On a Singular Limit for the Compressible Rotating Euler System
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Journal of Mathematical Fluid Mechanics
On a Singular Limit for the Compressible Rotating Euler System ˇarka Neˇcasov´a and Tong Tang S´ Communicated by M. Feistauer
Abstract. The work addresses a singular limit for a rotating compressible Euler system in the low Mach number and low Rossby number regime. Based on the concept of dissipative measure-valued solution, the quasi-geostrophic system is identified as the limit problem in the case of ill-prepared initial data. The ill-prepared initial data will cause rapidly oscillating acoustic waves. Using dispersive estimates of Strichartz type, the effect of the acoustic waves in the asymptotic limit is eliminated. Mathematics Subject Classification. 35Q30. Keywords. Compressible Euler equations, Singular limit, Low Mach number, Low Rossby number, Dissipative measure-valued solutions.
1. Introduction Earth’s graceful rotation is an unignorable factor at geophysical fluids models. These models play an important role in the analysis of complex Earth phenomena in meteorology, geophysical and astrophysics. In order to describe the effect of rotation, people introduce two factors: Coriolis acceleration and centrifugal acceleration. In many real world applications, the action of centrifugal force is neglected, as it is in equilibrium with stratification caused by the gravity of the Earth. Under the above assumptions, we consider the following scaled Euler equations in an infinite slap Ω = R2 × (0, 1): ∂t ρ + div(ρu) = 0, (1.1) 1 ∂t (ρu) + div(ρu ⊗ u) + M1a2 ∇x p(ρ) + Ro ρ(ω × u) = 0, where the unknown fields ρ = ρ(t, x) and u = u(t, x) represent the density and the velocity of an inviscid compressible fluid, ω = (0, 0, 1) is the rotation axis. The Mach number Ma, proportional to the characteristic velocity field divided by the sound speed, and the Rossby number Ro, defined as the ratio of the displacement due to Coriolis forces, play the role of singular (small) parameters. The symbol p = p(ρ) denotes the barotropic pressure (assumptions on the pressure see (3.1)). The system is supplemented by the far field conditions u → 0,
ρ → ρ, as |x| → ∞, where ρ > 0,
(1.2)
and boundary condition u · n|∂Ω = 0,
(1.3)
where n is outer normal vector to ∂Ω. ˇ arka Neˇ ˇ S´ casov´ a: The research of S.N. leading to these results has received funding from the Czech Sciences Foundation ˇ (GACR), P201-16-032308 and RVO 67985840. Final version of the paper was made under support the Czech Sciences ˇ Foundation (GACR), GA19-04243S. Tong Tang: The research of T.T. is supported by the NSFC Grant No. 11801138. 0123456789().: V,-vol
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JMFM
From modeling of geophysical fluids, the value of Mach number and Rossby number can be considered very small. It is well known that the compressible fluid flow becomes incompressible in the low Mach number limit, as the density distribution is constant and the velocity field becomes solenoidal. On the other hand, low Rossby number corresponds to fast rotation and the fast rotating fluids will lead to the so-called Taylor–
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