Abstract. The evolution of a planar vortex sheet is described by the Birkhott-Rott equation. Duchon and Robert [C.R. Acad

Consider a flat two-dimensional vortex sheet perturbed initially by a small analytic disturbance.By a formal perturbation analysis, Moore derived an approximate differential equation for the evolution of the vortex sheet. We present a simplified derivation of Moore's approximate equation and analyze errors in the approximation. The result is used to prove existence of smooth solutions for long time. If the initial perturbation is of size E and is analytic in a strip 19m yI < p , existence of a smooth solution of Birkhofl's equation is shown for time I < n 2 p , if E is sufficiently small, withFor the particular case of sinusoidal data of wave length 7r and amplitude E, Moore's analysis and independent numerical results show singularity development at time r, = [log €1 + O(log(log ED. Our results prove existence for t < rcllog €1, if E is sufficiently small, with I( -+ 1 as E -+ 0. Thus our existence results are nearly optimal.

Abstract. An approximation method of Moore for Kelvin-Helmholtz instability is formulated as a general method for two-dimensional, incompressible, inviscid flows generated by a vortex sheet. In this method the nonlocal equations describing evolution of the sheet are approximated by a system of (local) differential equations. These equations are useful for predicting singularity formation on the sheet and for analyzing the initial value problem before singularity formation. The general method is applied to a number of problems: Kelvin-Helmholtz instability for periodic vortex sheets, motion of an interface in Hele-Shaw flow, Rayleigh-Taylor instability for stratified flow, and Krasny's desingularized vortex sheet equation. A new physically desingularized vortex sheet equation is proposed, which agrees with the finite thickness vortex layer equations in the localized approximation.

No abstract

Perturbative QCD in mass independent schemes leads in general to running coupling $a(Q^2)$ which is nonanalytic (nonholomorphic) in the regime of low spacelike momenta $|Q^2| \lesssim 1 \ {\rm GeV}^2$. Such (Landau) singularities are inconvenient in the following sense: evaluations of spacelike physical quantities ${\cal D}(Q^2)$ with such a running coupling $a(\kappa Q^2)$ ($\kappa \sim 1$) give us expressions with the same kind of singularities, while the general principles of local quantum field theory require that the mentioned physical quantities have no such singularities. In a previous work, certain classes of perturbative mass independent beta functions were found such that the resulting coupling was holomorphic. However, the resulting perturbation series showed explosive increase of coefficients already at ${\rm N}^4{\rm LO}$ order, as a consequence of the requirement that the theory reproduce the correct value of the $\tau$ lepton semihadronic strangeless decay ratio $r_{\tau}$. In this work we successfully extend the construction to specific classes of perturbative beta functions such that the perturbation series do not show explosive increase of coefficients, the perturbative coupling is holomorphic, and the correct value of $r_{\tau}$ is reproduced. In addition, we extract, with Borel sum rule analysis of the V+A channel of the semihadronic strangeless decays of $\tau$ lepton, reasonable values of the corresponding D=4 and D=6 condensates.Comment: v3: 27 pages; 12 figures; includes new Sec.V - sum rule analysis for semihadronic decays of tau lepton in a proposed QCD scheme; new Refs.: [114,115,118-132]; to appear in Int.J.Mod.Phys.