rigorous remarks about scaling laws in turbulent fluids a definition of scaling law for suitable families of measures is given and investigated first a number of necessary conditions are proved they imply the absence of scaling laws for 2d stochastic navier stokes equations and for the stochastic stokes linear problem in any dimension while they imply a lower bound on the mean vortex stretching in 3d second for the 3d stochastic navier stokes equations necessary and sufficient conditions for scaling laws to hold are given translating the problem into bounds for energy and enstrophy of high and low modes respectively unlike in the 2d case the validity or invalidity of such conditions in 3d remains open

hearing pseudoconvexity in lipschitz domains with holes via bar partial let omega widetilde omega setminus overline d where widetilde omega is a bounded domain with connected complement in mathbb c n or more generally in a stein manifold and d is relatively compact open subset of widetilde omega with connected complement in widetilde omega we obtain characterizations of pseudoconvexity of widetilde omega and d through the vanishing or hausdorff property of the dolbeault cohomology groups of omega on various function spaces in particular we show that if the boundaries of widetilde omega and d are lipschitz and c 2 smooth respectively then both widetilde omega and d are pseudoconvex if and only if 0 is not in the spectrum of the overline partial neumann laplacian of omega on 0 q forms for 1 le q le n 2 when n ge 3 or 0 is not a limit point of the spectrum of the overline partial neumann laplacian on 0 1 forms when n 2

holomorphic approximation via dolbeault cohomology the purpose of this paper is to study holomorphic approximation and approximation of overline partial closed forms in complex manifolds of complex dimension n ge 1 we consider extensions of the classical runge theorem and the mergelyan property to domains in complex manifolds for the mathcal c infty smooth and the l 2 topology we characterize the runge or mergelyan property in terms of certain dolbeault cohomology groups and some geometric sufficient conditions are given

l 2 serre duality on domains in complex manifolds and applications an l 2 version of the serre duality on domains in complex manifolds involving duality of hilbert space realizations of the operator is established this duality is used to study the solution of the equation with prescribed support applications are given to closed extension of forms as well to bochner hartogs type extension of cr functions comp e with the quotient topology where we endow spaces of compactly supported forms with the natural inductive limit topology in fact condition that the two maps in 1 have closed range is also necessary for the duality theorem to hold see 9 also see 26 27 28 for further results of this type serre s original proof 33 is based on sheaf theory and the theory of topological vector spaces a different approach to this result in the case when is a compact complex manifold was given by kodaira using hodge theory see 23 or 7 in this note we extend kodaira s method to non compact hermitian manifolds to obtain an l 2 analog of the serre duality special cases of serre duality using l 2 methods have appeared before in many contexts see 25 or 11 theorem 5 1 7 and 19 20 for example our treatment aims to streamline and systematize these results with emphasis on non compact manifolds and point out its close relation with the choice of l 2 realizations of the cauchy riemann operator or alternatively choice of boundary conditions for the l 2 realizations of the formal complex laplacian e e e e the l 2 duality can be interpreted in many ways at one level it is a duality between the standard laplacian with neumann boundary conditions and thec laplacian with dual dirichlet boundary conditions using another approach results regarding solution of the equation in l 2 can be converted to statements regarding the solution of the c equation this leads to a solution of the cauchy problem i e solution of the equation with prescribed support at the heart of the matter lies the existence of

canonical quantization of lattice higgs yang mills fields krein essential selfadjointness of the hamiltonian using a krein indefinite metric in fock space the hamiltonian for cut off models of canonically quantized higgs yang mills fields interpolating between the gupta bleuler feynman and landau gauges is shown to be essentially maximal accretive and essentially krein selfadjoint

the yang mills heat semigroup on three manifolds with boundary long time existence and uniqueness of solutions to the yang mills heat equation is proven over a compact 3 manifold with smooth boundary the initial data is taken to be a lie algebra valued connection form in the sobolev space h1 three kinds of boundary conditions are explored dirichlet type neumann type and marini boundary conditions the last is a nonlinear boundary condition specified by setting the normal component of the curvature to zero on the boundary the yang mills heat equation is a weakly parabolic nonlinear equation we use gauge symmetry breaking to convert it to a parabolic equation and then gauge transform the solution of the parabolic equation back to a solution of the original equation apriori estimates are developed by first establishing a gauge invariant version of the gaffney friedrichs inequality a gauge invariant regularization procedure for solutions is also established uniqueness holds upon imposition of boundary conditions on only two of the three components of the connection form because of weak parabolicity this work is motivated by possible applications to quantum field theory

bounds on kolmogorov spectra for the navier stokes equations let u x t be a possibly weak solution of the navier stokes equations on all of r 3 or on the torus r 3 z 3 denoting the fourier transform by u fu the energy spectrum of u t is the spherical integral

critical strong feller regularity for markov solutions to the navier stokes equations abstract the main purpose of this paper is to show that markov solutions to the 3d navier stokes equations driven by gaussian noise have the strong feller property up to the critical topology given by the domain of the stokes operator to the power one fourth

existence of densities for the 3d navier stokes equations driven by gaussian noise we prove three results on the existence of densities for the laws of finite dimensional functionals of the solutions of the stochastic navier stokes equations in dimension 3 in particular under very mild assumptions on the noise we prove that finite dimensional projections of the solutions have densities with respect to the lebesgue measure which have some smoothness when measured in a besov space this is proved thanks to a new argument inspired by an idea introduced in fournier and printems 2010

uniqueness and blow up for a stochastic viscous dyadic model we consider the dyadic model with viscosity and additive gaussian noise as a simplified version of the stochastic navier stokes equations with the purpose of studying uniqueness and emergence of singularities we prove pathwise uniqueness and absence of blow up in the intermediate intensity of the non linearity morally corresponding to the 3d case and blow up for stronger intensity moreover blow up happens with probability one for regular initial data

analysis of equilibrium states of markov solutions to the 3d navier stokes equations driven by additive noise we prove that every markov solution to the three dimensional navier stokes equations with periodic boundary conditions driven by additive gaussian noise is uniquely ergodic the convergence to the unique invariant measure is exponentially fast

path space formulas for covariant gauges in krein spaces path space representations in terms of feynman gauge stochastic integrals are given for the free maxwell field in all covariant gauges interpolating between the feynman and landau gauges these are applied to quantum electrodynamics and yang mills theories with spatial lattice cutoffs

stochastic tamed 3d navier stokes equations existence uniqueness and ergodicity in this paper we prove the existence of a unique strong solution to a stochastic tamed 3d navier stokes equation in the whole space as well as in the periodic boundary case then we also study the feller property of solutions and prove the existence of invariant measures for the corresponding feller semigroup in the case of periodic conditions moreover in the case of periodic boundary and degenerated additive noise using the notion of asymptotic strong feller property proposed by hairer and mattingly ann math 164 993 1032 2006 we prove the uniqueness of invariant measures for the corresponding transition semigroup

convergence of u 1 3 lattice gauge theory to its continuum limit it is shown that in three space time dimensions the pure u 1 lattice gauge theory with villain action and fixed coupling constant converges to the free electromagnetic field as the lattice spacing approaches zero the same holds for the wilson action on the electric sector

canonical quantization of lattice higgs maxwell chern simons fields krein self adjointness it is shown how techniques from constructive quantum field theory may be applied to indefinite metric gauge theories in hilbert space for the case of a higgs maxwell chern simons theory on a lattice the hamiltonian operator is shown to be krein essentially self adjoint by means of unbounded but krein unitary transformations relating the hamiltonian to an essentially maximal accretive operator

convergence of compact lattice scalar field theory to its continuum limit it is shown that the compact lattice scalar field theory converges in the sense of the correlation functions to its continuum limit when simultaneously the lattice spacing tends to zero and the amplitude cutoff and the lattice volume go to infinity this is achieved in any number of dimensions provided that the last two parameters go to infinity not too rapidly besides it is proved the right upper bound to the limit with no condition on the way it is performed

a path space formula for gauss vectors in chern simons quantum electrodynamics canonical quantization of a chern simons gauge field minimally coupled to a spinor field is studied as an indefinite metric quantum field theory in the usual covariant gauges by using a lattice cutoff for this model we show that positivity for the indefinite metric gauss law gauge invariance and osterwalder schrader positivity for a self adjoint hamiltonian are equivalent in addition the path space formula for the osterwalder schrader semigroup is constructed in terms of a euclidean scalar massive gaussian random field

convergence of theu 1 4 lattice gauge theory to its continuum limit it is shown that in four space time dimensions the compactu 1 lattice gauge theory with general energy function converges to a renormalized free electromagnetic field on the current sector as the lattice spacing approaches zero provided the coupling constant is sufficiently large for the wilson energy function it is possible by judicious choice of the gibbs state to get convergence for arbitrary coupling strengths furthermore for all but a countable number of values of the coupling constant the limit exists and is independent of the particular state chosen to define the lattice model

a convergent lattice approximation for nonlinear electromagnetic fields in four dimensions a discrete version of the quaternionic cauchy riemann equation a f where f is gauss poisson white noise is discussed on the lattice z4 random variables f n and a n are constructed which approximate the corresponding random fields f and a respectively in the limit 0 in the gaussian case the random variables a n can be interpreted as the lattice approximation of the free electromagnetic euclidean potential field whereas in the non gaussian case one obtains an approximation of nonlinear interacting electromagnetic quantum fields convergence to the continuum limit is proven

u 1 gauge theory on a torus u 1 gauge theory with the villain action on a cubic lattice approximation of three and four dimensional torus is considered as the lattice spacing approaches zero provided the coupling constant correspondingly approaches zero the naturally chosen correlation functions converge to the correlation functions of the r gauge electrodynamics on three and four dimensional torus when the torus radius tends to infinity these correlation functions converge to the correlation functions of the r gauge euclidean electrodynamics

the leading term of the yang mills free energy abstract this article gives an explicit formula for the leading term of the free energy of three dimensional u n lattice gauge theory for any n as the lattice spacing tends to zero the proof is based on a novel technique that avoids phase cell renormalization the technique also yields a similar formula for the four dimensional theory but only in the weak coupling limit

exponential mixing for the 3d stochastic navier stokes equations we study the navier stokes equations in dimension 3 ns3d driven by a noise which is white in time we establish that if the noise is at the same time sufficiently smooth and non degenerate in space then the weak solutions converge exponentially fast to equilibrium

martingale and stationary solutions for stochastic navier stokes equations we prove the existence of martingale solutions and of stationary solutions of stochastic navier stokes equations under very general hypotheses on the diffusion term the stationary martingale solutions yield the existence of invariant measures when the transition semigroup is well defined the results are obtained by a new method of compactness

markov selections for the 3d stochastic navier stokes equations we investigate the markov property and the continuity with respect to the initial conditions strong feller property for the solutions to the navier stokes equations forced by an additive noise first we prove by means of an abstract selection principle that there are markov solutions to the navier stokes equations due to the lack of continuity of solutions in the space of finite energy the markov property holds almost everywhere in time then depending on the regularity of the noise we prove that any markov solution has the strong feller property for regular initial conditions we give also a few consequences of these facts together with a new sufficient condition for well posedness

on the hausdorff property of some dolbeault cohomology groups let x be a complex manifold the study of the closed range property of the cauchy riemann equations is of fundamental importance both from the sheaf theoretic point of view and the pde point of view in terms of the associated cohomology it means that the corresponding cohomology is hausdorff hence separated there are many known results for the hausdorff property of such cohomologies in complex manifolds see in particular 16 18 for instance it is well known that for a bounded pseudoconvex domain d incn the dolbeault cohomology h p q d in the frechet space c p q d vanishes for all q 0 it is also known that the l2 cohomology also vanishes much less is known about the cohomologies whose topology does not have the hausdorff property even for domains in cn an example is given in 13 section 14 in this paper we study duality of the cauchy riemann complex in various function spaces and the hausdorff property of the corresponding cohomologies such duality is classical if

exponential ergodicity for stochastic burgers and 2d navier stokes equations abstract it is shown that transition measures of the stochastic navier stokes equation in 2 d converge exponentially fast to the corresponding invariant measures in the distance of total variation as a corollary we obtain the existence of spectral gap for a related semigroup obtained by a sort of ground state transformation analogous results are proved for the stochastic burgers equation