Well-Posedness of Strong Solutions to the Anelastic Equations of Stratified Viscous Flows
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Journal of Mathematical Fluid Mechanics
Well-Posedness of Strong Solutions to the Anelastic Equations of Stratified Viscous Flows Xin Liu
and Edriss S. Titi
Communicated by G. P. Galdi
Abstract. We establish the local and global well-posedness of strong solutions to the two- and three-dimensional anelastic equations of stratified viscous flows. In this model, the interaction of the density profile with the velocity field is taken into account, and the density background profile is permitted to have physical vacuum singularity. The existing time of the solutions is infinite in two dimensions, with general initial data, and in three dimensions with small initial data. Mathematics Subject Classification. 35Q30, 35Q86, 76D03, 76D05. Keywords. Anelastic approximation, Well-posedness, Physical vacuum, Stratified flows.
1. Introduction The anelastic Navier–Stokes system for stratified flows, ρ(∂t u + u · ∇u) + ρ∇p = Δu div(ρu) = 0
in Ω, in Ω,
(1)
is derived as the limiting system of the compressible Navier–Stokes system after filtering out the acoustic waves for strong stratified flows. Here the velocity field u and the pressure p are the unknowns while the background density ρ is given as a time-independent, non-negative function. The rigorous derivation of (1) can be found in [23]. Comparing to the incompressible Navier–Stokes system (see, e.g., [5,28]), the main difference is the incompressible condition divu = 0 is replaced by the anelastic relation div(ρu) = 0 with the background density profile ρ, which represents the strong stratification owing to the balance of the gravity and the pressure (see, e.g., [11]). Such an approximation preserves slight compressibility while filtering out the acoustic waves, which significantly simplifies the original compressible Navier–Stokes system, and enables more efficient computation applications to relevant model flows in physical reality. In particular, the anelastic approximation is used to describe the semi-compressible ocean dynamics (see, e.g., [7,8]), as well as the tornado-hurricane dynamics (see, e.g., [24,26]). We refer the readers to [1,2,10,15–17,22,25] for related topics and comparisons of various models of the atmospheric and oceanic dynamics. We remark that the background density profile ρ in the anelastic relation div(ρu) = 0 is given by the resting state ∇P (ρ) = ρg ez , where P (ρ) denotes the pressure potential and g is the gravity acceleration. For the sake of simplifying the presentation, we have choosed the gravity to point upwards, which can be done after performing a vertical reflection of the coordinates. In the case when the flow connects to vacuum continuously, the resting state yields a degenerate density profile. For an isentropic flow with P (ρ) = ργ , γ > 1, this implies ργ−1 z, referred to as the physical vacuum in the study of compressible flows (see, e.g., [14,19]). The main characteristics of the physical vacuum is the H¨ older continuity of the background density profile, whose derivatives are singular at z = 0. While there are some recent 0123456789().: V,-vol
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