localization of analytic regularity criteria on the vorticity and balance between the vorticity magnitude and coherence of the vorticity direction in the 3d nse the first part of the paper provides spatio temporal localization of a family of analytic regularity classes for the 3d nse obtained by beirao da veiga space time integrability of the gradient of the velocity on mathbb r 3 times 0 t which is out of the range of the sobolev embedding theorem reduction to the classical foias ladyzhenskaya prodi serrin space time integrability conditions on the velocity as well as the localization of the beale kato majda regularity criterion time integrability of the l norm of the vorticity the second part introduces a family of local scaling invariant hybrid geometric analytic classes in which coherence of the vorticity direction serves as a weight in the local spatio temporal integrability of the vorticity magnitude

a condition for finite blow up time for a volterra integral equation abstract the paper is devoted to the study of the equation u x x 0 x s 1 g u s ds 0 x 0 where g is continuous and nondecreasing some estimates of u and the interval of the existence of a nontrivial solution u are given

weighted composition operators ofc0 x s abstract in this paper we prove that into isometries and disjointness preserving linear maps from c 0 x into c 0 y are essentially weighted composition operators tf h f for some continuous map and some continuous scalar valued function h

an extension and simpler proof of berselli cordoba s geometric regularity condition for the navier stokes system abstract in this short note we provide an simpler proof of berselli cordoba s geometric regularity condition for the navier stokes system moreover some interesting extensions are obtained

a strong regularity result for parabolic equations we consider a parabolic equation with a drift term u b u ut 0 under the condition divb 0 we prove that solutions possess dramatically better regularity than those provided by standard theory for example we prove continuity of solutions when not even boundedness is expected

on vorticity directions near singularities for the navier stokes flows with infinite energy we give a geometric nonblow up criterion on the direction of the vorticity for the three dimensional navier stokes flow whose initial data is just bounded and may have infinite energy we prove that under a restriction on behavior in time type i condition the solution does not blow up if the vorticity direction is uniformly continuous at the place where the vorticity magnitude is large this improves the regularity condition for the vorticity direction first introduced by p constantin and c fefferman 1993 for finite energy weak solution our method is based on a simple blow up argument which says that the situation looks like two dimensional under continuity of the vorticity direction we also discuss boundary value problems

space time localization of a class of geometric criteria for preventing blow up in the 3d nse a class of local in the space time conditions on the vorticity directions implying local regularity of weak solutions to the 3d navier stokes equations is established in all the preceding results the relevant geometric conditions although being local in nature have been assumed uniformly throughout the spatial regions of high vorticity magnitude and uniformly in time in addition similar results are obtained assuming a less restrictive integral condition on the vorticity directions

existence and uniqueness of nontrivial collocation solutions of implicitly linear homogeneous volterra integral equations we analyse collocation methods for nonlinear homogeneous volterra hammerstein integral equations with non lipschitz nonlinearity we present different kinds of existence and uniqueness of nontrivial collocation solutions and give conditions for such existence and uniqueness in some cases finally we illustrate these methods with an example of a collocation problem and give some examples of collocation problems that do not fit in the cases studied previously

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

hermitian weighted composition operators on h 2 weighted composition operators have been related to products of composition operators and their adjoints and to isometries of hardy spaces in this paper we identify the hermitian weighted composition operators on h 2 and compute their spectral measures some relevant semigroups are studied the resulting ideas can be used to find the polar decomposition the absolute value and the aluthge transform of some composition operators on h 2

complex symmetric weighted composition operators on h2 d abstract in this paper we study complex symmetric weighted composition operators on the hardy space we provide some characterizations of and when a weighted composition operator w is complex symmetric we investigate which combinations of weights and maps of the open unit disk give rise to complex symmetric weighted composition operators with a special conjugation as some applications we obtain several examples for nonnormal complex symmetric operators in addition we give spectral properties of complex symmetric weighted composition operators we examine eigenvalues and eigenvectors of such operators and find some conditions for which a complex symmetric weighted composition operator is hilbert schmidt finally we consider cyclicity hypercyclicity and the single valued extension property for complex symmetric weighted composition operators

aspects of the behaviour of solutions of nonlinear abel equations abstract in this paper we study properties about the behaviour uniqueness locally boundedness attracting behaviour etc of positive solutions for nonlinear abel integral equations and we present examples of nonlinear abel integral equations with two nontrivial solutions these examples are related to the existence of nonlocally bounded solutions

localization and geometric depletion of vortex stretching in the 3d nse vortex stretching and consequently the evolution of the vorticity is localized on an arbitrarily small space time cylinder this yields a complete localization of the geometric condition s for the regularity involving coherence of the vorticity direction in particular it implies the regularity of any geometrically constrained leray solution independently of the type of the spatial domain or the boundary conditions

blow up rate estimates for weak solutions of the navier stokes equations the interior regularity problem for the leray weak solutions u of the navier stokes equations in a domain rn with n 3 is investigated it is shown that u is regular in a neighbourhood of a point x t omega 0 t if there exist constants 0 1 and small 0 such that with q 1 k i x t x r n x x 1 k t 1 k 2 t 1 k 2 if x t is an irregular point of u there exists a sequence of non zero measure sets e ki q 1 ki x t for i 1 2 such that the blow up rate estimate holds

attraction principle for nonlinear integral operators of the volterra type 0 a x y u y dy 1 all function appearing here are nonnegative and defined for 0 y x the eq 1 has the trivial solution u x 0 it can have also other solutions we prove using the method due to okrasinski that under certain conditions upon a x y and x there can be at most one solution which does not vanish identically in a neighborhood of 0 our main result is the attraction property of this nonnegative solution provided that it exists namely we show that the iterations tu of the operator tu x x

on depletion of the vortex stretching term in the 3d navier stokes equations certain new cancellation properties in the vortex stretching term are detected leading to new geometric criteria for preventing finite time blow up in the 3d navier stokes equations

navier stokes equations and area of interfaces we present new a priori estimates for the vorticity of solutions of the three dimensional navier stokes equations these estimates imply that thel 1 norm of the vorticity is a priori bounded in time and that the time average of the 4 3 e power of thel 4 3 e spatial norm of the gradient of the vorticity is a priori bounded using these bounds we construct global leray weak solutions of the navier stokes equations which satisfy these inequalities in particular it follows that vortex sheet vortex line and even more general vortex structures with arbitrarily large vortex strengths are initial data which give rise to global weak solutions of this type of the navier stokes equations next we apply these inequalities in conjunction with geometric measure theoretical arguments to study the two dimensional hausdorff measure of level sets of the vorticity magnitude we obtain a priori bounds on an average such measure when expressed in terms of the reynolds number and the kolmogorov dissipation length these bounds are left langle mu right rangle leqq frac l 3 eta left 1 operatorname re 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 right 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 the right hand side of this inequality has a simple geometrical interpretation it represents the area of a union of non overlapping spheres of radii which fill a fraction of the spatial domain as the reynolds number increases this fraction decreases we study also the area of level sets of scalars and in particular isotherms in rayleigh benard convection we define a quantity r t x 0 describing an average value of the area of a portion of a level set contained in a small ball of radiusr about the pointx 0 we obtain the inequality left langle mu right rangle r t left x 0 right leqq c kappa 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 r 5 mathord left vphantom 5 2 right kern nulldelimiterspace 2 left langle v left x 0 right right rangle 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 wherek is the diffusivity constant and is the velocity the inequality is valid forr small but larger than a small scale k 1 in the case of turbulent velocities this scale is smaller than the smallest significant physical scale suggesting that 2 5 is a lower bound for the fractal dimension of an ensemble average interface in turbulent flow a fact which agrees with the experimental lower bound of 2 35 we define a similar quantity t representing an average value of the area of a portion of a level set contained in the region of spaced x d dist x d whered is the domain of aspect ratio of order one and diameterl where the convection takes place we obtain the inequality left langle mu right rangle delta t leqq c left l 3 delta 1 l 5 mathord left vphantom 5 2 right kern nulldelimiterspace 2 delta 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 ra 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 right where ra is the rayleigh number

nonconvolution nonlinear integral volterra equations with monotone operators abstract some results about existence uniqueness and attractive behaviour of solutions for nonlinear volterra integral equations with nonconvolution kernels are presented in this paper these results are based on similar ones about nonlinear volterra integral equations with convolutions kernels and some comparison techniques therefore this paper is devoted to find a wide class of nonconvolutions volterra integral equations where their solutions behave like those of volterra equations with convolution kernels

geometric statistics in turbulence the author presents results regarding certain average properties of incompressible fluids derived from the equations of motion the author estimates the average dissipation rate the average dimension of level sets the role played by the field of direction of vorticity in the three dimensional euler and navier stokes equations is discussed and a class of two dimensional equations that are useful models of incompressible dynamics is described the author presents results concerning scaling exponents in turbulence