Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping

Abstract

In this article, we consider the well-posedness of the local smooth solutions to the physical vacuum free boundary problem of the cylindrical symmetric Euler equations with time-dependent damping −μ(1+t)λρu-\frac{\mu }{{(1+t)}^{\lambda }}\rho {\bf{u}}, μ>0\mu \gt 0, and λ>0\lambda \gt 0. In this case, the free boundary is moving in the radial direction and the radial velocity will affect the angular velocity, but the axial velocity can be explicitly expressed by ω=ω0(1+t)−μ\omega ={\omega }_{0}{\left(1+t)}^{-\mu } when λ=1\lambda =1 and ω=ω0eμ1−λ[1−(1+t)1−λ]\omega ={\omega }_{0}{e}^{\tfrac{\mu }{1-\lambda }\left[1-{\left(1+t)}^{1-\lambda }]} when λ≠1\lambda \ne 1. Thus, when 01\lambda \gt 1, ω\omega converges to ω0eμ1−λ{\omega }_{0}{e}^{\tfrac{\mu }{1-\lambda }} as t→+∞t\to +\infty . This is essentially different from the case of μ=1\mu =1, λ=0\lambda =0 in (Meng-Mai-Mei, JDE, 2022), whose axial velocity is only expressed by ω=ω0e−t\omega ={\omega }_{0}{e}^{-t}. Moreover, at the moving boundary, the compressible Euler equations with time-dependent damping become a degenerate hyperbolic system. Thanks to the Hardy-type inequality and cut-off functions, we choose a suitable weighted Sobolev spaces to construct a priori estimates to overcome the degeneracy of the system in the Lagrangian coordinates, which confirms the existence of local smooth solution. Gronwall’s inequality guarantees the uniqueness of local smooth solution.

Keywords

• compressible euler equations
• time-dependent damping
• cylindrical symmetry
• physical vacuum
• free boundary
• local smooth solutions
• 35r35
• 35l60
• 35q35
• 35q31