Multiple scales and singular limits of perfect fluids

Abstract

AbstractIn this article, our goal is to study the singular limits for a scaled barotropic Euler system modeling a rotating, compressible and inviscid fluid, where Mach number $$=\epsilon ^m $$ = ϵ m , Rossby number $$=\epsilon $$ = ϵ and Froude number $$=\epsilon ^n $$ = ϵ n are proportional to a small parameter $$\epsilon \rightarrow 0$$ ϵ → 0 . The fluid is confined to an infinite slab, the limit behavior is identified as the incompressible Euler system or a damped incompressible Euler system depending on the relation between m and n. For well-prepared initial data, the convergence is shown on the lifespan time interval of the strong solutions of the target system, whereas a class of generalized dissipative solutions is considered for the primitive system. The technique can be adapted to the compressible Navier–Stokes system in the subcritical range of the adiabatic exponent $$\gamma $$ γ with $$1<\gamma \le \frac{3}{2}$$ 1 < γ ≤ 3 2 , where weak solutions are not known to exist.