Non-uniqueness for the compressible Euler–Maxwell equations

Abstract

We consider 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, we could construct infinitely many \(\alpha \)-Hölder continuous entropy solutions emanating from the same initial data for \(\alpha <\frac{1}{7}\). Especially, the electromagnetic field belongs to the Hölder class \(C^{1,\alpha }\). Furthermore, we provide a continuous entropy solution satisfying the entropy inequality strictly. The proof relies on the convex integration scheme. Due to the constrain of the Maxwell equations, we propose a method of Mikado potential and construct new building blocks.