, then the solution to the Cauchy problem for the NSP system converges, up to a dynamical shift, to a superposition of the corresponding shock profile and the rarefaction wave as time tends to infinity. Our proof is based on the method of $a$-contraction with shifts, which has recently been applied to the Navier--Stokes equations to establish the asymptotic stability of composite waves. To adapt this method to the NSP system, we employ a modulated relative functional introduced in our previous work on the stability of single shock profiles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08059 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic stability of composite waves of shock profile and rarefaction for the Navier-Stokes-Poisson system Shim, Wanyong Analysis of PDEs 35Q35, 35B35, 35B40 We study the stability of composite waves consisting of a shock profile and a rarefaction wave for the one-dimensional isothermal Navier--Stokes--Poisson (NSP) system, which describes the ion dynamics in a collision-dominated plasma. More precisely, we prove that if the initial data are sufficiently close in the $H^2$ norm to the Riemann data corresponding to a solution consisting of a shock and a rarefaction wave of the associated quasi-neutral Euler system, then the solution to the Cauchy problem for the NSP system converges, up to a dynamical shift, to a superposition of the corresponding shock profile and the rarefaction wave as time tends to infinity. Our proof is based on the method of $a$-contraction with shifts, which has recently been applied to the Navier--Stokes equations to establish the asymptotic stability of composite waves. To adapt this method to the NSP system, we employ a modulated relative functional introduced in our previous work on the stability of single shock profiles. |
| title | Asymptotic stability of composite waves of shock profile and rarefaction for the Navier-Stokes-Poisson system |
| topic | Analysis of PDEs 35Q35, 35B35, 35B40 |
| url | https://arxiv.org/abs/2508.08059 |