Low Mach and thin domain limit for the compressible Euler system
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Low Mach and thin domain limit for the compressible Euler system Matteo Caggio1 · Bernard Ducomet2 · Šárka Nečasová1 · Tong Tang3 Received: 24 January 2020 / Accepted: 26 September 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We consider the compressible Euler system describing the motion of an ideal fluid confined to a straight layer Ω𝛿 = (0, 𝛿) × ℝ2 , 𝛿 > 0 . In the framework of dissipative measurevalued solutions, we show the convergence to the strong solution of the 2D incompressible Euler system when the Mach number tends to zero and 𝛿 → 0. Keywords Compressible Euler equations · Dissipative measure-valued solutions · Low Mach number · Thin domain Mathematics Subject Classification 35Q31 · 76N10
1 Introduction The present paper is devoted to the problem of the limit passage from three-dimensional to two-dimensional geometry, and from compressible to incompressible inviscid fluid. In the infinite straight layer (1.1)
Ω𝛿 = (0, 𝛿) × ℝ2 , 𝛿 > 0,
we consider the compressible Euler system describing the motion of a barotropic fluid, ) ( 𝜕t 𝜚𝜖 + divx 𝜚𝜖 𝐮𝜖 = 0, (1.2)
) 1 ( ) ( ( ) 𝜕t 𝜚𝜖 𝐮𝜖 + divx 𝜚𝜖 𝐮𝜖 ⊗ 𝐮𝜖 + 2 ∇x p 𝜚𝜖 = 0, 𝜖
(1.3)
supplemented with the initial conditions * Šárka Nečasová [email protected] 1
Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 11567 Praha 1, Czech Republic
2
LAMA, Université Paris-Est, 61 Avenue du Général de Gaulle, 94010 Créteil, France
3
Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098, People’s Republic of China
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𝜌𝜖 (0, ⋅) = 𝜌0,𝜖 , 𝜌𝜖 𝐮𝜖 (0, ⋅) = 𝐦0,𝜖 ,
(1.4)
𝐮𝜀 → 0, 𝜌𝜖 → 𝜌̃ as |x| → ∞,
(1.5)
and the far field conditions
with 𝜌� > 0 constant. The above system is written in its non-dimensional form, with 𝜖 the Mach number (the Strouhal number allowed by the scale analysis is set equal to one). Here, 𝜚𝜖 = 𝜚𝜖 (x, t) 𝐮𝜖 = 𝐮𝜖 (x, t) and p = p(𝜚𝜖 (x, t)) represents the mass density, the velocity vector and the pressure respectively (for assumptions on the pressure see Theorem 3.1). In the context of the low Mach number limit, the convergence of the solution of the compressible Euler system to the solution of the incompressible system was shown in several papers for well-prepared initial data of the compressible system, namely data for which the acoustic waves are not allowed, and for smooth solutions of the compressible flow (see [2, 22, 32, 35, 39]). Indeed, it is known that solutions of the compressible Euler system develop singularities in a finite time independently how smooth and/or small the initial data are. Moreover, it was shown by Feireisl et al. [24], it is very hard to prove that the life span of the smooth solutions is independent of the Mach number. Consequently, smooth solutions are quite restrictive for compressible inviscid flows. Precisely speaking, in [29, (𝛾−1)∕2 ∈ H m is small. 30] it is assumed that the initial density 𝜌0 has compact support and 𝜌0 Moreover, the crucial ass
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