An Eulerian-Lagrangian formulation for compressible Euler Equations

Abstract: In this paper, we present a novel Eulerian-Lagrangian formulation for the compressible isentropic Euler equations with a general pressure law. Using the developed Lagrangian formulation, we show a short-time solution which is unique in the framework of Bessel space $H_{p}^{\beta}(\mathbb{T}^{d})$ for $\beta>\frac{d}{p}+1$. The proof is precise and relatively simple, relying on the application of the Implicit Function Theorem in Banach spaces combined with a carefully designed regularization procedure. To the best of our knowledge, this is the first time the Lagrangian formulation has been employed to show local wellposedness for compressible Euler equations. Moreover, it can be adaptableto $C^{k,\alpha}$ spaces, and notably Besov type spaces.