On Non-uniqueness of continuous entropy solutions to the isentropic compressible Euler equations

We consider the Cauchy problem for the isentropic compressible Euler equations in a three-dimensional periodic domain under general pressure laws. For any smooth initial density away from the vacuum, we construct infinitely many entropy solutions with no presence of shock. In particular, the constructed density is smooth and the momentum is $\alpha$-H\"older continuous for $\alpha<1/7$. Also, we provide a continuous entropy solution satisfying the entropy inequality strictly.