Global existence and convergence to the modified Barenblatt solution for the compressible Euler equations with physical
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Calculus of Variations
Global existence and convergence to the modified Barenblatt solution for the compressible Euler equations with physical vacuum and time-dependent damping Xinghong Pan1 Received: 12 July 2020 / Accepted: 2 November 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, the smooth solution of the physical vacuum problem for the one dimensional compressible Euler equations with time-dependent damping is considered. Near the vacuum boundary, the sound speed is C 1/2 -Hölder continuous. The coefficient of the damping depends μ on time, given by this form (1+t) λ , λ, μ > 0, which decays by order −λ in time. Under the assumption that 0 < λ < 1, 0 < μ or λ = 1, 2 < μ, we will prove the global existence of smooth solutions and convergence to the modified Barenblatt solution of the related porous media equation with time-dependent dissipation and the same total mass when the initial data of the Euler equations is a small perturbation of that of the Barenblatt solution. The pointwise convergence rates of the density, velocity and the expanding rate of the physical vacuum boundary are also given. The proof is based on space-time weighted energy estimates, elliptic estimates and Hardy inequality in the Lagrangian coordinates. Our result is an extension of that in Luo–Zeng (Commun Pure Appl Math 69(7):1354–1396, 2016), where the authors considered the physical vacuum free boundary problem of the compressible Euler equations with constant-coefficient damping. Mathematics Subject Classification 35A01 · 35Q31
1 Introduction In this paper, we investigate the global existence of smooth solutions for the physical vacuum boundary problem of the following 1-d compressible Euler equations with time-dependent
Communicated by F.-H. Lin. X. Pan is supported by Natural Science Foundation of Jiangsu Province (No. SBK2018041027) and National Natural Science Foundation of China (No. 11801268).
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Xinghong Pan [email protected] Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China 0123456789().: V,-vol
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damping.
X. Pan
⎧ ρt + (ρu)x = 0 in I (t) := {(x, t)|x− (t) < x < x+ (t), t > 0}, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (ρu)t + ( p(ρ) + ρu 2 )x = − μ ρu in I (t), (1 + t)λ ⎪ ⎪ ⎪ ρ > 0 in I (t), ρ = 0 on x± (t), ⎪ ⎪ ⎩ (ρ, u) = (ρ0 , u 0 ) on I (0) := {x|x− (0) < x < x+ (0)},
(1.1)
where the boundary x± (t) satisfies x˙± (t) = u(x± (t), t). Here (x, t) ∈ R × [0, ∞), ρ, u, and p denote the space and time variable, density, velocity, and pressure, respectively. I(t), x± (t), and x˙± (t) represent the changing domain occupied by the gas, the moving vacuum boundary and the velocity of x b (t), respectively. μ − (1+t) λ ρu, appearing on the right-hand side of (1.1)2 describes the frictional damping which will decay by order −λ in time. We assume the gas is the isentropic flow and the pressure satisfies the γ law: 1 p(ρ) = ρ γ for γ > 1. γ (Here the adiabatic constant is set to be γ1 .) Let c = p (ρ) be the sound speed. A vacuum bo
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