Non-degeneracy, Mean Field Equations and the Onsager Theory of 2D Turbulence

Abstract

The understanding of some large energy, negative specific heat states in the Onsager description of 2D turbulence, seems to require the analysis of a subtle open problem about bubbling solutions of the mean field equation. Motivated by this application we prove that, under suitable non degeneracy assumptions on the associated m-vortex Hamiltonian, the mpoint bubbling solutions of the mean field equation are non degenerate as well. Then we deduce that the Onsager mean field equilibrium entropy is smooth and strictly convex in the high energy regime on domains of second kind. function of the flow, ρ λ (ψ) := e λψ Ω e λψ , is the vorticity density and −λ = β = 1 κTstat is minus the inverse statistical temperature, κ being the Boltzmann constant. This result has been more recently generalized to cover the case of any smooth, bounded and connected domain in [8].Definition 1.1. Let Ω ⊂ R 2 be a smooth and bounded domain. We say that Ω is of second kind if (P λ ) admits a solution for λ = 8π. Otherwise Ω is said to be of first kind.It follows from the results in [19] and [8] that Ω is of second kind if and only if E 8π < +∞, see Definition 1.3 for the definition of E 8π . As first proved in [18], for E ≥ E 8π the Onsager intuitions lead to another amazing result, which is the non concavity of the equilibrium entropy. Although the statistical temperature in this model has nothing to do with the physical temperature, this is still a very interesting phenomenon since, if the entropy S(E) is convex in a certain interval, then the system surprisingly "cools down" when the energy increases in that range. In other words the (statistical) specific heat is negative. With the exception of the results in [18] and more recently in [4], we do not know of any progress in the rigorous analysis of this problem. Actually, it has been shown in [18] that S(E) is not concave in (E 8π , +∞) while, under a suitable set of assumptions, it has been shown in [4] that it is strictly convex in (E * , ∞) for some E * > E 8π on any strictly star-shaped domain of second kind. One of our aims is to remove the assumptions in [4] and prove that in fact S(E) is convex on any convex domain of second kind for any E > E 8π large enough.The definition of domains of first/second kind was first introduced in [17] with an equivalent but different formulation. We refer to [19] and [8] for a complete discussion about the characterization of domains of first/second kind and related examples.Remark 1.2. It is well known that any disk, say B R = B R (0), is of first kind and that in this case (P λ ) admits a solution if and only if λ < 8π. Any regular polygon is of first kind [19]. However, there exist domains of first kind with non trivial topology where (P λ ) admits solutions also for λ > 8π. For example Ω = B R \ B r (x 0 ), with x 0 = 0 is of first kind if r is small enough [8]. In this case it is well known that for any N ≥ 2 there are solutions concentrating (see (1.1)) at N distinct points as λ → 8πN [26,37], as well as other solutions for any λ / ∈ 8πN [5...

Local uniqueness of m-bubbling sequences for the Gel’fand equation

“…Therefore all those projections vanish which yield the desired result. We point out that this is different from the previous works [24] and [6,7], where one is bound to use the assumptions about l(q) and D(q) to show that the projection along Y 0 vanishes. This paper is organized as follows.…”

“…Actually, Lin and Yan in [24] have initiated the study of the local uniqueness of m-bubbling sequences for the Chern-Simons-Higgs equation. Inspired by that approach, in [6,7] the authors of this work proved local uniqueness and nondegeneracy of m-bubbling sequences of (1.3). More precisely, we have the following:…”

“…Theorem 1A ( [6,7]). Let q = (q 1 , · · · , q m ) be a critical point of f m such that q j ∈ Ω \ {p 1 , · · · , p ℓ }, j = 1, · · · , m and det(D 2 Ω f m (q)) = 0.…”

Local uniqueness and non-degeneracy of blow up solutions of mean field equations with singular data

“…For what concerns regular bubbling solutions, for q = (q 1 , · · · , q m ) ∈ Ω × · · · × Ω, we let G * j (x) = 8πR(x, q j ) + 8π ∑ For (x 1 , · · · , x m ) ∈ Ω × · · · Ω, we also define the m-vortex Hamiltonian, G(x l , x j ). (1.1) Then, by assuming suitable non-degeneracy conditions the authors in [8,9] proved that regular m-bubbling solutions are unique and non-degenerate (see also [10] for an analogous result for the Gelfand equation).…”

“…Theorem A ( [8,9]). Let u (1) n and u (2) n be two regular m-bubbling solutions of (P ρ n ), with ρ (1) n = ρ n = ρ (2) n , blowing up at the points q j / ∈ {p 1 , · · · , p N }, j = 1, · · · , m, where q = (q 1 , · · · , q m ) is a critical point of H m .…”