nonlinear vibration of a coating fgm substrate cylindrical panel subjected to a temperature gradient this paper deals with the linear and nonlinear vibration analysis of a three layer coating fgm substrate cylindrical panel with general boundary conditions and subjected to a temperature gradient across the thickness due to steady heat conduction the theoretical formulation is based on geometric nonlinearity in von karman sense and the first order shear deformation theory which accounts for the effects of transverse shear strains and rotary inertia a nonlinear pre vibration analysis is conducted to determine the thermally induced deformation and pre stress state of the cylindrical panel due to the bending and stretching coupling effect being introduced by the graded layer the nonlinear governing differential equations of motion are obtained by adding an incremental dynamic state to the pre vibration state a numerical method that makes use of differential quadrature approximation together with iterative algorithms is employed to model the linear and nonlinear vibration behavior of the cylindrical panels numerical results are presented in both tabular and graphical forms showing that the linear and nonlinear vibration frequencies are significantly influenced by vibration amplitude temperature change out of plane boundary support and the values of a b a h and r a the effects of the in plane displacement constraints and the volume fraction index of the intermediate fgm thin layer are not very prominent

combined breathing kink modes in the fpu lattice abstract combined breather kink modes have been observed in several systems here we show that in the fpu lattice small amplitude breathing kink modes are three soliton solutions of an associated mkdv equation since this is integrable the modes can be constructed using the backlund transform as well as finding explicit solutions we consider the stability of the combined mode using variational techniques and illustrate stability for some parameter values and instability for others

l2 contraction of large planar shock waves for multi dimensional scalar viscous conservation laws abstract we consider a l 2 contraction a l 2 type stability of large viscous shock waves for the multi dimensional scalar viscous conservation laws up to a suitable shift by using the relative entropy methods quite different from the previous results we find a new way to determine the shift function which depends both on the time and space variables and solves a viscous hamilton jacobi type equation with source terms moreover we do not impose any conditions on the anti derivative variables of the perturbation around the shock profile more precisely it is proved that if the initial perturbation around the viscous shock wave is suitably small in l 2 norm then the l 2 contraction holds true for the viscous shock wave up to a suitable shift function note that bv norm or the l norm of the initial perturbation and the shock wave strength can be arbitrarily large furthermore as the time t tends to infinity the l 2 contraction holds true up to a spatially homogeneous time dependent shift function in particular if we choose some special initial perturbations then l 2 convergence of the solutions towards the associated shock profile can be proved up to a time dependent shift

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

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

acoustic limit for the boltzmann equation in optimal scaling based on a recent l 2 l 1 framework we establish the acoustic limit of the boltzmann equation for general collision kernels the scaling of the fluctuations with respect to knudsen number is optimal our approach is based on a new analysis of the compressible euler limit of the boltzmann equation as well as refined estimates of euler and acoustic solutions

relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non bv perturbations we develop a theory based on relative entropy to show the uniqueness and l 2 stability up to a translation of extremal entropic rankine hugoniot discontinuities for systems of conservation laws typically 1 shocks n shocks 1 contact discontinuities and n contact discontinuities of large amplitude among bounded entropic weak solutions having an additional trace property the existence of a convex entropy is needed no bv estimate is needed on the weak solutions considered the theory holds without smallness conditions the assumptions are quite general for instance strict hyperbolicity is not needed globally for fluid mechanics the theory handles solutions with vacuums

diffusive expansion for solutions of the boltzmann equation in the whole space this paper is concerned with the diffusive expansion for solutions of the rescaled boltzmann equation in the whole space 0 1 tf v xf 1 q f f x rn v rn t 0 with prescribed initial data 0 2 f t x v t 0 f0 x v x rn v rn our main purpose is to justify the global validity of the diffusive expansion 0 3 f t x v f1 t x v 2f2 t x v n 1fn 1 t x v nfn t x v for a solution f t x v of the rescaled boltzmann equation 0 1 in the whole space rn for all t 0 with initial data f0 x v satisfying the initial expansion 0 4 f0 x v f1 0 x v 2f2 0 x v n 1fn 1 0 x v nfn 0 x v x v r2n here v 2 n2exp v 22 is a normalized global maxwellian under the assumption that the fluid components of the coefficients fm 0 x v 1 m n of the initial expansion f0 x v have divergence free velocity fields um0 x as well as temperature fields m0 x if we assume further that the velocity temperature fields u10 x 10 x of f1 0 x v have small amplitude in hs rn s 2 n n 2 we can determine these coefficients fm t x v 1 m n in the diffusive expansion 0 3 uniquely by an iteration method and energy method the hydrodynamic component of these coefficients fm t x v 1 m n satisfies the incompressible condition the boussinesq relations and or the navier stokes fourier system respectively while the microscopic component of these coefficients is determined by a recursive formula compared with the corresponding problem inside a periodic box studied in y guo 2006 18 the main difficulty here is due to the fact that poincare s inequality is not valid in the whole space rn and this difficulty is overcome by using the lp lq estimate on the riesz potential moreover by exploiting the energy method we can also deduce certain the space time energy estimates on these coefficients fm t x v 1 m n once the coefficients fm t x v 1 m n in the diffusive expansion 0 3 are uniquely determined and some delicate estimates have been obtained the uniform estimates with respect to on the remainders fn t x v are then established via a unified nonlinear energy method and such an estimate guarantees the validity of the diffusive expansion 0 3 in the large provided that 0 5 n 2n 2 notice that for m 2 um t x is no longer a divergence free vector and it is worth to pointing out that for m 3 it was in deducing certain estimates on pm t x by the lp lq estimate on the riesz potential that we need to require that n 2n 2

a discontinuous galerkin finite element method for a 1d prototype of the boltzmann equation to develop and analyze new computational techniques for the boltzmann equation based on model or approximation adaptivity it is imperative to have disposal of a compliant model problem that displays the essential characteristics of the boltzmann equation and that admits the extraction of highly accurate reference solutions for standard collision processes the boltzmann equation itself fails to meet the second requirement for d 2 3 spatial dimensions on account of its setting in 2d while for d 1 the first requirement is violated because the boltzmann equation then lacks the convergence to equilibrium property that forms the substructure of simplified models in this article we present a numerical investigation of a new one dimensional prototype of the boltzmann equation the underlying molecular model is endowed with random collisions which have been fabricated such that the corresponding boltzmann equation exhibits convergence to maxwell boltzmann equilibrium solutions the new boltzmann model is approximated by means of a discontinuous galerkin dg finite element method to validate the one dimensional boltzmann model we conduct numerical experiments and compare the results with monte carlo simulations of equivalent molecular dynamics models

stability of contact discontinuity for the boltzmann equation the boltzmann equation which describes the time evolution of a large number of particles through the binary collision in statistics physics has close relation to the systems of fluid dynamics that is euler equations and navier stokes equations as for a basic wave pattern to euler equations we consider the nonlinear stability of contact discontinuities to the boltzmann equation even though the stability of the other two nonlinear waves i e shocks and rarefaction waves has been extensively studied there are few stability results on the contact discontinuity because unlike shock waves and rarefaction waves its derivative has no definite sign and decays slower than a rarefaction wave moreover it behaves like a linear wave in a nonlinear setting so that its coupling with other nonlinear waves reveals a complicated interaction mechanism based on the new definition of contact waves to the boltzmann equation corresponding to the contact discontinuities for the euler equations we succeed in obtaining the time asymptotic stability of this wave pattern with a convergence rate in our analysis an intrinsic dissipative mechanism associated with this profile is found and used for closing the energy estimates

contact discontinuity with general perturbations for gas motions the contact discontinuity is one of the basic wave patterns in gas motions the stability of contact discontinuities with general perturbations for the navier stokes equations and the boltzmann equation is a long standing open problem general perturbations of a contact discontinuity may generate diffusion waves which evolve and interact with the contact wave to cause analytic difficulties in this paper we succeed in obtaining the large time asymptotic stability of a contact wave pattern with a convergence rate for the navier stokes equations and the boltzmann equation in a uniform way one of the key observations is that even though the energy norm of the deviation of the solution from the contact wave may grow at the rate it can be compensated by the decay in the energy norm of the derivatives of the deviation which is of the order of thus this reciprocal order of decay rates for the time evolution of the perturbation is essential to close the a priori estimate containing the uniform bounds of the l8 norm on the lower order estimate and then it gives the decay of the solution to the contact wave pattern

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

weak strong uniqueness for the isothermal navier stokes equations in this paper we are concerned with weak strong uniqueness results for the isothermal navier stokes equations in two space dimension using the methods of relative entropy we obtain some conditions on a weak solution such as the ones built up by plotnikov and weigant siam j math anal 47 626 653 2015 so that it is unique

asymptotic analysis of the lattice boltzmann equation in this article we analyze the lattice boltzmann equation lbe by using the asymptotic expansion technique we first relate the lbe to the finite discrete velocity model fdvm of the boltzmann equation with the diffusive scaling the analysis of this model directly leads to the incompressible navier stokes equations as opposed to the compressible navier stokes equations obtained by the chapman enskog analysis with convective scaling we also apply the asymptotic analysis directly to the fully discrete lbe as opposed to the usual practice of analyzing a continuous equation obtained through the taylor expansion of the lbe this leads to a consistency analysis which provides order by order information about the numerical solution of the lbe the asymptotic technique enables us to analyze the structure of the leading order errors and the accuracy of numerically derived quantities such as vorticity it also justifies the use of richardson s extrapolation method as an example a two dimensional taylor vortex flow is used to validate our analysis the numerical results agree very well with our analytic predictions

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

global existence of weak solutions to the fene dumbbell model of polymeric flows systems coupling fluids and polymers are of great interest in many branches of sciences one of the most classical models to describe them is the fene finite extensible nonlinear elastic dumbbell model we prove global existence of weak solutions to the fene dumbbell model of polymeric flows the main difficulty is the passage to the limit in a nonlinear term that has no obvious compactness properties the proof uses many weak convergence techniques in particular it is based on the control of the propagation of strong convergence of some well chosen quantity by studying a transport equation for its defect measure in addition this quantity controls a rescaled defect measure of the gradient of the velocity

moment closure approximations of the boltzmann equation based on varphi divergences this paper is concerned with approximations of the boltzmann equation based on the method of moments we propose a generalization of the setting of the moment closure problem from relative entropy to varphi divergences and a corresponding closure procedure based on minimization of varphi divergences the proposed description encapsulates as special cases grad s classical closure based on expansion in hermite polynomials and levermore s entropy based closure we establish that the generalization to divergence based closures enables the construction of extended thermodynamic theories that avoid essential limitations of the standard moment closure formulations such as inadmissibility of the approximate phase space distribution potential loss of hyperbolicity and singularity of flux functions at local equilibrium the divergence based closure leads to a hierarchy of tractable symmetric hyperbolic systems that retain the fundamental structural properties of the boltzmann equation

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

an entropy stable discontinuous galerkin finite element moment method for the boltzmann equation this paper presents a numerical approximation technique for the boltzmann equation based on a moment system approximation in velocity dependence and a discontinuous galerkin finite element approximation in position dependence the closure relation for the moment systems derives from minimization of a suitable divergence this divergence based closure yields a hierarchy of tractable symmetric hyperbolic moment systems that retain the fundamental structural properties of the boltzmann equation the approximation in position dependence is based on the discontinuous galerkin finite element method the resulting combined discontinuous galerkin moment method corresponds to a galerkin approximation of the boltzmann equation in renormalized form the new momentclosure formulation engenders a new upwind numerical flux function based on half space integrals of the approximate distribution we establish that the proposed upwind flux ensures entropy dissipation of the discontinuous galerkin finite element approximation numerical results are presented for a one dimensional test case

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

from the boltzmann equation to the euler equations in the presence of boundaries the fluid dynamic limit of the boltzmann equation leading to the euler equations for an incompressible fluid with constant density in the presence of material boundaries shares some important features with the better known inviscid limit of the navier stokes equations the present paper slightly extends recent results from c bardos f golse l paillard the incompressible euler limit of the boltzmann equation with accommodation boundary condition comm math sci 10 2012 159 190 to the case of boundary conditions for the boltzmann equation more general than maxwell s accommodation condition