well posedness and ill posedness for a fifth order shallow water wave equation abstract in this paper we establish a new bilinear estimate in suitable bourgain spaces by using a fundamental estimate on dyadic blocks for the kawahara equation which was obtained by the k z multiplier norm method of tao 2001 2 then the local well posedness of the cauchy problem for a fifth order shallow water wave equation in h s r with s 5 4 is obtained by the fourier restriction norm method and some ill posedness in h s r with s 5 4 is derived from a general principle of bejenaru and tao

hopf algebra of the planar binary trees is a sub hopf algebra there is a basis qn of soln such that the composite map k qn soln k sn has the following property its linear dual k sn k qn is induced by a set theoretic map sn qn in between sn and qn there is an intermediate set yn which is made of planar binary trees with n vertices for instance for n 3 the number of elements in s3 y3 and q3 is 6 5 and 4 respectively the projection from sn to qn is the composite of two maps sn w yn w qn article no ai981759

multilinear estimates for periodic kdv equations and applications we prove an endpoint multilinear estimate for the xs b spaces associated to the periodic airy equation as a consequence we obtain sharp local well posedness results for periodic generalized kdv equations as well as some global well posedness results below the energy norm

second microlocal ellipticity and propagation of conormality for semilinear wave equations we revisit three problems on the propagation of conormal singularities for semilinear wave equations the evolution of two simply tangent waves the interaction of three conormal waves and the evolution of one wave with a cusp singularity we use quasihomogeneous blow ups to reduce the geometries to normal crossing and show that the lift of the operator by these blow down maps is elliptic in some directions of the compressed cotangent bundle of the blown up manifold this leads to both a strengthening of the previously known results and to simplification of their proofs

inverse problems for lorentzian manifolds and non linear hyperbolic equations we study two inverse problems on a globally hyperbolic lorentzian manifold m g the problems are 1 passive observations in spacetime consider observations in an open set v subset m the light observation set corresponding to a point source at q in m is the intersection of v and the light cone emanating from the point q let w subset m be an unknown open relatively compact set we show that under natural causality conditions the family of light observation sets corresponding to point sources at points q in w determine uniquely the conformal type of w 2 active measurements in spacetime we develop a new method for inverse problems for non linear hyperbolic equations that utilizes the non linearity as a tool this enables us to solve inverse problems for non linear equations for which the corresponding problems for linear equations are still unsolved to illustrate this method we solve an inverse problem for semilinear wave equations with quadratic non linearities we assume that we are given the neighborhood v of the time like path mu and the source to solution operator that maps the source supported on v to the restriction of the solution of the wave equation to v when m is 4 dimensional we show that these data determine the topological differentiable and conformal structures of the spacetime in the maximal set where waves can propagate from mu and return back to mu

brouwer s degree without properness a degree deg f y is defined for every continuous function f rn rn which possesses all the properties of brouwer s degree provided that y is restricted to the complement of some closed set a f of asymptotic values sufficient conditions are given for a f to be nowhere dense it is also shown that in the opposite direction a f having nonempty interior has a direct impact on the solutions of f x y which cannot be discovered by degree arguments

on the existence of pareto solutions for polynomial vector optimization problems we are interested in the existence of pareto solutions to the vector optimization problem begin aligned mathrm min mathbb r m f x x in mathbb r n end aligned where f mathbb r n rightarrow mathbb r m is a polynomial map by using the tangency variety of f we first construct a semi algebraic set of dimension at most m 1 containing the set of pareto values of the problem then we establish connections between the palais smale conditions m tameness and properness for the map f based on these results we provide some sufficient conditions for the existence of pareto solutions of the problem we also introduce a generic class of polynomial vector optimization problems having at least one pareto solution

yang baxter bases of 0 hecke algebras and representation theory of 0 ariki koike shoji algebras after reformulating the representation theory of 0 hecke algebras in an appropriate family of yang baxter bases we investigate certain specializations of the ariki koike algebras obtained by setting q 0 in a suitably normalized version of shoji s presentation we classify the simple and projective modules and describe restrictions induction products cartan invariants and decomposition matrices this allows us to identify the grothendieck rings of the towers of algebras in terms of certain graded hopf algebras known as the mantaci reutenauer descent algebras and poirier quasi symmetric functions we also describe the ext quivers and conclude with numerical tables

sharp null form estimates for the wave equation null form estimates from dot h alpha 1 times dot h alpha 2 to l q t l r x for the wave equation in bbb r n 1 are studied for n ge4 we obtain the sharp null form estimates except for the endpoints for n 2 3 we obtain the estimates under the additional assumption 4 n q when 2 r

rough solutions of the fifth order kdv equations abstract we consider the cauchy problem of the fifth order equation arising from the korteweg de vries kdv hierarchy t u x 5 u c 1 x u x 2 u c 2 u x 3 u 0 x t r u 0 x u 0 x u 0 h s r we prove a priori bound of solutions for h s r with s 5 4 and the local well posedness for s 2 the method is a short time x s b space which was first developed by ionescu kenig and tataru 13 in the context of the kp i equation in addition we use a weight on localized x s b structures to reduce the contribution of high low frequency interaction where the low frequency has large modulation as an immediate result from a conservation law we obtain that the fifth order equation in the kdv hierarchy t u x 5 u 30 u 2 x u 20 x u x 2 u 10 u x 3 u 0 is globally well posed in the energy space h 2

foliations and polynomial diffeomorphisms of mathbb r 3 let y f g h mathbb r 3 to mathbb r 3 be a c 2 map and let spec y denote the set of eigenvalues of the derivative dy p when p varies in mathbb r 3 we begin proving that if for some 0 spec y cap epsilon epsilon emptyset then the foliation mathcal f k with k in f g h made up by the level surfaces k constant consists just of planes as a consequence we prove a bijectivity result related to the three dimensional case of jelonek s jacobian conjecture for polynomial maps of mathbb r n

cyclotomic solomon algebras this paper introduces an analogue of the solomon descent algebra for the complex reflection groups of type g r 1 n as with the solomon descent algebra our algebra has a basis given by sums of distinguished coset representatives for certain reflection subgroups we explicitly describe the structure constants with respect to this basis and show that they are polynomials in r this allows us to define a deformation or q analogue of these algebras which depends on a parameter q we determine the irreducible representations of all of these algebras and give a basis for their radicals finally we show that the direct sum of cyclotomic solomon algebras is canonically isomorphic to a concatenation hopf algebra

spreading of singularities at the boundary in semilinear hyperbolic mixed problems ii crossing and self spreading on etudie comment des singularites anormales sont produites dans des problemes mixtes pour des equations d onde semilineaires u f u sur le demi espace r n 1 u h s loc s n 2 2 a cause du croisement et de l auto etalement aux points limites

remarks on chern simons gauged o 3 sigma model in one space dimension we find the stationary solutions of the chern simons gauged o 3 sigma model in one space dimension and study an initial value problem under the gauge condition a1 0 we also study low regularity local well posedness using bilinear estimates for wave sobolev space h1 as a by product we construct a finite energy global solution by considering the conservation of energy we find the stationary solutions of the chern simons gauged o 3 sigma model in one space dimension and study an initial value problem under the gauge condition a1 0 we also study low regularity local well posedness using bilinear estimates for wave sobolev space h1 as a by product we construct a finite energy global solution by considering the conservation of energy

a solomon descent theory for the wreath products _ we propose an analogue of solomon s descent theory for the case of a wreath product g g n where g is a finite abelian group our construction mixes a number of ingredients mantaci reutenauer algebras specht s theory for the representations of wreath products okada s extension to wreath products of the robinson schensted correspondence and poirier s quasisymmetric functions we insist on the functorial aspect of our definitions and explain the relation of our results with previous work concerning the hyperoctaedral group

conormal and piecewise smooth solutions to quasilinear wave equations in this paper we show first that if a solution u of the equation p2 t x u du d u f t x u du where p2 t x u du d is a second order strictly hyperbolic quasilinear operator is conormal with respect to a single characteristic hypersurface i of p2 in the past and i is smooth in the past then i is smooth and u is conormal with respect to i for all time second let z0 and z1 be characteristic hypersurfaces of p2 which intersect transversally and let r 0 n ii if 4o and ii are smooth in the past and u is conormal with repect to o 11 in the past then r is smooth and u is conormal with respect to 1o ii locally in time outside of r even though z0 and ii are no longer necessarily smooth across r finally we show that if u o x and tu o x are in an appropriate sobolev space and are piecewise smooth outside of r then u is piecewise smooth locally in time outside of iouz1

the generation of semilinear singularities by a swallowtail caustic we construct examples of bounded solutions to a semilinear system p u f z u z inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i f smooth p p id with p a strictly hyperbolic differential operator of second order with smooth coefficients such that for a time function t of p the following properties are satisfied 1 u is conormal to a smooth characteristic hypersurface in t t 0 2 for t 0 u is singular not only at but also on the forward characteristic cone over the swallowtail tip

combinatorial hopf algebras noncommutative hall littlewood functions and permutation tableaux abstract we introduce a new family of noncommutative analogues of the hall littlewood symmetric functions our construction relies upon tevlin s bases and simple q deformations of the classical combinatorial hopf algebras we connect our new hall littlewood functions to permutation tableaux and also give an exact formula for the q enumeration of permutation tableaux of a fixed shape this gives an explicit formula for the steady state probability of each state in the partially asymmetric exclusion process pasep the polynomial enumerating permutations with a fixed set of weak excedances according to crossings the polynomial enumerating permutations with a fixed set of descent bottoms according to occurrences of the generalized pattern 2 31

a decomposition of the descent algebra of the hyperoctahedral group elements of the hyperoctahedral group bn can be represented as lists of integers 1 2 n where the list of absolute value of the k s is a permutation of the integers from 1 to n the descent set of such a is des k 0 n 1 k k 1 where k 0 we set 0 0 a descent class as of bn is the formal sum of all elements of bn with a given descent set s 0 1 n 1 in this paper we shall give a combinatorial proof of a result of solomon 6 stating that the product in the group algebra q bn of two descent classes is a linear combination of descent classes we refer to the linear span of these descent classes it is a subalgebra bn of q bn as solomon s hyperoctahedral descent algebra we derive from our combinatorial multiplication rule for bn that there is a family of ideals jv n of bn such that j0 n j1 n jn n with bn v 1 bnjv n in 4 garsia and reutenauer relate a similar decomposition of the multiplicative structure of the descent algebra of the symmetric group sn to the action of the symmetric group on the free lie algebra it develops that one can introduce a bn analog of the free lie algebra and give similar results for bn

spreading of singularities at the boundary in semilinear hyperbolic mixed problems i microlocal hs s regularity on etudie la propagation de la regularite d ordre eleve 2s n 2 pour des equations hyperboliques semi lineaires d ordre 2 avec des conditions de dirichlet on decrit la creation et l etalement de singularites anormales dues aux interactions a la frontiere

local well posedness of the yang mills equation in the temporal gauge below the energy norm abstract we show that the yang mills equation in three dimensions in the temporal gauge is locally well posed in hs for s 3 4 if the hs norm is sufficiently small the temporal gauge is slightly less convenient technically than the more popular coulomb gauge but has the advantage of uniqueness even for large initial data and does not require solving a nonlinear elliptic problem to handle the temporal gauge correctly we project the connection into curl and divergence free components and develop some new bilinear estimates of xs b type which can handle integration in the time direction

singular solutions for semilinear hyperbolic equations i on construit des solutions avec singularites de type choc pour des equations semilineaires strictement hyperboliques

the peak algebra and the descent algebras of types b and d we show the existence of a unital subalgebra n of the symmetric group algebra linearly spanned by sums of permutations with a common peak set which we call the peak algebra we show that n is the image of the descent algebra of type b under the map to the descent algebra of type a which forgets the signs and also the image of the descent algebra of type d the algebra n contains a two sided ideal n which is defined in terms of interior peaks this object was introduced in previous work by nyman 2003 we find that it is the image of certain ideals of the descent algebras of types b and d we derive an exact sequence of the form 0 n n n 2 0 we obtain this and many other properties of the peak algebra and its peak ideal by first establishing analogous results for signed permutations and then forgetting the signs in particular we construct two new commutative semisimple subalgebras of the descent algebra of dimensions n and n 2 1 by grouping permutations according to their number of peaks or interior peaks we discuss the hopf algebraic structures that exist on the direct sums of the spaces n and n over n 0 and explain the connection with previous work of stembridge 1997 we also obtain new properties of his descents to peaks map and construct a type b analog

justification of multidimensional single phase semilinear geometric optics for semilinear strictly hyperbolic systems lu f x u we construct and justify high frequency nonlinear asymptotic expansions of the form u e x e j u j x x e lu e f x u e 0 the study of the principal term of such expansions is called nonlinear geometric optics in the applied literature we show 1 formal expansions with periodic profiles u j can be computed to all orders ii the equations for the profiles from i are solvable and iii there are solutions of the exact equations which have the formal series as high frequency asymptotic expansion

a decomposition of solomon s descent algebra abstract a descent class in the symmetric group sn is the collection of permutations with a given descent set it was shown by l solomon j algebra41 1976 255 264 that the product in the group algebra q sn of two descent classes is a linear combination of descent classes thus descent classes generate a subalgebra of q sn we refer to it here as solomon s descent algebra and denote it by n this algebra is not semisimple but it has a faithul representation in terms of upper triangular matrices the main goal of this paper is a decomposition of its multiplicative structure it develops that n acts in a natural way on the so called lie monomials this action has a purely combinatorial description and is a crucial tool in the construction of a complete set of indecomposable representations of n in particular we obtain a natural basis of irreducible orthogonal idempotents indexed by partitions of n for the quotient n n natural bases of nilpotents and idempotents for the subspaces e ne for two arbitrary partitions and are also constructed and the dimensions of these spaces are given a combinatorial interpretation in terms of the so called decreasing factorization of an arbitary word into a product of lyndon words

multi valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems multi valued solutions are constructed for 2 2 first order systems using a generalization of the hodograph transformation the solution is found as a complex analytic function on a complex riemann surface for which the branch points move as part of the solution the branch point singularities are envelopes for the characteristics and thus move at the characteristic speeds we perform an analysis of stability of these singularities with respect to perturbations of the initial data the generic singularity types are folds cusps and nondegenerate umbilic points with non zero 3 jet an isolated singularity is generically a square root branch point corresponding to a fold two types of collisions between singularities are generic at a tangential collision between two singularities moving at the same characteristic speed a cube root branch point is formed corresponding to a cusp a non tangential collision between two square root branch points moving at different characteristic speeds remains a square root branch point at the collision and corresponds to a nondegenerate umbilic point these results are also valid for a diagonalizable n th order system for which there are exactly two speeds 1993 john wiley sons inc

probleme de cauchy semi lineaire en 3 dimensions d espace un resultat de finitude soit x x 0 x le point courant de l espace temps r 4 x x 1 x 2 x 3 l operateur des ondes un ouvert de r 4 x 0 0 on suppose que est un domaine d influence pour soit u x 0 x un element de h s s 2 3 1 2 qui verifie l equation des ondes semi lineaire u p u dans u x 0 0 u 0 h loc s u x 0 x 0 0 u 1 h loc s 1 on suppose que p u j 0 d p j x u j ou les p j x sont des fonctions c sur et que les traces u 0 et u 1 sont des distributions integrales de fourier c sur ou ct est une lagrangienne analytique reelle lisse pour tout er il existe un ensemble sous analytique homogene isotrope l 6 dans t tel que wf u cl

reflection and interaction of progressing waves for semilinear wave equations recently many works have been devoted to the discussion of the propagation and interaction of singularities for solutions of nonlinear hyperbolic equations among them the study of the interaction of nonlinear progressing waves is noticeable first in 11 j m bony discussed the behavior of the interaction of two progressing waves he showed that for semilinear hyperbolic equations of second order when two progressing waves intersect transversally no anomalous singularities are caused by the interaction and for semilinear hyperbolic equations of higher order anomalous progressing waves appear issuing from the intersection however the interaction is much more complicated when three progressing waves intersect transversally with each other at one point j rauch and m reed constructed an example showing that in the triple intersection case besides the linear predicted singularities a new progressing wave caused by interaction issuing from the intersection point propagates along the characteristic cone recently r melrose and n ritter 3 j m bony 141 and m beals 5 discussed the general triple intersection of progressing waves for semilinear wave equations with two space variables by different methods they pointed out that in the generalized case the singularities have the same structure as that in rauchhreed s example on the other hand m beals and g metivier considered reflection of progressing waves for boundary value problems of semilinear hyperbolic equations on the basis of these works we study in this paper the interaction of two progressing waves with reflection on the boundary for boundary value problems of semilinear wave equations in two space variables our result shows that in this case beside the linear predicted singularities there appears an anomalous reflected wave which issues from the intersection

multilinear weighted convolution of l 2 functions and applications to nonlinear dispersive equations the x s b spaces as used by beals bourgain kenig ponce vega klainerman machedon and others are fundamental tools to study the low regularity behavior of nonlinear dispersive equations it is of particular interest to obtain bilinear or multilinear estimates involving these spaces by plancherel s theorem and duality these estimates reduce to estimating a weighted convolution integral in terms of the l 2 norms of the component functions in this paper we systematically study weighted convolution estimates on l 2 as a consequence we obtain sharp bilinear estimates for the kdv wave and schrodinger x s b spaces

analytic k theory on manifolds with corners let y be a compact manifold in 1 atiyah proposed an analytic delinition of the k homology groups k y defined abstractly as the dual theory to the atiyah hirzebruch k cohomology theory these ideas were developed by kasparov 15 and brown douglas and fillmore lo cycles for this analytic k homology theory are abstract elliptic operators i e bounded linear operators between hilbert spaces satisfying additional conditions derived from the properties of elliptic pseudo differential operators in particular elliptic pseudodifferential operators between sections of vector bundles define such cycles the homology class of the cycle depends only on the principal symbol and indeed only on the k cohomology class it represents on the cotangent bundle in fact it had already been shown in essence by atiyah that the resulting map

the shifted poirier reutenauer algebra based on the shifted schensted correspondence and the shifted knuth equivalence a shifted analog of the poirier reutenauer algebra is introduced it is a right coideal subalgebra of the poirier reutenauer algebra and turns out to be a higher lift of schur s p functions its close relations with the peak subalgebra and the stembridge algebra of peak functions are also uncovered

the cauchy problem for semilinear hyperbolic systems with discontinuous data on demontre l existence a t court des solutions pour le probleme de cauchy strictement hyperbolique lu t u l 1 n a l t x xt u f t x u t x u t 0 g ou g est lisse par morceaux avec des sauts seulement sur une hypersurface t r n

ondes semi lineaires conormales par rapport a deux hypersurfaces transverses on considere un systeme semi lineaire d ordre 1 en dimension 2 ou une equation scalaire d ordre 2 dans un ouvert de r n soient 1 2 deux hypersurfaces caracteristiques se coupant transversalement selon une sous variete de codimension 2 les bicaracteristiques sur 1 et 2 etant supposees transverses a l arete on etudie les symboles et des developpements asymptotiques d une solution u conormale par rapport a 1 2

duality between quasi symmetrical functions and the solomon descent algebra abstract the ring qsym of quasi symmetric functions is naturally the dual of the solomon descent algebra the product and the two coproducts of the first extending those of the symmetric functions correspond to a coproduct and two products of the second which are defined by restriction from the symmetric group algebra a consequence is that qsym is a free commutative algebra

a bilinear approach to the restriction and kakeya conjectures the purpose of this paper is to investigate bilinear variants of the restriction and kakeya conjectures to relate them to the standard formulations of these conjectures and to give applications of this bilinear approach to existing conjectures the methods used are based on several observations and results of bourgain see 2 6 together with some refinements by moyua vargas and vega 17 18 this paper is organized as follows in the first section we discuss bilinear restriction estimates and show how one can pass back and forth between these estimates and the standard restriction estimates we also generalize the 12 7 bilinear restriction estimate of 18 to higher dimensions in the second section we give analogues of the above results for the kakeya operator in particular we give a bilinear improvement to wolff s kakeya theorem in arbitrary dimension in the third section we give applications of these bilinear estimates in three dimensions for example we are able to improve the 42 11 exponent in wolff s restriction theorem to 34 9 we are also able to prove a sharp lp lq restriction theorem which improves on the classical l2 l4 tomas stein theorem and also give some concrete progress on a bilinear restriction conjecture of klainerman and machedon we also give a non bilinear approach to these estimates which gives weaker results but is more direct and probably has a wider range of application

skew schubert functions and the pieri formula for flag manifolds we show the equivalence of the pieri formula for flag manifolds with certain identities among the structure constants for the schubert basis of the polynomial ring this gives new proofs of both the pieri formula and of these identities a key step is the association of a symmetric function to a finite poset with labeled hasse diagram satisfying a symmetry condition this gives a unified definition of skew schur functions stanley symmetric functions and skew schubert functions defined here we also use algebraic geometry to show the coefficient of a monomial in a schubert polynomial counts certain chains in the bruhat order obtainng a combinatorial chain construction of schubert polynomials

nonlinear geometric optics for short pulses abstract this paper studies the propagation of pulse like solutions of semilinear hyperbolic equations in the limit of short wavelength the pulses are located at a wavefront 0 where satisfies the eikonal equation and d lies on a regular sheet of the characteristic variety the approximate solutions are u e approx u t x t x e where u t x r is a smooth function with compact support in r when u satisfies a familiar nonlinear transport equation from geometric optics it is proved that there is a family of exact solutions u e exact such that u e approx has relative error o e as e 0 while the transport equation is familiar the construction of correctors and justification of the approximation are different from the analogous problems concerning the propagation of wave trains with slowly varying envelope

interaction of nonlinear progressing waves for semilinear wave equations on etudie l interaction transversale de 3 ondes planes progressives pour une equation d onde semi lineaire a 3 dimensions

equations des ondes semi lineaires ii controle des singularites et caustiques non lineaires nous etudions le phenomene d interaction des singularites pour les equations des ondes non lineaires nous traitons en particulier l apparition generique de caustiques en dimension 2 d espace

progressing wave solutions to certain nonlinear mixed problems on considere des equations non lineaires hyperboliques on etudie la propagation de la regularite quand la solution du probleme mixte est anormale par rapport a une seule hypersurface caracteristique dans le passe

singularities produced in conormal wave interactions three problems on the interactions of conormal waves are considered two are examples which demonstrate that nonlinear spreading of singularities can occur when the waves are conormal in one case two of the waves are tangential and the other wave is transversal to the first two the third result is a noninteraction theorem it is shown that under certain conditions no nonlinear spreading of the singularities will occur

testing sets for properness of polynomial mappings in this paper we characterize testing sets for properness of polynomial mappings f mathbb c n rightarrow mathbb c m we also study the set of points at which such mappings are not proper as the first application we give a proof of the complementary conjecture of mckay and wang conjecture 12 in 16 the second application is an answer to the problem of kraft russell about a characterization of mathbb c n the third application is given in 13 a solution of the problem of van den essen and shpilrain about endomorphism of the ring mathbb c x 1 x n the fourth application is a theorem about extensions of affine varieties

a symmetry of the descent algebra of a finite coxeter group abstract the descent algebra d w of a finite coxeter group w discovered by solomon in 1976 is a subalgebra of the group algebra of w due to solomon it is intimately linked to the representation theory of w by means of a homomorphism of algebras mapping d w into the algebra of class functions of w for w of type a jollenbeck and reutenauer derived the identity x y y x for all x y d w where class functions of w have been extended to the group algebra of w linearly they conjectured that this symmetry property of d w holds for arbitrary finite coxeter groups w this conjecture actually a combinatorial refinement is proven here as a consequence several properties of the characters of w afforded by the primitive idempotents of d w may be derived at once including a symmetry of the corresponding character table and a combinatorial description of their intertwining numbers with the descent characters of w this recovers and extends results of gessel reutenauer and scharf thibon on the symmetric group and of poirier on the hyperoctahedral group

self spreading and strength of singularities for solutions to semilinear wave equations algebres microlocales propagation de la regularite solutions lineaires conduisant a l etalement auto etalement des singularites

global singularity structure of weak solutions to 3 d semilinear dispersive wave equations with discontinuous initial data abstract we study the global singularity structure of solutions to 3 d semilinear wave equations with discontinuous initial data more precisely using strichartz inequality we show that the solutions stay conormal after nonlinear interaction if the cauchy data are conormal along a circle

microlocal analysis in nonlinear thermoelasticity abstract this paper is devoted to the study of the propagation of singularities for semilinear hyperbolic parabolic coupled systems in thermoelasticity first a linear transformation involving pseudodifferential operators for unknowns is introduced to decouple the hyperbolic and the parabolic operators by using the decoupled system we obtain that the semilinear system of thermoelasticity has finite speeds for the propagation of singularities furthermore it is proved that the microlocal singularities of solutions to this semilinear hyperbolic parabolic coupled system are propagated along null bicharacteristics of the hyperbolic operator by using the theory of nonsmooth pseudodifferential operators finally the cauchy problem for the semilinear system of thermoelasticity is investigated when the initial data have singularities

global rough solutions to the critical generalized kdv equation abstract we prove that the initial value problem ivp for the critical generalized kdv equation u t u x x x u 5 x 0 on the real line is globally well posed in h s r if s 3 5 with the appropriate smallness assumption on the initial data

global singularity structures of weak solutions to 4 d semilinear dispersive wave equations in this paper we are concerned with the global singularity structures of weak solutions to 4 d semilinear dispersive wave equations whose initial data are chosen to be discontinuous on the unit sphere combining strichartz s inequality with the commutator argument techniques we show that the weak solutions are c 2 regular away from the focusing cone surface x t 1 and the outgoing cone surface x t 1

interactions of conormal waves for fully semilinear wave equations an existence theorem is proved for the continuation form of the cauchy problem pu z u z u z where p is a second order strictly hyperbolic differential operator with initial data strongly conormal to three characteristic hypersurfaces intersecting at 0 and the light cone for p over 0 we assume that the hypersurfaces are either pairwisely transversal or two of them are tangent of finite order along the line of intersection by reduction to a model case conormal energy estimates are obtained for the linear equation with zero initial data the proof of the theorem follows from an extension lemma and a contraction mapping argument

calcul paradifferentiel precise et applications a des equations aux derivees partielles non semilineaires on demontre des theoremes de regularite pour une solution u suffisamment reguliere de l equation f x u u m 0 ou f est une fonction c de ses arguments dans le cas de l ordre 1 ou de la dimension 2

descent representations and multivariate statistics combinatorial identities on weyl groups of types a and b are derived from special bases of the corresponding coinvariant algebras using the garsia stanton descent basis of the coinvariant algebra of type a we give a new construction of the solomon descent representations an extension of the descent basis to type b using new multivariate statistics on the group yields a refinement of the descent representations these constructions are then applied to refine well known decomposition rules of the coinvariant algebra and to generalize various identities

the cauchy problem for higher order modified camassa holm equations on the circle abstract in this paper we investigate the cauchy problem for the shallow water type equation u t x 2 n 1 u 1 2 x u 2 x 1 x 2 1 u 2 1 2 u x 2 0 with low regularity data in the periodic settings firstly we proved that the bilinear estimate related to the nonlinear term of the equation in space w s defined in page 5 is invalid with s n 2 1 then the locally well posed of the cauchy problem for the periodic shallow water type equation is obtained in h s t with s n 3 2 n 2 for arbitrary initial data thus our result improves the result of himonas and misiolek commun partial differ equ 23 1998 123 139 where they have proved that the problem is locally well posed for small initial data in h s t with s n 2 1 n n with the aid of the standard fourier restriction norm method