The Cauchy problem for the Pavlov equation

Commutation of multidimensional vector fields leads to integrable nonlinear dispersionless PDEs arising in various problems of mathematical physics and intensively studied in the recent literature. This report is aiming to solve the scattering and inverse scattering problem for integrable dispersionless PDEs, recently introduced just at a formal level, concentrating on the prototypical example of the Pavlov equation, and to justify an existence theorem for global bounded solutions of the associated Cauchy problem with small data.Comment: In the new version the analytical technique was essentially revised. The previous version contained a wrong statement about the solvability of the inverse problem for large data. This problem remains ope

On the critical dissipative quasi-geostrophic equation

The 2D quasi-geostrophic (QG) equation is a two dimensional model of the 3D incompressible Euler equations. When dissipation is included in the model then solutions always exist if the dissipation's wave number dependence is super-linear. Below this critical power the dissipation appears to be insufficient. For instance, it is not known if the critical dissipative QG equation has global smooth solutions for arbitrary large initial data. In this paper we prove existence and uniqueness of global classical solutions of the critical dissipative QG equation for initial data that have small $L^\infty$ norm. The importance of an $L^{\infty}$ smallness condition is due to the fact that $L^{\infty}$ is a conserved norm for the non-dissipative QG equation and is non-increasing on all solutions of the dissipative QG., irrespective of size.Comment: 12 page

A Wake Model for Free-Streamline Flow Theory, Part II. Cavity Flows Past Obstacles of Arbitrary Profile

In Part I of this paper a free-streamline wake model was introduced to treat the fully and partially developed wake flow or cavity flow past an oblique flat plate. This theory is generalized here to investigate the cavity flow past an obstacle of arbitrary profile at an arbitrary cavitation number. Consideration is first given to the cavity flow past a polygonal obstacle whose wetted sides may be concave towards the flow and may also possess some gentle convex corners. The general case of curved walls is then obtained by a limiting process. The analysis in this general case leads to a set of two functional equations for which several methods of solution are developed and discussed. As a few typical examples the analysis is carried out in detail for the specific cases of wedges, two-step wedges, flapped hydrofoils, and inclined circular arc plate. For these cases the present theory is found in good agreement with the experimental results available

A wake model for free-streamline flow theory. Part 2. Cavity flows past obstacles of arbitrary profile

In Part 1 of this paper a free-streamline wake model mas introduced to treat the fully and partially developed wake flow or cavity flow past an oblique flat plate. This theory is generalized here to investigate the cavity flow past an obstacle of arbitrary profile at an arbitrary cavitation number. Consideration is first given to the cavity flow past a polygonal obstacle whose wetted sides may be concave towards the flow and may also possess some gentle convex corners. The general case of curved walls is then obtained by a limiting process. The analysis in this general case leads to a set of two funnctional equations for which several methods of solutioii are developed and discussed. As a few typictbl examples the analysis is carried out in detail for the specific cases of wedges, two-step wedges, flapped hydrofoils, and inclined circular arc plates. For these cases the present theory is found to be in good agreement with the experimental results available

Small-Time Behavior of Unsteady Cavity Flows

A perturbation theory is applied to investigate the small-time behavior of unsteady cavity flows in which the time-dependent part of the flow may be taken as a small-time expansion superimposed on an established steady cavity flow of an ideal fluid. One purpose of this paper is to study the effect of the initial cavity size on the resulting flow due to a given disturbance. Various existing steady cavity-flow models have been employed for this purpose to evaluate the initial reaction of a cavitated body in an unsteady motion. Furthermore, a physical model is proposed here to give a proper representation of the mechanism by which the cavity volume may be changed with time; the initial hydrodynamic force resulting from such change is calculated based on this model

An Approximate Numerical Scheme for the Theory of Cavity Flows Past Obstacles of Arbitrary Profile

Recently an exact theory for the cavity flow past an obstacle of arbitrary profile at an arbitrary cavitation number has been developed by adopting a free-streamline wake model. The analysis in this general case leads to a set of two functional equations for which several numerical methods have been devised; some of these methods have already been successfully carried out for several typical cases on a high speed electronic computer. In this paper an approximate numerical scheme, somewhat like an engineering principle, is introduced which greatly shortens the computation of the dual functional equations while still retaining a high degree of accuracy of the numerical result. With such drastic simplification, it becomes feasible to carry out this approximate mrmerical scheme even with a hand computing machine

Model checking probabilistic and stochastic extensions of the pi-calculus

We present an implementation of model checking for probabilistic and stochastic extensions of the pi-calculus, a process algebra which supports modelling of concurrency and mobility. Formal verification techniques for such extensions have clear applications in several domains, including mobile ad-hoc network protocols, probabilistic security protocols and biological pathways. Despite this, no implementation of automated verification exists. Building upon the pi-calculus model checker MMC, we first show an automated procedure for constructing the underlying semantic model of a probabilistic or stochastic pi-calculus process. This can then be verified using existing probabilistic model checkers such as PRISM. Secondly, we demonstrate how for processes of a specific structure a more efficient, compositional approach is applicable, which uses our extension of MMC on each parallel component of the system and then translates the results into a high-level modular description for the PRISM tool. The feasibility of our techniques is demonstrated through a number of case studies from the pi-calculus literature

k-dependent SU(4) model of high-temperature superconductivity and its coherent-state solutions

We extend the SU(4) model [1-5] for high-Tc superconductivity to an SU(4)k model that permits explicit momentum (k) dependence in predicted observables. We derive and solve gap equations that depend on k, temperature, and doping from the SU(4)k coherent states, and show that the new SU(4)k model reduces to the original SU(4) model for observables that do not depend explicitly on momentum. The results of the SU(4)k model are relevant for experiments such as ARPES that detect explicitly k-dependent properties. The present SU(4)k model describes quantitatively the pseudogap temperature scale and may explain why the ARPES-measured T* along the anti-nodal direction is larger than other measurements that do not resolve momentum. It also provides an immediate microscopic explanation for Fermi arcs observed in the pseudogap region. In addition, the model leads to a prediction that even in the underdoped regime, there exist doping-dependent windows around nodal points in the k-space, where antiferromagnetism may be completely suppressed for all doping fractions, permitting pure superconducting states to exist.Comment: 10 pages, 7 figure