A General Convex Integration Scheme for the Isentropic Compressible Euler Equations

Abstract

We prove via convex integration a result that allows to pass from a so-called subsolution of the isentropic Euler equations (in space dimension at least 2) to exact weak solutions. The method is closely related to the incompressible scheme established by De Lellis-Székelyhidi, in particular we only perturb momenta and not densities. Surprisingly, though, this turns out not to be a restriction, as can be seen from our simple characterization of the Λ-convex hull of the constitutive set. An important application of our scheme will be exhibited in forthcoming work by Gallenmüller-Wiedemann.

Which measure-valued solutions of the monoatomic gas equations are generated by weak solutions?

“…Clearly, I ≤ 0 with equality if and only if | m(0,x)| 2 dρ(0,x) + ρ(0, x) γ = Q(0, x). The proof of the perturbation property (Proposition 3.1) in [23] then yields the following statement: Proposition 4.8. With the previous notation, let T > ε > 0 and α > 0.…”

“…The notation above indicating the kinetic energy density is taken from [23] which itself is adopted from [6]. The following theorem is a consequence of the genuine compressible convex integration method developed in [23], which is a generalization of the results in [15] to non-constant energies. This will be used as a black-box for convex integration in the proof of Proposition 4.9.…”

“…The proof proceeds as the one of Proposition 3.1 in [23], where the perturbations do not affect ρ and Q. The only difference to said proof is that we include perturbations only up to time ε to ensure the support condition.…”

“…For the latter we will make use of a compressible version of convex integration from [23], see Theorem 4.1 below, which is a generalization to non-constant energies of the results in [15]. Note that the only parts in which the particular form of the Euler equations plays a role are this last convex integration step and the step using the perturbation property.…”

Measure-valued low Mach number limits of ideal fluids

“…[3] and [7]. But only recently a fully compressible generalization has been worked out in [5], see also [21]. Other singular limits like the limit of weak solutions or the vanishing viscosity limit have been studied more extensively in the past years.…”