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

time periodic solutions of the compressible magnetohydrodynamic equations abstract the compressible magnetohydrodynamic equations with a time periodic external force in r n are studied in this paper we establish the existence uniqueness and time asymptotic stability of time periodic solutions when n 5 our proof is essentially based on a combination of the energy method and the time decay estimates of solutions to the linearized system

long range cohesive interactions of non local continuum faced by fractional calculus abstract a non local continuum model including long range forces between non adjacent volume elements has been studied in this paper the proposed continuum model has been obtained as limit case of two fully equivalent mechanical models i a volume element model including contact forces between adjacent volumes as well as long range interactions distance decaying between non adjacent elements ii a discrete point spring model with local springs between adjacent points and non local springs with distance decaying stiffness connecting non adjacent points under the assumption of fractional distance decaying interactions between non adjacent elements a fractional differential equation involving marchaud type fractional derivatives has been obtained for unbounded domains it is shown that for unbounded domains the two mechanical models revert to lazopoulos and eringen model with fractional distance decaying functions it has also been shown that for a confined bar the stress strain relation is substantially different from that obtained simply using the truncated marchaud derivatives since a double integral instead of convolution integral appears moreover in the analysis of bounded domains the governing equations turn out to an integro differential equation including only the integral part of marchaud fractional derivatives on finite interval the mechanical boundary condition for the proposed model has been introduced consistently on the basis of mechanical considerations and the constitutive law of the proposed continuum model has been reported by mathematical induction several numerical applications have been reported to show verify and assess the concepts listed in this paper

the boltzmann equation besov spaces and optimal time decay rates in rxn abstract we prove that k th order derivatives of perturbative classical solutions to the hard and soft potential boltzmann equation without the angular cut off assumption in the whole space r x n with n 3 converge in large time to the global maxwellian with the optimal decay rate of o t 1 2 k n 2 n r in the l x r l v 2 norm for any 2 r these results hold for any 0 n 2 as long as initially f 0 b 2 l v 2 in the hard potential case we prove faster decay results in the sense that if p f 0 b 2 l v 2 and i p f 0 b 2 1 l v 2 for n 2 n 2 2 then the solution decays the global maxwellian in l v 2 l x 2 with the optimal large time decay rate of o t 1 2

exponential decay of the landau equation with potential forces abstract the cauchy problem of the landau equation with potential forces on torus is investigated the global existence of solutions with the symmetry of origin and the exponential convergence rate in time to the steady state are obtained for any initially smooth periodic origin symmetric small perturbation

rate of convergence of non stationary flow to the steady flow of compressible viscous fluid we consider a compressible viscous fluid affected by external forces of general form which are small and smooth enough in suitable norms in r 3 in shibata and tanaka y shibata k tanaka on the steady flow of compressible viscous fluid and its stability with respect to initial disturbance j math soc japan 55 2003 797 826 we proved the unique existence and some regularity of the steady flow and its globally in time stability with respect to a small initial disturbance in the h 3 framework in this paper we investigate the rate of the convergence of the non stationary flow to the corresponding steady flow when the initial data are small enough in the h 3 and also belong to l 6 5

some aspects of fractional diffusion equations of single and distributed order abstract the time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order 0 1 the fundamental solution for the cauchy problem is interpreted as a probability density of a self similar non markovian stochastic process related to a phenomenon of sub diffusion the variance grows in time sub linearly a further generalization is obtained by considering a continuous or discrete distribution of fractional time derivatives of order less than one then the fundamental solution is still a probability density of a non markovian process that however is no longer self similar but exhibits a corresponding distribution of time scales

on the trend to global equilibrium for spatially inhomogeneous kinetic systems the boltzmann equation as part of our study of convergence to equilibrium for spatially inhomogeneous kinetic equations started in 21 we derive estimates on the rate of convergence to equilibrium for solutions of the boltzmann equation like o t our results hold conditionally to some strong but natural estimates of smoothness decay at large velocities and strict positivity which at the moment have only been established in certain particular cases among the most important steps in our proof are 1 quantitative variants of boltzmann s h theorem as proven in 52 60 based on symmetry features hypercontractivity and information theoretical tools 2 a new quantitative version of the instability of the hydrodynamic description for non small knudsen number 3 some functional inequalities with geometrical content in particular the korn type inequality which we established in 22 and 4 the study of a system of coupled differential inequalities of second order by a treatment inspired from 21 we also briefly point out the particular role of conformal velocity fields when they are allowed by the geometry of the problem

on the cauchy problem for the boltzmann equation in the whole space global existence and uniform stability in lξ2 hxn abstract based on a refined energy method in this paper we prove the global existence and uniform in time stability of solutions in the space l 2 h x n to the cauchy problem for the boltzmann equation around a global maxwellian in the whole space r 3 compared with the solution space used by the spectral analysis and the classical energy method the velocity weight functions or time derivatives need not be included in the norms of l 2 h x n which is realized by introducing some temporal interactive energy functionals to estimate the macroscopic dissipation rate the key proof is carried out in terms of the macroscopic equations together with the local conservation laws it is also found that the perturbed macroscopic variables actually satisfy the linearized compressible navier stokes equations with remaining terms only related to the microscopic part

navier stokes equations for compressible fluids global existence and qualitative properties of the solutions in the general case we consider the equations which describe the motion of a viscous compressible fluid taking into consideration the case of inflow and or outflow through the boundary by means of some a priori estimates we prove the existence of a global in time solution moreover as a consequence of a stability result we show that there exist a periodic solution and a stationary solution

stability of the boltzmann equation with potential forces on torus abstract in this paper we are concerned with the stability of solutions to the cauchy problem of the boltzmann equation with potential forces on torus it is shown that the natural steady state with the symmetry of origin is asymptotically stable in the sobolev space with exponential rate in time for any initially smooth periodic origin symmetric small perturbation which preserves the same total mass momentum and mechanical energy for the non symmetric steady state it is also shown that it is stable in l 1 norm for any initial data with the finite total mass mechanical energy and entropy

the vlasov poisson boltzmann system without angular cutoff this paper is concerned with the vlasov poisson boltzmann system for plasma particles of two species in three space dimensions the boltzmann collision kernel is assumed to be angular non cutoff with 3 2s and 1 2 s 1 where s are two parameters describing the kinetic and angular singularities respectively we establish the global existence and convergence rates of classical solutions to the cauchy problem when initial data is near maxwellians this extends the results in duan et al j diff eqs 252 12 6356 6386 2012 math models methods appl sci 23 6 927 2013 for the cutoff kernel with 2 1 to the case 3 2 as long as the angular singularity exists instead and is strong enough i e s is close to 1 the proof is based on the time weighted energy method building also upon the recent studies of the non cutoff boltzmann equation in gressman and strain j amer math soc 24 3 771 847 2011 and the vlasov poisson landau system in guo j amer math soc 25 759 812 2012

space time fractional zener wave equation the space time fractional zener wave equation describing viscoelastic materials obeying the time fractional zener model and the space fractional strain measure is derived and analysed this model includes waves with finite speed as well as non propagating disturbances the existence and the uniqueness of the solution to the generalized cauchy problem are proved special cases are investigated and numerical examples are presented

a new anisotropic fractional model of diffusion suitable for applications of diffusion tensor imaging in biological tissues an anomalous anisotropic diffusion equation is constructed in which the order of the spatial pseudo differential operator is generalized to be distributed with a directionally dependent distributio

erratum to time periodic solutions of the navier stokes equations in unbounded cylindrical domains leray s problem for periodic flows poiseuille flows in infinite cylindrical pipes in spite of their enormous simplicity have a main role in many theoretical and applied problems as is well known the poiseuille flow is a stationary solution of the stokes and the navier stokes equations with a given constant flux time periodic flows in channels and pipes have a comparable importance however the problem of the existence of time periodic flows in correspondence to any given time periodic total flux is still an open problem a solution is known only in some very particular cases for instance the womersley flows our aim is to solve this problem in the general case the above existence result opens the way to further investigations as an example of this possibility we consider the extension of the classical leray s problem for poiseuille flows to arbitrary time periodic flows

optimal large time behavior of the vlasov maxwell boltzmann system in the whole space in this paper we study the large time behavior of classical solutions to the two species vlasov maxwell boltzmann system in the whole space input amssym bbb r 3 the existence of global in time nearby maxwellian solutions is known from strain in 2006 however the asymptotic behavior of these solutions has been a challenging open problem building on our previous work on time decay for the simpler vlasov poisson boltzmann system we prove that these solutions converge to the global maxwellian with the optimal decay rate of o t 3 2 3 2r in the l l norm for any 2 r if initial perturbation is smooth enough and decays in space velocity fast enough at infinity moreover some explicit rates for the electromagnetic field tending to 0 are also provided 2011 wiley periodicals inc

on the cauchy problem for the nonlinear boltzmann equation global existence uniqueness and asymptotic stability the analysis of the initial value problem for the nonlinear boltzmann equation is considered in this paper a theorem defining global existence and uniqueness for initial data which decay at infinity with an inverse power law is the main result of this work and is obtained by suitable application of fixed point theorems in banach spaces the theorem also defines the asymptotic stability of the solutions

time distributed order diffusion wave equation ii applications of laplace and fourier transformations a cauchy problem for a time distributed order multi dimensional diffusion wave equation containing a forcing term is reinterpreted in the space of tempered distributions and a distributional diffusion wave equation is obtained the distributional equation is solved in the general case of weight function or distribution solutions are given in terms of solution kernels green s functions which are studied separately for two cases the first case is when the order of the fractional derivative is in the interval 0 1 while in the second case the order of the fractional derivative is in the interval 0 2 solutions of fractional diffusionwave and fractional telegraph equations are obtained as special cases numerical experiments are also performed an analogue of the maximum principle is also presented

optimal time decay of the boltzmann system for gas mixtures abstract in this paper we are concerned with the boltzmann equation for the mixture of vapors of two gases in the whole space given the initial data of one gas near vacuum and the other near a global maxwellian state we obtain the global existence and optimal large time behaviors of the boltzmann system for such binary mixture the proof is based on kawashima s compensating function technique and a refined energy method

distributed order calculus and equations of ultraslow diffusion abstract we consider equations of the form d u t x u t x f t x t 0 x r n where d is a distributed order derivative that is d t 0 1 d t d d is the caputo dzhrbashyan fractional derivative of order is a positive weight function the above equation is used in physical literature for modeling diffusion with a logarithmic growth of the mean square displacement in this work we develop a mathematical theory of such equations study the derivatives and integrals of distributed order

spectral properties of the linearized boltzmann operator in l p for 1 p the spectrum of both classical and relativistic boltzmann operator for hard interactions in a whole space is shown to be independent of p in l p for 1 p it consists of a half plane operatorname re lambda leq nu 0 and countably many branches in the strip nu 0 operatorname re lambda leq0 moreover the resolvent set is independent of p for all 1 p but in l this operator possesses in addition to the continuous also an uncountable set of point spectrum

fractional differential equations and related exact mechanical models the aim of the paper is the description of fractional order differential equations in terms of exact mechanical models this result will be archived in the paper for the case of linear multiphase fractional hereditariness involving linear combinations of power laws in relaxation creep functions the mechanical model corresponding to fractional order differential equations is the extension of a recently introduced exact mechanical representation di paola and zingales 2012 33 and di paola et al 2012 34 of fractional order integrals and derivatives some numerical applications have been reported in the paper to assess the capabilities of the model in terms of a peculiar arrangement of linear springs and dashpots

time distributed order diffusion wave equation i volterra type equation a single order time fractional diffusion wave equation is generalized by introducing a time distributed order fractional derivative and forcing term while a laplacian is replaced by a general linear multi dimensional spatial differential operator the obtained equation is in the case of the laplacian called a time distributed order diffusion wave equation we analyse a cauchy problem for such an equation by means of the theory of an abstract volterra equation the weight distribution occurring in the distributed order fractional derivative is specified as the sum of the dirac distributions and the existence and uniqueness of solutions to the cauchy problem and the corresponding volterra type equation were proven for a general linear spatial differential operator as well as in the special case when the operator is laplacian

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

multi dimensional solutions of space time fractional diffusion equations green9s functions and propagators for the multi dimensional anisotropic space time fractional diffusion equation are obtained in integral form uniqueness of the solutions is studied in the framework of abstract volterra equations unimodality of propagator functions is demonstrated in the centrally symmetric case the transition from diffusion to wave propagation is discussed numerical examples are constructed

hypocoercivity of the relativistic boltzmann and landau equations in the whole space abstract we study the hypocoercivity property for some kinetic equations in the whole space and obtain the optimal convergence rates of solutions to the equilibrium state in some function spaces the analysis relies on the basic energy method and the compensating function introduced by kawashima to the classical boltzmann equation and developed by glassey and strauss in the relativistic setting it is also motivated by the recent work duan et al 2008 8 on the boltzmann equation by combining the spectrum analysis and energy method the advantage of the method introduced in this paper is that it can be applied to some complicated system whose detailed spectrum is not known in fact only some estimates through the fourier transform on the conservative transport operator and the dissipation of the linearized operator on the subspace orthogonal to the collision invariants are needed

force implication a new approach to human reasoning abstract as it is well known the logical material implication is not natural from the point of view of human reasoning besides in fuzzy logic it is possible to classify the ply operators refering either to the boolean implication or to the boolean conjunction mamdani s operator combining the aim to modelize human reasoning in a more natural way subsequently in the spirit of mamdani with the necessity to get an implication that is to say a non symmetric operator a new ply operator is introduced here named the force implication then conditional objects and generalized modus ponens are evoked through their links with the force implication

optimal decay estimates on the linearized boltzmann equation with time dependent force and their applications although the decay in time estimates of the semi group generated by the linearized boltzmann operator without forcing have been well established there is no corresponding result for the case with general external force this paper is mainly concerned with the optimal decay estimates on the solution operator in some weighted sobolev spaces for the linearized boltzmann equation with a time dependent external force no time decay assumption is made on the force the proof is based on both the energy method through the macro micro decomposition and the lp lq estimates from the spectral analysis the decay estimates thus obtained are applied to the study on the global existence of the cauchy problem to the nonlinear boltzmann equation with time dependent external force and source precisely for space dimension n 3 the global existence and decay rates of solutions to the cauchy problem are obtained under the condition that the force and source decay in time with some rates this time decay restriction can be removed for space dimension n 5 moreover the existence and asymptotic stability of the time periodic solution are given for space dimension n 5 when the force and source are time periodic with the same period

the relativistic boltzmann equation for soft potentials abstract the paper concerns the cauchy problem on the relativistic boltzmann equation for soft potentials in a periodic box we show that the global in time solutions around relativistic maxwellians exist in the weighted l perturbation framework and also approach equilibrium states in large time in the weighted l 2 framework at the rate of exp t for some 0 and 0 1 the proof is based on the nonlinear l 2 energy method and nonlinear l pointwise estimate with appropriate exponential weights in momentum the results extend those on the classical boltzmann equation by caflisch 2 3 and strain and guo 31 to the relativistic version and also improve the recent result on almost exponential time decay by strain 28 to the exponential rate moreover we study the propagation of spatial regularity for the obtained solutions and also the large time behavior in the corresponding regular sobolev space provided that the spatial derivatives of initial data are bounded not necessarily small

global stability of boltzmann equation with large external potential for a class of large oscillation data abstract in this paper we investigate the stability of boltzmann equation with large external potential in t 3 for a class of initial data with large oscillations in l x v around the local maxwellian we prove the existence of a global solution to the boltzmann equation provided the initial perturbation is suitably small in l 2 norm the large time behavior of the boltzmann solution with exponential decay rate is also obtained this seems to be the first result on the perturbation theory of large amplitude non constant equilibriums for large amplitude initial data

a kinetic flocking model with diffusion we study the stability of the equilibrium states and the rate of convergence of solutions towards them for the continuous kinetic version of the cucker smale flocking in presence of diffusion whose strength depends on the density this kinetic equation describes the collective behavior of an ensemble of organisms animals or devices which are forced to adapt their velocities according to a certain rule implying a final configuration in which the ensemble flies at the mean velocity of the initial configuration our analysis takes advantage both from the fact that the global equilibrium is a maxwellian distribution function and on the contrary to what happens in the cucker smale model ieee trans autom control 52 852 862 2007 the interaction potential is an integrable function precise conditions which guarantee polynomial rates of convergence towards the global equilibrium are found

the boltzmann equation global existence for a rare gas in an infinite vacuum solutions of the boltzmann equation are proved to exist globally in time under conditions that include the case of a finite volume of gas in an infinite vacuum when the mean free path of the gas is large enough it is also proved as might be expected in this case that the density of the gas at each point in space goes to zero as time goes to infinity

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

waves in fractional zener type viscoelastic media classical wave equation is generalized for the case of viscoelastic materials obeying fractional zener model instead of hooke s law cauchy problem for such an equation is studied existence and uniqueness of the fundamental solution is proven and solution is calculated

global existence of classical solutions to the vlasov poisson boltzmann system the time evolution of the distribution function for the charged particles in a dilute gas is governed by the vlasov poisson boltzmann system when the force is self induced and its potential function satisfies the poisson equation in this paper we give a satisfactory global existence theory of classical solutions to this system when the initial data is a small perturbation of a global maxwellian moreover the convergence rate in time to the global maxwellian is also obtained through the energy method the proof is based on the theory of compressible navier stokes equations with forcing and the decomposition of the solutions to the boltzmann equation with respect to the local maxwellian introduced in 23 and elaborated in 31

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

compressible navier stokes approximation to the boltzmann equation abstract even though the system of the compressible navier stokes equations is not a limiting system of the boltzmann equation when the knudsen number tends to zero it is the second order approximation by applying the chapman enskog expansion the purpose of this paper is to justify this approximation rigorously in mathematics that is if the difference between the initial data for the compressible navier stokes equations and the boltzmann equation is of the second order of the knudsen number so is the difference between two solutions for all time the analysis is based on a refined energy method for a fluid type system using the techniques for the system of viscous conservation laws

asymptotic analysis of the non steady navier stokes equations in a tube structure i the case without boundary layer in time abstract the non steady navier stokes equations with dirichlet boundary conditions are considered in thin tube structures these domains are connected finite unions of thin finite cylinders in the 2 d case respectively thin rectangles the complete asymptotic expansion of the solution is constructed in the case without boundary layer in time the estimates for the difference of the exact solution and its j th asymptotic approximation is proved the method of asymptotic partial domain decomposition is formulated and justified for the non steady navier stokes equations in a tube structure it gives the asymptotically exact interface conditions of coupling of the 1d and 3d models of the flow note that the obtained results hold true for the important in applications case of time periodic flows

on mittag leffler type functions in fractional evolution processes abstract we review a variety of fractional evolution processes so defined being governed by equations of fractional order whose solutions turn out to be related to mittag leffler type functions the chosen equations are the simplest of the fractional calculus and include the abel integral equations of the second kind which are relevant in typical inverse problems and the fractional differential equations which govern generalized relaxation and oscillation phenomena