l 1 stability estimates for n x n conservation laws let u t f u x 0 be a strictly hyperbolic n times n system of conservation laws each characteristic field being linearly degenerate or genuinely nonlinear in this paper we explicitly define a functional phi phi u v equivalent to the l 1 distance which is almost decreasing i e phi big u t v t big phi big u s v s big leq o ve cdot t s quad hbox for all t s geq 0 for every pair of e approximate solutions u v with small total variation generated by a wave front tracking algorithm the small parameter e here controls the errors in the wave speeds the maximum size of rarefaction fronts and the total strength of all non physical waves in u and in v from the above estimate it follows that front tracking approximations converge to a unique limit solution depending lipschitz continuously on the initial data in the vec l 1 norm this provides a new proof of the existence of the standard riemann semigroup generated by a n n system of conservation laws

the broadwell model for initial values in l 1 _ bf r the cauchy problem for the broadwell model is shown to have a global mild solution for initial data inl 1 with smalll1 norm and a local solution for arbitrary initial data inl 1 for data which are small inl1 the asymptotic behaviour of the solutions ast is determined moreover it is shown that a global solution exists for all initial values inl 1 with finite entropy if theh theorem holds

existence of uncountably many bounded nonoscillatory solutions and their iterative approximations for second order nonlinear neutral delay difference equations in this paper the banach fixed point theorem is employed to establish several existence results of uncountably many bounded nonoscillatory solutions for the second order nonlinear neutral delay difference email protected n dx n b n x email protected email protected n x h 1 n x h 2 n x h k n f n x f 1 n x f 2 n x f k n c n n n 0 where t email protected n n email protected n 0 a b c n n 0 r with a n 0 for email protected n n 0 h f n n 0xr k r and h l f l n n 0 z withlimn h l n limn f l n email protected 1 2 k a few mann type iterative approximation schemes with errors are suggested and the errors estimates between the iterative approximations and the nonoscillatory solutions are discussed seven nontrivial examples are given to illustrate the advantages of our results

on diffuse reflection at the boundary for the boltzmann equation and related equations the paper considers diffuse reflection at the boundary with nonconstant boundary temperature and unbounded velocities the solutions obtained are proved to conserve mass at the boundary after a preliminary study of the collisionless case the main results obtained are existence for the boltzmann equation in a diperna lions framework with the above boundary conditions in a bounded measure sense and existence together with uniqueness for the bgk equation with maxwellian diffusion on the boundary in anl framework

metastability and the ising model we discuss a recent theorem which establishes a precise connection between i the approximate degeneracy of the zero eigenvalue for the generator of the glauber dynamics of the ising model in a small nonzero field and below the critical temperature ii the existence of a partition of the configuration space into a normal region and a metastable region this enables us to demonstrate that the recent approach to metastability of davies and martin may be viewed as a simple although in some ways fairly crude approximation to the conventional approach we also obtain what appear to be the first results concerning the stability of metastable states under small perturbations

l 1 stability of the boltzmann equation for the hard sphere model we study the l1 stability of classical solutions to the boltzmann equation for a hard sphere model when initial datum is a small perturbation of a vacuum and tends to zero exponentially fast at infinity in the phase space for this we introduce nonlinear functionals measuring potential interactions between particles with different velocities and l1 distance between classical solutions we use pointwise estimates for a solution and the gain term of a collision operator to control the time evolution of nonlinear functionals

broadwell model and conservative supersonic boundary in this paper we consider an initial boundary value problem for the broadwell model with a supersonic physical boundary here we pose a conservative boundary condition which preserves the total number of particles we show that the solution converges pointwise to a boundary layer exponentially when the perturbations of the initial data to the equilibrium state are sufficiently small by using green s function as established in lan et al netw heterog media 1 167 183 2006 weighted energy estimates and by constructing a pair of anti derivatives to convert the conservative boundary condition into a dissipative boundary condition with conservation laws together with an a priori chosen boundary layer

linearized oscillation of first order nonlinear neutral delay differential equations abstract in this paper we establish a equivalence of oscillation for the first order nonlinear neutral delay differential equation x t px t i 1 m q i f i x t i 0 and its corresponding linear equation x t px t i 1 m q i x t i 0 under a weaker condition on fi

lyapunov functionals and l 1 stability for discrete velocity boltzmann equations we devise lyapunov functionals and prove uniform l 1 stability for one dimensional semilinear hyperbolic systems with quadratic nonlinear source terms these systems encompass a class of discrete velocity models for the boltzmann equation the lyapunov functional is equivalent to the l 1 distance between two weak solutions and non increasing in time they result from computations of two point interactions in the phase space for certain models with only transversal collisional terms there exist generalizations for three and multi point interactions

nonlinear functionals of the boltzmann equation and uniform stability estimates abstract we present nonlinear functionals measuring physical space variation and l 1 distance between two classical solutions for the boltzmann equation with a cut off inverse power potential in the case that initial datum is a small smooth perturbation of vacuum and decays fast enough in the phase space we show that these functionals satisfy stability estimates which lead to bv type estimates and a uniform l 1 stability

the discrete boltzmann equation with multiple collisions global existence and stability for the initial value problem this paper deals with the analysis of the initial value problem in all space rd d 2 for the discrete boltzmann equation with multiple collisions after a preliminary analysis of the mathematical modeling of the evolution equations global existence and asymptotic behavior are proven using suitable stability criteria some applications verify the analysis for specific models

a new contribution to nonlinear stability of a discrete velocity model for a class of discrete velocity models of kinetic theory we prove exponential nonlinear conditional stability of the constant basic state in the slab 0 1

well posedness theory for hyperbolic conservation laws the paper presents a well posedness theory for the initial value problem for a general system of hyperbolic conservation laws we will start with the refinement of glimm s existence theory and discuss the principle of nonlinear through wave tracing our main goal is to introduce a nonlinear functional for two solutions with the property that it is equivalent to the l1 x distance between the two solutions and is time decreasing moreover the functional is constructed explicitly in terms of the wave patterns of the solutions through the nonlinear superposition it consists of a linear term measuring the l1 x distance a quadratic term measuring the coupling of waves and distance and a generalized entropy functional 1999 john wiley sons inc

a random discrete velocity model and approximation of the boltzmann equation an approximation procedure for the boltzmann equation based on random choices of collision pairs from a fixed velocity set and on discrete velocity models is designed in a suitable limit the procedure is shown to converge to the time discretized and spatially homogeneous boltzmann equation

bounded oscillation for second order nonlinear delay differential equation in a critical state in this paper we establish sufficient conditions for bounded oscillation of the second order nonlinear delay differential equation x t i 1m pix t i f t x t 1 t x t n t under a critical state by the second order ordinary differential equation y t 1 e e 0t f t e 0 t 1 t y t e 0 t t y t 0 at the same time necessary conditions for the above bounded oscillation equation are obtained

sharp asymptotics for fixation times in stochastic population genetics models at low mutation probabilities this article describes new models in population genetics that extend the neutral wright fisher model by including strong selection and mutation fixation times are studied in the limit of small mutation probabilities within the framework of markov chains with rare transitions the main result outlines the role of the discrete geometry of the fitness landscape and provides a mean for estimating the expected number of generations for an individual with better fitness value to appear some connections to evolutionary algorithms are discussed as well

dynamical stability of metastable states abstract we continue our study of symmetric markov semigroups for which the zero eigenvalue is almost but not quite degenerate we show that the metastable states of the semigroup are dynamically stable under small perturbations without any of the technical restrictions or approximations of our earlier papers

approximation of the boltzmann equation by discrete velocity models two convergence results related to the approximation of the boltzmann equation by discrete velocity models are presented first we construct a sequence of deterministic discrete velocity models and prove convergence as the number of discrete velocities tends to infinity of their solutions to the solution of a spatially homogeneous boltzmann equation second we introduce a sequence of markov jump processes interpreted as random discrete velocity models and prove convergence as the intensity of jumps tends to infinity of these processes to the solution of a deterministic discrete velocity model

global weak solutions of the boltzmann equation a new definition of the concept of weak solution of the nonlinear boltzmann equation is introduced it is proved that without any truncation on the collision kernel the boltzmann equation in the one dimensional case has a global weak solution in this sense global conservation of energy follows

discrete velocity models of the boltzmann equation a survey on the mathematical aspects of the theory discrete velocity models of the boltzmann equation are of considerable conceptual interest in the kinetic theory of gases and at the same time a fascinating mathematical subject the last decade has seen a lot of work on this subject which is reviewed here we put an emphasis on mathematical aspects in particular the global existence question qualitative properties of solutions explicit solutions and the fluid dynamical limit the mechanical background is discussed more concisely in part i numerous contributions concerning other questions are outlined we try to give a complete reference list up to and including 1985

nonlinear functionals of multi d discrete velocity boltzmann equations in this paper we study nonlinear functionals measuring potential interactions and l1 distance between two mild solutions for the multi dimensional discrete velocity boltzmann equations when the initial data are a small perturbation of a vacuum we employ bony s dispersion estimates to show that these functionals satisfy lyapunov type estimates which are useful for the study of time asymptotics and l1 stability of mild solutions

measure solutions of the steady boltzmann equation in a slab it is shown that the steady boltzmann equation in a slab 0 a has solutionsx x such that the ingoing boundary measures 0 0 and 0 can be prescribed a priori the collision kernel is truncated such that particles with smallx component of the velocity have a reduced collision rate

the broadwell model in a box strong l 1 convergence to equilibrium the global solution of the one dimensional broadwell model in the interval 0 1 with reflecting boundary conditions at 0 and 1 is shown to converge strongly in l 1 0 1 to the constant equilibrium solution

weak asymptotic decay via a relaxed invariance principle for a wave equation with nonlinear non monotone damping this paper considers the problem of asymptotic decay as t of solutions of the wave equation u tt u a x u t u t x a bounded open connected set in n n 1 with smooth boundary u 0 on the nonlinear function is not assumed to be globally lipschitz continuous 0 y 2 y n 1 0 y 1 y 1 y n 1 0 for all y n 1 is not assumed to be monotone in y 1 under additional restrictions on the kernel of conditions are given which imply that u u t converges to 0 0 weakly in h h 1 0 l 2 as t the work generalises earlier results of dafermos and haraux where strong decay in h as t was obtained in the case y 1 y n 1 q y 1 q monotone on

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

on the cauchy problem for the discrete boltzmann equation with initial values inl1 ℝ we prove a global existence theorem for a discrete velocity model of the boltzmann equation when the initial values i x have finite entropy and for some constant 0 1 x i x el1

global weak solutions and uniform lp stability of the boltzmann enskog equation abstract we present the regularity theory of renormalized solutions and uniform l p stability estimates of the boltzmann enskog model we use a multi dimensional bony type functional to control the time phase space integral of a truncated collision operator we employ cercignani s arguments and a bony type functional to show that the renormalized solutions to the boltzmann enskog model are weak solutions in usual distributional sense for the uniform l p stability estimate we use lu s trick together with the holder inequality

large time behavior of the broadwell model of a discrete velocity gas with specularly reflective boundary conditions this paper studies the behavior of the one dimensional broadwell model of a discrete three velocity gas on a bounded domain 0 x 1 with specularly reflective boundary condition at x 0 1 for smooth initial data we show that the initial boundary value problem possesses a unique smooth solution which tends as t to a free state consisting of traveling waves f 1e x ct f 2e x ct f 3e x where each f ie is 2 periodic the convergence is in the weak topology of an appropriate orlicz banach state space no smallness assumptions are made on the data

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

large time behavior of discrete velocity boltzmann equations we study the asymptotic behavior of equations representing one dimensional motions in a fictitious gas with a discrete set of velocities the solutions considered have finite mass but arbitrary amplitude with certain assumptions every solution approaches a state in which each component is a traveling wave without interaction the techniques are similar to those in an earlier treatment of the broadwell model 1

boundary value problems for the steady boltzmann equation we discuss steady boundary value problems for the boltzmann equation with inflow and diffusive boundary conditions in one two and three dimensions with suitable truncations of the collision kernel general existence and uniqueness results are obtained if the domain is sufficiently small in one dimension the existence of solutions on general intervals is obtained by abstract fixed point theory

trend to equilibrium of weak solutions of the boltzmann equation in a slab with diffusive boundary conditions recently r illner and the author proved that under a physically realistic truncation on the collision kernel the boltzmann equation in the one dimensional slab 0 1 with general diffusive boundary conditions at 0 and 1 has a global weak solution in the traditional sense here it is proved that when the maxwellians associated with the boundary conditions atx 0 andx 1 are the same maxwellianmw then the solution is uniformly bounded and tends tomw fort

a global existence theorem for the initial boundary value problem for the boltzmann equation when the boundaries are not isothermal we extend the existence theorem recently proved by hamdache for the initial boundary value problem for the nonlinear boltzmann equation in a vessel with isothermal boundaries to more general situations including the case when the boundaries are not isothermal in the latter case a cut off for large speeds is introduced in the collision term of the boltzmann equation

global weak solutions of the boltzmann equation in a slab with diffusive boundary conditions it is proved that under a physically realistic truncation of the collision kernel the boltzmann equation in the one dimensional slab 0 1 with general diffusive boundary conditions at 0 and 1 has a global weak solution in the traditional sense this solution satisfies the boundary conditions almost everywhere and has at worst exponentially growing total energy

bounds on the spectrum for markov chains and markov processes a generalization of cheeger s inequality we prove a general version of cheeger s inequality for discrete time markov chains and continuous time markovian jump processes both reversible and nonreversible with general state space we also prove a version of cheeger s inequality for markov chains and processes with killing as an application we prove l 2 exponential convergence to equilibrium for random walk with inward drift on a class of countable rooted graphs

dynamics of 2 1 dimensional sos surfaces above a wall slow mixing induced by entropic repulsion we study the glauber dynamics for the 2 1 d solid on solid model above a hard wall and below a far away ceiling on an l times l box of z 2 with zero boundary conditions at large inverse temperature beta it was shown by bricmont el mellouki and frohlich 1986 that the floor constraint induces an entropic repulsion effect which lifts the surface to an average height h asymp 1 beta log l as an essential step in understanding the effect of entropic repulsion on the glauber dynamics we determine the equilibrium height h to within an additive constant h 1 4 beta log l o 1 we then show that starting from zero initial conditions the surface rises to its final height h through a sequence of metastable transitions between consecutive levels the time for a transition from height h ah a in 0 1 to height h 1 is roughly exp c l a for some constant c 0 in particular the mixing time of the dynamics is exponentially large in l i e t mix geq e c l we also provide the matching upper bound t mix leq e c l requiring a challenging analysis of the statistics of height contours at low temperature and new coupling ideas and techniques finally to emphasize the role of entropic repulsion we show that without a floor constraint at height zero the mixing time is no longer exponentially large in l