On the global solubility of the Cauchy problem for hyperbolic Monge–Ampére systems

Singularities in one-dimensional Euler flows

“…such that ω ∧ dω = 0 for any solution f (u, ρ) to (12). Integrals of the form ω give us solutions to E .…”

“…Let us apply the following separation ansatz to (12): f (u, ρ) = ρµ(u) + ν(ρ). Then, we get two ODEs on µ(u) and ν(ρ):…”

“…The tendency of studying such phenomena is remaining nowadays as well, see, for example [7], where the case of Chaplygin gases is considered, [8,9], where the authors discuss weak shock waves, which is actually the case considered in the present paper, and also it is worth mentioning [10,11], where the influence of turbulence on detonations is investigated. The properties of global solvability for Euler equations and singularities of their solutions were also studied in a series of works [12,13,14,15].…”

“…In this case we are able to find all 3-dimensional manifolds N0 of the form N0 = {t − f (ρ, u)/ρ = 0}. Taking any of solutions to (12) we get the corresponding integrable distribution h = h+, h− , and according to Theorem 6, its integrals give us multivalued solutions.…”