Almost Global Solutions to the Three-Dimensional Isentropic Inviscid Flows with Damping in a Physical Vacuum Around Bare
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Almost Global Solutions to the Three-Dimensional Isentropic Inviscid Flows with Damping in a Physical Vacuum Around Barenlatt Solutions Huihui Zeng Communicated by K. Ballstadt
Abstract For the three-dimensional vacuum free boundary problem with physical singularity where the sound speed is C 1/2 -Hölder continuous across the vacuum boundary of the compressible Euler equations with damping, without any symmetry assumptions, we prove the almost global existence of smooth solutions when the initial data are small perturbations of the Barenblatt self-similar solutions to the corresponding porous media equations simplified via Darcy’s law. It is proved that if the initial perturbation is of the size of ε, then the existing time for smooth solutions is at least of the order of exp(ε−2/3 ). The key issue for the analysis is the slow sub-linear growth of vacuum boundaries of the order of t 1/(3γ −1) , where γ > 1 is the adiabatic exponent for the gas. This is in sharp contrast to the currently available global-in-time existence theory of expanding solutions to the vacuum free boundary problems with physical singularity of compressible Euler equations for which the expanding rate of vacuum boundaries is linear. The results obtained in this paper are closely related to the open question in multiple dimensions framed by T.-P. Liu’s construction of particular solutions in 1996.
1. Introduction Consider the following three-dimensional vacuum free boundary problem for compressible Euler equations with damping: ∂t ρ + div(ρu) = 0 ∂t (ρu) + div(ρu ⊗ u) + ∇x p(ρ) = −ρu
in (t), in (t),
(1.1a) (1.1b)
ρ>0 ρ=0
in (t), on (t) = ∂(t),
(1.1c) (1.1d)
V((t)) = u · N ,
(1.1e)
H. Zeng
(ρ, u) = (ρ0 , u 0 )
on (0),
(1.1f)
where (t, x) ∈ [0, ∞) × R3 , ρ, u, and p denote, respectively, the time and space variable, density, velocity and pressure; (t) ⊂ R3 , (t), V((t)) and N represent, respectively, the changing volume occupied by the gas at time t, moving vacuum boundary, normal velocity of (t), and exterior unit normal vector to (t). We are concerned with the polytropic gas for which the equation of state is given by p(ρ) = ρ γ , where γ > 1 is the adiabatic exponent. Let c(ρ) = p (ρ) be the sound speed, and let the condition −∞ < ∇N c2 (ρ) < 0 on (t)
(1.2)
define a physical vacuum boundary (cf. [5,7,19,23,25,26]), which is also called a vacuum boundary with physical singularity in contrast to the case that ∇N c2 (ρ) = 0 on (t). The physical vacuum singularity plays the role of pushing vacuum boundaries, which can be seen by restricting the momentum Equation (1.1b) on (t): Dt u · N = −(γ − 1)−1 ∇N c2 (ρ) − u · N , where Dt u = (∂t + u · ∇x )u is the acceleration of (t), and the term −(γ − 1)−1 ∇N c2 (ρ) > 0 serves as a force due to the pressure effect to accelerate vacuum boundaries. In order to capture this physical singularity, the initial density is supposed to satisfy ρ0 > 0 in (0), ρ0 = 0 on (0), ρ0 (x) dx = M, (0) (1.3) − ∞ < ∇N c2 (ρ0 ) < 0 on (0), where M ∈ (0, ∞) is the initi
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