Global well-posedness and stability of three-dimensional isothermal Euler equations with damping
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909499547189248 |
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| author | Huang, Feimin Tang, Houzhi Zhang, Shuxing Zou, Weiyuan |
| author_facet | Huang, Feimin Tang, Houzhi Zhang, Shuxing Zou, Weiyuan |
| contents | The global well-posedness and stability of solutions to the three-dimensional compressible Euler equations with damping is a longstanding open problem. This problem was addressed in \cite{WY, STW} in the isentropic regime (i.e. $γ>1$) for small smooth solutions. In this paper, we prove the global well-posedness and stability of smooth solutions to the three-dimensional isothermal Euler equations ($γ=1$) with damping for some partially large initial values, i.e., $\|(ρ_0-ρ_*,u_0)\|_{L^2}$ could be large, but $\|D^3(ρ_0-ρ_*,u_0)\|_{L^2}$ is necessarily small. Moreover, the optimal algebraic decay rate is also obtained.
The proof is based on the observation that the isothermal Euler equations with damping possess a good structure so that the equations can be reduced into a symmetrically hyperbolic system with partial damping, i.e., \eqref{au}. In the new system, all desired a priori estimates can be obtained under the assumption that $\int_0^T(\|\nabla \mathrm{ln}ρ\|_{L^{\infty}}+\|\nabla u\|_{L^{\infty}}) \mathrm{d}t $ is small. The assumption can be verified through the low-high frequency analysis via Fourier transformation under the condition that $\|D^3(ρ_0-ρ_*,u_0)\|_{L^2}$ is small, but $\|(ρ_0-ρ_*,u_0)\|_{L^2}$ could be large. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_12457 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global well-posedness and stability of three-dimensional isothermal Euler equations with damping Huang, Feimin Tang, Houzhi Zhang, Shuxing Zou, Weiyuan Analysis of PDEs 35Q31, 35B40, 35B44, 76N15 The global well-posedness and stability of solutions to the three-dimensional compressible Euler equations with damping is a longstanding open problem. This problem was addressed in \cite{WY, STW} in the isentropic regime (i.e. $γ>1$) for small smooth solutions. In this paper, we prove the global well-posedness and stability of smooth solutions to the three-dimensional isothermal Euler equations ($γ=1$) with damping for some partially large initial values, i.e., $\|(ρ_0-ρ_*,u_0)\|_{L^2}$ could be large, but $\|D^3(ρ_0-ρ_*,u_0)\|_{L^2}$ is necessarily small. Moreover, the optimal algebraic decay rate is also obtained. The proof is based on the observation that the isothermal Euler equations with damping possess a good structure so that the equations can be reduced into a symmetrically hyperbolic system with partial damping, i.e., \eqref{au}. In the new system, all desired a priori estimates can be obtained under the assumption that $\int_0^T(\|\nabla \mathrm{ln}ρ\|_{L^{\infty}}+\|\nabla u\|_{L^{\infty}}) \mathrm{d}t $ is small. The assumption can be verified through the low-high frequency analysis via Fourier transformation under the condition that $\|D^3(ρ_0-ρ_*,u_0)\|_{L^2}$ is small, but $\|(ρ_0-ρ_*,u_0)\|_{L^2}$ could be large. |
| title | Global well-posedness and stability of three-dimensional isothermal Euler equations with damping |
| topic | Analysis of PDEs 35Q31, 35B40, 35B44, 76N15 |
| url | https://arxiv.org/abs/2502.12457 |