the milne and kramers problems for the boltzmann equation of a hard sphere gas existence uniqueness and properties of asymptotic behavior are proved for solutions of the milne and kramers problems for the linearized boltzmann equation for a gas of hard spheres the proof uses energy like estimates and follows the analysis of bardos santos and sentis for the equations of neutron transport

critical exponent for evolution equations in modulation spaces abstract in this paper we propose a method to find the critical exponent for certain evolution equations in modulation spaces we define an index s q and use it to determine the critical exponent of the fractional heat equation as an example we prove that when s q is greater than the critical exponent this equation is locally well posed in the space c 0 t m p q s and when s q is less than the critical exponent this equation is ill posed in the space c 0 t m 2 q s our method may further be applied to some other evolution equations

the global cauchy problem for the nls and nlkg with small rough data abstract by using the unit cube decomposition to the frequency spaces we study the cauchy problem for the nonlinear schrodinger equation and the nonlinear klein gordon equation some global well posedness results are obtained for the small cauchy data in some modulation spaces m p 1 s

nonlinear boundary layers of the boltzmann equation i existence we study the half space problem of the nonlinear boltzmann equation assigning the dirichlet data for outgoing particles at the boundary and a maxwellian as the far field we will show that the solvability of the problem changes with the mach number of the far maxwellian if 1 respectively we also show that the same is true for the linearized problem at the far maxwellian and the manifold is then a hyperplane the proof is essentially based on the macro micro or hydrodynamics kinetic decomposition of solutions combined with an artificial damping term and a spatially exponential decay weight

frequency uniform decomposition method for the generalized bo kdv and nls equations abstract sufficient and necessary conditions for the embeddings between besov spaces b p q s 1 and modulation spaces m p q s 2 are obtained moreover using the frequency uniform decomposition method we study the cauchy problem for the generalized bo kdv and nls equations for which the global well posedness of solutions with the small rough data in certain modulation spaces m 2 1 s is shown

polymer gas approach to n body lattice systems we give a simple proof based only on combinatorial arguments of the koteck preiss condition for the convergence of the cluster expansion then we consider spin systems with long range n body interactions we prove directly using the polymer gas representation that the pressure may be written in terms of an absolutely convergent series uniformly in the volume when the interaction is summable in a suitable sense we also give an estimate of this radius of convergence in order to get the proof we use a method introduced by cassandro and olivieri in the early 1980s we apply this method to various concrete examples

propagation of a stationary shock layer in the presence of a boundary we are interested in the large time behavior of a stationary shock layer in the presence of a boundary effect it is shown that the stationary shock propagates away from the boundary with speed inversely proportional to the time t consequently the location of the wave front is of order log thinspace 1 t we show therefore that the zero viscosity and time asymptotic limits do not commute our analysis involves detailed point wise estimates which allow us to trace the shock

well posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces abstract in this paper we prove the local and global well posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces p m q s n m q s 2 under certain conditions of s q q and

sharp well posedness and ill posedness results for a quadratic non linear schrodinger equation abstract we establish that the quadratic non linear schrodinger equation iu t u xx u 2 where u r r c is locally well posed in h s r when s 1 and ill posed when s 1 previous work in c kenig g ponce l vega quadratic forms for the 1 d semilinear schrodinger equation trans amer math soc 346 1996 3323 3353 had established local well posedness for s 3 4 the local well posedness is achieved by an iteration using a modification of the standard x s b spaces the ill posedness uses an abstract and general argument relying on the high to low frequency cascade present in the non linearity and a computation of the first non linear iterate

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

a new proof of ruelle s superstability bounds for a slightly stronger assumptions on the interaction we give a very transparent proof of ruelle s result in the language of poisson integral measure representation for the correlation functions on the configuration space using some kind of cluster expansion in the densities of configurations

navier stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices abstract the cauchy problems for navier stokes equations and nonlinear heat equations are studied in modulation spaces m q s r n though the case of the derivative index s 0 has been treated in our previous work the case s 0 is also treated in this paper our aim is to reveal the conditions of s q and of m q s r n for the existence of local and global solutions for initial data u 0 m q s r n

remark on uniqueness of mild solutions to the navier stokes equations abstract we investigate a limiting uniqueness criterion to the navier stokes equations we prove that the mild solution is unique under the class c 0 t bmo 1 l loc 0 t l where bmo 1 is the critical space including l n as an application of uniqueness theorem we also consider the local well posedness of navier stokes equations in bmo 1

continuity properties for modulation spaces with applications to pseudo differential calculus i we discuss continuity for weighted modulation spaces andprove that many such spaces can be obtained in a canonicalway from the corresponding standard modulation spaces we also discussthe trace operator a a 0 acting on modulationspaces the results are used to get inclusions betweenmodulation spaces and besov spaces and proving continuityfor pseudo differential operators and toeplitz operators

the dilation property of modulation spaces and their inclusion relation with besov spaces abstract we consider the dilation property of the modulation spaces m p q let d f t f t be the dilation operator and we consider the behavior of the operator norm d m p q m p q with respect to our result determines the best order for it and as an application we establish the optimality of the inclusion relation between the modulation spaces and besov spaces which was proved by toft j toft continuity properties for modulation spaces with applications to pseudo differential calculus i j funct anal 207 2004 399 429

initial boundary value problem for one dimensional wave solutions of the boltzmann equation tai ping liu and his collaborator shih hsien yu solve the initial boundary value problem for the boltzmann equation because of the inclusion of microscopic velocity in the kinetic theory the boltzmann equation can model richer and physically more realistic boundary effects more so than the fluid equations such as the navier stokes this motivates our study the reflection of fluid waves off the boundary is indicated in the figure for the case only one of the euler wave velocities is positive

isometric decomposition operators function spaces ep qλ and applications to nonlinear evolution equations abstract by using the isometric decomposition to the frequency spaces we will introduce a new class of function spaces e p q which is a subspace of gevrey 1 class g 1 r n c r n for 0 and we will study the cauchy problem for the nonlinear schrodinger equation the complex ginzburg landau equation and the navier stokes equation some well posed results are obtained for the cauchy data in e 2 1 0 and the regularity behavior in e 2 1 ct g 1 r n for the complex ginzburg landau equation and the navier stokes equation is also obtained as time t 0

the cauchy problem for the generalized zakharov kuznetsov equation on modulation spaces abstract we consider the cauchy problem for the generalized zakharov kuznetsov equation t u x 1 u x 1 u m 1 on three and higher dimensions we mainly study the local well posedness and the small data global well posedness in the modulation space m 2 1 0 r n for m 4 and n 3 we also investigate the quartic case i e m 3

poisson measure representation and cluster expansion in classical statistical mechanics a new representation for distribution functions of the grand canonical ensemble by the poisson measure functional integral is obtained due to the ultralocal nature of the measure the construction of the cluster expansion is very simple for the convergence of the cluster expansion the requirement of exponential decay of the interaction potential is not necessary

scattering problem for local perturbations newline of the free quantum gas scattering theory for perturbations of the intrinsic dirichlet laplace beltrami operator h 0 div on l 2 z i e the space of z square integrable functions on the configuration space over d is studied here z denotes poisson measure with intensity z we show that for an arbitrary regular non zero potential v the standard wave operators w h 0 h 0 v do not exist and propose to consider dirichlet operators of perturbed poisson measures instead of potential perturbations of the hamiltonian h 0 as case studies cylindric smooth densities and finite volume gibbs perturbations of the poisson measure are considered in these cases the existence of the corresponding wave operators is proved

boltzmann equation micro macro decompositions and positivity of shock profiles we introduce an elementary energy method for the boltzmann equation based on a decomposition of the equation into macroscopic and microscopic components the decomposition is useful for the study of time asymptotic stability of nonlinear waves the wave location is determined by the macroscopic equation the microscopic component has an equilibrating property the coupling of macroscopic and microscopic components gives rise naturally to the dissipations similar to those obtained by the chapman enskog expansion our main result is the establishment of the positivity of shock profiles for the boltzmann equation this is shown by the time asymptotic approach and the maximal principle for the collision operator

navier stokes equations with external forces in lorentz spaces and its application to the self similar solutions abstract we show existence theorem of global mild solutions with small initial data and external forces in lorentz spaces with scaling invariant norms if the initial data have more regularity in another scaling invariant class then our mild solution is actually the strong solution the result on local existence of solutions for large data is also discussed our method is based on the maximal regularity theorem on the stokes equations in lorentz spaces then we apply our theorem to prove existence of self similar solutions provided both initial data and external forces are homogeneous functions since we construct the global solution by means of the implicit function theorem as a byproduct its stability with respect to the given data is necessarily obtained

the gas phase of continuous systems of hard spheres interacting via n body potential we show that a system of classical continuous hard spheres interacting through a general n body potential satisfying suitable integrability conditions admits a high temperature low activity gas phase we find explicitly a condition on the activity and the inverse temperature which ensures that the mayer series for the pressure is absolutely convergent uniformly in the volume

a classification of well posed kinetic layer problems in the first part of this paper we study the half space boundary value problem for the boltzmann equation with an incoming distribution obtained when considering the boundary layer arising in the kinetic theory of gases as the mean free path tends to zero we linearize it about a drifting maxwellian and prove that as conjectured by cercignani 4 the problem is well posed when the drift velocity u exceeds the sound speed c but that one respectively four five additional conditions must be imposed when 0 u c respectively c u 0 and u c in the second part we show that the well posedness and the asymptotic behavior results for kinetic layers equations with prescribed incoming flux can be extended to more general and realistic boundary conditions

pointwise convergence to shock waves for viscous conservation laws we are interested in the pointwise behavior of the perturbations of shock waves for viscous conservation laws it is shown that besides a translation of the shock waves and of linear and nonlinear diffusion waves of heat and burgers equations a perturbation also gives rise to algebraically decaying terms which measure the coupling of waves of different characteristic families our technique is a combination of time asymptotic expansion construction of approximate green functions and analysis of nonlinear wave interactions the pointwise estimates yield optimal lp convergence of the perturbation to the shock and diffusion waves 1 p the new approach of obtaining pointwise estimates based on the green functions for the linearized system and the analysis of nonlinear wave interactions is also useful for studying the stability of waves of distinct types and nonclassical shocks these are being explored elsewhere 1997 john wiley sons inc

generalized poisson functionals with the aim of treating nonlinear systems with inputs being discrete and outputs being generalized functions generalized poisson functional are defined and analysed where the transforms and the renormalizational play essential roles for poisson functionals the differential operators with respect to a poisson white noise dot p t their adjoint operators and the multiplication operators by dot p t are defined since these operators involve the time parameter explicitly they can be used to obtain information concerning the poisson functional at each point in time as an example a new method for measuring the wiener kernels of such functionals is outlined

α modulation spaces ii derivative nls abstract the current paper is a continuation of our early work 11 we use the techniques and properties of modulation spaces developed in 11 to study the cauchy problem for the derivative nonlinear schrodinger equations by applying some new type of inhomogeneous linear estimate to control the high low frequency interaction appearing from the nonlinearity we obtain the critical regularity index for the global well posedness with small initial data in modulation spaces for one type of derivative nonlinear schrodinger equations we also study the generalized derivative schrodinger equations

ill posedness for the navier stokes equations in critical besov spaces b q 1 abstract we study the cauchy problem for the incompressible navier stokes equations in two and higher spatial dimensions 0 1 u t u u u p 0 div u 0 u 0 x u 0 for arbitrarily small 0 we show that the solution map u 0 u in critical besov spaces b q 1 q 1 2 is discontinuous at origin it is known that the navier stokes equation is globally well posed for small data in bmo 1 20 taking notice of the embedding b q 1 bmo 1 q 2 we see that for sufficiently small 0 u 0 b q 1 q 2 can guarantee that 0 1 has a unique global solution in bmo 1 however this solution is instable in b q 1 and the solution can have an inflation in b q 1 for certain initial data

on the poisson integrals representation in the classical statistical mechanics of continuous systems description of the grand canonical gibbs ensemble for classical continuous systems in terms of the nonlocally perturbed infinite divisible generalized random fields is presented the equivalence of the traditional description with the ones presented here on the level of dlr equations is established the antiferromagnetism for the purely repulsive interactions has been observed finally the usefulness of our description for an analysis of the high temperature cluster expansion has been demonstrated

the convergence of cluster expansion for continuous systems with many body interaction the existence of a unique thermodynamic state for dilute classical systems is proved for a class of multi particle potentials under ordinary assumptions of stability and integrability thus we do not use the cumbersome conditions of regularity needed in previous publications for the many body analysis the method relies on the poisson measure representation and cluster expansion for distribution functions

invariant manifolds for steady boltzmann flows and applications we develop a theory of invariant manifolds for the steady boltzmann equation and apply it to the study of boundary layers and nonlinear waves the steady boltzmann equation is an infinite dimensional differential equation so the standard center manifold theory for differential equations based on spectral information does not apply here instead we employ a time asymptotic approach using the pointwise information of green s function for the construction of the linear invariant manifolds at the resonance cases when the mach number at the far field is around one of the critical values of 1 0 or 1 the truly nonlinear theory arises in such a case there are wave patterns combining the fast decaying knudsen type and slow varying fluid like waves the key knudsen manifolds consisting of only knudsentype layers are constructed through delicate analysis of identifying the singular behavior around the critical mach numbers around mach number 1 the fluidlike waves are compressive and expansive waves and around the mach number 0 they are linear thermal layers the quantitative analysis of the fluid like waves is done using the reduction of dimensions to the center manifolds two scale nonlinear dynamics based on those on the knudsen and center manifolds are formulated for the study of the global dynamics of the combined wave patterns there are striking bifurcations in the transition of evaporation to condensation and in the transition of the milne s problem with a subsonic far field to one with a supersonic far field the analysis of these wave patterns allows us to understand the sone diagram for the study of the complete condensation boundary value problem the monotonicity of the boltzmann shock profiles a problem that initially motivated the present study is shown as a consequence of the quantitative analysis of the nonlinear fluid like waves

de rham cohomology of configuration spaces with poisson measure the space x of all locally finite configurations in a riemannian manifold x of infinite volume is considered the derham complex of square integrable differential forms over x equipped with the poisson measure and the corresponding derham cohomology are studied the latter is shown to be unitarily isomorphic to a certain hilbert tensor algebra generated by the l2 cohomology of the underlying manifold x