analytic continuation and boundary continuity of functions of several complex variables this note treats some questions about analytic continuation in several variables the first theorem in effect determines the envelops of holomorphy of certain domains in n the second main result is a continuity theorem if a bounded holomorphic function f on a convex domain in n has boundary values that are continuous on the complement in b of a set of the form int b where is a real hyperplane in n that misses then f is continuous on in addition we obtain what may be regarded as a local version of the theorem in our earlier paper concerning the one dimensional extension property our methods depend on hartogs theorem n 3 and directly on the bochnermartinelli formula n 2

smooth solutions of the euler and navier stokes equations from the a posteriori analysis of approximate solutions abstract the main result of morosi and pizzocchero 2012 is presented in a variant based on a c formulation of the cauchy problem in this approach the a posteriori analysis of an approximate solution gives a bound on the sobolev distance of any order between the exact and the approximate solution

power series for solutions of the 3 mathcal d navier stokes system on r 3 in this paper we study the fourier transform of the 3 mathcal d navier stokes system without external forcing on the whole space r 3 the properties of solutions depend very much on the space in which the system is considered in this paper we deal with the space phi alpha alpha of functions v k frac c k k alpha where alpha 2 epsilon epsilon 0 and c k is bounded sup k in r 3 smallsetminus 0 c k infty we construct the power series which converges for small t and gives solutions of the system for bounded intervals of time these solutions can be estimated at infinity in k space by exp rm const sqrt t k

on a magnetohydrodynamic problem of euler type abstract the magnetohydrodynamic or mhd equations of an incompressible and homogeneous plasma are considered in the limit of vanishing viscosity and resistivity the plasma spreading over a smoothly bounded domain g of r n n 2 the problem is posed and solved as an equation of evolution in a space of divergence free vector fields parallel to the boundary of g main results are existence uniqueness and regularity of maximal solutions and continuous dependence on forcing terms and initial data the proof of existence is based on galerkin s method and a priori estimates the continuity result is obtained by use of a regularization procedure the approach is inspired by the work of r temam and t kato and c y lai on the euler equations of ideal fluid flow

on some class of sets with multi sheeted envelope of holomorphy let m be a smooth cr manifold of dimension 2n 2 contained in the unit sphere open image in new window we prove that if m is the finite union of boundaries of n 1 dimensional analytic varieties and m is globally minimal then the envelope of holomorphy of open image in new window is multi sheeted in some cases we prove that the number of sheets is exactly two it is also proved for 4 dimensional cr manifolds contained in open image in new window that the property of being globally minimal even being of finite type at each point is generic

on the distributional divergence of vector fields vanishing at infinity the equation div v f has a solution v in the space of continuous vector fields vanishing at infinity if and only if f acts linearly on bv m m 1 rm the space of functions in l m m 1 rm whose distributional gradient is a vector valued measure and satisfies the following continuity condition f uj converges to zero for each sequence uj such that the measure norms of uj are uniformly bounded and uj 0 weakly in l m m 1 rm

conditioning and dichotomy in differential algebraic equations the relationship between the dichotomy of the solution space for a differential algebraic system and the well conditioning of associated boundary value problems is investigated this work extends results that are known for regular explicit ordinary differential equations and it follows the presentation by de hoog and mattheij siam j numer anal 24 1987 pp 89 105

new answers to an old question in the theory of differential algebraic equations in the context of linear differential algebraic equations daes one finds different associated explicit ordinary differential equations odes among them essential underlying and inherent explicit regular ones abbreviated euodes and ierodes euodes have been introduced in 1991 for index 2 daes in hessenberg form by means of special transformations ierodes result within the framework of the projector based decoupling each such explicit ode is occasionally considered to rule the flow of the dae the question to which extend euodes and ierodes are related to each other has been asked promptly after 1991 for index 2 hessenberg form daes answers have been given in 2005 saying that euodes represent somehow condensed ierodes recently euodes have been indicated for general arbitrary index daes and it has been proved that they are condensed ierodes the understanding of the relation between the ierode and the euodes enables to uncover the stability behavior of the dae flow in the present paper we show that both the ierodes and euodes of a dae with arbitrary high index do not at all depend on derivatives of the right hand side we consider adjoint pairs of daes and provide generalizations of the classical lagrange identity furthermore we address lyapunov spectra and lyapunov regularity

adjoint sensitivity analysis for differential algebraic equations the adjoint dae system and its numerical solution an adjoint sensitivity method is presented for parameter dependent differential algebraic equation systems daes the adjoint system is derived along with conditions for its consistent initialization for daes of index up to two hessenberg for stable linear daes stability of the adjoint system for semi explicit daes or of an augmented adjoint system for fully implicit daes is shown in addition it is shown for these systems that numerical stability is maintained for the adjoint system or for the augmented adjoint system

some global results on extension of cr objects in complex manifolds we prove some results concerning the holomorphic extendability of cr objects defined on real hypersurfaces of a complex manifold after a preliminary generalization of the classic theorem on the extendability from the boundary of a relatively compact domain we discuss the extendability from a part of the boundary of such a domain the one side extendability from a piece of hypersurface and the extendability from the boundary of an unbounded domain let m be a noncompact connected complex analytic manifold with countable topology of complex dimension n 2 the present paper deals with holomorphic extension of cr objects defined on real hypersurfaces of m what in fact we do for the most part is to discuss some generalizations in different directions of the well known global extension theorem which asserts the existence and uniqueness of a holomorphic extension on an open domain d c m for a cr function defined on the boundary bd of d under the condition h1 m 0 cf 9 the cr objects considered here are cr sections cr distribution sections and cr hyperfunction sections of a holomorphic complex vector bundle on m our first result theorem 1 1 relates to the holomorphic extension of these kinds of cr objects from the boundary bd of a domain d c m which need not be relatively compact but the closure of which is supposed to belong to some paracompactifying family in m not the whole family of closed sets such a generalization of the global extension theorem does not seem to be already found in the literature actually although it does not require any essentially new ideas is worth being discussed as it allows one to derive other results of some independent interest on extension of cr objects later in 2 we discuss the question of holomorphic extendability on all of a domain d c m for cr objects which need not be defined on the whole boundary bd but just on the complement bd k of a closed set k c bd this question has already been investigated in more particular settings cf 11 13 14 16 17 22 23 by using integral representations techniques here we received by the editors september 10 1987 and in revised form march 3 1988 and november 15 1988 1980 mathematics subject classification 1985 revision primary 32d99 1990 american mathematical society 0002 9947 90 1 00 25 per page

lower bounds on blow up solutions of the three dimensional navier stokes equations in homogeneous sobolev spaces suppose that u t is a solution of the three dimensional navier stokes equations either on the whole space or with periodic boundary conditions that has a singularity at time t in this paper we show that the norm of u t t in the homogeneous sobolev space hs must be bounded below by cst 2s 1 4 for 1 2 5 2 the result for 1 2 s 3 2 follows from well known lower bounds on blowup in lp spaces we show in particular that the local existence time in hs r3 depends only on the hs norm for 1 2 s 5 2 s 3 2

transfer of boundary conditions for daes of index 1 in this paper the concept of abramov s method for transferring boundary conditions posed for regular ordinary differential equations odes is applied to index 1 differential algebraic equations daes having discussed the reduction of inhomogeneous problems to homogeneous ones and analyzed the underlying ideas of abramov s method we consider boundary value problems for index 1 linear daes both with constant and varying leading matrices we describe the relations defining the subspaces of solutions satisfying the prescribed boundary conditions at one end of the interval the index 1 daes which realize the transfer are given and their properties are studied the results are reformulated for inhomogeneous index 1 daes as well

an a posteriori condition on the numerical approximations of the navier stokes equations for the existence of a strong solution in their 2006 paper chernyshenko et al j math phys 48 2007 065204 15 pp prove that a sufficiently smooth strong solution of the 3d navier stokes equations is robust with respect to small enough changes in initial conditions and forcing function they also show that if a regular enough strong solution exists then galerkin approximations converge to it they then use these results to conclude that the existence of a sufficiently regular strong solution can be verified using sufficiently refined numerical computations in this paper we study the strong solutions with less regularity than those considered in chernyshenko et al j math phys 48 2007 065204 15 pp we prove a similar robustness result and show the validity of the results relating convergent numerical computations and the existence of the strong solutions

on the stability of the abramov transfer for differential algebraic equations of index 1 the transfer of boundary conditions for ordinary differential equations developed by abramov zh vychisl mat mat fiz 1 1961 pp 542 545 is a stable method for representing the solution spaces of linear boundary value problems instead of boundary value problems matrix valued initial value problems are solved when integrating these differential equations the inner independence of the columns of the solution matrix and hence of the solutions of the resulting linear system of equations remains valid balla and marz have generalized abramov s transfer for homogenized index 1 differential algebraic equations siam j numer anal 33 1996 pp 2318 2332 in this article a direct version of the abramov transfer for inhomogeneous linear mbox index 1 differen linebreak 4 tial algebraic equations is developed and the numerical stability of this method is proved

the gauss green theorem and removable sets for pdes in divergence form abstract applying a very general gauss green theorem established for the generalized riemann integral we obtain simple proofs of new results about removable sets of singularities for the laplace and minimal surface equations we treat simultaneously singularities with respect to differentiability and continuity

on certain integrals associated to cr functions we construct explicit 0 primitives of cauchy fantappie type kernels on analytic subvarieties of domains in cn outside the zero set of a holomorphic map defined on the variety then we use these primitives to derive by means of a process of passing to a residue integral formulas for cr functions defined on subsets of the boundary of a variety certain geometric restrictions on the varieties are imposed these primitives apply in the particular case of the bochner martinelli kernel in domains in cn and we use these primitives in this case to prove a criterion for holomorphic extendability of cr functions defined on certain subsets of the boundary of a domain in cn introduction let d g c cn be bounded domains with smooth boundary and d c g let h g c be a holomorphic map and set z h c e g h o in 2 we constructed kernels ah 4 z and h so that if f is a holomorphic function on z h and z e z h n d then 1 f z f 4c a 4 z a fh c jez h n od the differential forms a h and a h were explicitly defined in terms of h a hefer decomposition hi of h i e holomorphic functions hij c z in gx g with n hi 4 hi z e hij c z j zj i 1 p c z e g x g j 1 and in terms of a function y od x d cn so that n e yj c z cj zj 1 for c z e od x d j l certain geometric restrictions transversality conditions were imposed which guarantee that z h n 3d is a smooth manifold in this paper we pursue further these constructions by proving an integral formula satisfied by cr functions which are defined on certain parts of the boundary of a variety first we construct another kernel i h z which is a received by the editors february 15 1988 1980 mathematics subject classification 1985 revision primary 32a25 32a27 32d1 5 1989 american mathematical society 0002 9947 89 1 00 25 per page

the divergence theorem for unbounded vector fields in the context of lebesgue integration we derive the divergence theorem for unbounded vector elds that can have singularities at every point of a compact set whose minkowski content of codimension greater than two is nite the resulting integration by parts theorem is applied to removable sets of holomorphic and harmonic functions

gauss green theorem for weakly differentiable vector fields sets of finite perimeter and balance laws we analyze a class of weakly differentiable vector fields f w r n r n with the property that f 2 l 1 and div f is a signed radon measure these fields are called bounded divergence measure fields the primary focus of our inves tigation is to introduce a suitable notion of the normal trace of any divergence measure field f over the boundary of an arbitrary set of finite perimeter that en sures the validity of the gauss green theorem to achieve this we first establish a fundamental approximation theorem which states that given a signed radon measurethat is absolutely continuous with respect to h n 1 on r n a ny set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure theoretic interior of the set with respect to the mea sure kk the total variation measure we employ this approximation theorem to derive the normal trace of f on the boundary of any set of finite perimeter e as the limit of the normal traces of f on the boundaries of the approximate sets with smooth boundary so that the gauss green theorem for f holds one with these results we analyze the cauchy flux that is bounded by a nonnegative radon measure over any oriented surface i e an n 1 dimensional surface that is a part of the boundary of a set of finite perimeter and thereby develop a general mathematical formulation of the physical principle of the balance law through the cauchy flux finally we apply this framework to the derivation of systems of balance laws with measure valued source terms from the formulation of the balance law this framework also allows the recovery of cauchy entropy flux through the lax entropy inequality for entropy solutions of hyperbolic con servation laws c 2008 wiley periodicals inc

on the reynolds number expansion for the navier stokes equations abstract in a previous paper of ours morosi and pizzocchero 2012 1 we have considered the incompressible navier stokes ns equations on a d dimensional torus t d in the functional setting of the sobolev spaces h 0 n t d of divergence free zero mean vector fields n d 2 1 in the cited work we have presented a general setting for the a posteriori analysis of approximate solutions of the ns cauchy problem given any approximate solution u a this allows to infer a lower bound t c on the time of existence of the exact solution u and to construct a function r n such that u t u a t n r n t for all t 0 t c in certain cases it is t c so global existence is granted for u in the present paper the framework of morosi and pizzocchero 2012 1 is applied using as an approximate solution an expansion u n t j 0 n r j u j t where r is the reynolds number this allows amongst else to derive the global existence of u when r is below some critical value r increasing with n in the examples that we analyze after a general discussion about the reynolds expansion and its a posteriori analysis we consider the expansions of orders n 1 2 5 in dimension d 3 with the initial datum of behr necas and wu 2001 11 computations of order n 5 yield a quantitative improvement of the results previously obtained for this initial datum in morosi and pizzocchero 2012 1 where a galerkin approximate solution was employed in place of the reynolds expansion

projected implicit runge kutta methods for differential algebraic equations in this paper a new class of numerical methods projected implicit runge kutta methods is introduced for the solution of index 2 hessenberg systems of initial and boundary value differential algebraic equations daes these types of systems arise in a variety of applications including the modeling of singular optimal control problems and parameter estimation for differential algebraic equations such as multibody systems the new methods appear to be particularly promising for the solution of dae boundary value problems where the need to maintain stability in the differential part of the system often necessitates the use of methods based on symmetric discretizations previously defined symmetric methods have severe limitations when applied to these problems including instability oscillation and loss of accuracy the new methods overcome these difficulties for linear problems we define an essential underlying boundary value ode and prove well conditioning of the differential or state space solutio

linear subspaces for linear daes of index 1 abstract in this paper the representation of linear subspaces of solutions of index 1 daes a x y b x y 0 is given it is shown that the solutions of adjoint systems are strongly related to the subspaces

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

the multidimensional luzin theorem each measurable map of an open set u subset of r n to r n is equal almost everywhere to the gradient of a continuous almost everywhere differentiable function defined on r n that vanishes together with its gradient outside u c 2007 elsevier inc all rights reserved

limiting case of the sobolev inequality in bmo with application to the euler equations we shall prove a logarithmic sobolev inequality by means of the bmo norm in the critical exponents as an application we shall establish a blow up criterion of solutions to the euler equations

bilinear estimates in bmo and the navier stokes equations we prove that the bmo norm of the velocity and the vorticity controls the blow up phenomena of smooth solutions to the navier stokes equations our result is applied to the criterion on uniqueness and regularity of weak solutions in the marginal class

a posteriori regularity of the three dimensional navier stokes equations from numerical computations in this paper we consider the role that numerical computations in particular galerkin approximations can play in problems modeled by the three dimensional 3d navier stokes equations for which no rigorous proof of the existence of unique solutions is currently available we prove a robustness theorem for strong solutions from which we derive an a posteriori check that can be applied to a numerical solution to guarantee the existence of a strong solution of the corresponding exact problem we then consider galerkin approximations and show that if a strong solution exists the galerkin approximations will converge to it thus if one is prepared to assume that the navier stokes equations are regular one can justify this particular numerical method rigorously combining these two results we show that if a strong solution of the exact problem exists then this can be verified numerically using an algorithm that can be guaranteed to terminate in a finite time we thus introduce the possibility of rigorous com

rigorous numerical verification of uniqueness and smoothness in a surface growth model abstract based on numerical data and a posteriori analysis we verify rigorously the uniqueness and smoothness of global solutions to a scalar surface growth model with striking similarities to the 3d navier stokes equations for certain initial data for which analytical approaches fail the key point is the derivation of a scalar ode controlling the norm of the solution whose coefficients depend on the numerical data instead of solving this ode explicitly we explore three different numerical methods that provide rigorous upper bounds for its solution

new results on the constants in some inequalities for the navierstokes quadratic nonlinearity we give fully explicit upper and lower bounds for the constants in two known inequalities related to the quadratic nonlinearity of the incompressible euler or navierstokes equations on the torus td these inequalities are tame generalizations in the sense of nashmoser of the ones analyzed in the previous works morosi and pizzocchero 2013 6

remarks on singularities dimension and energy dissipation for ideal hydrodynamics and mhd for weak solutions of the incompressible euler equations there is energy conservation if the velocity is in the besov space b 3 s with s greater than 1 3 b 3 s consists of functions that are lip s i e holder continuous with exponent s measured in the l p norm here this result is applied to a velocity field that is lip 0 except on a set of co dimension on which it is lip agr 1 with uniformity that will be made precise below we show that the frisch parisi multifractal formalism is valid at least in one direction for such a function and that there is energy conservation if analogous conservation results are derived for the equations of incompressible ideal mhd i e zero viscosity and resistivity for both energy and helicity in addition a necessary condition is derived for singularity development in ideal mhd generalizing the beale kato majda condition for ideal hydrodynamics

topological properties of q convex sets we discuss the topological properties of a certain class of compact sets in a q complete complex manifold m these sets which we call q convex in m include for q 0 the o m convex compact sets in a stein manifold then we show applications of the topological results to the subjects of removable singularities for b

local existence for the non resistive mhd equations in nearly optimal sobolev spaces this paper establishes the local in time existence and uniqueness of solutions to the viscous non resistive magnetohydrodynamics mhd equations in rd where d 2 3 with initial data b0 hs rd and u0 hs 1 e rd for s d 2 and any 0 e 1 the proof relies on maximal regularity estimates for the stokes equation the obstruction to taking e 0 is explained by the failure of solutions of the heat equation with initial data u0 hs 1 to satisfy u l1 0 t hs 1 we provide an explicit example of this phenomenon

on approximate solutions of the equations of incompressible magnetohydrodynamics abstract inspired by an approach proposed previously for the incompressible navier stokes ns equations we present a general framework for the a posteriori analysis of the equations of incompressible magnetohydrodynamics mhd on a torus of arbitrary dimension d this setting involves a sobolev space of infinite order made of c vector fields with vanishing divergence and mean on the torus given any approximate solution of the mhd cauchy problem its a posteriori analysis with the method of the present work allows one to infer a lower bound on the time of existence of the exact solution and to bound from above the sobolev distance of any order between the exact and the approximate solution in certain cases the above mentioned lower bound on the time of existence is found to be infinite so one infers the global existence of the exact mhd solution we present some applications of this general scheme the most sophisticated one lives in dimension d 3 with the abc flow perturbed magnetically as an initial datum and uses for the cauchy problem a galerkin approximate solution in 124 fourier modes we illustrate the conclusions arising in this case from the a posteriori analysis of the galerkin approximant these include the derivation of global existence of the exact mhd solution with the abc datum when the dimensionless viscosity and resistivity are equal and stay above a critical value

an h1 setting for the navier stokes equations quantitative estimates abstract we consider the incompressible navier stokes ns equations on a torus in the setting of the spaces l 2 and h 1 our approach is based on a general framework for semi linear or quasi linear parabolic equations proposed in the previous work morosi and pizzocchero 2008 5 we present some estimates on the linear semigroup generated by the laplacian and on the quadratic ns nonlinearity these are fully quantitative i e all the constants appearing therein are given explicitly as an application we show that on a three dimensional torus t 3 the mild solution of the ns cauchy problem is global for each h 1 initial datum u 0 with zero mean such that curl u 0 l 2 0 407 this improves the bound for global existence curl u 0 l 2 0 00724 derived recently by robinson and sadowski 2008 3 we announce some future applications based again on the h 1 framework and on the general scheme of 5

topologies for the space of bv integrable functions in r n we introduce two topologies on the space of bv integrable functions in r n among the studied properties are barrelledness and integral representation of linear continuous functionals in one case we charctcterize the dual space we show the relations with the study of multipliers finally we also introduce and study a topology on the space of bounded bv functions in r n c 1997 academic press

higher order commutator estimates and local existence for the non resistive mhd equations and related models this paper establishes the local in time existence and uniqueness of strong solutions in hs for s n 2 to the viscous non resistive magnetohydrodynamics mhd equations in rn n 2 3 as well as for a related model where the advection terms are removed from the velocity equation the uniform bounds required for proving existence are established by means of a new estimate which is a partial generalisation of the commutator estimate of kato ponce comm pure appl math 41 7 891 907 1988

on dichotomy and well conditioning in bvp we investigate the relationships between the stability bounds of the problem on the one hand and the growth behaviour of the fundamental solution on the other hand it is shown that if these stability bounds are moderate i e if the problem is well conditioned then the homogeneous solution space is dichotomic which means that it can be split into a subspace of nondecreasing and a complementary subspace of nonincreasing modes this is done by carefully examining the green s functions if these exhibit an exponential behaviour then the solution space is also exponentially dichotomic on the other hand we also show that exponential dichotomy implies moderate stability constants i e well conditioning from this it follows that both concepts are more or less equivalent

linear differential algebraic equations in spaces of integrable functions on developpe une l 2 theorie pour des equations differentielles algebriques lineaires t x t a t x t b t x t f t t a b x t r m

the gauss green theorem abstract in the m dimensional euclidean space we establish the gauss green theorem for any bounded set of bounded variation and any bounded vector field continuous outside a set of m 1 dimensional hausdorff measure zero and almost differentiable outside a set of finite m 1 dimensional hausdorff measure

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

distributions for which div v f has a continuous solution the equation div upsilon f has a continuous weak solution in an open set u subset of r m if and only if the distribution f satisfies the following condition the f phi i converge to 0 for every sequence phi i of test functions such that the support of each i is contained in a fixed compact subset of u and in the l 1 norm phi i converges to 0 and del phi i is bounded c 2007 wiley periodicals inc

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references

on numerical differential algebraic problems with application to semiconductor device simulation this paper considers questions regarding conditioning of and numerical methods for certain differential algebraic equations subject to initial and boundary conditions the approach taken is that of separating differential and algebraic solution components at least theoretically this yields conditioning results for differential algebraic boundary value problems in terms of pure differential problems for which existing theory is well developed the process is carried out for problems with global index 1 or 2 for semi explicit boundary value problems of index 1 where solution components are separated a convergence theorem is given for a special class of collocation methods for general index 1 problems advantages and disadvantages of certain symmetric difference schemes are discussed for initial value problems with index 2 the use of bdf schemes is examined with a summary of conditions for their successful and stable utilization finally the present considerations and analysis are applied to

boundary conditions and their transfer for differential algebraic equations of index 1 abstract for the homogeneous linear index 1 tractable dae we give the characterization of the linear subspace defined by a homogeneous linear relationship prescribed at an arbitrarily fixed point

on approximate solutions of the incompressible euler and navier stokes equations abstract we consider the incompressible euler or navier stokes ns equations on a torus t d in the functional setting of the sobolev spaces h 0 n t d of divergence free zero mean vector fields on t d for n d 2 1 we present a general theory of approximate solutions for the euler ns cauchy problem this allows to infer a lower bound t c on the time of existence of the exact solution u analyzing a posteriori any approximate solution u a and also to construct a function n such that u t u a t n n t for all t 0 t c both t c and n are determined solving suitable control inequalities depending on the error of u a the fully quantitative implementation of this scheme depends on some previous estimates of ours on the euler ns quadratic nonlinearity morosi and pizzocchero 2010 in press 7 8 to keep in touch with the existing literature on the subject our results are compared with a setting for approximate euler ns solutions proposed in chernyshenko et al 2007 1 as a first application of the present framework we consider the galerkin approximate solutions of the euler ns cauchy problem with a specific initial datum considered in behr et al 2001 9 in this case our methods allow amongst else to prove global existence for the ns cauchy problem when the viscosity is above an explicitly given bound