about the regularizing properties of the non cut off kac equation we prove in this work that under suitable assumptions the solution of the spatially homogeneous non cut off kac equation or of the spatially homogeneous non cut off 2d boltzmann equation with maxwellian molecules in the radial case becomes very regular with respect to the velocity variable as soon as the time is strictly positive

entropy dissipation and long range interactions we study boltzmann s collision operator for long range interactions i e without grad s angular cut off assumption we establish a functional inequality showing that the entropy dissipation controls smoothness of the distribution function in a precise sense our estimate is optimal and gives a unified treatment of both the linear and the nonlinear cases we also give simple and self contained proofs of several useful results that were scattered in previous works as an application we obtain several helpful estimates for the cauchy problem and for the landau approximation in plasma physics

the cauchy problem for the radially symmetric homogeneous boltzmann equation with shubin class initial datum and gelfand shilov smoothing effect abstract in this paper we study the cauchy problem for the radially symmetric homogeneous non cutoff boltzmann equation with maxwellian molecules the initial datum belongs to shubin space of the negative index which can be characterized by spectral decomposition of the harmonic oscillator and it is a small perturbation of maxwellian distribution the shubin space of the negative index contains the probability measures based on this spectral decomposition we construct the weak solution with shubin class initial datum we also prove that the cauchy problem enjoys gelfand shilov smoothing effect meaning that the smoothing properties are the same as the cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator

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

regularity of renormalized solutions in the boltzmann equation with long range interactions it is well established that renormalized solutions of the boltzmann equation enjoy some kind of regularity or at least compactness in the velocity variable when the angular collision kernel is nonintegrable however obtaining explicit estimates in convenient and natural functional settings proves rather difficult in this work we derive a velocity smoothness estimate from the a priori control of the renormalized dissipation as a direct consequence of our result we show that in the presence of long range interactions any renormalized solution f t x v to the boltzmann equation satisfies locally for every and for some s 0 depending on p we also provide an application of this new estimate to the hydrodynamic limit of the boltzmann equation without cutoff 2012 wiley periodicals inc

maps between non commutative spaces let j be a graded ideal in a not necessarily commutative graded k algebra a a 0 a 1 in which dim k a i for all i we show that the map a a j induces a closed immersion i proj nc a j proj nc a between the non commutative projective spaces with homogeneous coordinate rings a and a j we also examine two other kinds of maps between non commutative spaces first a homomorphism a b between not necessarily commutative n graded rings induces an affine map proj nc b u proj nc a from a non empty open subspace u c proj nc b second if a is a right noetherian connected graded algebra not necessarily generated in degree one and a n is a veronese subalgebra of a there is a map proj nc a proj nc a n we identify open subspaces on which this map is an isomorphism applying these general results when a is a quotient of a weighted polynomial ring produces a non commutative resolution of a closed subscheme of a weighted projective space

well posedness and large time behaviour for the non cutoff kac equation with a gaussian thermostat we consider here a kac equation with a gaussian thermostat in the case of a non cutoff cross section under the sole assumptions of finite mass and finite energy for the initial data we prove the existence of a global in time solution for which mass and energy are preserved then via fourier transform techniques we show that this solution is smooth unique and converges to the corresponding stationary state

gelfand shilov smoothing properties of the radially symmetric spatially homogeneous boltzmann equation without angular cutoff we prove that the cauchy problem associated to the radially symmetric spatially homogeneous non cutoff boltzmann equation with maxwellian molecules enjoys the same gelfand shilov regularizing effect as the cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator

smoothing effect and cauchy problem for radially symmetric homogeneous boltzmann equation with debye yukawa potential of shubin class initial datum abstract in this paper we study the cauchy problem for the radially symmetric homogeneous non cutoff boltzmann equation with debye yukawa potential the initial datum belongs to shubin type space of the negative index which can be characterized by spectral decomposition of the harmonic oscillator and it is a small perturbation of maxwellian distribution the shubin type space of negative index contains the probability measures based on the spectral decomposition we construct the weak solution with shubin type class initial datum and prove the smoothing effect for the solution to this cauchy problem

regularizing effect and local existence for the non cutoff boltzmann equation the boltzmann equation without grad s angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non integrable angular singularity of the cross section however even though this has been justified satisfactorily for the spatially homogeneous boltzmann equation it is still basically unsolved for the spatially inhomogeneous boltzmann equation in this paper by sharpening the coercivity and upper bound estimates for the collision operator establishing the hypo ellipticity of the boltzmann operator based on a generalized version of the uncertainty principle and analyzing the commutators between the collision operator and some weighted pseudo differential operators we prove the regularizing effect in all time space and velocity variables on the solutions when some mild regularity is imposed on these solutions for completeness we also show that when the initial data has this mild regularity and a maxwellian type decay in the velocity variable there exists a unique local solution with the same regularity so that this solution acquires the c regularity for any positive time

the geometry of arithmetic noncommutative projective lines abstract let k be a perfect field and let k k be a finite extension of fields an arithmetic noncommutative projective line is a noncommutative space of the form proj s k v where v be a k central two sided vector space over k of rank two and s k v is the noncommutative symmetric algebra generated by v over k defined by m van den bergh 26 we study the geometry of these spaces more precisely we prove they are integral we classify vector bundles over them we classify them up to isomorphism and we classify isomorphisms between them using the classification of isomorphisms we compute the automorphism group of an arithmetic noncommutative projective line

the landau equation in a periodic box the landau equation which was proposed by landau in 1936 is a fundamental equation to describe collisions among charged particles interacting with their coulombic force in this article global in time classical solutions near maxwellians are constructed for the landau equation in a periodic box our result also covers a class of generalized landau equations which describes grazing collisions in a dilute gas

phase space analysis of semilinear parabolic equations abstract we present a wave packet analysis of a class of possibly degenerate parabolic equations with variable coefficients as a consequence we prove local wellposedness of the corresponding cauchy problem in spaces of low regularity namely the modulation spaces assuming a nonlinearity of analytic type as another application we deduce that the corresponding phase space flow decreases the global wave front set we also consider the action on spaces of analytic functions provided the coefficients are analytic themselves

a translation principle for the four dimensional sklyanin algebras abstract in this paper we prove a translation principle for the central quotients of four dimensional sklyanin algebras which is analogous to the translation principle for semi simple lie algebras in the course of the proof we construct an elliptic analog of the sheaf of differential operators on2 1 the translation principle may be used to construct the fat points of a four dimensional non pi sklyanin algebra

noetherian connected graded algebras of global dimension 3 abstract let a be a noetherian connected graded algebra of global dimension 3 we show that a is regular in the sense of artin and schelter if one of the following conditions holds 1 a is generated by two elements 2 the graded simple module has a standard resolution 3 the degrees of minimal relations are the same this includes the quadratic case some general properties of a are studied without assuming regularity

ultra analytic effect of cauchy problem for a class of kinetic equations abstract the smoothing effect of the cauchy problem for a class of kinetic equations is studied we firstly consider the spatially homogeneous nonlinear landau equation with maxwellian molecules and inhomogeneous linear fokker planck equation to show the ultra analytic effects of the cauchy problem those smoothing effect results are optimal and similar to heat equation in the second part we study a model of spatially inhomogeneous linear landau equation with maxwellian molecules and show the analytic effect of the cauchy problem

algebras associated to elliptic curves this paper completes the classification of artin schelter regular algebras of global dimension three for algebras generated by elements of degree one this has been achieved by artin schelter tate and van den bergh we are therefore concerned with algebras which are not generated in degree one we show that there exist some exceptional algebras each of which has geometric data consisting of an elliptic curve together with an automorphism just as in the case where the algebras are assumed to be generated in degree one in particular we study the elliptic algebras a a and a a where a p2 which were first defined in an earlier paper we omit a set s p2 consisting of 11 specified points where the algebras a a become too degenerate to be regular theorem let a represent a a or a a where a p2 s then a is an artin schelter regular algebra of global dimension three moreover a is a noetherian domain with the same hilbert series as the appropriately graded commutative polynomial ring in three variables this combined with our earlier results completes the classification

global solutions in the critical besov space for the non cutoff boltzmann equation abstract the boltzmann equation is studied without the cutoff assumption under a perturbative setting a unique global solution of the cauchy problem of the equation is established in a critical chemin lerner space in order to analyze the collisional term of the equation a chemin lerner norm is combined with a non isotropic norm with respect to a velocity variable which yields an a priori estimate for an energy estimate together with local existence following from commutator estimates and the hahn banach extension theorem the desired solution is obtained

gevrey regularity of spatially homogeneous boltzmann equation without cutoff abstract in this paper we study the gevrey regularity of spatially homogeneous boltzmann equation without angular cutoff we prove the propagation of gevrey regularity for c solutions with the maxwellian decay to the cauchy problem of spatially homogeneous boltzmann equation the idea we use here is based on the framework of morimoto ukai s recent paper see y morimoto s ukai gevrey smoothing effect of solutions for spatially homogeneous nonlinear boltzmann equation without angular cutoff j pseudo differ oper appl 1 2010 139 159 but we extend the range of the index satisfying 2 s 1 1 s 0 1 2 and in this case we consider the kinetic factor in the form of v v instead of v as morimoto and ukai did before

basic properties of generalized down up algebras abstract we introduce a large class of infinite dimensional associative algebras which generalize down up algebras let k be a field and fix f k x and r s k define l l f r s to be the algebra generated by d u and h with defining relations d h r d 0 h u r u 0 d u s f h 0 included in this family are smith s class of algebras similar to u sl 2 le bruyn s conformal sl 2 enveloping algebras and the algebras studied by rueda the algebras l have gelfand kirillov dimension 3 and are noetherian domains if and only if rs 0 we calculate the global dimension of l and for rs 0 classify the simple weight modules for l including all finite dimensional simple modules simple weight modules need not be classical highest weight modules

blowing up quantum weighted projective planes abstract quantum weighted projective planes of degree 1 1 n were classified by stephenson we show that the blowups of some of these spaces are quantum ruled surfaces as defined by patrick phd thesis mit 1997 this is a quantum version of the fact that projective cones over rational normal curves blowup to rational ruled surfaces the class of algebras which we can exhibit as blowdowns of quantum ruled surfaces are those which can be twisted to ore extensions of k x y

quantum planes of weight 1 1 n 1 in this article we study the categories tails a arising from artin schelter regular algebras of global dimension 3 of weight 1 1 n where n is an integer greater than 1 we determine generators and relations for all such algebras and this information is used to classify the points and lines in tails a

global well posedness in spatially critical besov space for the boltzmann equation the unique global strong solution in the chemin lerner type space to the cauchy problem on the boltzmann equation for hard potentials is constructed in a perturbation framework such a solution space is of critical regularity with respect to the spatial variable and it can capture the intrinsic properties of the boltzmann equation for the proof of global well posedness we develop some new estimates on the nonlinear collision term through the littlewood paley theory

ill posedness of the 3d navier stokes equations in a generalized besov space near bmo 1 abstract the ill posedness of the 3d navier stokes equations in a generalized besov space which is smaller than b q 1 q 2 is considered in 2008 bourgain pavlovic proved that the 3d navier stokes equation is ill posed in b 1 by showing norm inflation phenomena of the solution for some initial data on the other hand in 2008 germain proved that the flow map is not c 2 in the space b q 1 for q 2 however he did not treat ill posed problem in such spaces thus our result is an extension of these previous results

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

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

gevrey hypoellipticity for linear and non linear fokker planck equations abstract this paper studies the gevrey regularity of weak solutions of a class of linear and semi linear fokker planck equations

the cauchy problem for the inhomogeneous non cutoff kac equation in critical besov space abstract in this work we investigate the cauchy problem for the spatially inhomogeneous non cutoff kac equation if the initial datum belongs to the spatially critical besov space we can prove the well posedness of weak solution under a perturbation framework furthermore it is shown that the solution enjoys gelfand shilov regularizing properties with respect to the velocity variable and gevrey regularizing properties with respect to the position variable in comparison with the recent result in 18 the gelfand shilov regularity index is improved to be optimal to the best of our knowledge our work is the first one that exhibits smoothing effect for the kinetic equation in besov spaces

existence and regularity of a solution of a kac equation without cutoff using the stochastic calculus of variations we consider a generalized kac equation without cutoff with which we associate a non standard nonlinear stochastic differential equation we adapt the techniques in bichteler and jacod 2 to prove that the law of a solution of the stochastic differential equation has a density which is a solution of the kac equation the initial law is very general we only assume it has second order moments and is not the dirac mass at 0 we thus generalize the analytical results of existence of a solution of this equation if we furthermore assume existence of all moments for the initial law we obtain as a corollary using the proof in desvillettes 6 that the density is smooth we prove a slightly better regularity result under more stringent assumptions using the stochastic calculus of variations adapting the methods in 1

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

from semigroups to subelliptic estimates for quadratic operators using an approach based on the techniques of fbi transforms we give a new simple proof of the global subelliptic estimates for non selfadjoint non elliptic quadratic differential operators under a natural averaging condition on the weyl symbols of the operators established by the second author the loss of the derivatives in the subelliptic estimates depends directly on algebraic properties of the hamilton maps of the quadratic symbols using the fbi point of view we also give accurate smoothing estimates of gelfand shilov type for the associated heat semigroup in the limit of small times

uncertainty principle and kinetic equations abstract a large number of mathematical studies on the boltzmann equation are based on the grad s angular cutoff assumption however for particle interaction with inverse power law potentials the associated cross sections have a non integrable singularity corresponding to the grazing collisions smoothing properties of solutions are then expected on the other hand the uncertainty principle established by heisenberg in 1927 has been developed so far in various situations and it has been applied to the study of the existence and smoothness of solutions to partial differential equations this paper is the first one to apply this celebrated principle to the study of the singularity in the cross sections for kinetic equations precisely we will first prove a generalized version of the uncertainty principle and then apply it to justify rigorously the smoothing properties of solutions to some kinetic equations in particular we give some estimates on the regularity of solutions in sobolev spaces w r t all variables for both linearized and nonlinear space inhomogeneous boltzmann equations without angular cutoff as well as the linearized space inhomogeneous landau equation

wave packet analysis of schrodinger equations in analytic function spaces abstract we consider a class of linear schrodinger equations in r d with analytic symbols we prove a global in time integral representation for the corresponding propagator as a generalized gabor multiplier with a window analytic and decaying exponentially at infinity which is transported by the hamiltonian flow we then provide three applications of the above result the exponential sparsity in phase space of the corresponding propagator with respect to gabor wave packets a wave packet characterization of fourier integral operators with analytic phases and symbols and the propagation of analytic singularities

noncommutative symmetric algebras of two sided vector spaces abstract we generalize the definition of the symmetric algebra of a vector space in order to define noncommutative symmetric algebras of two sided vector spaces not all two sided vector spaces have noncommutative symmetric algebras the ones that do are called admissible and conditions for admissibility are given further for some classes of admissible two sided vector spaces the skew fields of fractions of their noncommutative symmetric algebras are computed the degree 0 components of these skew fields correspond to function fields of certain noncommutative ruled surfaces and hence allow us to determine birational equivalence classes for such surfaces

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

propagation of gevrey regularity for solutions of the boltzmann equation for maxwellian molecules we prove that gevrey regularity is propagated by the boltzmann equation with maxwellian molecules with or without angular cut off the proof relies on the wild expansion of the solution to the equation and on the characterization of gevrey regularity by the fourier transform

species and non commutative p1 s over non algebraic bimodules abstract we study non commutative projective lines over not necessarily algebraic bimodules in particular we give a complete description of their categories of coherent sheaves and show they are derived equivalent to certain bimodule species this allows us to classify modules over these species and thus generalize and give a geometric interpretation for results of c ringel 21

the boltzmann equation without angular cutoff in the whole space qualitative properties of solutions this is a continuation of our series of works for the inhomogeneous boltzmann equation we study qualitative properties of classical solutions the full regularization in all variables uniqueness non negativity and convergence rate to the equilibrium to be precise together with the results of parts i and ii about the well posedness of the cauchy problem around the maxwellian we conclude this series with a satisfactory mathematical theory for the boltzmann equation without angular cutoff

gelfand shilov and gevrey smoothing effect of the cauchy problem for fokker planck equation abstract in this work we study the cauchy problem for fokker planck equation with the external potential we show that the solution of the cauchy problem enjoys gelfand shilov s 1 1 r 3 regularizing property with respect to the velocity variable and gevrey g 3 2 r 3 regularizing property with respect to the position variable

global existence and full regularity of the boltzmann equation without angular cutoff we prove the global existence and uniqueness of classical solutions around an equilibrium to the boltzmann equation without angular cutoff in some sobolev spaces in addition the solutions thus obtained are shown to be non negative and c in all variables for any positive time in this paper we study the maxwellian molecule type collision operator with mild singularity one of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator this norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated h theorem

compactness in kinetic transport equations and hypoellipticity abstract we establish improved hypoelliptic estimates on the solutions of kinetic transport equations using a suitable decomposition of the phase space our main result shows that the relative compactness in all variables of a bounded family of nonnegative functions f x v l 1 satisfying some appropriate transport relation v x f 1 x 2 1 v 2 g may be inferred solely from additional integrability and compactness with respect to v in a forthcoming work the authors make a crucial application of this new approach to the study of the hydrodynamic limit of the boltzmann equation with a rough force field arsenio and saint raymond in preparation 4

sharp regularity and cauchy problem of the spatially homogeneous boltzmann equation with debye yukawa potential abstract in this paper we study the cauchy problem for the linear spatially homogeneous boltzmann equation with debye yukawa potential using the spectral decomposition of the linear operator we prove that for an initial datum in the sense of distribution which contains the dual of the sobolev spaces there exists a unique solution which belongs to a more regular sobolev space for any positive time we also study the sharp regularity of the solution

weak solutions for the navier stokes equations with b 1 ln bxr 1 r 21 r l2 initial data one of the major topics in the study of nonlinear partial differential equations of the evolutionary type is to look for as large as possible initial value spaces so that as many as possible solutions of such equations can be obtained in the book recent developments in the navier stokes problems lemarie rieusset proved that the navier stokes equations have global weak solutions for initial data in the space bxr 1 r 21 r rn l2 rn 0 r 1 where xr is the space of functions whose pointwise products with hr functions belong to l2 xr denotes the closure of c0 rn in xr and bxr 1 r 21 r rn is the besov space over xr in this paper we partially extend this result of lemarie rieusset to the larger initial value space b 1 ln rn bxr 1 r 21 r rn l2 rn 0 r 1 where b 1 ln rn is a logarithmically modified version of the usual besov space b 1 rn

gelfand shilov and gevrey smoothing effect for the spatially inhomogeneous non cutoff kac equation abstract we consider the spatially inhomogeneous non cutoff kac s model of the boltzmann equation we prove that the cauchy problem for the fluctuation around the maxwellian distribution enjoys gelfand shilov regularizing properties with respect to the velocity variable and gevrey regularizing properties with respect to the position variable

global classical solutions for partially dissipative hyperbolic system of balance laws the basic existence theory of kato and majda enables us to obtain local in time classical solutions to generally quasilinear hyperbolic systems in the framework of sobolev spaces in x with higher regularity however it remains a challenging open problem whether classical solutions still preserve well posedness in the case of critical regularity this paper is concerned with partially dissipative hyperbolic system of balance laws under the entropy dissipative assumption we establish the local well posedness and blow up criterion of classical solutions in the framework of besov spaces with critical regularity with the aid of the standard iteration argument and friedrichs regularization method then we explore the theory of function spaces and develop an elementary fact that indicates the relation between homogeneous and inhomogeneous chemin lerner spaces mixed space time besov spaces this fact allows us to capture the dissipation rates generated from the partial dissipative source term and further obtain the global well posedness and stability by assuming at all times the shizuta kawashima algebraic condition as a direct application the corresponding well posedness and stability of classical solutions to the compressible euler equations with damping are also obtained

decompositions of gelfand shilov kernels into kernels of similar class we prove that any linear operator with kernel in a gelfand shilov space is a composition of two operators with kernels in the same gelfand shilov space we also give links on numerical approximatio