This paper investigates the well-posedness of local smooth solutions for the vacuum free boundary problem of the spherically symmetric non-isentropic compressible Euler equations. At the free boundary, the system degenerates into a hyperbolic system while exhibiting a singularity at the spherical center. To overcome these difficulties, a Lagrangian coordinate transformation is introduced, converting the problem into an initial-boundary value problem with fixed boundaries. Within this framework, the degeneracy of the system can be explicitly expressed by the function $ \rho_0^{\gamma-1}(\gamma > 1) $ of the initial density, whose value equals the distance function near the boundaries. Consequently, by employing the distance-dependent Hardy's inequality and weighted Sobolev spaces, the degeneracy and singularity of the system are successfully resolved, and a priori estimates for the local smooth solutions are established. Specifically, by building upon high-order time derivative estimates, the high-order spatial derivative estimates are rigorously proved via mathematical induction. Crucially, the influence of entropy introduces additional higher-order terms, necessitating boundedness of initial entropy derivatives up to a certain order. Finally, we establish the existence and uniqueness of local smooth solutions.
Citation: La-Su Mai, Jiarong Dong. The vacuum free boundary problem for the spherically symmetric non-isentropic compressible Euler equations[J]. Communications in Analysis and Mechanics, 2026, 18(3): 553-592. doi: 10.3934/cam.2026023
This paper investigates the well-posedness of local smooth solutions for the vacuum free boundary problem of the spherically symmetric non-isentropic compressible Euler equations. At the free boundary, the system degenerates into a hyperbolic system while exhibiting a singularity at the spherical center. To overcome these difficulties, a Lagrangian coordinate transformation is introduced, converting the problem into an initial-boundary value problem with fixed boundaries. Within this framework, the degeneracy of the system can be explicitly expressed by the function $ \rho_0^{\gamma-1}(\gamma > 1) $ of the initial density, whose value equals the distance function near the boundaries. Consequently, by employing the distance-dependent Hardy's inequality and weighted Sobolev spaces, the degeneracy and singularity of the system are successfully resolved, and a priori estimates for the local smooth solutions are established. Specifically, by building upon high-order time derivative estimates, the high-order spatial derivative estimates are rigorously proved via mathematical induction. Crucially, the influence of entropy introduces additional higher-order terms, necessitating boundedness of initial entropy derivatives up to a certain order. Finally, we establish the existence and uniqueness of local smooth solutions.
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La-Su Mai, Jiarong Dong. The vacuum free boundary problem for the spherically symmetric non-isentropic compressible Euler equations[J]. Communications in Analysis and Mechanics, 2026, 18(3): 553-592. doi: 10.3934/cam.2026023
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