Non-uniqueness of Hölder continuous solutions for Inhomogeneous Incompressible Euler flows

We consider the inhomogeneous (or density dependent) incompressible Euler equations in a three-dimensional periodic domain. We construct density $\varrho$ and velocity $u$ such that, for any $\alpha <1\mathrm{/}7$, both of them are $\alpha$-Hölder continuous and $(\varrho ,u)$ is a weak solution to the underlying equations. The proof is based on typical convex integration techniques using Mikado flows as building blocks. As a main novelty with respect to the related literature, our result produces a Hölder continuous density.