Global existence and optimal time decay rate to one-dimensional two-phase flow model
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866917849297059840 |
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| author | Huang, Xushan Wang, Yi |
| author_facet | Huang, Xushan Wang, Yi |
| contents | We investigate the global existence and optimal time decay rate of solution to the one-dimensional (1D) two-phase flow described by compressible Euler equations coupled with compressible Navier-Stokes equations through the relaxation drag force on the momentum equations (Euler-Navier-Stokes system). First, we prove the global existence of a strong solution and the stability of the constant equilibrium state to 1D Cauchy problem of compressible Euler-Navier-Stokes system by using the standard continuity argument for small $H^{1}$ data while its second order derivative can be large. Then we derive the optimal time decay rate to the constant equilibrium state. Compared with the multi-dimensional case, it is much harder to get the optimal time decay rate by the direct spectrum method due to a slower convergence rate of the fundamental solution in the 1D case. To overcome this main difficulty, we need to first carry out time-weighted energy estimates (not optimal) for higher order derivatives, and based on these time-weighted estimates, we can close a priori assumptions and get the optimal time decay rate by spectral analysis method. Moreover, due to the non-conserved form and insufficient decay rate of the coupled drag force terms between the two-phase flows, we essentially need to use momentum variables $(m= ρu, M=nω)$, rather than velocity variables $(u, ω)$ in the spectrum analysis, to fully cancel out those non-conserved and insufficient time decay drag force terms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_01992 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Global existence and optimal time decay rate to one-dimensional two-phase flow model Huang, Xushan Wang, Yi Analysis of PDEs We investigate the global existence and optimal time decay rate of solution to the one-dimensional (1D) two-phase flow described by compressible Euler equations coupled with compressible Navier-Stokes equations through the relaxation drag force on the momentum equations (Euler-Navier-Stokes system). First, we prove the global existence of a strong solution and the stability of the constant equilibrium state to 1D Cauchy problem of compressible Euler-Navier-Stokes system by using the standard continuity argument for small $H^{1}$ data while its second order derivative can be large. Then we derive the optimal time decay rate to the constant equilibrium state. Compared with the multi-dimensional case, it is much harder to get the optimal time decay rate by the direct spectrum method due to a slower convergence rate of the fundamental solution in the 1D case. To overcome this main difficulty, we need to first carry out time-weighted energy estimates (not optimal) for higher order derivatives, and based on these time-weighted estimates, we can close a priori assumptions and get the optimal time decay rate by spectral analysis method. Moreover, due to the non-conserved form and insufficient decay rate of the coupled drag force terms between the two-phase flows, we essentially need to use momentum variables $(m= ρu, M=nω)$, rather than velocity variables $(u, ω)$ in the spectrum analysis, to fully cancel out those non-conserved and insufficient time decay drag force terms. |
| title | Global existence and optimal time decay rate to one-dimensional two-phase flow model |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2307.01992 |