Non-uniqueness for the compressible Euler-Maxwell equations (Q6583650)

This article is concerned with the Cauchy problem for the isentropic compressible Euler-Maxwell equations under general pressure laws in a three-dimensional periodic domain. For any smooth initial electron density away from the vacuum and smooth equilibrium-charged ion density, the authors construct infinitely many \(\alpha\)-Hölder continuous entropy solutions emanating from the same initial data for \(\alpha<\frac17\). Especially, the electromagnetic field belongs to the Hölder class \(C^{1,\alpha}\). Furthermore, the authors provide a continuous entropy solution satisfying the entropy inequality strictly. The proof relies on the convex integration scheme originating from De Lellis-Székelyhidi's breakthrough work. In particular, the authors adapt the convex integration scheme proposed by \textit{C. De Lellis} and \textit{H. Kwon} [Anal. PDE 15, No. 8, 2003--2059 (2022; Zbl 1509.35196)] and \textit{V. Giri} and \textit{H. Kwon} [Arch. Ration. Mech. Anal. 245, No. 2, 1213--1283 (2022; Zbl 1504.35254)] to the compressible Euler-Maxwell system. Due to the constraint of the Maxwell equations, the authors propose a new method of Mikado's potential and construct new building blocks which satisfy the Maxwell equations and can be used to construct the perturbation.