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