Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

In this paper, we show that the only solution of the vortex sheet equation, either stationary or uniformly rotating with negative angular velocity $$\Omega $$ Ω , such that it has positive vorticity and is concentrated in a finite disjoint union of smooth curves with finite length is the trivial one: constant vorticity amplitude supported on a union of nested, concentric circles. The proof follows a desingularization argument and a calculus of variations flavor.

Exponential instability for a class of dispersing billiards

The billiard in the exterior of a finite disjoint union $K$ of strictly convex bodies in ${\mathbb R}^d$ with smooth boundaries is considered. The existence of global constants $0 < \delta < 1$ and $C > 0$ is established such that if two billiard trajectories have $n$ successive reflections from the same convex components of $K$, then the distance between their $j$th reflection points is less than $C(\delta^j + \delta^{n-j})$ for a sequence of integers $j$ with uniform density in $\{1,2,\dots,n\}$. Consequently, the billiard ball map (although not continuous in general) is expansive. As applications, an asymptotic of the number of prime closed billiard trajectories is proved which generalizes a result of Morita [Mor], and it is shown that the topological entropy of the billiard flow does not exceed $\log (s-1)/a$, where $s$ is the number of convex components of $K$ and $a$ is the minimal distance between different convex components of $K$.

A formula for the number of branches for one-dimensional semianalytic sets

Let F = (F1, …, Fn-1): (ℝn, 0)→(ℝn-1, 0) and G:(ℝn, 0)→(ℝ, 0) be germs of analytic mappings, and let X = F-1(0). Assume that 0 ∈ ℝn is an isolated singular point in X, i.e. 0 ∈ ℝn is isolated in {x ∈ X|rank[DF(x)] < n-1}. Hence a germ of X/{0} at the origin is either void or a finite disjoint union of analytic curves. Let b denote the number of branches, i.e. connected components, of X/{0} and let b+ (resp. b-, b0) denote the number of branches of X/{0} on which G is positive (resp. G is negative, G vanishes). The problem is to calculate the numbers b, b+, b-, b0 in terms of F and G.