Non-uniqueness of global-in-time admissible weak solutions to the isentropic compressible Euler equations for a dense set of initial data

Abstract

In recent years, the technique of convex integration has demonstrated that for some initial data (also referred to as 'wild initial data'), many PDE models of mathematical fluid mechanics allow for a multitude of admissible (weak) solutions. This paper is concerned with the question regarding how large the set of such wild initial data is for the isentropic Euler equations. We prove that wild initial data form a dense set in $L^r$ for any $r \in [1,\infty)$. In contrast to existing results in the literature, in this paper 'wild initial data' are data which give rise to infinitely many global-in-time weak solutions which are admissible in the sense that the local energy inequality holds. In other words, the set of initial data with infinitely many admissible weak solutions (independent of the choice of time interval) is dense. A novel part of the construction is that we use a measure-valued (dissipative) solution as the ansatz for the subsolution. This requires several new ideas, in order to ensure the required regularity of the subsolution and to obtain a lower bound for the density. Another crucial ingredient of the proof is that the (local) energy density and the energy flux are constructed as part of the convex integration scheme, in order to obtain solutions which adhere to the local energy inequality.