quantum variance and ergodicity for the baker s map we prove an egorov theorem or quantum classical correspondence for the quantised baker s map valid up to the ehrenfest time this yields a logarithmic upper bound for the decay of the quantum variance and as a corollary a quantum ergodic theorem for this map

asymptotic behavior for the viscous burgers equation with a stationary source long time asymptotic behavior for the viscous burgers equation on the real line is considered when there is a non negative and compactly supported radon measure as a stationary source we prove that solutions of the viscous burgers equation converge to a positive bounded and nondecreasing steady state by finding an almost optimal convergence order the non integrability of the steady state only allows local convergence on compact subsets hence a veron type argument must be modified by adopting a proper weight function

asymptotic agreement of moments and higher order contraction in the burgers equation the purpose of this paper is to investigate the relation between the moments and the asymptotic behavior of solutions to the burgers equation the burgers equation is a special nonlinear problem that turns into a linear one after the cole hopf transformation our asymptotic analysis depends on this transformation in this paper an asymptotic approximate solution is constructed which is given by the inverse cole hopf transformation of a summation of n heat kernels the k th order moments of the exact and the approximate solution are contracting with order o t k 2n 1 1 p in lp norm as t this asymptotics indicates that the convergence order is increased by a similarity scale whenever the order of controlled moments is increased by one the theoretical asymptotic convergence orders are tested numerically

blowup result for a type of generalized zakharov system abstract we study the blowup for a type of generalized zakharov system in this paper it is proved that the solution of such system either blows up in finite time or blows up in infinite time provided that the initial energy is negative

quantum ergodicity of c dynamical systems we define a notion of quantum or non commutative ergodicity for a class ofc dynamical systems a g which we callquantized gns systems such a system possesses a natural classical limit state which induces a classical limit system by the gns construction the criterion for quantum ergodicity is that the time average a of an observable a a equals the space average a i plus an errork which is negligible in the classical limit we prove that ergodicity of is a sufficient condition for quantum ergodicity of a g if the classical limit system is abelian give a conditional converse and discuss a number of applications

scarred eigenstates for quantum cat maps of minimal periods in this paper we construct a sequence of eigenfunctions of the quantum arnold s cat map that in the semiclassical limit shows a strong scarring phenomenon on the periodic orbits of the dynamics more precisely those states have a semiclassical limit measure that is the sum of 1 2 the normalized lebesgue measure on the torus plus 1 2 the normalized dirac measure concentrated on any a priori given periodic orbit of the dynamics it is known the schnirelman theorem that most sequences of eigenfunctions equidistribute on the torus the sequences we construct therefore provide an example of an exception to this general rule our method of construction and proof exploits the existence of special values of h for which the quantum period of the map is relatively short and a sharp control on the evolution of coherent states up to this time scale we also provide a pointwise description of these states in phase space which uncovers their hyperbolic structure in the vicinity of the fixed points and yields more precise localization estimates

blow up of h1 solution for the nonlinear schrodinger equation we consider the blow up of the solution in h1 for the following nonlinear schrodinger equation i tu u u p 1u x rn t 0 u 0 x u0 x x rn t 0 where n 2 and 1 4 n p min n 2 n 2 5 we prove that if the initial data u0 in h1 are radially symmetric and have negative energy then the solution of in h1 blows up in finite time we do not assume that xu0 l2 and therefore our result is the generalization of the results of glassey 4 and m tsutsumi 18 for the radially symmetric case

large time behavior for convection diffusion equations in rn abstract we describe the large time behavior of solutions of the convection diffusion equation u t u a u q 1 u in 0 r n with a r n and q 1 1 n n 1 when q 1 1 n we prove that the large time behavior of solutions with initial data in l 1 r n is given by a uniparametric family of self similar solutions the relevant parameter is the mass of the solution that is conserved for all t our result extends to dimensions n 1 well known results on the large time behavior of solutions for viscous burgers equations in one space dimension the proof is based on la salle s invariance principle applied to the equation written in its self similarity variables when q 1 1 n the convection term is too weak and the large time behavior is given by the heat kernel in this case the result is easily proved applying standard estimates of the heat kernel on the integral equation related to the problem

hecke theory and equidistribution for the quantization of linear maps of the torus 1 1 background one of the key issues of quantum chaos is the nature of the semiclassical limit of eigenstates of classically chaotic systems when the classical system is given by the geodesic flow on a compact riemannian manifoldm or rather on its cotangent bundle one can formulate the problem as follows the quantum hamiltonian is in suitable units represented by the positive laplacian on m to measure the distribution of its eigenstates we start with a smooth classical observable that is a smooth function on the unit cotangent bundle s m via some choice of quantization from symbols to pseudodifferential operators we form its quantization op f this is a zero order pseudodifferential operator with principal symbol f the expectation value of op f in the eigenstate is op f let j be a sequence of normalized eigenfunctions j j j 0 m j 2 1 the problem then is to understand the possible limits as j of the distributions f c s m op f j j 1 1

on the distribution of matrix elements for the quantum cat map for many classically chaotic systems it is believed that the quantum wave functions become uniformly distributed that is the matrix elements of smooth observables tend to the phase space average of the observable in this paper we study the fluctuations of the matrix elements for the desymmetrized quantum cat map we present a conjecture for the distribution of the normalized matrix elements namely that their distribution is that of a certain weighted sum of traces of independent matrices in su 2 this is in contrast to generic chaotic systems where the distribution is expected to be gaussian we compute the second and fourth moment of the normalized matrix elements and obtain agreement with our conjecture

on the cauchy problem for the zakharov system abstract we study the local cauchy problem in time for the zakharov system 1 1 and 1 2 governing langmuir turbulence with initial data u 0 n 0 t n 0 h k h lscr h l 1 in arbitrary space dimension we define a natural notion of criticality according to which the critical values of k l are 2 3 2 2 2 using a method recently developed by bourgain we prove that the zakharov system is locally well posed for a variety of values of k l the results cover the whole subcritical range for 4 for 3 they cover only part of it and the lowest admissible values are k l 1 2 0 for 2 3 and k l 0 1 2 for 1 as a by product of the one dimensional result we prove well posedness of the benney system 1 14 and 1 15 governing the interaction of short and long waves for the same values of k l

the posedness of the periodic initial value problem for generalized zakharov equations abstract in this paper firstly we study the periodic initial value problem of generalized zakharov equations if 0 p 4 we obtain the existence and uniqueness of the global classical solution by means of galerkin method and integral estimates secondly we study the periodic initial boundary value problem of generalized zakharov equations if p 2 1 a 2 e 0 2 1 3 a 2 e 0 1 a 2 e 0 and initial data satisfy some conditions we prove that these have blowup of local solution

classical limit of the quantized hyperbolic toral automorphisms the canonical quantization of any hyperbolic symplectomorphisma of the 2 torus yields a periodic unitary operator on an dimenional hilbert space n 1 h we prove that this quantum system becomes ergodic and mixing at the classical limit n n prime which can be interchanged with the time average limit the recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the lebesgue measure the wigner functions of the eigenstates supported on the classical periodic orbits are indeed proved to become uniformly speread in phase space

blow up and nonlinear instability for the magnetic zakharov system abstract this study deals with the generalized zakharov system with magnetic field first of all we construct a kind of blow up solutions and establish the existence of blow up solutions to the system through considering an elliptic system next we show the nonlinear instability for a kind of periodic solutions in addition we consider the concentration properties of blow up solutions for the system under study at the end of this paper we establish the global existence of weak solutions to the cauchy problem of the system under consideration

a local riemann hypothesis i daniel bump1 kwok kwong choi2 par kurlberg3 jeffrey vaaler4 1 department of mathematics stanford university stanford ca 94305 2125 usa e mail bump math stanford edu 2 department of mathematics the university of hong kong pokfulam road hong kong hong kong e mail choi maths hku hk 3 department of mathematics university of georgia athens ga 30602 usa e mail kurlberg math uga edu 4 university of texas at austin department of mathematics rlm 8 100 austin tx 78712 1082 usa e mail vaaler math utexas edu

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus we look at the expectation values for quantized linear symplectic maps on the multidimensional torus and their distribu tion in the semiclassical limit we construct super scars that are stable under the arithmetic symmetries of the system and localize on invariant manifolds we show that these super scars exist only when there are isotropic rational subspaces invariant under the linear map in the case where there are no such scars we com pute the variance of the fluctuations of the matrix elements for the desymmetrized system and present a conjecture for their limiting distributions

asymptotic behavior of solutions to viscous conservation laws with slowly varying external forces this paper considers the existence and large time behavior of solutions to the convection diffusion equation ut u b x u u q 1 f x t in n 0 where f x t is slowly decaying and q 1 1 n or in some particular cases q 1 the initial condition u0 is supposed to be in an appropriate lp space uniform and nonuniform decay of the solutions will be established depending on the data and the forcing term

an explicit solution of burgers equation with stationary point source existence uniqueness and regularity of the global weak solution to the burgers equation with a reaction term is shown when the reaction term is given as a time independent point source and produces heat constantly an explicit solution is obtained and used to show the long time asymptotic convergence of the solution to a steady state for the heat equation case without any convection the solution diverges everywhere as time increases and hence it is the first order convection term that gives the compactness of the solution trajectory of the burgers equation with reaction

nonexistence of positive solutions of nonlinear elliptic systems with potentials vanishing at infinity abstract we study the nonexistence of positive solutions for nonlinear elliptic systems with potentials vanishing at infinity and establish the optimal vanishing order of potentials for the nonexistence of positive supersolutions in exterior domains