classical solutions for a generalized euler equation in two dimensions abstract it is well known that the euler equations in two spatial dimensions have global classical solutions we provide a new proof which is analytic rather than geometric it is set in an abstract framework that applies to the so called lake and the great lake equations describing weakly non hydrostatic effects of bottom topography on the motion of shallow water the key ingredient is a new lpestimate on the nonlinear term the estimate is used to develop a global hmtheory for bounded domains in r 2which is similar in spirit to a 1975 paper by r temam it also provides explicit bounds on the hmnorm which grow like exp exp t

gevrey regularity for navier stokes equations under lions boundary conditions abstract the navier stokes system is considered in a compact riemannian manifold gevrey class regularity is proven under lions boundary conditions in 2d for the rectangle cylinder and hemisphere and in 3d for the rectangle the cases of the 2d sphere and 2d and 3d torus are also revisited

supersymmetric sigma models twistors and the atiyah hitchin metric the legendre transform and its generalizations originally found in supersymmetric models are techniques that can be used to givelocal constructions of hyperkahler metrics we give a twistor space interpretation to the generalizations of the legendre transform construction the atiyah hitchin metric on the moduli space of two monopoles is used as a detailed example

global well posedness for the lake equations abstract we prove global well posedness for the lake equations these equations arise to leading order in a low aspect ratio low froude number i e low wave speed and very small wave amplitude expansion of the three dimensional incompressible euler equations in a basin with a free upper surface and a spatially varying bottom topography well posedness means that there exists a solution that it is unique and that it depends continuously on the data i e the initial condition and the bottom topography our approach follows the works of yudovitch and bardos in constructing the solutions as the inviscid limit of solutions to a system with artificial viscosity which is the analog of the navier stokes with respect to the euler equations one of the main assumptions in the present work is a nondegenerate bottom topography

global well posedness for the averaged euler equations in two dimensions abstract we prove global well posedness of the two dimensional averaged euler or euler equations for initial potential vorticity of class l2 this model generalizes the one dimensional fokas fuchssteiner camassa holm equation which describes the propagation of unidirectional waves on the surface of shallow water as such it can be realized as a geodesic equation for the h1 metric on the lie algebra of vector fields moreover in two dimensions the model obeys an advection equation for the so called potential vorticity in close analogy to the vorticity form of the euler equations we construct solutions to the weak form of the potential vorticity equation by taking the inviscid limit of solutions to a system regularized by artificial viscosity since the streamfunction vorticity relation is of order four we can show uniqueness even for potential vorticities in l2

n 2 super yang mills theory in projective superspace we constructn 2 yang mills theory in projective superspace by exploiting the analogy to ward s twistor construction of self dual yang mills fields

new hyperkahler metrics and new supermultiplets we present new constructions of hyperkahler metrics along with the three new classes ofn 4 supermultiplets that they derive from further we provide a general setting for understanding the constructions and give a detailed description of the multiplets inn 2 andn 4 superspace

the real geometry of holomorphic four metrics the real geometry of holomorphic four metrics is investigated the almost product and complex structures associated with the real eight metrics corresponding to the real and imaginary parts of the holomorphic metrics are studied it is shown that half flat holomorphic metrics and the corresponding real eight metrics are associated with integrable almost product complex and hyper kahler structures real and complex local coordinate descriptions are presented

on match lengths zero entropy and large deviations with application to sliding window lempel ziv algorithm the sliding window lempel ziv swlz algorithm that makes use of recurrence times and match lengths has been studied from various perspectives in information theory literature in this paper we undertake a finer study of these quantities under two different scenarios 1 zero entropy sources that are characterized by strong long term memory and 2 the processes with weak memory as described through various mixing conditions for zero entropy sources a general statement on match length is obtained it is used in the proof of almost sure optimality of fixed shift variant of lempel ziv fslz and swlz algorithms given in literature through an example of stationary and ergodic processes generated by an irrational rotation we establish that for a window of size n w a compression ratio given by o log n w n w a where a depends on n w and approaches 1 as n w is obtained under the application of fslz and swlz algorithms in addition we give a general expression for the compression ratio for a class of stationary and ergodic processes with zero entropy next we extend the study of ornstein and weiss on the asymptotic behavior of the normalized version of recurrence times and establish the large deviation property for a class of mixing processes in addition an estimator of entropy based on recurrence times is proposed for which large deviation principle is proved for sources satisfying similar mixing conditions

on the lagrangian and eulerian analyticity for the euler equations abstract we address preservation of the lagrangian analyticity radius of solutions to the euler equations belonging to natural analytic space based on the size of taylor or gevrey coefficients we prove that if the solution belongs to such space then the solution also belongs to it for a positive amount of time we also prove the local analog of this result for a sufficiently large gevrey parameter however we show that the preservation holds independently of the size of the radius finally we construct a solution which shows that the first result does not hold in the eulerian setting

on the regularity and convergence of solutions to the 3d navier stokes voigt equations abstract in this paper we consider the 3d navier stokes voigt equations with periodic boundary conditions we first prove the higher order global regularity including both sobolev and gevrey regularity of solutions to the navier stokes voigt equations then we show the convergence of solutions of the 3d navier stokes voigt equations to the corresponding strong solution of the limit 3d navier stokes equations on the interval of existence of the latter as the parameter tends to zero

analyticity of solutions for a generalized euler equation abstract we consider the so called lake and great lake equations which are shallow water equations that describe the long time motion of an inviscid incompressible fluid contained in a shallow basin with a slowly spatially varying bottom a free upper surface and vertical side walls under the influence of gravity and in the limit of small characteristic velocities and very small surface amplitude if these equations are posed on a space periodic domain and the initial data are real analytic the solution remains real analytic for all times the proof is based on a characterization of gevrey classes in terms of decay of fourier coefficients in particular our result recovers known results for the euler equations in two and three spatial dimensions we believe the proof is new

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

long time effects of bottom topography in shallow water abstract we present and discuss new shallow water equations that provide an estimate of the long time asymptotic effects of slowly varying bottom topography and weak hydrostatic imbalance on the vertically averaged horizontal velocity of an incompressible fluid with a free surface which is moving under the force of gravity we consider the regime where the froude number is much smaller than the aspect ratio of the shallow domain the new equations are obtained at first order in an asymptotic expansion of the solutions of the euler equations for a shallow fluid by using the small parameter 2 the leading order equations in this expansion enforce hydrostatic balance while those obtained at first order retain certain nonhydrostatic effects both sets of equations conserve energy and circulation convect potential vorticity and have a hamiltonian formulation the corresponding energy and enstrophy are quadratic integrals with which we can bound the cumulative influence of the nonhydrostatic effects

hyperkahler manifolds and nonlinear supermultiplets we present a new construction of hyperkahler metrics that derives from the 3 dimensionaln 4 nonlinear supermultiplet further we give a detailed description of the nonlinear multiplet inn 2 and 4 superspace

space analyticity for the nonlinear heat equation in a bounded domain 1 introductionin fot foias and temam introduced a method for estimating thespace analyticity radius of solutions of the navier stokes equation withperiodic boundary conditions for the numerous applications seee g cees crt dth ft gr k lo in our previous paper gk we did not use the fourier series and were thus able to treat the navier stokes and other semilinear parabolic equations with singular l

the dissipative scale of the stochastics navier stokes equation regularization and analyticity we prove spatial analyticity for solutions of the stochastically forced navier stokes equation provided that the forcing is sufficiently smooth spatially we also give estimates which extend to the stationary regime providing strong control of both of the expected rate of dissipation and fluctuations about this mean surprisingly we could not obtain non random estimates of the exponential decay rate of the spatial fourier spectra