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.
Highlights
The anelastic Navier–Stokes system for stratified flows, ρ(∂tu + u · ∇u) + ρ∇p = Δu in Ω, (1)div(ρu) = 0 in Ω, is derived as the limiting system of the compressible Navier–Stokes system after filtering out the acoustic waves for strong stratified flows
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
Comparing to the incompressible Navier–Stokes system, 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
Summary
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.
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