an extension of the beale kato majda criterion for the euler equations the beale kato majda criterion asserts that smooth solutions to the euler equations remain bounded past t as long as is finite ohgr being the vorticity we show how to replace this by a weaker statement on where j is a frequency localization around

multipliers paramultipliers and weak strong uniqueness for the navier stokes equations abstract in this article we describe spaces p such that if u is a weak in the sense of leray j leray sur le mouvement d un fluide visqueux remplissant l espace acta math 63 1934 193 248 solution of the navier stokes system for some initial data u 0 and if u belongs to p then u is unique in the class of weak solutions we say then that weak strong uniqueness holds it turns out that the proof of such results relies on the boundedness of a trilinear functional f l 2 h l 2 h p r where belong to 0 1 in order to find optimal conditions for the boundedness of f we are led to describing spaces of multipliers and of paramultipliers that is functions which map by classical pointwise product or by paraproduct a given sobolev spaces in another given sobolev space the study of these spaces enables us to give conditions for weak strong uniqueness which generalise all previously known results from the famous serrin criterion j serrin the initial value problem for the navier stokes equations in r e langer ed nonlinear problems univ of wisconsin press 1963 pp 69 98 to the recent conditions formulated by lemarie rieusset p g lemarie rieusset recent developments in the navier stokes problem chapman and hall 2003

effect of mean on variance function estimation in nonparametric regression variance function estimation in nonparametric regression is considered and the minimax rate of convergence is derived we are particularly interested in the effect of the unknown mean on the estimation of the variance function our results indicate that contrary to the common practice it is not desirable to base the estimator of the variance function on the residuals from an optimal estimator of the mean when the mean function is not smooth instead it is more desirable to use estimators of the mean with minimal bias on the other hand when the mean function is very smooth our numerical results show that the residual based method performs better but not substantial better than the first order difference based estimator in addition our asymptotic results also correct the optimal rate claimed in hall and carroll j roy statist soc ser b51 1989 3 14

the regularity of weak solutions of the 3d navier stokes equations in b 1 _ infty infty we show that if a leray hopf solution u of the three dimensional navier stokes equation belongs to c 0 t b 1 infty infty or its jumps in the b 1 infty infty norm do not exceed a constant multiple of viscosity then u is regular for 0 t our method uses frequency local estimates on the nonlinear term and yields an extension of the classical ladyzhenskaya prodi serrin criterion

a boundedness criterion for generalized calderon zygmund operators in 1965 a p calderon showed the l2 boundedness of the so called first calderon commutator this is one of the first examples of a non convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the hilbert transform these properties together with the l2 boundedness imply the lp boundedness for all p s in 1 oo many operators in analysis such as certain classes of pseudo differential operators and the cauchy integral operator on a curve are associated with kernels having these properties for these operators one of the major questions is if they are bounded on l2 we are going to give necessary and sufficient conditions for such an operator to be bounded on l2 they are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in bmo in the case of the aforementioned first commutator this can be checked by an integration by parts in the first section we present some basic notions and state the theorem which is proved in sections 2 and 3 in section 4 we show how to recover some classical results in sections 5 and 6 we construct a functional calculus for small perturbations of a and in section 7 we show a connection between the theory of calderon zygmund operators and kato s conjecture it is a pleasure to express our thanks to r r coifman and y meyer for suggesting many elegant simplifications in our proofs and most of the applications we also wish to thank stephen semmes for several pertinent remarks

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

extension criterion via two components of vorticity on strong solutions to the 3 d navier stokes equations we shall show that only two components of vorticity play an essential role to determine possibility of extension of the time interval for the local strong solution to the navier stokes equations then we shall apply our extension theorem to regularity criterion on weak solutions due to serrin and beirao da veiga chae choe proved the same criterion as beirao da veiga only by means of the two components of vorticity we deal with the critical case which they excluded our criterion may be regarded as the generalization of the result of beal kato majda from l to bmo

on difference based variance estimation in nonparametric regression when the covariate is high dimensional summary we consider the problem of estimating the noise variance in homoscedastic nonparametric regression models for low dimensional covariates t d d 1 2 difference based estimators have been investigated in a series of papers for a given length of such an estimator difference schemes which minimize the asymptotic mean squared error can be computed for d 1 and d 2 however from numerical studies it is known that for finite sample sizes the performance of these estimators may be deficient owing to a large finite sample bias we provide theoretical support for these findings in particular we show that with increasing dimension d this becomes more drastic if d4 these estimators even fail to be consistent a different class of estimators is discussed which allow better control of the bias and remain consistent when d4 these estimators are compared numerically with kernel type estimators which are asymptotically efficient and some guidance is given about when their use becomes necessary

on nonparametric confidence intervals an inequality is given for the expected length of a confidence interval given that a particular distribution generated the data and assuming that the confidence interval has a given coverage probability over a family of distributions as a corollary attempts to adapt to the regularity of the true density within derivative smoothness classes cannot improve the rate of convergence of the length of the confidence interval over minimax fixed length intervals and still maintain uniform coverage probability however adaptive confidence intervals can attain improved rates of convergence in some other classes of densities such as those satisfying a shape restriction

extension criterion via partial components of vorticity on strong solutions to the navier stokes equations in higher dimensions abstract we consider the extension criterion of strong solutions to the navier stokes equations in r n it is proved that among n n 1 2 components of the vorticity n 2 components are negligible for the criterion whether the time local solutions can be extended beyond the critical time our result may be regarded as generalization to the higher dimensional case of chae choe 4 in the 3d case which showed that only two components in l q 3 2 q of the vorticity contribute to such an extension criterion furthermore the critical case q originally treated by kato ponce 8 in r n is also generalized in such a way that n 2 components of vortex matrix are redundant for the extension criterion

the beale kato majda criterion for the 3d magneto hydrodynamics equations we study the blow up criterion of smooth solutions to the 3d mhd equations by means of the littlewood paley decomposition we prove a beale kato majda type blow up criterion of smooth solutions via the vorticity of velocity only namely sup j in mathbb z int 0 t delta j nabla times u infty dt where j is the frequency localization operator in the littlewood paley decomposition

navier stokes equations with vorticity in besov spaces of negative regular indices abstract this paper studies the cauchy problem for the three dimensional navier stokes equations and shows that the condition u l 2 2 r 0 t b r 0 r 2 ensures the regularity of the solution on 0 t this improves and extends many previous results

on global infinite energy solutions to the navier stokes equations in two dimensions this paper studies the bidimensional navier stokes equations with large initial data in the homogeneous besov space as long as r q global existence and uniqueness of solutions are proved we also prove that weak strong uniqueness holds for the d dimensional equations with data in l 2 d for d r 2 q 1

the mckay conjecture and galois automorphisms the main problem of representation theory of finite groups is to find proofs of several conjectures stating that certain global invariants of a finite group g can be computed locally the simplest of these conjectures is the mckay conjecture which asserts that the number of irreducible complex characters of g of degree not divisible by p is the same if computed in a p sylow normalizer of g in this paper we propose a much stronger version of this conjecture which deals with galois automorphisms in fact the same idea can be applied to the celebrated alperin and dade conjectures

components reduction regularity results for the navier stokes equations in general dimensions abstract we consider the cauchy problem of the navier stokes equations in arbitrary dimensions and establish several new components reduction regularity criteria

new goodness of fit tests and their application to nonparametric confidence sets suppose one observes a process v on the unit interval where dv f 0 dw with an unknown parameter f 0 l 1 0 1 and standard brownian motion w we propose a particular test of one point hypotheses about f 0 which is based on suitably standardized increments of v this test is shown to have desirable consistency properties if for instance f 0 is restricted to various holder classes of functions the test is mimicked in the context of nonparametric density estimation nonparametric regression and interval censored data under shape restrictions on the parameter such as monotonicity or convexity we obtain confidence sets for f 0 adapting to its unknown smoothness

refinements of dade s projective conjecture for p solvable groups dade s projective conjecture is known to be true for finite p solvable groups thanks to work of g r robinson but remains open in general work of isaacs and navarro suggested to uno and boltje refinements of this conjecture these refinements were studied for finite p solvable groups by glesser in the present paper inspired by earlier work of turull we propose further refinements of the conjecture that take into account the schur indices and the elements of the brauer group we prove that all these refinements of dade s projective conjecture hold for all finite p solvable groups in particular we obtain that the version of dade s projective conjecture which involves character degree residues modulo p fields of definition and schur indices as well as the full strength of boltje s conjecture both hold for all finite p solvable groups the proof develops a clifford theory for normalizers of chains of p subgroups which allows one to reduce the calculation of the relevant sums to simpler groups

endoisomorphisms and character triple isomorphisms abstract this paper concerns aspects of clifford theory of finite groups in earlier papers turull proved that if two finite groups yielded the same element of the brauer clifford group then there was an endoisomorphism from one group to the other and furthermore that associated with each endoisomorphism there was an essentially unique correspondence of modules over many different fields from one group to the other the paper adapts the definition of character triple isomorphism so that it involves ordinary and brauer characters and it preserves fields of definition schur indices decomposition numbers and blocks it is proved that each endoisomorphism yields exactly one character triple isomorphism character triple isomorphisms can be composed restricted produced by direct sums extension of fields and these operations have their parallel for the endoisomorphisms one goal of the paper is to provide tools for the study of the character theory of finite groups in an accessible way suitable for applications

an adaptation theory for nonparametric confidence intervals a nonparametric adaptation theory is developed for the construction of confidence intervals for linear functionals a between class modulus of continuity captures the expected length of adaptive confidence intervals sharp lower bounds are given for the expected length and an ordered modulus of continuity is used to construct adaptive confidence procedures which are within a constant factor of the lower bounds in addition minimax theory over nonconvex parameter spaces is developed

on the euler equations in the critical triebel lizorkin spaces in this paper we study the initial value problem of the incompressible euler equations in n for initial data belonging to the critical triebel lizorkin spaces i e v0 fn 11 q q 1 we prove the blow up criterion of solutions in fn 11 q for n 2 3 for n 2 in particular we prove global well posedness of the euler equations in f31 q q 1 for the proof of these results we establish a sharp moser type inequality as well as a commutator type estimate in these spaces the key methods are the littlewood paley decomposition and the paradifferential calculus by j m bony

strengthening the mckay conjecture to include local fields and local schur indices abstract a central problem in the representation theory of finite groups is given a prime p the study of relationships between the representations of g and those of p local subgroups of g the simplest of these is perhaps the mckay conjecture which concerns the number of irreducible characters of degree prime to p even though this conjecture remains open in an effort to understand it a number of strengthenings of it have been proposed recently isaacs and navarro and later navarro have proposed interesting stronger conjectures indicating that congruences modulo p of the degree of the characters and the invariance under certain type of fixed galois automorphism could be taken into account as well we propose an even further strengthening of mckay s conjecture which takes into account in addition the p local field of definition of the irreducible characters as well as their p local schur indices

concave regression value constrained estimation and likelihood ratio based inference we propose a likelihood ratio statistic for forming hypothesis tests and confidence intervals for a nonparametrically estimated univariate regression function based on the shape restriction of concavity alternatively convexity dealing with the likelihood ratio statistic requires studying an estimator satisfying a null hypothesis that is studying a concave least squares estimator satisfying a further equality constraint we study this null hypothesis least squares estimator nlse here and use it to study our likelihood ratio statistic the nlse is the solution to a convex program and we find a set of inequality and equality constraints that characterize the solution we also study a corresponding limiting version of the convex program based on observing a brownian motion with drift the solution to the limit problem is a stochastic process we study the optimality conditions for the solution to the limit problem and find that they match those we derived for the solution to the finite sample problem this allows us to show the limit stochastic process yields the limit distribution of the finite sample nlse we conjecture that the likelihood ratio statistic is asymptotically pivotal meaning that it has a limit distribution with no nuisance parameters to be estimated which makes it a very effective tool for this difficult inference problem we provide a partial proof of this conjecture and we also provide simulation evidence strongly supporting this conjecture

regularity criteria for a lagrangian averaged magnetohydrodynamic α model abstract in this paper we study the n dimensional lagrangian averaged magnetohydrodynamic lamhd model with 5 n 7 throughout the whole space various regularity criteria are established for both the cases with and those without viscosity

on the deformations of the incompressible euler equations we consider systems of deformed system of equations which are obtained by some transformations from the system of incompressible euler equations these have similar properties to the original euler equations including the scaling invariance for one form of deformed system we prove that finite time blow up actually occurs for generic initial data while for the other form of the deformed system we prove the global in time regularity for smooth initial data moreover using the explicit functional relations between the solutions of those deformed systems and that of the original euler system we derive the condition of finite time blow up of the euler system in terms of solutions of one of its deformed systems as another application of those relations we deduce a lower estimate of the possible blow up time of the 3d euler equations

existence and partial regularity results for the heat flow for harmonic maps on considere le probleme d evolution t u m u n u 0 dans m r u 0 u 0 sur m ou m et n sont des varietes de riemann lisses compactes et u 0 l application harmonique de m dans n on demontre l existence d une solution globale faible

adaptive testing on a regression function at a point we consider the problem of inference on a regression function at a point when the entire function satisfies a sign or shape restriction under the null we propose a test that achieves the optimal minimax rate adaptively over a range of holder classes up to a log logn term which we show to be necessary for adaptation we apply the results to adaptive one sided tests for the regression discontinuity parameter under a monotonicity restriction the value of a monotone regression function at the boundary and the proportion of true null hypotheses in a multiple testing problem

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

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

a new regularity criterion of the 2d mhd equations abstract in this paper we establish a new regularity criterion for the two dimensional incompressible generalized magnetohydrodynamics 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

global well posedness of incompressible flow in porous media with critical diffusion in besov spaces abstract in this paper we study the model of heat transfer in a porous medium with a critical diffusion we obtain global existence and uniqueness of solutions to the equations of heat transfer of incompressible fluid in besov spaces b p 1 3 p r 3 with 1 p by the method of modulus of continuity and fourier localization technique

some refinements of dade s projective conjecture abstract in this paper a further refinement of dade s projective conjecture due to boltje is presented this new statement includes ideas first published by isaacs and navarro as well as the recent contractibility version of alperin s conjecture introduced by boltje leaning heavily on the work of robinson weaker forms of the conjecture are proved in the case of p solvable groups

the second iterate for the navier stokes equation we consider the iterative resolution scheme for the navier stokes equation and focus on the second iterate more precisely on the map from the initial data to the second iterate at a given time t we investigate boundedness properties of this bilinear operator this new approach yields very interesting results a new perspective on koch tataru solutions a first step towards weak strong uniqueness for koch tataru solutions and finally an instability result in b q 1 for q 2

about weak strong uniqueness for the 3d incompressible navier stokes system this article studies the problem of l2 stability and weak strong uniqueness of solutions of the incompressible navier stokes on the whole space input amssym bbb s 3 constructed by kato s approach in spaces coming from littlewood paley theory and using the l1 smoothing effect for the heat flow 2011 wiley periodicals inc

adaptive confidence intervals for regression functions under shape constraints adaptive confidence intervals for regression functions are constructed under shape constraints of monotonicity and convexity a natural benchmark is established for the minimum expected length of confidence intervals at a given function in terms of an analytic quantity the local modulus of continuity this bound depends not only on the function but also the assumed function class these benchmarks show that the constructed confidence intervals have near minimum expected length for each individual function while maintaining a given coverage probability for functions within the class such adaptivity is much stronger than adaptive minimaxity over a collection of large parameter spaces

statistical estimation and optimal recovery new formulas are given for the minimax linear risk in estimating a linear functional of an unknown object from indirect data contaminated with random gaussian noise the formulas cover a variety of loss functions and do not require the symmetry of the convex a priori class it is shown that affine minimax rules are within a few percent of minimax even among nonlinear rules for a variety of loss functions it is also shown that difficulty of estimation is measured by the modulus of continuity of the functional to be estimated the method of proof exposes a correspondence between minimax affine estimates in the statistical estimation problem and optimal algorithms in the theory of optimal recovery

well posedness for the navier stokes equations where u is the velocity and p is the pressure it is well known that the navierstokes equations are locally well posed for smooth enough initial data as long as one imposes appropriate boundary conditions on the pressure at for instance it is easy to see see 9 for much more general results that if s n 2 then for any h initial data there exists a unique c 0 t h r local solution with a pressure p c 0 t h r in the sequel we consider solutions for less regular initial data this has to be understood in the sense that the map from the initial data to the solution extends continuously to rougher function spaces the question we are interested in is the global well posedness for small data and local well posedness for large data with respect to a certain space of

partial regularity for stationary harmonic maps into spheres in an interesting recent paper 12 f hi lein has shown that any weakly harmonic mapping from a two dimensional surface into a sphere is smooth i present here a kind of generalization to higher dimensions asserting in effect that a stationary harmonic mapping from an open subset of nn n 3 into a sphere is smooth except possibly for a closed singular set of n 2 dimensional hausdorff measure zero my proof crucially depends upon several of h l in s observations as streamlined by p l lions to state the result precisely let us suppose that m n 2 u is a smooth open subset of nn and s m 1 denotes the unit sphere in nm a function u in the sobolev space hi u rm u u 1 urn belongs to h i u s ml provided t u 1 a e in u

endoisomorphisms yield module and character correspondences abstract in an earlier paper turull 2012 12 the author introduced the concept of endoisomorphisms and showed its natural connection with clifford theory of finite groups associated with clifford theory are module and character correspondences in the present paper the author shows that each endoisomorphism produces a unique module and character correspondence with excellent compatibility results

a remark on the blow up criterion of strong solutions to the navier stokes equations abstract in this note we prove that b 1 norm of the vorticity controls the blow up phenomena of strong solutions to the navier stokes equations our criterion is a limiting case in yuan zhang 11 and an improvement of giga s result 3

the strengthened alperin weight conjecture for p solvable groups abstract the alperin weight conjecture is a well known conjecture and it is central to the modern representation theory of finite groups it is known to be true for many types of finite groups but it remains open in particular the conjecture is known to be true for finite p solvable groups in this paper we prove that a very strong form of the conjecture holds for all finite p solvable groups it follows that the strengthening of the alperin weight conjecture that includes galois automorphisms over the p adic numbers holds for all finite p solvable groups this strengthening of the alperin weight conjecture was suggested in an earlier work of navarro on the mckay conjecture the present paper provides the first direct evidence beyond the calculation of individual cases and the case of groups of odd order for this strengthening

on critical cases of sobolev s inequalities abstract we present a new form of the trudinger type inequality which shows an explicit dependence moreover we give an alternative proof of the brezis gallouet wainger inequality

remarks on gagliardo nirenberg type inequality with critical sobolev space and bmo we consider the generalized gagliardo nirenberg inequality in mathbb r n in the homogeneous sobolev space dot h s r mathbb r n with the critical differential order s n r which describes the embedding such as l p mathbb r n cap dot h n r r mathbb r n subset l q mathbb r n for all q with p q n r which may be regarded as a generalization of the brezis gallouet wainger inequality

ill posedness of the navier stokes equations in a critical space in 3d abstract we prove that the cauchy problem for the three dimensional navier stokes equations is ill posed in b 1 in the sense that a norm inflation happens in finite time more precisely we show that initial data in the schwartz class s that are arbitrarily small in b 1 can produce solutions arbitrarily large in b 1 after an arbitrarily short time such a result implies that the solution map itself is discontinuous in b 1 at the origin

the optimal upper and lower bounds of convergence rates for the 3d navier stokes equations under large initial perturbation abstract this paper is concerned with the optimal algebraic convergence rates for leray weak solutions of the 3d navier stokes equations in morrey space it is shown that if the global leray weak solution u x t of the 3d navier stokes equations satisfies u l r 0 m p q r 3 2 r 3 p 2 3 2 p p q 2 then even for the large initial perturbation every weak solution v x t of the perturbed navier stokes equations converges algebraically to u x t with the optimal upper and lower bounds c 1 1 t 2 v t u t l 2 c 2 1 t 2 for large t 1 2 5 2 the findings are mainly based on the developed fourier splitting methods and iterative process

regularity criterion of axisymmetric weak solutions to the 3d navier stokes equations abstract we consider the regularity of axisymmetric weak solutions to the navier stokes equations in r 3 let u be an axisymmetric weak solution in r 3 0 t w curl u and w be the azimuthal component of w in the cylindrical coordinates chae lee d chae j lee on the regularity of axisymmetric solutions of the navier stokes equations math z 239 2002 645 671 proved the regularity of weak solutions under the condition w l q 0 t l r with 3 2 r 2 q 3 r 2 we deal with the marginal case r which they excluded it is proved that u becomes a regular solution if w l 1 0 t b 0

on weak strong uniqueness of solutions to the generalized incompressible navier stokes equations in the recent paper li and zhai 2010 proved the well posedness of the cauchy problem to the n dimensional generalized incompressible navier stokes equations with initial data u 0 belonging to the so called q space q 1 r n with 1 2 1 and 0 in this paper by using the littlewood paley theory we prove the weak strong uniqueness between weak solution and li zhai s strong solution for the n dimensional generalized incompressible navier stokes equations

uniformly local l p estimate for 2 d vorticity equation and its application to euler equations with initial vorticity in bmo we shall prove the global existence theorem for the 2 dimensional euler equations in open image in new window with the initial vorticity in bmo containing functions which do not decay at infinity and have logarithmic singularities

remarks on a paper by j t beale t kato and a majda we prove that the maximum norm of the deformation tensor controls the possible breakdown of smooth solutions for the 3 dimensional euler equations more precisely the loss of regularity in a local smooth solution of the euler equations implies the growth without bound of the deformation tensor as the critical time approaches equivalently if the deformation tensor remains bounded the existence of a smooth solution is guaranteed

the critical sobolev inequalities in besov spaces and regularity criterion to some semi linear evolution equations we show the critical sobolev inequalities in the besov spaces with the logarithmic form such as brezis gallouet wainger and beale kato majda as an application of those inequalities the regularity problem under the critical condition to the navier stokes equations the euler equations in mathbb r n and the gradient flow to the harmonic map to the sphere are discussed namely the serrin ohyama type regularity criteria are improved in the terms of the besov spaces

about local continuity with respect to l_ 2 l2 initial data for energy solutions of the navier stokes equations in this paper we consider classes of initial data that ensure local in time hadamard well posedness of the associated weak leray hopf solutions of the three dimensional navier stokes equations in particular for any solenodial l 2 initial data u 0 belonging to certain subsets of vmo 1 mathbb r 3 we show that weak leray hopf solutions depend continuously with respect to small divergence free l 2 perturbations of the initial data u 0 on some finite time interval our main result is inspired by and improves upon previous work of the author barker in j math fluid mech 20 1 133 160 2018 and work of jean yves chemin commun pure appl math 64 12 1587 1598 2011 our method builds upon 4 9 in particular our method hinges on decomposition results for the initial data inspired by calderon trans am math soc 318 1 179 200 1990 together with use of persistence of regularity results the persistence of regularity statement presented may be of independent interest since it does not rely upon the solution or the initial data being in the perturbative regime

on the weak strong uniqueness of koch tataruʼs solution for the navier stokes equations abstract we prove the weak strong uniqueness between koch tataru s solution and leray s weak solution for three dimensional incompressible navier stokes equations

a beale kato majda blow up criterion for a hydrodynamic system modeling vesicle and fluid interactions abstract in this paper we establish an analog of the beale kato majda type criterion for singularities of smooth solutions of a hydrodynamic system modeling vesicle and fluid interactions the result shows that the maximum norm of the vorticity alone controls the breakdown of smooth solutions

above the glauberman correspondence let s be a finite solvable group and suppose s acts on the finite group n and they have coprime orders then the celebrated glauberman correspondence provides a natural bijection from the set irrs n of irreducible characters of n which are invariant under the action of s to the set irr cn s of all irreducible characters of the centralizer of s in n suppose further that the semidirect product sn is a normal subgroup of a finite group g let irrs n and let irr cn s be its glauberman correspondent we prove that there is a bijection with good compatibility properties from the set irr g of the irreducible characters of g above to irr ng s such that in the case when s is a p group for some prime p it preserves fields of values and schur indices over qp the field of p adic numbers using this result we also prove a strengthening of the mckay conjecture for all p solvable groups

global well posedness for the 2 d boussinesq system with the temperature dependent viscosity and thermal diffusivity abstract we prove the global well posedness for the 2 d boussinesq system with the temperature dependent viscosity and thermal diffusivity

new refinements of the mckay conjecture for arbitrary finite groups let g be an arbitrary finite group and fix a prime number p the mckay conjecture asserts that g and the normalizer in g of a sylow p subgroup have equal numbers of irreducible characters with degrees not divisible by p the alperin mckay conjecture is version of this as applied to individual brauer p blocks of g we offer evidence that perhaps much stronger forms of both of these conjectures are true

optimal inference in a class of regression models we consider the problem of constructing confidence intervals cis for a linear functional of a regression function such as its value at a point the regression discontinuity parameter or a regression coefficient in a linear or partly linear regression our main assumption is that the regression function is known to lie in a convex function class which covers most smoothness and or shape assumptions used in econometrics we derive finite sample optimal cis and sharp efficiency bounds under normal errors with known variance we show that these results translate to uniform over the function class asymptotic results when the error distribution is not known when the function class is centrosymmetric these efficiency bounds imply thatminimax cis are close to efficient at smooth regression functions this implies in particular that it is impossible to form cis that are tighter using data dependent tuning parameters and maintain coverage over the whole function class we specialize our results to inference in a linear regression and inference on the regression discontinuity parameter and illustrate them in an empirical application

regularity criterion for solutions to the navier stokes equations in the whole 3d space based on two vorticity components abstract we prove among others the following regularity criterion for the solutions to the navier stokes equations if u is a global weak solution satisfying the energy inequality and u then u is regular on 0 t t 0 if two components of belong to the space l q 0 t b 3 p for p 3 and 2 q 3 p 2 this result is an improvement of the results presented by chae and choe 1999 7 or zhang and chen 2005 38 our method of the proof uses a suitable application of the bony decomposition and can also be used for the proofs of some other kin criteria one such example is presented in appendix

remarks on the blowup criteria for oldroyd models abstract we provide a new method to prove and improve the chemin masmoudi criterion for viscoelastic systems of oldroyd type in j y chemin n masmoudi about lifespan of regular solutions of equations related to viscoelastic fluids siam j math anal 33 1 2001 84 112 in two space dimensions our method is much easier than the one based on the well known losing a priori estimate and is expected to be easily adopted to other problems involving the losing a priori estimate

on the calculation of local invariants of irreducible characters abstract let k be a finite extension of the field q p of p adic numbers let be an irreducible character of a finite group whose values are in k associated with and k is an element of the brauer group of k and therefore by standard results a local invariant in q z the paper defines some classes of finite groups that the paper calls p basic groups and gives formulas to calculate the local invariant of their characters it also shows how one can calculate the local invariant of any irreducible character of any finite group that has values in k by reducing the problem to the case where the groups are p basic and using the explicit formulas given in the paper

strichartz estimates for the euler equations in the rotational framework abstract we consider the initial value problems of the incompressible euler equations in the rotational framework we obtain the optimal range of the strichartz estimate for the linear group associated with the coriolis forces as an application we prove that the lifespan of the solution can be taken arbitrarily large provided that the speed of rotation is sufficiently high

on blow up criteria of smooth solutions to the 3 d euler equations in a bounded domain abstract we show that a smooth solution of the 3 d euler equations in a bounded domain breaks down if and only if a certain norm of vorticity blows up at the same time here this norm is weaker than bmo norm

optimal spatial adaptation to inhomogeneous smoothness an approach based on kernel estimates with variable bandwidth selectors a new v ria e 1 s e 0r estimation is proposed the application of this bandwidth selector leads to kernel estimates that achieve optimal rates of convergence over b 3 sses this implies that the procedure adapts to spatially inhomogeneous s noothness in particu lar the estimates share optimality properties with wavelet estimates based on thresholding of empirical wavelet coefficients 1 introduction in nonparametric curve estimation the statistical anal ysis may focus on the inference of the qualitative structure of the analyzed curve often interesting features of the curve are connected with spatially inhomogeneous smoothness in this case curve estimates that are spatially adaptlve ar e appropriaie a variety of such procedures have been proposed in the literature in breiman friedman olshen and stone 1983 piecewise constant least squares estimates are considered with a data adaptive choice of the pieces cart more generally friedman and silverman 1989 friedman 1991 and luo and wahba 1995 use variable knot splines mars knot points are added removed and allocated recursively using cross validation techniques these methods have shown good performance in simulations and real data examples however no asymptotic theory is available mammen and van de geer 1997 discuss penalized least squares curve estimation for spatially inhomogeneous curves they propose penalty terms which allow more spatial inhomogeneity than the usual l2 norms of deriva tives of the curve the estimates turn out to be variable knot splines see also mammen 1991 results on rates of convergence and a pointwise asymptotic distribution theory are given miiller and stadtmiiller 1987 staniswalis 1989 and brockmann gasser and hermann 1993 propose kernel estimation with locally variable band width selectors the calculation of local bandwidths is based on pilot estima tion of local smoothness characteristics an asymptotic analysis is available

asymptotically optimal difference based estimation of variance in nonparametric regression summary we define and compute asymptotically optimal difference sequences for estimating error variance in homoscedastic nonparametric regression our optimal difference sequences do not depend on unknowns such as the mean function and provide substantial improvements over the suboptimal sequences commonly used in practice for example in the case of normal data the usual variance estimator based on symmetric second order differences is only 64 efficient relative to the estimator based on optimal second order differences the efficiency of an optimal mth order difference estimator relative to the error sample variance is 2m 2m 1 again this is for normal data and increases as the tails of the error distribution become heavier

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

logarithmically improved blow up criteria for a phase field navier stokes vesicle fluid interaction model a b s t r a c t in this paper we study a hydrodynamical system modeling the deformation of the vesicle membrane under external incompressible viscous flow fields the system is in the eulerian formulation and is governed by the coupling of the incompressible navier stokes equations with a phase field equation in the three dimensional case we establish two logarithmically improved blow up criteria for local smooth solutions of this system in terms of the vorticity field only in the homogeneous besov spaces

optimal pointwise adaptive methods in nonparametric estimation the problem of optimal adaptive estimation of a function at a given point from noisy data is considered two procedures are proved to be asymptotically optimal for different settings first we study the problem of bandwidth selection for nonparametric pointwise kernel estimation with a given kernel we propose a bandwidth selection procedure and prove its optimality in the asymptotic sense moreover this optimality is stated not only among kernel estimators with a variable bandwidth the resulting estimator is asymptotically optimal among all feasible estimators the important feature of this procedure is that it is fully adaptive and it works for a very wide class of functions obeying a mild regularity restriction with it the attainable accuracy of estimation depends on the function itself and is expressed in terms of the ideal adaptive bandwidth corresponding to this function and a given kernel the second procedure can be considered as a specialization of the first one under the qualitative assumption that the function to be estimated belongs to some holder class l with unknown parameters l this assumption allows us to choose a family of kernels in an optimal way and the resulting procedure appears to be asymptotically optimal in the adaptive sense in any range of adaptation with 2

a regularity criterion for the solution of nematic liquid crystal flows in terms of the b 1 norm abstract in this paper we investigate the regularity criterion for the solution of nematic liquid crystal flows in three dimensions n 3 and in two dimensions n 2 we prove that the solution u d is smooth up to time t provided that there exists a positive constant e 0 0 such that i for n 3 u d l 0 t b 1 e 0 and ii for n 2 d l 0 t b 1 e 0

liouville theorems for the navier stokes equations and applications we study bounded ancient solutions of the navier stokes equations these are solutions with bounded velocity defined in rn 1 0 in two space dimensions we prove that such solutions are either constant or of the form u x t b t depending on the exact definition of admissible solutions the general 3 dimensional problem seems to be out of reach of existing techniques but partial results can be obtained in the case of axisymmetric solutions we apply these results to some scenarios of potential singularity formation for axi symmetric solutions and obtain extensions of results in a recent paper by chen strain tsai and yau 4

local well posedness and regularity criterion for the density dependent incompressible euler equations in this paper local well posedness for the density dependent incompressible euler equations is established in besov spaces we also obtain a blow up criterion for the corresponding solution

global weak besov solutions of the navier stokes equations and applications we introduce a notion of global weak solution to the navier stokes equations in three dimensions with initial values in the critical homogeneous besov spaces dot b 1 frac 3 p p infty p 3 these solutions satisfy a certain stability property with respect to the weak ast convergence of initial conditions to illustrate this property we provide applications to blow up criteria minimal blow up initial data and forward self similar solutions our proof relies on a new splitting result in homogeneous besov spaces that may be of independent interest