Global smooth axisymmetric solutions to 2D compressible Euler equations of Chaplygin gases with non-zero vorticity
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2018
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| _version_ | 1866910944017252352 |
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| author | Hou, Fei Yin, Huicheng |
| author_facet | Hou, Fei Yin, Huicheng |
| contents | For 2D compressible isentropic Euler equations of polytropic gases, when the rotationally invariant data are a perturbation of size $\ve>0$ of a rest state, S.~Alinhac in \cite{Alinhac92} and \cite{Alinhac93} establishes that the smooth solution blows up in finite time and the lifespan $T_{\ve}$ satisfies $\ds\lim_{\ve\to 0}\ve^2 T_{\ve}=τ_{0}^2>0$. In the present paper, for 2D compressible isentropic Euler equations of Chaplygin gases, we shall show that the small perturbed smooth solution exists globally when the rotationally invariant data are a perturbation of size $\ve>0$ of a rest state. Near the light cone, 2D Euler equations of Chaplygin gases can be transformed into a second order quasilinear wave equation of potential, which satisfies both the first and the second null conditions. This will lead to that the corresponding second order quasilinear wave equation admits a global smooth solution near the light cone (see \cite{Alinhac01}). However, away from the light cone, the hydrodynamical waves of 2D Chaplygin gases have no decay in time and strongly affect the related acoustical waves. Thanks to introducing a nonlinear ODE and taking some delicate observations, we can distinguish the fast decay part and non-decay part explicitly so that the global energy estimates with different weights can be derived by involved analysis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1810_11615 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Global smooth axisymmetric solutions to 2D compressible Euler equations of Chaplygin gases with non-zero vorticity Hou, Fei Yin, Huicheng Analysis of PDEs For 2D compressible isentropic Euler equations of polytropic gases, when the rotationally invariant data are a perturbation of size $\ve>0$ of a rest state, S.~Alinhac in \cite{Alinhac92} and \cite{Alinhac93} establishes that the smooth solution blows up in finite time and the lifespan $T_{\ve}$ satisfies $\ds\lim_{\ve\to 0}\ve^2 T_{\ve}=τ_{0}^2>0$. In the present paper, for 2D compressible isentropic Euler equations of Chaplygin gases, we shall show that the small perturbed smooth solution exists globally when the rotationally invariant data are a perturbation of size $\ve>0$ of a rest state. Near the light cone, 2D Euler equations of Chaplygin gases can be transformed into a second order quasilinear wave equation of potential, which satisfies both the first and the second null conditions. This will lead to that the corresponding second order quasilinear wave equation admits a global smooth solution near the light cone (see \cite{Alinhac01}). However, away from the light cone, the hydrodynamical waves of 2D Chaplygin gases have no decay in time and strongly affect the related acoustical waves. Thanks to introducing a nonlinear ODE and taking some delicate observations, we can distinguish the fast decay part and non-decay part explicitly so that the global energy estimates with different weights can be derived by involved analysis. |
| title | Global smooth axisymmetric solutions to 2D compressible Euler equations of Chaplygin gases with non-zero vorticity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1810.11615 |