the cauchy problem for indefinite improper affine spheres and their hessian equation abstract we give a conformal representation for indefinite improper affine spheres which solve the cauchy problem for their hessian equation as consequences we can characterize their geodesics and obtain a generalized symmetry principle then we classify the helicoidal indefinite improper affine spheres and find a new family with geodesically complete non flat affine metric moreover we present interesting examples with singular curves and isolated singularities

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

the bernstein problem for affine maximal hypersurfaces in this paper we prove the validity of the chern conjecture in affine geometry 18 namely that an affine maximal graph of a smooth locally uniformly convex function on two dimensional euclidean space r 2 must be a paraboloid more generally we shall consider the n dimensional case r n showing that the corresponding result holds in higher dimensions provided that a uniform strict convexity condition holds we also extend the notion of affine maximal to non smooth convex graphs and produce a counterexample showing that the bernstein result does not hold in this generality for dimension n 10

collapse of a class of three dimensional euler vortices substitution of the flow field u x y z t u x y t v x y t z x y t into the three dimensional incompressible euler equations generates a closed system of evolution equations for the strain rate x y t and the two dimensional vorticity x y t which are uniform in the z direction the system models a class of dynamical stretched three dimensional vortex flows that include burgers9 vortices recent numerical simulations by ohkitani gibbon have revealed that the strain rate x y t appears to develop a finite time singularity from smooth initial data in the region where gamma is negative here we prove that for a large class of initial data the support of max 0 necessarily collapses to zero in a finite time while at the same time the l 1 norm of remains non zero hence must necessarily become singular before or at the time of collapse our vortex flow represents one of a subclass of euler solutions that have infinite energy the fundamental question of finite time singularity formation from smooth initial data for finite energy three dimensional euler solutions remains the important open question

a class of polynomial planar vector fields with polynomial first integral abstract we give an algorithm for deciding whether a planar polynomial differential system has a first integral which factorizes as a product of defining polynomials of curves with only one place at infinity in the affirmative case our algorithm computes a minimal first integral in addition we solve the poincare problem for the class of systems which admit a polynomial first integral as above in the sense that the degree of the minimal first integral can be computed from the reduction of singularities of the corresponding vector field

affine complete locally convex hypersurfaces an open problem in affine geometry is whether an affine complete locally uniformly convex hypersurface in euclidean n 1 space is euclidean complete for n 2 in this paper we give the affirmative answer as an application it follows that an affine complete affine maximal surface in r 3 must be an elliptic paraboloid

self consistent hamiltonian dynamics of wave mean flow interaction for a rotating stratified incompressible fluid abstract we derive a hierarchy of approximate models of wave mean flow interaction wmfi by using asymptotic expansions and phase averages either in euler s equations for a stratified rotating inviscid incompressible fluid or in hamilton s principle for these equations two small dimensionless parameters are used in these expansions the first parameter is the ratio of time scales between internal waves at most wave numbers and the mesoscale mean flow of the fluid this adiabatic ratio is small and is comparable to the corresponding ratio of space scales for the class of initial conditions that support internal waves the other small parameter used in these expansions is the ratio of the amplitude of the internal wave to its wavelength an application of noether s theorem to the phase averaged hamilton s principle shows that the resulting equations conserve the wave action convect a potentional vorticity and can depending on the order of approximation convect wave angular momentum passage from the phase averaged hamilton s principle to the hamiltonian formulation brings the wmfi theory into the lie poisson framework in which formal and nonlinear stability analysis may be applied this framework also suggests a two fluid model of the interaction of waves and mean flow similar to that for the superfluid and normal fluid components of liquid he 4 without vortices we also discuss the relations of these results to the charney drazin nonacceleration theorem whitham averaging wkb stability theory and lagrangian mean fluid equations for prescribed wave displacements

integrability degenerate centers and limit cycles for a class of polynomial differential systems we consider the class of polynomial differential equations email protected p n x y p n 1 x y p n 2 x y email protected q n x y q n 1 x y q n 2 x y for n 1 and where p i and q i are homogeneous polynomials of degree i these systems have a linearly zero singular point at the origin if n 2 inside this class we identify a new subclass of darboux integrable systems and some of them having a degenerate center i e a center with linear part identically zero moreover under additional conditions such darboux integrable systems can have at most one limit cycle we provide the explicit expression of this limit cycle

some new results on darboux integrable differential systems abstract we deal with complex planar differential systems having a darboux first integral h we present a definition of remarkable values and remarkable curves associated to h and characterize the existence of a polynomial inverse integrating factor for these systems furthermore we study the relation between the characteristic polynomial f and the inverse integrating factors of the system and show the importance of the numerator of the exponential factor of h in the construction of f

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

darboux theory of integrability in cn taking into account the multiplicity abstract darboux in 1878 provided a theory on the existence of first integrals of polynomial systems based on the existence of sufficient invariant algebraic hypersurfaces called now the darboux theory of integrability in 1979 jouanolou successfully improved the darboux theory of integrability characterizing the existence of rational first integrals for this he used sophisticated tools of algebraic geometry the aim of this paper is to improve the classical result of darboux and the new one of jouanolou taking into account the multiplicity of the invariant algebraic hypersurfaces additionally our proof of the improved result of jouanolou is much simpler and elementary than the original one some examples show that the improved darboux theory of integrability with multiplicity is much useful than the classical one

global well posedness for the generalized large scale semigeostrophic equations we prove existence and uniqueness of global classical solutions to the generalized large scale semigeostrophic equations with periodic boundary conditions this family of hamiltonian balance models for rapidly rotating shallow water includes the l1 model derived by r salmon in 1985 and its 2006 generalization by the second author the results are under the physical restriction that the initial potential vorticity is positive as strong as those available for the euler equations of ideal fluid flow in two dimensions moreover we identify a special case in which the velocity field is two derivatives smoother in sobolev space as compared to the general case our results are based on careful estimates which show that although the potential vorticity inversion is nonlinear bounds on the potential vorticity inversion operator remain linear in derivatives of the potential vorticity this permits the adaptation of an argument based on elliptic lp theory proposed by yudovich in 1963 for proving existence and uniqueness of weak solutions for the two dimensional euler equations to our particular nonlinear situation

cyclicity of a simple focus via the vanishing multiplicity of inverse integrating factors abstract first we provide new properties about the vanishing multiplicity of the inverse integrating factor of a planar analytic differential system at a focus after we use this vanishing multiplicity for studying the cyclicity of foci with pure imaginary eigenvalues and with homogeneous nonlinearities of arbitrary degree having either its radial or angular speed independent of the angle variable in polar coordinates after we study the cyclicity of a class of nilpotent foci in their analytic normal form

a topological characterization of the ω limit sets for analytic flows on the plane the sphere and the projective plane in this paper the limit sets for analytic and polynomial differential equations on the plane are characterized up to homeomorphisms the analogous problem is solved in full detail for analytic flows on the sphere and the projective plane we also explain how to carry on the same program for analytic flows defined on open subsets of these surfaces

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

configurations of limit cycles in lienard equations abstract we show that every finite configuration of disjoint simple closed curves in the plane is topologically realizable as the set of limit cycles of a polynomial lienard equation the related vector field x is morse smale moreover it has the minimum number of singularities required for realizing the configuration in a lienard equation we provide an explicit upper bound on the degree of x which is lower than the results obtained before obtained in the context of general polynomial vector fields

an anelastic approximation arising in astrophysics we identify the asymptotic limit of the compressible non isentropic navier stokes system in the regime of low mach low froude and high reynolds number the system is driven by a long range gravitational potential we show convergence to an anelastic system for ill prepared initial data the proof is based on frequency localized strichartz estimates for the acoustic equation based on the recent work of metcalfe and tataru

the indefinite affine cauchy problem abstract we solve the geometric cauchy problem for the class of affine maximal surfaces with indefinite affine metric in the affine space r 3 that is we find all the surfaces of this class which contain a given regular curve of r 3 with prescribed affine normal and affine conormal along it we prove the problem is well posed when the initial data are non characteristic and show that uniqueness of the solution can fail at characteristic asymptotic directions as application we obtain some results about geodesics and symmetries of indefinite affine maximal surfaces

algebraic limit cycles in polynomial systems of differential equations using elementary tools we construct cubic polynomial systems of differential equations with algebraic limit cycles of degrees 4 5 and 6 we also construct a cubic polynomial system of differential equations having an algebraic homoclinic loop of degree 3 moreover we show that there are polynomial systems of differential equations of arbitrary degree that have algebraic limit cycles of degree 3 as well as give an example of a cubic polynomial system of differential equations with two algebraic limit cycles of degree 4

the cauchy problem for improper affine spheres and the hessian one equation we give a conformal representation for improper affine spheres which is used to solve the cauchy problem for the hessian one equation with this representation we characterize the geodesics of an improper affine sphere study its symmetries and classify the helicoidal ones finally we obtain the complete classification of the isolated singularities of the hessian one monge ampere equation

note on a paper of j llibre and g rodriguez concerning algebraic limit cycles abstract in a recent paper of llibre and rodriguez j differential equations 198 2004 374 380 it is proved that every configuration of cycles in the plane is realizable up to homeomorphism by a polynomial vector field of degree at most 2 n r 1 where n is the number of cycles and r the number of primary cycles a cycle c is primary if there are no other cycles contained in the bounded region limited by c in this letter we prove the same theorem by using an easier construction but with a greater polynomial bound the vector field we construct has degree at most 4 n 1 by using the same technique we also construct r 3 polynomial vector fields realizing up to homeomorphism any configuration of limit cycles which can be linked and knotted in r 3 this answers a question of r sverdlove

the poincare problem algebraic integrability and dicritical divisors abstract we solve the poincare problem for plane foliations with only one dicritical divisor moreover in this case we give a simple algorithm that decides whether a foliation has a rational first integral and computes it in the affirmative case we also provide an algorithm to compute a rational first integral of prefixed genus g 1 of any type of plane foliation f when the number of dicritical divisors dic f is larger than 2 this algorithm depends on suitable families of invariant curves when dic f 2 it proves that the degree of the rational first integral can be bounded only in terms of g the degree of f and the local analytic type of the dicritical singularities of f

the affine cauchy problem abstract the aim of this paper is to solve the cauchy problem for locally strongly convex surfaces which are extremal for the equiaffine area functional these surfaces are called affine maximal surfaces and here we give a new complex representation which let us describe the solution to the corresponding cauchy problem as applications we obtain a generalized symmetry principle characterize when a curve in r 3 can be a geodesic or pre geodesic of a such surface and study the helicoidal affine maximal surfaces finally we investigate the existence and uniqueness of affine maximal surfaces with a given analytic curve in its singular set

using melnikov functions of any order for studying limit cycles abstract in this work we study a class of planar hamiltonian systems perturbed with general quadratic polynomials in order to identify the number of limit cycles employing melnikov functions of any order we show that the class of systems can have at most three limit cycles and the limit is reached

a class of vectorfields on s2 that are topologically equivalent to polynomial vectorfields abstract let x be a c1 vectorfield on s2 x y z r x2 y2 z2 1 such that no open subset of s2 is the union of closed orbits of x if x has only a finite number of singular orbits and satisfies one additional condition then it is shown that x is topologically equivalent to a polynomial vectorfield a corollary is that a foliation f of the plane is topologically equivalent to a foliation by orbits of a polynomial vectorfield if and only if f has only a finite number of inseparable leaves

on the possibility of soft and hard turbulence in the complex ginzburg landau equation abstract we present analytical methods which predict the occurrence of both soft weak and hard strong turbulence in the complex ginzburg landau cgl equation a t ra 1 i a 1 i a a 2 on a periodic domain 0 1 d in d spatial dimensions hard turbulence is characterised by large fluctuations away from spatial and temporal averages with a cascade of energy to small scales this form of hard turbulence appears to occur not in 1d but only in 2d and 3d in parameter regions which are bounded by hyperbolic curves in the second and fourth quadrants of the planes where the system is modulationally unstable 1 2n f n n 1 a 2 n a 2n dx for n 0 for large times and large r upper bounds exist for the infinite set of fn s constructed from the hierarchy of differential inequalities fn 2nr cn a 2 fn bnf2n fn 1 for cn bn 0 f0 1 estimates for the bottom rung f2 give upper bounds for the whole ladder long time upper bounds on f2 and a 2 and hence all fn are well controlled in the soft region but become much larger in the hard region whereas spati al and temporal averages remain comparatively small when the nonlinearity is a a 2q the critical case qd 2 gives parallel results

sufficient conditions for the existence of at least n or exactly n limit cycles for the lienard differential systems in this paper we study the limit cycles of the lienard differential system of the form x f x x g x 0 or its equivalent system x y f x y g x we provide sufficient conditions in order that the system exhibits at least n or exactly n limit cycles

the cauchy problem for the liouville equation and bryant surfaces abstract we give a construction that connects the cauchy problem for the 2 dimensional elliptic liouville equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3 space h 3 and solve both of them we construct the unique mean curvature one surface in h 3 that passes through a given curve with a given unit normal along it and provide diverse applications in particular topics such as period problems symmetries finite total curvature planar geodesics rigidity etc are treated for these surfaces

the geometric cauchy problem for the hyperbolic hessian one equation abstract we solve the problem of finding all indefinite improper affine spheres passing through a given regular curve of r 3 with a prescribed affine co normal vector field along this curve we prove the problem is well posed when the initial data are non characteristic and show that uniqueness of the solution can fail at characteristic directions as application we classify the indefinite improper affine spheres admitting a geodesic planar curve

darboux theory of integrability in the sparse case darboux s theorem and jouanolou s theorem deal with the existence of first integrals and rational first integrals of a polynomial vector field these results are given in terms of the degree of the polynomial vector field here we show that we can get the same kind of results if we consider the size of a newton polytope associated to the vector field furthermore we show that in this context the bound is optimal

morphisms and inverse problems for darboux integrating factors polynomial vector fields which admit a prescribed darboux integrat ing factor are quite well understood when the geometry of the underlying curve is nondegenerate in the general setting morphisms of the affine plane may remove degeneracies of the curve and thus allow more structural insight in the present paper we establish some properties of integrating factors subjected to morphisms and we discuss in detail one particular class of morphisms related to finite reflection groups the results indicate that degeneracies for the underlying curve generally impose restrictions on the nontrivial vector fields which admit a given integrating factor

a characterization of theω limit sets of planar continuous dynamical systems abstract if the family of curves g i i and the set of points s are given we find necessary and sufficient conditions for the existence of a c k map f from the plane into itself 0 k if k 0 we also assume that f is locally lipschitz such that the curves from g and the points from s are respectively nondegenerate trajectories and singular points of the system x f x and additionally s i i is the limit set of some trajectory of the system in such cases we also provide a detailed geometric description of s i i after contraction of the connected components of s

global well posedness and finite dimensional global attractor for a 3 d planetary geostrophic viscous model in this paper we consider a three dimensional planetary geostrophic viscous model of the gyre scale mid latitude ocean we show the global existence and uniqueness of the weak and strong solutions to this model moreover we establish the existence of a nite dimensional global attractor to this dissipative evolution system c 2003 wiley periodicals inc

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

asymptotic limit for rotational quantum compressible navier stokes equations with multiple scales abstract in this paper we consider the degenerate quantum compressible navier stokes equations giving rise to a variety of mathematical problems in many areas we study the asymptotic limit for the rotational compressible navier stokes equations with quantum term and the ill prepared initial data

singularities of improper affine maps and their hessian equation abstract we study improper affine spheres with some admissible singularities called improper affine maps and associated to the unimodular hessian equation in particular we characterize when a curve of r 3 is the singular curve of some improper affine map with prescribed cuspidal edges and swallowtails also we consider improper affine maps with isolated singularities and show some similarities and differences between the hessian 1 equation and the hessian 1 equation as a consequence we construct global examples with the desired singularities

realization problems for limit cycles of planar polynomial vector fields abstract we show that for any finite configuration of closed curves r 2 one can construct an explicit planar polynomial vector field that realizes up to homeomorphism as the set of its limit cycles with prescribed periods multiplicities and stabilities the only obstruction given on this data is the obvious compatibility relation between the stabilities and the parity of the multiplicities the constructed vector fields are darboux integrable and admit a polynomial inverse integrating factor

bjorling problem for timelike surfaces in the lorentz minkowski space abstract we solve the bjorling problem for timelike surfaces in the lorentz minkowski space through a split complex representation formula obtained for this kind of surface our approach uses the split complex numbers and natural split holomorphic extensions as applications we show that the minimal timelike surfaces of revolution as well as minimal ruled timelike surfaces can be characterized as solutions of certain adequate bjorling problems in the lorentz minkowski space

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

local existence and uniqueness of euler equation in weighted sobolev space abstract in this paper we study the existence and uniqueness of euler equation in the weighted sobolev space this is motivated by the study of inviscid limit of navier stokes equation in half plane the use of the weighted calderon zygmund inequality in the estimate of the pressure term is the improvement of our work

configurations of limit cycles and planar polynomial vector fields abstract we show that every finite configuration of disjoint simple closed curves of the plane is topologically realizable as the set of limit cycles of a polynomial vector field moreover the realization can be made by algebraic limit cycles and we provide an explicit polynomial vector field exhibiting any given finite configuration of limit cycles

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

inverse problems for dynamical systems in this paper we consider the problem of reconstructing a structurally stable vector field from its global phase portrait or from certain local pieces of the phase portrait we show that for arbitrary finite sets of points and of simple closed curves we can construct a structurally stable system having the elements of those sets as its critical points and limit cycles and a prescribed topological type in a neighborhood of each of them given a global structurally stable phase portrait we can construct a structurally stable polynomial system which is topologically equivalent to it some of our results were originally announced in 28 1 the general theory of dynamical systems orginated in the study of solution curves for systems of ordinary differential equations in a phase space this qualitative theory had its first major development in the work of poincare on the topology of integral curves a typical problem is to give a geometric description of the solutions of a particular system or class of systems with emphasis on the location of special features such as critical points and periodic orbits an inverse problem then is the construction of a system of equations given geometric properties of the solutions which may be in the form of a picture of the flow a schematic diagram of phase space or a list of critical orbits defined analytically a method for solving such problems is potentially useful in building models in the various fields in which the theory of dynamical systems is now being applied it could provide a basic model to which statistical identification techniques 141 could then be applied there is also an intrinsic interest in methods for constructing polynomial systems of prescribed type which is related to the lack of any progress toward a solution of hilbert s sixteenth problem the recent appearance of a counterexample 27 to the petrovsky landis conjecture for quadratic systems leaves the problem completely open one of the earliest results of this kind was obtained by forster in 1938

on the 16th hilbert problem for algebraic limit cycles abstract for a polynomial planar vector field of degree n 2 with generic invariant algebraic curves we show that the maximum number of algebraic limit cycles is 1 n 1 n 2 2 when n is even and n 1 n 2 2 when n is odd furthermore these upper bounds are reached

the 16th hilbert problem on algebraic limit cycles abstract for real planar polynomial differential systems there appeared a simple version of the 16th hilbert problem on algebraic limit cycles is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree m in j llibre r ramirez n sadovskaia on the 16th hilbert problem for algebraic limit cycles j differential equations 248 2010 1401 1409 llibre ramirez and sadovskaia solved the problem providing an exact upper bound in the case of invariant algebraic curves generic for the vector fields and they posed the following conjecture is 1 m 1 m 2 2 the maximal number of algebraic limit cycles that a polynomial vector field of degree m can have in this paper we will prove this conjecture for planar polynomial vector fields having only nodal invariant algebraic curves this result includes the llibre et al s as a special one for the polynomial vector fields having only non dicritical invariant algebraic curves we answer the simple version of the 16th hilbert problem

on the remarkable values of the rational first integrals of polynomial vector fields abstract the remarkable values for polynomial vector fields in the plane having a rational first integral were introduced by poincare he was mainly interested in their algebraic aspects here we are interested in their dynamic aspects i e how they contribute to the phase portrait of the system to its separatrices to its singular points etc the relationship between remarkable values and dynamics mainly takes place through the inverse integrating factor

darboux theory of integrability for a class of nonautonomous vector fields the goal of this paper is to extend the classical darboux theory of integrability from autonomous polynomial vector fields to a class of nonautonomous vector fields we also provide sufficient conditions for applying this theory of integrability and we illustrate this theory in several examples

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

length scales in solutions of the complex ginzburg landau equation abstract we generalise and in certain cases improve on previous a priori estimates of sobolev norms of solutions to the generalised complex ginzburg landau equation a set of dynamic length scales based on ratios of these norms is defined we are able to derive lower bounds for time averages and long time limits of these length scales the bounds scale like the inverses of our l bounds

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