global attractor for the kirchhoff type equation with a strong dissipation abstract the paper studies the longtime behavior of the kirchhoff type equation with a strong dissipation u t t m u 2 u u t h u t g u f x it proves that the related continuous semigroup s t possesses in the phase space with low regularity a global attractor which is connected and an example is shown

long time dynamics of kirchhoff wave models with strong nonlinear damping we study well posedness and long time dynamics of a class of quasilinear wave equations with a strong damping we accept the kirchhoff hypotheses and assume that the stiffness and damping coefficients are functions of the l2 norm of the gradient of the displacement we prove the existence and uniqueness of weak solutions and study their properties for a wide class of nonlinearities which covers the case of possible degeneration or even negativity of the stiffness coefficient and the case of a supercritical source term our main results deal with global attractors for strictly positive stiffness factors we prove that in the natural energy space endowed with a partially strong topology there exists a global finite dimensional attractor in the non supercritical case this attractor is strong in this case we also establish the existence of a fractal exponential attractor and give conditions that guarantee the existence of a finite number of determining functionals

singular limits in the cauchy problem for the damped extensible beam equation abstract we study the cauchy problem of the ball model for an extensible beam t 2 u t u x 4 u t x 4 u r x u 2 d x r t x u x u d x x 2 u the aim of this paper is to investigate singular limits as 0 for this problem in the authors previous paper 8 decay estimates of solutions u to the equation in the case 0 were shown with the help of the decay estimates we describe the singular limit in the sense of the following uniform in time estimate u u 0 l 0 h 2 r c

maximal theorems for the directional hilbert transform on the plane for a schwartz function f on the plane and a non zero v r 2 define the hilbert transform of f in the direction v to be h v f x p v r f x vy dy y let be a schwartz function with frequency support in the annulus 1 2 the l 2 estimate is sharp the method of proof is based upon techniques related to the pointwise convergence of fourier series indeed our main theorem implies this result on fourier series

approximation in higher order sobolev spaces and hodge systems abstract let d 2 be an integer 1 l d 1 and be a differential l form on r d with w 1 d coefficients it was proved by bourgain and brezis 5 theorem 5 that there exists a differential l form on r d with coefficients in l w 1 d such that d d in the same work bourgain and brezis also left as an open problem the extension of this result to the case of differential forms with coefficients in the higher order space w 2 d 2 or more generally in the fractional sobolev spaces w s p with s p d we give a positive answer to this question provided that d l d 1 where is the largest positive integer such that min p d the proof relies on an approximation result interesting in its own right for functions in w s p by functions in w s p l even though w s p does not embed into l in this critical case the proofs rely on some techniques due to bourgain and brezis but the context of higher order and or fractional sobolev spaces creates various difficulties and requires new ideas and methods

attractors for the degenerate kirchhoff wave model with strong damping existence and the fractal dimension abstract in this paper we study the long time behavior of the kirchhoff type wave equation in the space h 0 1 l 2 we prove the existence of the global attractor for the equation which covers the case of possible generation of the stiffness coefficient we also consider the geometrical property of the global attractor by means of the z 2 index we provide under suitable assumptions the fractal dimension of the global attractor is infinite

hyperbolic parabolic singular perturbation for nondegenerate kirchhoff equations with critical weak dissipation we consider the hyperbolic parabolic singular perturbation problem for a nondegenerate quasilinear equation of kirchhoff type with weak dissipation this means that the dissipative term is multiplied by a coefficient b t which tends to 0 as t the case where b t 1 t p with p 1 but they remain true in the critical case p 1

hyperbolic parabolic singular perturbation for quasilinear equations of kirchhoff type abstract we consider a hyperbolic parabolic singular perturbation problem for a quasilinear equation of kirchhoff type and obtain parameter dependent time decay estimates of the difference between the solutions of a quasilinear dissipative hyperbolic equation of kirchhoff type and the corresponding quasilinear parabolic equation for this purpose we show time decay estimates for hyperbolic parabolic singular perturbation problem for linear equations with a time dependent coefficient

multi parameter singular radon transforms iii real analytic surfaces abstract the goal of this paper is to study operators of the form t f x x f t x k t d t where is a real analytic function defined on a neighborhood of the origin in t x r n r n satisfying 0 x x is a cutoff function supported near 0 r n and k is a multi parameter singular kernel supported near 0 r n a main example is when k is a product kernel we also study maximal operators of the form m f x x sup 0 1 n 1 t 1 f 1 t 1 n t n x d t we show that m is bounded on l p 1 p we give conditions on under which t is bounded on l p 1 p these conditions hold automatically when k is a calderon zygmund kernel this is the final paper in a three part series the first two papers consider the more general case when is c

square functions for bi lipschitz maps and directional operators abstract first we prove a littlewood paley diagonalization result for bi lipschitz perturbations of the identity map on the real line this result entails a number of corollaries for the hilbert transform along lines and monomial curves in the plane second we prove a square function bound for a single scale directional operator as a corollary we give a new proof of part of a theorem of katz on direction fields with finitely many directions

interfacial waves and deformations in pre stressed elastic media the propagation of interfacial stoneley waves along the boundary between two half spaces of pre stressed incompressible isotropic elastic material is examined the underlying deformation in each half space corresponds to a pure homogeneous strain with one principal axis of strain normal to the interface and the others having a common orientation the secular equation governing the wave speed for propagation along a principal axis is obtained in respect of general strain energy functions detailed analysis of the secular equation reveals general sufficient conditions for the existence of a wave and in particular cases necessary and sufficient conditions for the existence of a unique interfacial wave it is also shown that when an interfacial wave exists its speed is greater than that of the least of the rayleigh wave speeds for the separate half spaces paralleling a result from the linear theory for the special case of quasi static interfacial deformations corresponding to vanishing wave speed an existence criterion is found moreover it is shown that inequalities that exclude surface deformations in each half space also exclude interfacial deformations dependence of the above results on the underlying homogeneous deformations and on material parameters is illustrated by numerical results for the neo hookean strain energy function

necessary condition on the dissipative term for the asymptotically free property of dissipative kirchhoff equations abstract it is known that the solutions of linear abstract dissipative wave equation with integrable coefficient of the dissipation term are asymptotically free namely the solutions have the same asymptotic behavior as that of the solutions of an abstract free wave equation in this paper we conclude that the integrability is not enough for asymptotic free property of kirchhoff equation by showing the following for the existence of a non trivial asymptotically free solution of abstract kirchhoff equation with initial data in some class an essentially stronger decay condition on the coefficient of dissipation term is necessary

global solvability for the kirchhoff equations in exterior domains of dimension three abstract we consider the initial boundary value problem for the kirchhoff equations in exterior domains or in the whole space of dimension three and show that these problems admit time global solutions provided the norms of the initial data in the usual sobolev spaces of appropriate order are sufficiently small we obtain uniform estimates of the l 1 r norms with respect to time variable at each point in the domain of solutions of initial boundary value problem for the linear wave equations we then show that the estimates above yield the unique global solvability for the kirchhoff equations

mildly degenerate kirchhoff equations with weak dissipation global existence and time decay abstract we consider the hyperbolic parabolic singular perturbation problem for a degenerate quasilinear kirchhoff equation with weak dissipation this means that the coefficient of the dissipative term tends to zero when t we prove that the hyperbolic problem has a unique global solution for suitable values of the parameters we also prove that the solution decays to zero as t with the same rate of the solution of the limit problem of parabolic type

finite dimensional attractors for the kirchhoff equation with a strong dissipation abstract the paper studies the existence of the finite dimensional global attractors and exponential attractors for the dynamical system associated with the kirchhoff type equation with a strong dissipation u t t m u 2 u u t h u t g u f x it proves that the above mentioned dynamical system possesses a global attractor which has finite fractal dimension and an exponential attractor for application the fact shows that for the concerned viscoelastic flow the permanent regime global attractor can be observed when the excitation starts from any bounded set in phase space and the dimension of the attractor that is the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow is finite

a maximal function for families of hilbert transforms along homogeneous curves let h u be the hilbert transform along the parabola t ut 2 where u in mathbb r for a set u of positive numbers consider the maximal function mathcal h u f sup h u f u in u we obtain an essentially optimal result for the l p operator norm of mathcal h u when 2 p infty the results are proved for families of hilbert transforms along more general nonflat homogeneous curves

on a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain soit un domaine non borne dans un espace euclidien a n dimensions r n a frontiere compacte et c on considere le probleme mixte u u f t x d 1 d 1 u f t x dans 0 u 0 sur 0 u 0 x u 0 x t u 0 x u 1 x dans f est une fonction non lineaire a valeur reelle et d 1 d 1 u t j x u 1 j 2

an attractor for the singularly perturbed kirchhoff equation with supercritical nonlinearity abstract we consider a quasilinear wave equation of kirchhoff type u t t 1 u 2 u u t f u g x where 0 is a small parameter without any growth restrictions on the nonlinearity f u we prove the existence of a finite dimensional global attractor on an appropriate bounded phase space the key step is the estimate of the difference between the solutions of a quasilinear dissipative hyperbolic equation of kirchhoff type and the corresponding quasilinear parabolic equation

maximal operators associated to families of flat curves in the plane bourgain showed that if c has non vanishing curvature the inequality 1 1 holds for p 2 see b in this paper we shall consider a situation when the curvature is allowed to vanish of finite order on a finite set of isolated points we shall need the following definition definition1 1 let c i r where i is a compact interval in r and c is smooth we say that c is finite type if c x c x0 does not vanish of infinite order for any x0 i and any unit vector we shall also need a more precise definition which would specify the order of vanishing at each point let a0 denote a point in the compact interval i we can always find a smooth function such that in a small neighborhood of a0 c s s s where s i definition1 2 let c be defined as before let c s s s in a small neighborhood of a0 we say that c is finite type m at a0 if a0 0 for 1 k m and a0 6 0 our main result is the following theorem 1 1 let c be a finite type curve which is finite type m at a0 let mtf x

hyperbolic parabolic singular perturbation for mildly degenerate kirchhoff equations time decay estimates we consider degenerate kirchhoff equations with a small parameter e in front of the second order time derivative it is well known that these equations admit global solutions when e is small enough and that these solutions decay as t with the same rate of solutions of the limit problem of parabolic type in this paper we prove decay error estimates for the difference between a solution of the hyperbolic problem and the solution of the corresponding parabolic problem these estimates show in the same time that the difference tends to zero both as e 0 and as t concerning the decay rates it turns out that the difference decays faster than the two terms separately as t proofs involve a nonlinear step where we separate fourier components with respect to the lowest frequency followed by a linear step where we exploit weighted versions of classical energies

singular and maximal radon transforms analysis and geometry part 2 geometric theory 8 curvature introduction 8 1 three notions of curvature 8 2 theorems 8 3 examples 9 curvature some details 9 1 the exponential representation 9 2 diffeomorphism invariance 9 3 curvature condition cy 9 4 two lemmas 9 5 double fibration formulation 10 equivalence of curvature conditions 10 1 invariant submanifolds and deficient lie algebras 10 2 vanishing jacobians 10 3 construction of invariant submanifolds

a gronwall type lemma with parameter and its application to kirchhoff type nonlinear wave equation abstract the paper investigates a new gronwall type lemma with parameter 0 and establishes the uniform in time estimates as an application the paper studies the asymptotic behavior of the kirchhoff type equations with a strong dissipation and proves that the related solution semigroup possesses the global attractor

wave front sets local smoothing and bourgain s circular maximal theorem the purpose of this paper is to improve certain known regularity results for the wave equation and to give a simple proof of bourgain s circular maximal theorem 1 we use easy wave front analysis along with techniques previously used in proofs of the carleson sj6lin theorem see 3 5 7 and in the proof of sharp regularity properties of fourier integral operators 13 the circular means operators are defined by

on the boundedness of the bilinear hilbert transform along non flat smooth curves we are proving l 2 bbb r times l 2 bbb r rightarrow l 1 bbb r bounds for the bilinear hilbert transform h gamma along curves gamma t gamma t with gamma being a smooth non flat curve near zero and infinity

a class of non analytic functions for the global solvability of kirchhoff equation abstract we consider the global solvability to the cauchy problem of kirchhoff equation with generalized classes of manfrin s class manfrin s class is a subclass of sobolev space but we shall extend this class as a subclass of the ultradifferentiable class and we succeed to prove the global solvability of kirchhoff equation with large data in wider classes from the previous works

a uniqueness result for kirchhoff equations with non lipschitz nonlinear term abstract we consider the second order cauchy problem u m a 1 2 u 2 a u 0 u 0 u 0 u 0 u 1 where m 0 0 is a continuous function and a is a self adjoint nonnegative operator with dense domain on a hilbert space it is well known that this problem admits local in time solutions provided that u 0 and u 1 are regular enough depending on the continuity modulus of m it is also well known that the solution is unique when m is locally lipschitz continuous in this paper we prove that if either a u 0 u 1 0 or a 1 2 u 1 2 m a 1 2 u 0 2 a u 0 2 then the local solution is unique even if m is not lipschitz continuous

the proof of the l 2 decoupling conjecture we prove the l 2 decoupling conjecture for compact hypersurfaces with positive denite second fundamental form and also for the cone this has a wide range of important consequences one of them is the validity of the discrete restriction conjecture which implies the full range of expected l p strichartz estimates for both the rational and up to n losses the irrational torus another one is an improvement in the range for the discrete restriction theorem for lattice points on the sphere various applications to additive combinatorics incidence geometry and number theory are also discussed our argument relies on the interplay between linear and multilinear restriction theory

optimal decay error estimates for the hyperbolic parabolic singular perturbation of a degenerate nonlinear equation abstract we consider a degenerate hyperbolic equation of kirchhoff type with a small parameter e in front of the second order time derivative in a recent paper under a suitable assumption on initial data we proved decay error estimates for the difference between solutions of the hyperbolic problem and the corresponding solutions of the limit parabolic problem these estimates show in the same time that the difference tends to zero both as e 0 and as t in particular in that case the difference decays faster than the two terms separately in this paper we consider the complementary assumption on initial data and we show that now the optimal decay error estimates involve a decay rate which is slower than the decay rate of the two terms in both cases the improvement or deterioration of decay rates depends on the smallest frequency represented in the fourier components of initial data

an attractor for a nonlinear dissipative wave equation of kirchhoff type abstract in this paper we prove the existence and some absorbing properties of an attractor in a local sense for the initial boundary value problem of a quasilinear wave equation of kirchhoff type with a standard dissipation u t

maximal operators and hilbert transforms along variable non flat homogeneous curves we prove that the maximal operator associated with variable homogeneous planar curves t ut t r 1 positive is bounded on lp r2 for each p 1 under the assumption that u r2 r is a lipschitz function furthermore we prove that the hilbert transform associated with t ut t r 1 positive is bounded on lp r2 for each p 1 under the assumption that u r2 r is a measurable function and is constant in the second variable our proofs rely on stationary phase methods tt arguments local smoothing estimates and a pointwise estimate for taking averages along curves

acceleration waves in elasto plasticity a spectral analysis of the acceleration wave problem in general elasto plastic materials is carried out whereby explicit expressions for the eigenvalues and eigenvectors are obtained in case of nonassociated plasticity all eigenvectors become nonorthogonal and one eigenvalue always remains unchanged and equal to the shear modulus for a very broad class of nonassociated plasticity models it is shown that the eigenvalues are always real implying that so called divergence instability can occur while flutter instability can never occur it is found that a certain value of the hardening modulus exists for which specific propagation directions will always imply that all wave speeds are identical and equal to the elastic distortion wave speed moreover in this situation the eigenvectors are arbitrary corresponding to a state of diffuse wave modes the criteria of von mises and rankine are used to illustrate some of the findings

long time existence and singular perturbation results for quasilinear hyperbolic equations with small parameter and dissipation term iii on considere des solutions lisses du probleme aux valeurs initiales hyperbolique quasilineaire eu tt u t ij 1 n a ij i j u 0 u 0 u 0 u t 0 u 1 ou e 0 0 u u x t r x r n on etudie l existence de solutions globales correspondant a des donnees arbitrairement grandes initiales u 0 et u 1 et a leur convergence quand e 0 vers la solution du probleme parabolique correspondant

an algorithm and a method to search bifurcation points in non linear problems this paper is devoted to a numerical method able to help the determination of the bifurcation threshold in non linear time independent continuum mechanic problems first some theoretical results about uniqueness are recalled in the framework of the large strain assumption the differences between the classical finite step problem and the rate problem are presented an iterative algorithm able to solve the rate problem is given using different initializations it is seen in some numerical experiments that it is possible with this algorithm to get different solutions when the underlying mathematical problem solved does not enjoy a uniqueness property the constitutive equations used have been chosen to be simple enough to deduce some theoretical knowledge about the corresponding uniqueness problems finally a method is given which is able in some case to give an upper bound of the bifurcation threshold copyright 2001 john wiley sons ltd

surface boundary conditions trigger flutter instability in non associative elastic plastic solids abstract it is shown that surface flutter instability may be triggered by the simultaneous influence of non associativity and boundary conditions even if taken independently neither the elastic plastic constitutive law satisfying deviatoric associativity nor the boundary conditions no applied traction rates would lead to flutter more specifically it is shown that for orthotropic elastic plastic constitutive tensors with an orthotropy axis tangent to the rate traction free boundary the onset of surface flutter instability coincides with the incipience of plasticity for any non associative flow rule whenever the normal to the boundary does not coincide with one of the other two orthotropy axes in the case of deviatoric associativity and coaxiality between the directions of orthotropy and the normal and tangent to the rate traction free boundary surface flutter instability may also occur but only for unusual values of material parameters additional results and discussion are also presented for the onset of stationary surface waves in associative and non associative elastic plastic bodies in contrast to the onset of surface flutter instabilities the condition for the onset of stationary surface waves involves material properties only that is it does not discriminate among different orientations of the rate traction free boundary with respect to the material orthotropy directions

l p regularity of averages over curves and bounds for associated maximal operators we prove that for a finite type curve in r 3 the maximal operator generated by dilations is bounded on if for sufficiently large p we also show the endpoint lp l p i p regularity result for the averaging operators for large p the proofs make use of a deep result of thomas wolff about decompositions of cone multipliers