Compatibility conditions allowing mono phasic oscillating solutions for the multidimensional incompressible Euler system

We are interested in Cauchy’s problem formed by a multidimensional incompressible Euler’s system and large amplitude oscillating initial data \(w(x,\varphi (x)/\varepsilon )\in \mathcal {C}^1(\Omega _r^0,\mathbb {R}^n)\), with \(\varepsilon \in ]0,1]\) is a parameter and \(\Omega ^0_r\subset \mathbb {R}^n\) the ball of centre zero and radius r. We determine the necessary and sufficient conditions that guarantee a solution on a domain of \(\mathbb {R}^+\times \mathbb {R}^n\) independent of \(\varepsilon \) for the Cauchy’s problem previously mentioned. These conditions are a system of nonlinear partial differential equations uniform in \(\varepsilon \) involving the couple \((\varphi ,w)\), we show the existence of this couple, and we discuss its propagation over time.