about the regularizing properties of the non cut off kac equation we prove in this work that under suitable assumptions the solution of the spatially homogeneous non cut off kac equation or of the spatially homogeneous non cut off 2d boltzmann equation with maxwellian molecules in the radial case becomes very regular with respect to the velocity variable as soon as the time is strictly positive

entropy dissipation and long range interactions we study boltzmann s collision operator for long range interactions i e without grad s angular cut off assumption we establish a functional inequality showing that the entropy dissipation controls smoothness of the distribution function in a precise sense our estimate is optimal and gives a unified treatment of both the linear and the nonlinear cases we also give simple and self contained proofs of several useful results that were scattered in previous works as an application we obtain several helpful estimates for the cauchy problem and for the landau approximation in plasma physics

regularity of renormalized solutions in the boltzmann equation with long range interactions it is well established that renormalized solutions of the boltzmann equation enjoy some kind of regularity or at least compactness in the velocity variable when the angular collision kernel is nonintegrable however obtaining explicit estimates in convenient and natural functional settings proves rather difficult in this work we derive a velocity smoothness estimate from the a priori control of the renormalized dissipation as a direct consequence of our result we show that in the presence of long range interactions any renormalized solution f t x v to the boltzmann equation satisfies locally for every and for some s 0 depending on p we also provide an application of this new estimate to the hydrodynamic limit of the boltzmann equation without cutoff 2012 wiley periodicals inc

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

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

well posedness and large time behaviour for the non cutoff kac equation with a gaussian thermostat we consider here a kac equation with a gaussian thermostat in the case of a non cutoff cross section under the sole assumptions of finite mass and finite energy for the initial data we prove the existence of a global in time solution for which mass and energy are preserved then via fourier transform techniques we show that this solution is smooth unique and converges to the corresponding stationary state

gelfand shilov smoothing properties of the radially symmetric spatially homogeneous boltzmann equation without angular cutoff we prove that the cauchy problem associated to the radially symmetric spatially homogeneous non cutoff boltzmann equation with maxwellian molecules enjoys the same gelfand shilov regularizing effect as the cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator

stokes fourier and acoustic limits for the boltzmann equation convergence proofs we establish a stokes fourier limit for the boltzmann equation considered over any periodic spatial domain of dimension 2 or more appropriately scaled families of diperna lions renormalized solutions are shown to have fluctuations that globally in time converge weakly to a unique limit governed by a solution of stokes fourier motion and heat equations provided that the fluid moments of their initial fluctuations converge to appropriate l 2 initial data of the stokesfourier equations both the motion and heat equations are recovered in the limit by controlling the fluxes and the local conservation defects of the diperna lions solutions with dissipation rate estimates the scaling of the fluctuations with respect to knudsen number is essentially optimal the assumptions on the collision kernel are little more than those required for the diperna lions theory and that the viscosity and heat conduction are finite for the acoustic limit these techniques also remove restrictions to bounded collision kernels and improve the scaling of the fluctuations both weak limits become strong when the initial fluctuations converge entropically to appropriate l 2 initial data c 2002 john wiley sons inc

from the boltzmann equations to the equations of incompressible fluid mechanics i we consider here the problem of deriving rigorously from boltzmann s equation globally in time and for general initial conditions fluid mechanics equations such as the navier stokes or euler equations

robust stability analysis of fractional order uncertain singular nonlinear system with external disturbance this paper investigates robust stability for fractional order fo singular nonlinear systems the fo system is disturbed by external uncertainty and disturbance a central analysis technique is enabled by proposing a fundamental boundedness lemma for the first time this lemma is used for robust stability analysis of fo systems especially for mittag leffler stability analysis of fo nonlinear systems more importantly how to obtain a more accurate bound is given to reduce conservative an fo proportional derivative pd controller is proposed to normalize the fo singular system furthermore a criterion for stability of the normalized fo nonlinear systems is provided by linear matrix inequalities lmis finally two illustrative simulation examples are presented to illustrate effectiveness of the proposed stability notion

robust decentralized control of perturbed fractional order linear interconnected systems the robust decentralized state feedback stabilization problem for a class of perturbed fractional order linear interconnected systems is investigated the systems considered are subject to structured or unstructured perturbations the decentralized state feedback controllers are designed such that the closed loop perturbed fractional order linear interconnected systems are asymptotically stable by using the definition of positive and negative definite matrices and stability theory of fractional order systems sufficient conditions for robust stabilization of closed loop perturbed fractional order linear interconnected systems are given simulation examples are given to demonstrate the effectiveness of the proposed theoretical results

smoothing estimates for boltzmann equation with full range interactions spatially homogeneous case in this work we are concerned with the regularities of the solutions to the boltzmann equation with physical collision kernels for the full range of intermolecular repulsive potentials r p 1 with p 2 we give new and constructive upper and lower bounds for the collision operator in terms of standard weighted fractional sobolev norms as an application we get the global entropy dissipation estimate which is a little stronger than that described by alexandre et al arch rational mech anal 152 4 327 355 2000 as another application we prove the smoothing effects for the strong solutions constructed by desvillettes and mouhot arch rational mech anal 193 2 227 253 2009 of the spatially homogeneous boltzmann equation with true hard potential and true moderately soft potential

a weak formulation of the boltzmann equation based on the fourier transform in this article we present an alternative formulation of the spatially homogeneous boltzmann equation rewriting the weak form of the equation with shifted test functions and using fourier techniques it turns out that the transformed problem contains only a three fold integral

regularizing effect and local existence for the non cutoff boltzmann equation the boltzmann equation without grad s angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non integrable angular singularity of the cross section however even though this has been justified satisfactorily for the spatially homogeneous boltzmann equation it is still basically unsolved for the spatially inhomogeneous boltzmann equation in this paper by sharpening the coercivity and upper bound estimates for the collision operator establishing the hypo ellipticity of the boltzmann operator based on a generalized version of the uncertainty principle and analyzing the commutators between the collision operator and some weighted pseudo differential operators we prove the regularizing effect in all time space and velocity variables on the solutions when some mild regularity is imposed on these solutions for completeness we also show that when the initial data has this mild regularity and a maxwellian type decay in the velocity variable there exists a unique local solution with the same regularity so that this solution acquires the c regularity for any positive time

some solutions of the boltzmann equation without angular cutoff we show the existence of local or global in time solutions for the non homogeneous boltzmann equation this is done under the assumptions that initial data are smaller than a suitable maxwellian and that collisional cross sections do not satisfy grad s angular cutoff partial regularity in space velocity of the solutions constructed herein is also 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

the acoustic limit for the boltzmann equation the acoustic equations are the linearization of the compressible euler equations about a spatially homogeneous fluid state we first derive them directly from the boltzmann equation as the formal limit of moment equations for an appropriately scaled family of boltzmann solutions we then establish this limit for the boltzmann equation considered over a periodic spatial domain for bounded collision kernels appropriately scaled families of diperna lions renormalized solutions are shown to have fluctuations that converge entropically and hence strongly in l 1 to a unique limit governed by a solution of the acoustic equations for all time provided that its initial fluctuations converge entropically to an appropriate limit associated to any given l 2 initial data of the acoustic equations the associated local conservation laws are recovered in the limit

stability preservation analysis for frequency based methods in numerical simulation of fractional order systems in this paper the frequency domain based numerical methods for simulation of fractional order systems are studied in the sense of stability preservation first the stability boundary curve is exactly determined for these methods then this boundary is analyzed and compared with an accurate ideal boundary in different frequency ranges also the critical regions in which the stability does not preserve are determined finally the analytical achievements are confirmed via some numerical illustrations

brief paper stability and resonance conditions of elementary fractional transfer functions elementary fractional transfer functions are studied in this paper some basic properties of elementary transfer functions of the first kind are recalled then two main results are presented regarding elementary fractional transfer functions of the second kind written in a canonical form and characterized by a commensurate order a pseudo damping factor and a natural frequency first stability conditions are established in terms of the pseudo damping factor and the commensurate order as a corollary to matignon s stability theorem they extend the previous result into conditions that are simpler to check then resonance conditions are established numerically in terms of the commensurate order and the pseudo damping factor and give interesting information on the frequency behavior of fractional systems it is shown that elementary transfer functions of the second kind might have up to two resonant frequencies moreover three abaci are given allowing to determine the pseudo damping factor and the commensurate order for respectively a desired normalized gain at each resonance a desired phase at each resonance and a desired normalized first or second resonant frequency

robust asymptotical stability of fractional order linear systems with structured perturbations this paper considers the robust asymptotical stability problem of fractional order linear systems with structured perturbations which are more general than the investigated fractional order interval systems based on the kronecker product and m analysis necessary and sufficient conditions for the robust asymptotical stability are established by transforming such a problem into checking the nonsingularity of a class of uncertain matrices furthermore the robustness bounds with respect to parametric perturbations to preserve the asymptotical stability are given in terms of the structured singular values finally illustrative examples are given to show the effectiveness of the proposed approach

cooling process for inelastic boltzmann equations for hard spheres part i the cauchy problem we develop the cauchy theory of the spatially homogeneous inelastic boltzmann equation for hard spheres for a general form of collision rate which includes in particular variable restitution coefficients depending on the kinetic energy and the relative velocity as well as the sticky particles model we prove local in time non concentration estimates in orlicz spaces from which we deduce weak stability and existence theorem strong stability together with uniqueness and instantaneous appearance of exponential moments are proved under additional smoothness assumption on the initial datum for a restricted class of collision rates concerning the long time behaviour we give conditions for the cooling process to occur or not in finite time

coprime factorizations and stability of fractional differential systems we give a frequency domain approach to stabilization for a large class of systems with transfer functions involving fractional powers of s a necessary and sufficient criterion for bibo stability is given and it is shown how to construct coprime factorizations and associated bezout factors in order to parametrize all stabilizing controllers of these systems

symmetrized compact scheme for receptivity study of 2d transitional channel flow in many physical problems one must maintain symmetry of numerical schemes the symmetry arising due to geometry and or physical conditions defining the flow the direct simulation of transitional plane poiseuille flow presents one such case that motivated the present study it has been shown here using the method of analyzing non periodic problems in sengupta et al 2003 that compact and other high accuracy schemes display strong directionality of the algorithm and such methods cannot be used for direct simulation of the physical flow this prevalent but unacceptable situation for dns is rectified in the present work and a new scheme is introduced that prevents this asymmetry from contaminating results the simulated transitional flow past 2d channel here using the new scheme are in agreement with experimental results and other recent views of sub critical instabilities

on the well posedness of the spatially homogeneous boltzmann equation with a moderate angular singularity we prove an inequality on the kantorovich rubinstein distance which can be seen as a particular case of a wasserstein metric between two solutions of the spatially homogeneous boltzmann equation without angular cutoff but with a moderate angular singularity our method is in the spirit of 7 we deduce some well posedness and stability results in the physically relevant cases of hard and moderately soft potentials in the case of hard potentials we relax the regularity assumption of 6 but we need stronger assumptions on the tail of the distribution namely some exponential decay we thus obtain the first uniqueness result for measure initial data in the case of moderately soft potentials we prove existence and uniqueness assuming only that the initial datum has finite energy and entropy for very moderately soft potentials plus sometimes an additionnal moment condition we thus improve significantly on all previous results where weighted sobolev spaces were involved

brief paper rational approximations in the simulation and implementation of fractional order dynamics a descriptor system approach this paper deals with issues related to the use of rational approximations in the simulation of fractional order systems and practical implementations of fractional order dynamics and controllers based on the mathematical formulation of the problem a descriptor model is found to describe the rational approximating model this model is analyzed and compared with the original fractional order system under the aspects which are important in their simulation and implementation from the results achieved one can determine in what applications the use of rational approximations would be unproblematic and in what applications it would lead to fallacious results in order to clarify this point some examples are presented in which the effects of using rational approximations are investigated

finite difference schemes for long time integration a general method for constructing finite difference schemes for longtime integration problems is presented it is demonstrated for discretizations of first and second space derivatives however the approach is not limited to these cases the schemes are constructed so as to minimize the global truncation error taking into account the initial data the resulting second order compact schemes can be used for integration times fourfold or more longer than previously studied schemes with similar computational complexity a similar approach was used to obtain improved integration schemes

analysis of central and upwind compact schemes central and upwind compact schemes for spatial discretization have been analyzed with respect to accuracy in spectral space numerical stability and dispersion relation preservation a von neumann matrix spectral analysis is developed here to analyze spatial discretization schemes for any explicit and implicit schemes to investigate the full domain simultaneously this allows one to evaluate various boundary closures and their effects on the domain interior the same method can be used for stability analysis performed for the semi discrete initial boundary value problems ibvp this analysis tells one about the stability for every resolved length scale some well known compact schemes that were found to be g k s and time stable are shown here to be unstable for selective length scales by this analysis this is attributed to boundary closure and we suggest special boundary treatment to remove this shortcoming to demonstrate the asymptotic stability of the resultant schemes numerical solution of the wave equation is compared with analytical solution furthermore some of these schemes are used to solve two dimensional navier stokes equation and a computational acoustic problem to check their ability to solve problems for long time it is found that those schemes that were found unstable for the wave equation are unsuitable for solving incompressible navier stokes equation in contrast the proposed compact schemes with improved boundary closure and an explicit higher order upwind scheme produced correct results the numerical solution for the acoustic problem is compared with the exact solution and the quality of the match shows that the used compact scheme has the requisite drp property

from boltzmann s equation to the incompressible navier stokes fourier system with long range interactions we establish a rigorous demonstration of the hydrodynamic convergence of the boltzmann equation towards a navier stokes fourier system under the presence of long range interactions this convergence is obtained by letting the knudsen number tend to zero and has been known to hold at least formally for decades it is only more recently that a fully rigorous mathematical derivation of this hydrodynamic limit was discovered however these results failed to encompass almost all physically relevant collision kernels due to a cutoff assumption which requires that the cross sections be integrable indeed as soon as long range intermolecular forces are present non integrable collision kernels have to be considered because of the enormous number of grazing collisions in the gas in this long range setting the boltzmann operator becomes a singular integral operator and the known rigorous proofs of hydrodynamic convergence simply do not carry over to that case in fact the diperna lions renormalized solutions do not even make sense in this situation and the relevant global solutions to the boltzmann equation are the so called renormalized solutions with a defect measure developed by alexandre and villani our work overcomes the new mathematical difficulties coming from the consideration of long range interactions by proving the hydrodynamic convergence of the alexandre villani solutions towards the leray solutions

on the fokker planck boltzmann equation we consider the boltzmann equation perturbed by fokker planck type operator to overcome the lack of strong a priori estimates and to define a meaningful collision operator we introduce a notion of renormalized solution which enables us to establish stability results for sequences of solutions and global existence for the cauchy problem with large data the proof of stability and existence combines renormalization with an analysis of a defect measure

global renormalized solutions and navier stokes limit of the boltzmann equation with incoming boundary condition for long range interaction abstract for the long range interaction we prove the global existence of renormalized solutions to the boltzmann equation with incoming boundary condition furthermore as knudsen number goes to zero the limit to the incompressible navier stokes limit with homogeneous dirichlet boundary condition is justified when the boundary data of the scaled boltzmann equation is close to the maxwellian with order o 3 in the sense of boundary relative entropy

explicit runge kutta nystro m methods with reduced phase errors for computing oscillating solutions we construct explicit runge kutta nystrom methods for the integration of first and second order differential equations having an oscillatory solution special attention is paid to the phase errors or dispersion of the dominant components in the numerical oscillations when these methods are applied to a linear homogeneous test model rk n methods are constructed which are dispersive of orders up to 10 whereas the algebraic order of accuracy is only 2 or 3 application of these methods to equations describing free and weakly forced oscillations and to semidiscretized hyperbolic equations reveals that the phase errors can significantly be reduced

analysis of anisotropy of numerical wave solutions by high accuracy finite difference methods fourier analysis is used to quantitatively assess the resolution and in particular the isotropy of wave solution using finite difference spatial discretization schemes along with fourth order runge kutta temporal scheme aspect ratio of the grid in two dimension along with the angle of wave propagation are the parameters varied to qualitatively and quantitatively assess the anisotropy of the solutions for a a skewed one dimensional wave convecting in two dimensions following the standard convection equation and b a wave propagating following the two dimensional linearized rotating shallow water equations results show the effect of changing the aspect ratio and the propagation angle on the directional nature of the solution as obtained by different methods for the above non dispersive and dispersive wave system

propagation of gevrey regularity for solutions of the boltzmann equation for maxwellian molecules we prove that gevrey regularity is propagated by the boltzmann equation with maxwellian molecules with or without angular cut off the proof relies on the wild expansion of the solution to the equation and on the characterization of gevrey regularity by the fourier transform

high accuracy solution of bi directional wave propagation in continuum mechanics solution of partial differential equations by numerical method is strongly affected due to numerical errors which are caused mainly by deviation of numerical dispersion relation from the physical dispersion relation to quantify and control such errors and obtain high accuracy solutions we consider a class of problems which involve second derivative of unknowns with respect to time here we analyse numerical metrics such as the numerical group velocity numerical phase speed and the numerical amplification factor for different methods in solving the model bi directional wave equation bdwe such equations can be solved directly for example by runge kutta nystrom rkn method alternatively the governing equation can be converted to a set of first order in time equations and then using four stage fourth order runge kutta rk4 method for time integration spatial discretisation considered are the classical second and fourth order central difference schemes along with lele s central compact scheme for evaluating second derivatives in another version we have used lele s scheme for evaluating first derivatives twice to obtain the second derivative as bdwe represents non dissipative non dispersive dynamics we also consider the canonical problem of linearised rotating shallow water equation lrswe in a new formulation involving second order derivative in time which represents dispersive waves along with a stationary mode the computations of lrswe with rk4 and rkn methods for temporal discretisation and lele s compact schemes for spatial discretisation are compared with computations performed with rk4 method for time discretisation and staggered compact scheme scs for spatial discretisation by treating it as a set of three equations as reported in rajpoot et al 2012 1

the geometry of binary collisions and generalized radon transforms elastic collisions are characterized by the conservation of momentum and energy we consider some geometrical aspects of such collisions when the energy of one particle can be expressed in terms of the moments as the geometry of elastic collisions is essential for the regularizing property of the gain term in the boltzmann equation which was proved by p l lions we show how such results can be deduced from a regularity theorem for generalized radon transforms by sogge stein this is possible for and for we also show that the same technique cannot be used with other choices of

the boltzmann equation without angular cutoff in the whole space qualitative properties of solutions this is a continuation of our series of works for the inhomogeneous boltzmann equation we study qualitative properties of classical solutions the full regularization in all variables uniqueness non negativity and convergence rate to the equilibrium to be precise together with the results of parts i and ii about the well posedness of the cauchy problem around the maxwellian we conclude this series with a satisfactory mathematical theory for the boltzmann equation without angular cutoff

entropy dissipation estimates for the landau equation in the coulomb case and applications abstract we present in this paper an estimate which bounds from below the entropy dissipation d f of the landau operator with coulomb interaction by a weighted h 1 norm of the square root of f as a consequence we get a weighted l t 1 l v 3 estimate for the solutions of the spatially homogeneous landau equation with coulomb interaction and the propagation of l 1 moments of any order for this equation we also present an application of our estimate to the landau equation with moderately soft potentials providing thus a new proof of some recent results of 30

global existence and full regularity of the boltzmann equation without angular cutoff we prove the global existence and uniqueness of classical solutions around an equilibrium to the boltzmann equation without angular cutoff in some sobolev spaces in addition the solutions thus obtained are shown to be non negative and c in all variables for any positive time in this paper we study the maxwellian molecule type collision operator with mild singularity one of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator this norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated h theorem

a two level time stepping method for layered ocean circulation models this paper describes a numerical method with two time levels for solving partial differential equations that represent layered models of ocean circulation the method is designed to be used with a barotropic baroclinic splitting that separates the fast and slow motions into subsystems that are solved by different techniques with this method some of the dependent variables are predicted and then corrected after the initial prediction steps all steps involve centered differencing and averaging about the midpoint of the time interval in question the coriolis and pressure terms are evaluated at the same time levels to avoid a first order truncation error in the geostrophic balance between those terms compared to the three level leapfrog method that is widely used in geophysical fluid dynamics the present method does not admit a computational mode and the maximum permissible time step is at least twice as large in addition with the two level method it is possible to use a nearly nonoscillatory advection algorithm to solve the equations for mass and momentum in a simple test problem the present method gives less phase error than the leapfrog method and two other methods and it does not give any amplitude error the differences are especially large when comparing the present method to the leapfrog method the two level method also gives good results in a nonlinear test problem involving vanishing layers at the top and bottom of the fluid domain and an interface moving along sloping bottom topography

a direct method for the regularity of the gain term in the boltzmann equation this paper presents an essentially simple and direct method to prove the regularizing properties of the collision gain term in the first equation of the boltzmann hierarchy some corollaries of the main results of this paper are given to improve the previous results on the regularity of the gain term in the boltzmann equation an extension of a new fundamental result concerning separated variable kernels is also given to some general kernels

chaotic attractors in incommensurate fractional order systems in this paper based on the stability theorems in fractional differential equations a necessary condition is given to check the existence of 1 scroll 2 scroll or multi scroll chaotic attractors in a fractional order system this condition is proposed for incommensurate order systems in general but in the special case it converts to the condition given in the previous works for the commensurate fractional order systems though the presented condition is only a necessary and not sufficient condition for the existence of chaos it can be used as a powerful tool to distinguish for what parameters and orders of a given fractional order system chaotic attractors can not be observed and for what parameters and orders the system may generate chaos it can also be used as a tool to confirm or reject results of a numerical simulation some of the numerical results reported in the previous literature are confirmed by this tool

convolution inequalities for the boltzmann collision operator we study integrability properties of a general version of the boltzmann collision operator for hard and soft potentials in n dimensions a reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution where the weight is given by an operator invariant under rotations using a symmetrization technique in l p we prove a young s inequality for hard potentials which is sharp for maxwell molecules in the l 2 case further we find a new hardy littlewood sobolev type of inequality for boltzmann collision integrals with soft potentials the same method extends to radially symmetric non increasing potentials that lie in some l s weak or l s the method we use resembles a brascamp lieb and luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting consequently it is closely connected to the classical analysis of young and hardy littlewood sobolev inequalities in all cases the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section in the spirit of grad cut off as an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions

regularity of solutions to the spatially homogeneous boltzmann equation for non maxwellian molecules without angular cutoff abstract in this paper we use the pseudo differential calculus to analyze the smoothing property of weak solutions to the spatially homogeneous boltzmann equation precisely we show that for the non maxwellian molecules with debye yukawa potential if the positive weak solution is lipschitz continuous in the velocity variable then it lies in the sobolev space h loc r 3 and hence it is automatically smooth

on the boltzmann equation for long range interactions we study the boltzmann equation without grad s angular cutoff assumption we introduce a suitable renormalized formulation that allows the cross section to be singular in both the angular and the relative velocity variables angular singularities occur as soon as one is interested in long range interactions while singularities in the relative velocity variable occur in the study of soft potentials in particular coulomb interaction together with several new estimates this new formulation enables us to prove existence of weak solutions and to give a proof of a conjecture by lions appearance of strong compactness under general fully realistic assumptions 2001 john wiley sons inc

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

the boltzmann equation without angular cutoff in the whole space i global existence for soft potential it is known that the singularity in the non cutoff cross section of the boltzmann equation leads to the gain of regularity and gain of weight in the velocity variable by defining and analyzing a non isotropy norm which precisely captures the dissipation in the linearized collision operator we first give a new and precise coercivity estimate for the non cutoff boltzmann equation for general physical cross sections then the cauchy problem for the boltzmann equation is considered in the framework of small perturbation of an equilibrium state in this part for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator we derive some new functional estimates on the nonlinear collision operator together with the coercivity estimates we prove the global existence of classical solutions for the boltzmann equation in weighted sobolev spaces

fractional order systems and pi sup spl lambda d sup spl mu controllers dynamic systems of an arbitrary real order fractional order systems are considered the concept of a fractional order pi sup spl lambda d sup spl mu controller involving fractional order integrator and fractional order differentiator is proposed the laplace transform formula for a new function of the mittag leffler type made it possible to obtain explicit analytical expressions for the unit step and unit impulse response of a linear fractional order system with fractional order controller for both open and closed loops an example demonstrating the use of the obtained formulas and the advantages of the proposed pi sup spl lambda d sup spl mu controllers is given

shallow water model on cubed sphere by multi moment finite volume method a global numerical model for shallow water flows on the cubed sphere grid is proposed in this paper the model is constructed by using the constrained interpolation profile multi moment finite volume method cip mm fvm two kinds of moments i e the point value pv and the volume integrated average via are defined and independently updated in the present model by different numerical formulations the lax friedrichs upwind splitting is used to update the pv moment in terms of a derivative riemann problem and a finite volume formulation derived by integrating the governing equations over each mesh element is used to predict the via moment the cubed sphere grid is applied to get around the polar singularity and to obtain uniform grid spacing for a spherical geometry highly localized reconstruction in cip mm fvm is well suited for the cubed sphere grid especially in dealing with the discontinuity in the coordinates between different patches the mass conservation is completely achieved over the whole globe the numerical model has been verified by williamson s standard test set for shallow water equation model on sphere the results reveal that the present model is competitive to most existing ones

fast and slow convergence to equilibrium for maxwellian molecules via wild sums we consider the spatially homogeneous boltzmann equation for maxwellian molecules and general finite energy initial data positive borel measures with finite moments up to order 2 we show that the coefficients in the wild sum converge strongly to the equilibrium and quantitatively estimate the rate we show that this depends on the initial data f essentially only through on the behavior near r 0 of the function j f r v 1 r v 2 df v these estimates on the terms in the wild sum yield a quantitative estimate in the strongest physical norm on the rate at which the solution converges to equilibrium as well as a global stability estimate we show that our upper bounds are qualitatively sharp by producing examples of solutions for which the convergence is as slow as permitted by our bounds these are the first examples of solutions of the boltzmann equation that converge to equilibrium more slowly than exponentially

solution of linearized rotating shallow water equations by compact schemes with different grid staggering strategies high accuracy solution of pdes requires proper error analysis previous analysis for a non dispersive system t k sengupta a dipankar p sagaut error dynamics beyond von neumann analysis j comput phys 226 2007 1211 1218 identified sources of error correctly here the aim is to extend the spectral analysis for the model linearized rotating shallow water equations lrswe as an example of dispersive system we perform the analysis when high accuracy compact schemes are used to solve the lrswe relevant to geophysical fluid dynamics using different grid arrangements proposed in mesinger and arakawa f mesinger a arakawa numerical methods used in atmospheric models garp publ ser no 17 vol 1 wmo geneva 1976 pp 43 64 and randall d a randall geostrophic adjustment and the finite difference shallow water equations mon wea rev 122 1994 1371 1377 compact schemes are used for fluid dynamical problem as these afford near spectral accuracy in solving non periodic problems however higher accuracy methods also suffer from errors those are often filtered by low order methods for example dispersion error is present in all numerical methods and extreme form of it leads to q waves which appear at higher wavenumbers for compact schemes as compared to lower order method we also evaluate a compact scheme specifically designed for use with staggered grids here two and four time level temporal discretization methods have been compared for solving lrswe by considering classical fourth order four stage runge kutta rk 4 two time level forward backward fb and four time level generalized fb temporal integration schemes

exponential convergence to equilibrium for the homogeneous boltzmann equation for hard potentials without cut off this paper deals with the long time behavior of solutions to the spatially homogeneous boltzmann equation the interactions considered are the so called non cut off with a moderate angular singularity and non mollified hard potentials we prove an exponential in time convergence towards the equilibrium improving results of villani commun math phys 234 3 455 490 2003 where a polynomial decay to equilibrium is proven the basis of the proof is the study of the linearized equation for which we prove a new spectral gap estimate in a l 1 space with a polynomial weight by taking advantage of the theory of enlargement of the functional space for the semigroup decay developed by gualdani et al http hal archives ouvertes fr ccsd 00495786 2013 we then get our final result by combining this new spectral gap estimate with bilinear estimates on the collisional operator that we establish

regularity theory for the spatially homogeneous boltzmann equation with cut off we develop the regularity theory of the spatially homogeneous boltzmann equation with cut off and hard potentials for instance hard spheres by i revisiting the lp theory to obtain constructive bounds ii establishing propagation of smoothness and singularities iii obtaining estimates on the decay of the singularities of the initial datum our proofs are based on a detailed study of the regularity of the gain operator an application to the long time behavior is presented

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

regularity of the moments of the solution of a transport equation let u u x v satisfy the transport equation u v xu f x rn v rnwhere f belongs to some space of type lp dx d v where is a positive bounded measure on rn we study the resulting regularity of the moment u x v d v in terms of fractional sobolev spaces for example counter examples are given in order to test the optimality of our results

chaos and hyperchaos in the fractional order rossler equations abstract the dynamics of fractional order systems have attracted increasing attentions in recent years in this paper we numerically study the chaotic behaviors in the fractional order rossler equations we found that chaotic behaviors exist in the fractional order rossler equation with orders less than 3 and hyperchaos exists in the fractional order rossler hyperchaotic equation with order less than 4 the lowest orders we found for chaos and hyperchaos to exist in such systems are 2 4 and 3 8 respectively period doubling routes to chaos in the fractional order rossler equation are also found