fluid dynamic limits of kinetic equations i formal derivations the connection between kinetic theory and the macroscopic equations of fluid dynamics is described in particular our results concerning the incompressible navier stokes equations are based on a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of hilbert or chapman enskog the moment formalism shows that the limit leading to the incompressible navier stokes equations like that leading to the compressible euler equations is a natural one in kinetic theory and is contrasted with the systematics leading to the compressible navier stokes equations some indications of the validity of these limits are given more specifically the connection between the diperna lions renormalized solution of the classical boltzmann equation and the leray solution of the navier stokes equations is discussed

boltzmann diffusive limit beyond the navier stokes approximation given a normalized maxwellian and n 1 we establish the global in time validity of a diffusive expansion 0 1 f e t x v ef 1 t x v e 2 f 2 t x v e n f e n t x v for a solution f e to the rescaled boltzmann equation diffusive scaling 0 2 e 1 f e v x fe 1 eq f e f e inside a periodic box t 3 we assume that in the initial expansion 0 1 at t 0 the fluid parts of these f m 0 x v have arbitrary divergence free velocity fields as well as temperature fields for all 1 m n while f 1 0 x v has small amplitude in h 2 for m 2 these f m t x v are determined by a sequence of linear navier stokes fourier systems iteratively more importantly the remainder f e n t x v is proven to decay in time uniformly in e via a unified nonlinear energy method in particular our results lead to an error estimate for f 1 t x v the well known navier stokes fourier approximation and beyond the collision kernel q includes hard sphere the cutoff inverse power as well as the coulomb interactions

global well posedness for a boussinesq navier stokes system with critical dissipation abstract in this paper we study a fractional diffusion boussinesq model which couples a navier stokes type equation with fractional diffusion for the velocity and a transport equation for the temperature we establish global well posedness results with rough initial data

the navier stokes limit of the boltzmann equation for bounded collision kernels the present work establishes a navier stokes limit for the boltzmann equation considered over the infinite spatial domain r3 appropriately scaled families of diperna lions renormalized solutions are shown to have fluctuations whose limit points in the w l1 topology are governed by leray solutions of the limiting navier stokes equations this completes the arguments in bardos golse levermore commun pure appl math 46 5 667 753 1993 for the steady case and in lions masmoudi arch ration mech anal 158 3 173 193 2001 for the time dependent case

energy method for boltzmann equation abstract a basic simple energy method for the boltzmann equation is presented here it is based on a new macro micro decomposition of the boltzmann equation as well as the h theorem this allows us to make use of the ideas from hyperbolic conservation laws and viscous conservation laws to yield the direct energy method as an illustration we apply the method for the study of the time asymptotic nonlinear stability of the global maxwellian states previous energy method starting with grad and finishing with ukai involves the spectral analysis and regularity of collision operator through sophisticated weighted norms

classical solutions to the boltzmann equation for molecules with an angular cutoff an important class of collision kernels in the boltzmann theory are governed by the inverse power law in which the intermolecular potential between two particles is an inverse power of their distance under the grad angular cutoff assumption global in time classical solutions near maxwellians are constructed in a periodic box for all soft potentials with 3 0

global regularity for the supercritical dissipative quasi geostrophic equation with large dispersive forcing we consider the 2d quasi geostrophic equation with supercritical dis sipation and dispersive forcing in the whole space when the dispersive amplitude parameter is large enough we prove the global well posedness of strong solution to the equation with large initial data we also show the strong convergence result as the amplitude parameter goes to 1 both results rely on the strichartz type estimates for the corresponding linear equation

global well posedness for the critical 2d dissipative quasi geostrophic equation we give an elementary proof of the global well posedness for the critical 2d dissipative quasi geostrophic equation the argument is based on a non local maximum principle involving appropriate moduli of continuity

a new bernstein s inequality and the 2d dissipative quasi geostrophic equation we show a new bernstein s inequality which generalizes the results of cannone planchon danchin and lemarie rieusset as an application of this inequality we prove the global well posedness of the 2d quasi geostrophic equation with the critical and super critical dissipation for the small initial data in the critical besov space and local well posedness for the large initial data

global well posedness of smooth solution to the supercritical sqg equation with large dispersive forcing and small viscosity abstract it is known that global well posedness for the supercritical sqg equation remains open even the smooth data in this paper motivated by the recent work cannone et al 2013 we provide a simpler approach to the proof of global well posedness for the supercritical sqg equation with large dispersive forcing even with small viscosity as an application we prove global well posedness for the 2d dissipative boussinesq equations with small viscosity and large background data

on the linearized relativistic boltzmann equation i existence of solutions the linearized relativistic boltzmann equation inl 2 r p is investigated the detailed analysis of the collision operatorl is carried out for a wide class of scattering cross sections l is proved to have a form of the multiplication operatorv p plus the compact inl 2 p perturbationk the collisional frequencyv p is analysed to discriminate between relativistic soft and hard interactions finally the existence and uniqueness of the solution to the linearized relativistic boltzmann equation is proved

fluid dynamic limits of kinetic equations ii convergence proofs for the boltzmann equation using relative entropy estimates about an absolute maxwellian it is shown that any properly scaled sequence of diperna lions renormalized solutions of some classical boltzmann equations has fluctuations that converge to an infinitesimal maxwellian with fluid variables that satisfy the incompressibility and boussinesq relations moreover if the initial fluctuations entropically converge to an infinitesimal maxwellian then the limiting fluid variables satisfy a version of the leray energy inequality if the sequence satisfies a local momentum conservation assumption the momentum densities globaly converge to a solution of the stokes equation a similar discrete time version of this result holds for the navier stokes limit with an additional mild weak compactness assumption the continuous time navier stokes limit is also discussed 01993 john wiley sons inc

compressive imaging via approximate message passing with image denoising we consider compressive imaging problems where images are reconstructed from a reduced number of linear measurements our objective is to improve over existing compressive imaging algorithms in terms of both reconstruction error and runtime to pursue our objective we propose compressive imaging algorithms that employ the approximate message passing amp framework amp is an iterative signal reconstruction algorithm that performs scalar denoising at each iteration in order for amp to reconstruct the original input signal well a good denoiser must be used we apply two wavelet based image denoisers within amp the first denoiser is the amplitude scale invariant bayes estimator abe and the second is an adaptive wiener filter we call our amp based algorithms for compressive imaging amp abe and amp wiener numerical results show that both amp abe and amp wiener significantly improve over the state of the art in terms of runtime in terms of reconstruction quality amp wiener offers lower mean square error mse than existing compressive imaging algorithms in contrast amp abe has higher mse because abe does not denoise as well as the adaptive wiener filter

exponential decay for soft potentials near maxwellian we consider both soft potentials with angular cutoff and landau collision kernels in the boltzmann theory inside a periodic box we prove that any smooth perturbation near a given maxwellian approaches zero at the rate of e lambda t p for some 0 and 0 p 1 our method is based on an unified energy estimate with appropriate exponential velocity weight our results extend the classical result caflisch of 2 to the case of very soft potential and coulomb interactions and also improve the recent almost exponential decay results by 5 14

erratum yan guo boltzmann diffusive limit beyond the navier stokes approximation communications on pure and applied mathematics 2006 59 5 626 687 the original article to which this erratum refers was published in communications on pure and applied mathematicscomm pure appl math 2006 59 5 626

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

global hilbert expansion for the vlasov poisson boltzmann system the dynamics of an electron gas in a constant ion background can be decribed by the vlasov poisson boltzmann system at the kinetic level or by the compressible euler poisson system at the fluid level we prove that any solution of the vlasov poisson boltzmann system near a smooth local maxwellian with a small irrotational velocity converges global in time to the corresponding solution to the euler poisson system as the mean free path e goes to zero we use a recent l2 l framework in the boltzmann theory to control the higher order remainder in the hilbert expansion uniformly in e and globally in time

a frequency localized maximum principle applied to the 2d quasi geostrophic equation in this paper we prove a maximum principle for a frequency localized transport diffusion equation as an application we prove the local well posedness of the supercritical quasi geostrophic equation in the critical besov spaces mathring b 1 alpha infty q and global well posedness of the critical quasi geostrophic equation in mathring b 0 infty q for all 1 q here mathring b s infty q is the closure of the schwartz functions in the norm of b s infty q

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

hilbert expansion from the boltzmann equation to relativistic fluids we study the local in time hydrodynamic limit of the relativistic boltzmann equation using a hilbert expansion more specifically we prove the existence of local solutions to the relativistic boltzmann equation that are nearby the local relativistic maxwellians the maxwellians are constructed from a class of solutions to the relativistic euler equations that includes a large subclass of near constant non vacuum fluid states in particular for small knudsen number these solutions to the relativistic boltzmann equation have dynamics that are effectively captured by corresponding solutions to the relativistic euler equations

global regularity for the 2d boussinesq equations with partial viscosity terms abstract in this paper we prove the global in time regularity for the 2d boussinesq system with either the zero diffusivity or the zero viscosity we also prove that as diffusivity viscosity tends to zero the solutions of the fully viscous equations converge strongly to those of zero diffusion viscosity equations our result for the zero diffusion system in particular solves the problem no 3 posed by moffatt in r l ricca ed kluwer academic publishers dordrecht the netherlands 2001 pp 3 10

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

from the boltzmann equation to an incompressible navier stokes fourier system we establish a navier stokes fourier limit for solutions of the boltzmann equation considered over any periodic spatial domain of dimension two or more we do this for a broad class of collision kernels that relaxes the grad small deflection cutoff condition for hard potentials and includes for the first time the case of soft potentials appropriately scaled families of diperna lions renormalized solutions are shown to have fluctuations that are compact every limit point is governed by a weak solution of a navier stokes fourier system for all time