Continuous dependence of the pressure field with respect to endpoints for ideal incompressible fluids

Abstract

In the Brenier variational model for perfect fluids, the datum is the joint law of the initial and final positions of the particles. In this paper, we show that both the optimal action and the pressure field are Hölder continuous with respect to this datum metrized in Monge-Kantorovic distance.

Small Noise Limit and Convexity for Generalized Incompressible Flows, Schrödinger Problems, and Optimal Transport

“…To prove this statement, one needs a result corresponding to [7,Thm. 1] in that setting but we did not pursue in this direction.…”

“…Finally, and as a byproduct of the previous analysis, our last set of contributions will focus on the time-convexity of the entropy functional along the relevant dynamical interpolations: the optimal transport and the Schrödinger problems on the one hand (Proposition 2. 7), and on the other hand the multiphase Brenier and multiphase Brödinger problems (Proposition 2. 8).…”

Nonlinear instability in Vlasov type equations around rough velocity profiles

“…In the smooth case (considered by Arnold), the geodesic equation is nothing but the Euler equation, whereas in general, as shown by Shnirelman in [30], we cannot prevent particles from crossing each other, and we obtain solutions to the kinetic Euler equation (at least in a weak sense). A study of (2) with PDE techniques provides information on the optimization problem: using the present paper, the author shows in [4] that the optimal action in the Brenier model, although continuous (see [5]) cannot be Lipschitz continuous with respect to the data.…”

“…Let us present this condition. For given n ∈ Z d and ω ∈ C d , the linearization of the Vlasov-Poisson equation (1) around a smooth profile µ admits a solution of the form (5) a(v) exp in · (x − ωt)…”

“…In the two last ones, as soon as we can find n ∈ Z d and ω ∈ C d such that (n · ω) > 0 and satisfying (7) or (8), then for all k ∈ N * , kn and ω satisfy the same properties (this is a consequence of the scale invariance of these equations as explained in Subsection 7.2). In view of (5), it means that for any unstable profile, we can find exponential growing modes with arbitrary large frequency n and with growing rate (n · ω) proportional to this large frequency. The instability is therefore far more violent in these cases.…”