On 3D slightly compressible Euler equations (Q1769831)

The authors study three-dimensional Euler equations for a barotropic inviscid compressible fluid for small Mach numbers. Mathematically, this is equivalent to adding the small perturbation term \({1\over 2}\varepsilon^2\cdot\nabla p\) to the incompressible Euler equation. Following the method of decomposition, the authors show the existence of a smooth compressible flow on an arbitrary time interval \([0, T]\) for sufficiently small Mach number and for almost constant initial density, when the incompressible limit flow is assumed to exist up to \(T\) as well. The life span \(T\) is estimated as \(O(1/\varepsilon^{\mu-1})\) for suitable \(\mu> 1\). The proof is essentially based on a priori energy estimates and interpolation inequalities of Nirenberg type.