brezis gallouet wainger type inequalities and blow up criteria for navier stokes equations in unbounded domains we shall find the weakest norm that satisfies the brezis gallouet wainger type inequality under some conditions as an application of the brezis gallouet wainger type inequality we shall establish beale kato majda type blow up criteria of smooth solutions to the 3 d navier stokes equations in unbounded domains

remark on the stability of the large stationary solutions to the navier stokes equations under the general flux condition abstract consider stationary weak solutions of the navier stokes equations in a bounded domain in r 3 under the nonhomogeneous boundary condition we give a new approach for the stability of the stationary flow in the l 2 framework furthermore we give some examples of stable solutions which may be large in l 3 or w 1 3 2

existence uniqueness and l q estimates for the stokes problem in an exterior domain as is well known the study of the properties of solutions to the stokes system plays an essential role in the mathematical theory of viscous fluid flows governed by the navier stokes equations it is not art overstatement to say that existence uniqueness and a priori estimates in sobolev spaces for the stokes system are fundamental in every basic question related to the navier stokes equations such as existence regularity asymptotic behavior in time etc see e g ladyzhenskaya 1969 solonni ov 1977 and te am 1977 so far however the stokes system has been fully analyzed only for bounded domains see sobolevskn 1960 ladvznenskaya 1959 solonnu ov 1960 vorovlch yot oowcr 1961 cattabrma 1961 in particular in the paper of cattabriga a complete treatment is given in the sobotev spaces w l 1 1 q c for space dimension n 3 cattabriga s results essentially state that if 2 is a bounded domain in r a of class c m m max l 2 2 l 1 then given f e wt q y2 3 gc wt l q o q e wl 2 1 q q 8o a 1 q there exists one and only one distributional solution v c w1 z q q 3 p e wt l q o to the stokes system

the poisson boundary of teichmuller space abstract for a markov operator on teichmuller space commuting with the action of the mapping class group we prove convergence of sample paths of the associated markov chain in the thurston compactification and show that the poisson boundary of the markov operator can be identified with the space of projective measured foliations the approach consists in using the authors results on the poisson boundary of the mapping class group in combination with a discretization procedure based on a harnack inequality for markov operators on teichmuller space

boundary value problems for first order elliptic wedge operators we develop an elliptic theory based in l 2 of boundary value problems for general wedge differential operators of first order under only mild assumptions on the boundary spectrum in particular we do not require the indicial roots to be constant along the base of the boundary fibration our theory includes as a special case the classical theory of elliptic boundary value problems for first order operators with and without the shapiro lopatinskii condition and can be thought of as a natural extension of that theory to the geometrically and analytically relevant class of wedge operators wedge operators arise in the global analysis on manifolds with incomplete edge singularities our theory settles in the first order case the long standing open problem to develop a robust elliptic theory of boundary value problems for such operators

an algebra of boundary value problems not requiring shapiro lopatinskij conditions abstract we construct an algebra of pseudo differential boundary value problems that contains the classical shapiro lopatinskij elliptic problems as well as all differential elliptic problems of dirac type with aps boundary conditions together with their parametrices global pseudo differential projections on the boundary are used to define ellipticity and to show the fredholm property in suitable scales of spaces

source coding and graph entropies a sender wants to accurately convey information to a receiver who has some possibly related data we study the expected number of bits the sender must transmit for one and for multiple instances in two communication scenarios and relate this number to the chromatic and korner 1973 entropies of a naturally defined graph

the method of types information theory the method of types is one of the key technical tools in shannon theory and this tool is valuable also in other fields in this paper some key applications are presented in sufficient detail enabling an interested nonspecialist to gain a working knowledge of the method and a wide selection of further applications are surveyed these range from hypothesis testing and large deviations theory through error exponents for discrete memoryless channels and capacity of arbitrarily varying channels to multiuser problems while the method of types is suitable primarily for discrete memoryless models its extensions to certain models with memory are also discussed

geometric bvps hardy spaces and the cauchy integral and transform on regions with corners abstract in this paper we give a new perspective on the cauchy integral and transform and hardy spaces for dirac type operators on manifolds with corners of codimension two instead of considering banach or hilbert spaces we use polyhomogeneous functions on a geometrically blown up version of the manifold called the total boundary blow up introduced by mazzeo and melrose r r mazzeo r b melrose analytic surgery and the eta invariant geom funct anal 5 1 1995 14 75 these polyhomogeneous functions are smooth everywhere on the original manifold except at the corners where they have a taylor series with possible log terms in polar coordinates the main application of our analysis is a complete fredholm theory for boundary value problems of dirac operators on manifolds with corners of codimension two

maximal regularity of the stokes operator in an exterior domain with moving boundary and application to the navier stokes equations consider the navier stokes system on an exterior domain with moving boundary and dirichlet boundary conditions in 2003 saal proved that the stokes operator in a domain with moving boundary has the property of maximal regularity provided that the operator is invertible hence his result can be applied if the domain is bounded or by adding a shift to the stokes operator if the domain is unbounded or the time interval is finite in this paper we will generalize his result to a result global in time if the reference domain is an exterior domain finally we will apply this result to the navier stokes equations to obtain a global in time existence theorem for small data

on the analyticity of the semigroup generated by the stokes operator with neumann type boundary conditions on lipschitz subdomains of riemannian manifolds we study the analyticity of the semigroup generated by the stokes operator equipped with neumann type boundary conditions on l p spaces in lipschitz domains our strategy is to regularize this operator by considering the hodge laplacian which has the additional property that it commutes with the leray projection

on a class of stable steady flows to the exterior convection problem abstract a class of stable steady convection flows in an exterior domain is given it is proved that the class is l 3 w l m w with m m m 1 when the gravitational field is in l m w for some 1 m

convergence of euler stokes splitting of the navier stokes equations we consider approximation by partial time steps of a smooth solution of the navier stokes equations in a smooth domain in two or three space dimensions with no slip boundary condition for small k 0 we alternate the solution for time k of the inviscid euler equations with tangential boundary condition and the solution of the linear stokes equations for time k with the no slip condition imposed we show that this approximation remains bounded in h2 p and is accurate to order k in lp for p the principal difficulty is that the initial state for each stokes step has tangential velocity at the boundary generated during the euler step and thus does not satisfy the boundary condition for the stokes step the validity of such a fractional step method or splitting is an underlying principle for some computational methods 1994 john wiley sons inc

resolvent estimates in l p for the stokes operator in lipschitz domains we establish the l p resolvent estimates for the stokes operator in lipschitz domains in mathbb r d d geqq 3 for frac 1 p frac 1 2 frac 1 2d varepsilon the result implies that the stokes operator in a three dimensional lipschitz domain generates a bounded analytic semigroup in l p for 3 2 e p 3 e this gives an affirmative answer to a conjecture of m taylor progr nonlinear differential equations appl vol 42 pp 320 334

existence and asymptotic behavior for a convection problem abstract we prove global existence and exponential decay of solutions for a system which arise in thermal convection flow for sufficiently small initial data these results improve previous ones in funkcial ekvac 34 1991 449 further we investigate the behavior of solutions for arbitrarily large initial data in particular we show that the length of the interval on which we have existence and exponential decay is inverse proportional to the size of the initial data

new thoughts on old results of r t seeley in this paper we show that the lp realization of a vector valued elliptic boundary value problem j admits a bounded calculus on lp g e 1 p provided the top order coefficients of are holder continuous here g denotes a domain in n with compact c2m boundary and e a banach space of class our proof is based on an abstract perturbation result for operators admitting bounded calculus and kernel estimates for the solution of j

stochastic volterra equations in banach spaces and stochastic partial differential equation abstract in this paper we study the existence uniqueness and large deviation estimate for stochastic volterra integral equations with singular kernels in 2 smooth banach spaces then we apply them to a large class of semilinear stochastic partial differential equations spde and obtain the existence of unique maximal strong solutions in the sense of sde and pde under local lipschitz conditions moreover stochastic navier stokes equations are also investigated

the stokes operator in weighted l q spaces ii weighted resolvent estimates and maximal l p regularity in this paper we establish a general weighted l q theory of the stokes operator mathcal a q omega in the whole space the half space and a bounded domain for general muckenhoupt weights omega in a q we show weighted l q estimates for the stokes resolvent system in bounded domains for general muckenhoupt weights these weighted resolvent estimates imply not only that the stokes operator mathcal a q omega generates a bounded analytic semigroup but even yield the maximal l p regularity of mathcal a q omega in the respective weighted l q spaces for arbitrary muckenhoupt weights omega in a q this conclusion is archived by combining a recent characterisation of maximal l p regularity by mathcal r bounded families due to weis operator valued fourier multiplier theorems and maximal l p regularity preprint 1999 with the fact that for l q spaces mathcal r boundedness is implied by weighted estimates

peristaltic transport of a heat conducting viscous fluid as an application of abstract differential equations and semigroup of operators abstract a recent abstract result of rankin trans amer math soc to appear is applied to the peristaltic transport of a heat conducting fluid in a flexible tube the oberbeck boussinesq equations are used as the governing equations a pressure drop over a wavelength of the tube is prescribed to ensure uniqueness and newton s cooling law for the temperature is imposed on the boundary with the notion of a weak solution introduced local existence uniqueness regularity and continuation of solutions are considered in particular w r 2 regularity of the solutions is established

dilworth rate a generalization of witsenhausen s zero error rate for directed graphs we investigate a communication setup where a source output is sent through a free noisy channel first and an additional codeword is sent through a noiseless but expensive channel later with the help of the second message the decoder should be able to decide with zero error whether its decoding of the first message was error free this scenario leads to the definition of a digraph parameter that generalizes witsenhausen s zero error rate for directed graphs we investigate this new parameter for some specific directed graphs and explore its relations to other digraph parameters such as sperner capacity and dichromatic number we also look at the natural variant of the above problem where the decoder should decode the first message with zero error not only decide whether its earlier decoding was correct in this case the witsenhausen rate of an appropriately defined undirected graph turns out to be the relevant parameter

on the uniqueness of time periodic solutions to the navier stokes equations in unbounded domains we present a uniqueness theorem for time periodic solutions to the navier stokes equations in unbounded domains thus far results on the uniqueness of time periodic solutions to the navier stokes equations in unbounded domain roughly speaking have only found that a small time periodic ln solution is unique within the class of solutions which have sufficiently small l ln norm in this paper we show that a small time periodic ln solution is unique within the class of all time periodic ln solutions which contains large solutions we also consider the uniqueness of solutions in weak ln space the proof of the present uniqueness theorem is based on the method of dual equations

the zero undetected error capacity approaches the sperner capacity ahlswede cai and zhang proved that in the noise free limit the zero undetected error capacity is lower bounded by the sperner capacity of the channel graph and they conjectured equality here we derive an upper bound that proves the conjecture

l p theory of the navier stokes flow in the exterior of a moving or rotating obstacle in this paper we describe two recent approaches for the lp theory of the navier stokes flow in the exterior of a moving or rotating obstacle

on the initial boundary value problem for a bingham fluid in a three dimensional domain the initial boundary value problem associated with the motion of a bingham fluid is considered the existence and uniqueness of strong solution is proved under a certain assumption on the data it is also shown that the solution exists globally in time when the data are small and that the solution converges to a periodic solution if the external force is time periodic o introduction the purpose of this paper is to establish the existence of strong solutions to a variational inequality which describes the motion of a bingham fluid in a bounded three dimensional domain a bingham fluid is a rigid viscoplastic fluid which is governed by a special constitutive law such that it moves like a rigid body if a certain function of the stresses does not reach the yield limit and it behaves like a viscous fluid when the yield limit is reached since the motion is governed by two entirely different stress strain relations depending on the state of stresses the conservation of momentum is expressed in terms of a variational inequality so that one can avoid the difficulty of separating the fluid zone and the 1 rlglcl zone the initial boundary value problem we shall study is formulated as 0 l it uj u a u w u b u u w j w j u f w a in 0 t for each test function w such that v w 0 in q and w 0 on aq 0 2 v u 0 in q x 0 t 0 3 u 0 on aq x 0 t 0 4 8 x 0 vo x in q here q is a bounded domain in r3 with smooth boundary aq a x t denotes the velocity of the fluid and f x t stands for external force we assume that the density the yield limit and the viscosity are positive constants in particular the received by the editors august 15 1986 1980 mathematics subject classijication 1985 revision primary 35q20 35b10 35b40 35b65 secondary 76a05 76d99 this research was supported by the air force office of scientific research under grant afosr86 0085 and by nsf grant dms 8521848 r 1987 american mathematical society 0002 9947 87 1 00 25 per page

on time periodic solutions to the boussinesq equations in exterior domains abstract we construct time periodic solutions to the boussinesq equations in a three dimensional exterior domain to this end we estimate the l 1 norm of buoyancy and give some estimates of the heat semigroup in the lorentz space

fredholm realizations of elliptic symbols on manifolds with boundary we show that the existence of a fredholm element of the zero calculus of pseudodifferential operators on a compact manifold with boundary with a given elliptic symbol is determined up to stability by the vanishing of the atiyah bott obstruction it follows that up to small deformations and stability the same symbols have fredholm realizations in the zero calculus in the scattering calculus and in the transmission calculus of boutet de monvel

l q estimates for the stationary oseen equations on exterior domains we study the dirichlet problem for the stationary oseen equations on exterior smooth domains in rn n 2 our main results are the existence and uniqueness of weak and very weak solutions of the oseen equations satisfying appropriate lq estimates the uniqueness of very weak solutions is shown by utilizing recent results for very weak solutions of the stokes equations on bounded domains then for n 1 n 1 q the existence and lq estimates of very weak solutions in lq n is deduced by a duality argument from our existence result and d2 r estimates for strong solutions these results combined with some regularity results for very weak solutions enable us to prove the existence uniqueness and d1 r estimates of weak solutions in d1 r n l n 1 r n 1 r n where n 1 n r n 1

a class of solutions to stationary stokes and navier stokes equations with boundary data in w 1 q q we develop a theory for a general class of very weak solutions to stationary stokes and navier stokes equations in a bounded domain with boundary of class c2 1 corresponding to boundary data in the distribution space w 1 q q 1 q these solutions exist and are unique for small data in the nonlinear case in their class of existence and satisfy a correponding estimate in terms of the data moreover they become regular if the data are regular to our knowledge the only existence result for solutions attaining such boundary data is due to giga 16 proposition 2 2 for the stokes case however the methods and the approach used in the present paper are different than giga s and cover more general issues including the nonlinear navier stokes equations and the precise way in which the boundary data are attained by the solutions we also introduce in the last section a further generalization of the solution class

morera theorems for complex manifolds abstract we prove morera theorems for the radon transform integrating on geodesic spheres on complex analytic manifolds of arbitrary dimension to avoid pathologies we assume that the radius of each sphere of integration is less than the injectivity radius at its center the proofs of the main results are local and they involve the microlocal properties of associated radon transforms and a theorem of hormander kawai and kashiwara on microlocal singularities we consider morera theorems for spheres of fixed radius and spheres of arbitrary radius

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

on stokes operators with variable viscosity in bounded and unbounded domains we consider a generalization of the stokes resolvent equation where the constant viscosity is replaced by a general given positive function such a system arises in many situations as linearized system when the viscosity of an incompressible viscous fluid depends on some other quantities we prove that an associated stokes like operator generates an analytic semi group and admits a bounded h calculus which implies the maximal l q regularity of the corresponding parabolic evolution equation the analysis is done for a large class of unbounded domains with w 2 frac1r r boundary for some r d with r q q in particular the existence of an l q helmholtz projection is assumed

weakly parametric pseudodifferential operators and atiyah patodi singer boundary problems this paper introduces a class of pseudodifferential operators depending on a parameter in a particular way the main application is a complete expansion of the trace of the resolvent of a dirac type operator with nonlocal boundary conditions of the kind introduced by atiyah patodi and singer aps this extends the partial expansion in g2 to a complete one and extends the complete expansion in gs 1 to the case where the dirac operator does not have a product structure near the boundary a secondary application is to obtain a complete expansion of the resolvent of a bdo on a compact manifold essentially reproving a result of agranovich agr the resolvent expansion yields immediately an expansion of the trace of the heat kernel and determines the singularities of the zeta function moreover a pseudodifferential factor can be allowed a major motive for these expansions is to obtain index formulas for elliptic operators there are many such applications in the physics and geometry literature the index formula comes from one particular term in the expansion but each term is a spectral invariant and they have been used for other purposes as well as for the index in particular branson and gilkey have a number of papers e g bg and gi analyzing these invariants and drawing geometric consequences interest in the asymptotic behavior of the resolvent goes back to carleman c more recently agmon agm developed it extensively for analytic applications he introduced the fundamental idea of treating the resolvent parameter essentially as another cotangent variable this idea was developed in s1 to analyze the singularities of the zeta function of an elliptic odo on a compact manifold and in 3 to analyze the resolvent of a differential operator with differential boundary conditions the technique works smoothly for differential operators producing so called local invariants integrals over the underlying

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

a dynamical zeta function for pseudo riemannian foliations we investigate a generalization of geodesic random walks to pseudo riemannian foliations the main application we have in mind is to consider the logarithm of the associated zeta function as grand canonical partition function in a theory unifying aspects of general relativity quantum mechanics and dynamical systems

regularity of solutions to a non homogeneous boundary value problem for general stokes systems in r n we give a simple and very complete proof of the existence of a strong h2 solution to the non homogeneous problem 1 1 under the non homogeneous boundary conditions 1 6 here we consider the half space case rn n 3 see theorem 1 2 this regularity result was previously obtained by solonnikov and scadilov in reference 33 for the classical stokes system 0 g x 0 in the 3 d homogenous case a 0 b 0 and a suitable open subset of r3

existence of strong solutions and decay of turbulent solutions of navier stokes flow with nonzero dirichlet boundary data abstract recently leray s problem of the l 2 decay of a special weak solution to the navier stokes equations with nonhomogeneous boundary values was studied by the authors exploiting properties of the approximate solutions converging to this solution in this paper this result is generalized to the case of an arbitrary weak solution satisfying the strong energy inequality

global well posedness of the two dimensional exterior navier stokes equations for non decaying data we prove global well posedness of the two dimensional exterior navier stokes equations for bounded initial data with a finite dirichlet integral subject to the non slip boundary condition as an application we construct global solutions for asymptotically constant initial data and arbitrary large reynolds numbers

ellipticity in pseudodifferential algebras of toeplitz type abstract let l be a filtered algebra of abstract pseudodifferential operators equipped with a notion of ellipticity and t be a subalgebra of operators of the form p 1 a p 0 where p 0 p 1 l are projections i e p j 2 p j the elements of l act as linear continuous operators in a scale of abstract sobolev spaces those of t in the corresponding subspaces determined by the projections we study how the ellipticity in l descends to t focusing on parametrix construction equivalence with the fredholm property characterisation in terms of invertibility of principal symbols and spectral invariance applications concern sg pseudodifferential operators operators on manifolds with conical singularities and boutet de monvel s algebra for boundary value problems in particular we derive invertibility of the stokes operator with dirichlet boundary conditions in a subalgebra of boutet de monvel s algebra we indicate how the concept generalizes to parameter dependent operators

graph capacities and zero error transmission over compound channels this correspondence investigates the behavior of the compound channel under a zero error constraint we derive expressions for the capacity when a neither the encoder nor decoder has side information about the channel b when only the encoder has such side information these expressions are given in terms of capacities of appropriately defined sets of graphs we clarify that an earlier treatment of the zero error capacity of a compound channel corresponds to the case where the decoder has side information about the channel we also characterize the minimum asymptotic rate for the source coding dual of the compound channel problem namely source coding with compound side information finally we contrast the zero error and asymptotically vanishing error capacities of the compound channel

liouville theorems for the stokes equations with applications to large time estimates abstract we study liouville theorems for the non stationary stokes equations in exterior domains in r n under decay conditions for spatial variables as applications we prove that the stokes semigroup is a bounded analytic semigroup on l of angle 2 for n 3 we also prove large time estimates for n 2 with zero net force