On 3D slightly compressible euler equations

Alessandro Morando and Paolo Secchi

Abstract: This paper is concerned with the Euler equations of a barotropic inviscid compressible fluid in the three dimensional space $\mathbb{R}^3$.
Following the method of decomposition in [4], [5], we show the existence of a smooth compressible flow on an arbitrary time interval $[0,T]$ for any Mach number sufficiently small and almost constant initial densities, when the incompressible limit flow is assumed to exist up to $T$ as well.
The life span $O(1/\epsilon^{\mu-1})$ for the compressible solution is obtained assuming also that the incompressible part of the solution itself has a life span of order $O(1/\epsilon^{\mu-1})$ and is $O(\epsilon^{\mu-1})$ for suitable $\mu>1$.

Keywords: compressible Euler equations; incompressible Euler equations; life span; incompressible limit; Mach number.

Classification (MSC2000): 35Q35, 76N10, 35L60.

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