a blow up criterion for two dimensional compressible viscous heat conductive flows abstract we establish a blow up criterion in terms of the upper bound of the density and temperature for the strong solution to 2d compressible viscous heat conductive flows the initial vacuum is allowed

multi mode trimming of imperfect thin rings using masses at pre selected locations abstract this paper presents a practical method for trimming the natural frequencies of an initially imperfect ring to simultaneously eliminate certain of the frequency splits present compared with previous work the novel feature of this method is that the trimming masses are positioned at pre selected locations on the ring the basis for the proposed method is the concept of equivalent imperfection mass which allows any imperfect ring to be considered as a perfect ring with equivalent imperfection masses attached by considering this trimming problem it is deduced that it is possible to trim n pairs of modes simultaneously by removing a minimum of 2 n trimming masses at particular locations around the ring by positioning the trimming masses at pre selected locations it is shown that a simple set of trimming masses can be calculated easily and from this set an infinite number of solution sets can be found methods for generating these sets are outlined for the trimming of both a single and a dual pair of modes in practice it is likely that the trimming masses will be spaced regularly for this special case it is found that it is not possible to trim all single and dual mode pairs with any arrangement of masses validation of the derived simple solution set and the proposed procedure to generate further sets is achieved by studying a number of theoretical examples

free vibrations of elliptical rings with circumferentially variable thickness abstract this paper deals with the in plane free vibration of rings with a nominally elliptical centreline results are presented for rings of constant axial length that have a rectangular cross section the radial thickness of which is constant or has a simple analytically defined circumferential variation additionally and for the first time the effects of small variations in in plane profile such as those arising in practical rings due to manufacturing tolerances are considered the problem is tackled using an approach in which the true middle surface is determined numerically from the outer and inner surface profiles which can be defined either by exact analytical expressions or in a more general way using fourier series the rayleigh ritz method is used to obtain the natural frequencies and mode shapes results are presented for a range of cases including some that have previously been studied by other authors and some that have not the effects on frequency splitting due to profile variations and the aspect ratio of the ellipse are emphasized results obtained using the developed numerical approach show excellent agreement with finite element predictions

existence and uniqueness of weak solutions of the compressible spherically symmetric navier stokes equations abstract one of the most influential fundamental tools in harmonic analysis is the riesz transforms it maps l p functions to l p functions for any p 1 which plays an important role in singular operators as an application in fluid dynamics the norm equivalence between u l p and div u l p curl u l p is well established for p 1 however since riesz operators sent bounded functions only to bmo functions there is no hope to bound u l in terms of div u l curl u l as pointed out by hoff 2006 11 this is the main obstacle to obtain uniqueness of weak solutions for isentropic compressible flows fortunately based on new observations see lemma 2 2 we derive an exact estimate for u l 2 1 n div u l for any n dimensional radially symmetric vector functions u as a direct application we give an affirmative answer to the open problem of uniqueness of some weak solutions to the compressible spherically symmetric flows in a bounded ball

well posedness for compressible mhd systems with highly oscillating initial data in this paper a unique local solution for compressible magnetohydrodynamics systems has been constructed in the critical besov space framework by converting the system in euler coordinates to a system in lagrange coordinates our results improve the range of the lebesgue exponent in the besov space from 2 n to 2 2n where n denotes the space dimension then we give a lower bound for the maximal existence time which is important for our construction of global solutions based on the lower bound we use the effective viscous flux and hoff s energy method to obtain the unique global solution which allows the initial velocity field and the magnetic field to have large energies and allows the initial density to exhibit large oscillations on a set of small measure

discontinuous solutions of the navier stokes equations for multidimensional flows of heat conducting fluids we prove the global existence of weak solutions of the navier stokes equations for compressible heat conducting fluids in two and three space dimensions when the initial density is close to a constant in l 2 l the initial temperature is close to a constant in l 2 and the initial velocity is small in h s l 4 where s 0 when n 2 and when n 3 the l p norms must be weighted slightly when n 2 in particular the initial data may be discontinuous across a hypersurface of n a great deal of qualitative information about the solution is obtained for example we show that the velocity vorticity and temperature are relatively smooth in positive time as is the effective viscous flux f which is the divergence of the velocity minus a certain multiple of the pressure we find that f plays a central role in the entire analysis particularly in closing the required energy estimates and in understanding rates of regularization near the initial layer moreover f is precisely the quantity through which the hyperbolicity of the corresponding equations for inviscid fluids shows itself an effect which is crucial for obtaining time independent pointwise bounds for the density

the analysis of a finite element method with streamline diffusion for the compressible navier stokes equations this paper deals with the rigorous error analysis of finite element solutions of the two and three dimensional nonstationary compressible navier stokes equations the streamline diffusion technique is applied to the continuity equation to obtain a stabilized finite element scheme it is proved that the numerical formulation has a unique solution and the solution is convergent an a priori error estimate uniform in time is obtained

divergence measure fields and hyperbolic conservation laws we analyze a class of l infty vector fields called divergence measure fields we establish the gauss green formula the normal traces over subsets of lipschitz boundaries and the product rule for this class of l infty fields then we apply this theory to analyze l infty entropy solutions of initial boundary value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data the examples of conservation laws include multidimensional scalar equations the system of nonlinear elasticity and a class of m x m systems with affine characteristic hypersurfaces the analysis in l infty also extends to l p

multi mode trimming of imperfect rings abstract this paper proposes a method for trimming the natural frequencies of an imperfect ring to simultaneously eliminate certain of the frequency splits present initially the effect of the addition of a number of imperfection masses on a perfect ring is considered this is achieved by using a rayleigh ritz approach in which it is assumed that the mode shapes are identical to those of a perfect ring by considering the inverse the so called trimming problem it is deduced that it is possible to trim n pairs of modes simultaneously by removing a minimum of n trimming masses at particular locations around the ring to calculate the trimming mass locations it is necessary to solve n non linear algebraic equations once this has been achieved the magnitude of the trimming masses can be calculated easily for the special case of trimming a single pair of modes analytic solutions for the magnitude and position of the single required trimming mass are available to trim two pairs of modes it is shown that a simple analytic relationship exists between the angular positions of the two required trimming masses and that the magnitude of these masses can be obtained easily to trim more pairs of modes numerical techniques are required and for this purpose a numerical procedure is proposed validation of the derived analytic results and the proposed numerical procedure is achieved by studying a number of theoretical examples

global existence for systems of parabolic conservation laws in several space variables abstract we prove the global existence of solutions of the cauchy problem for certain systems of conservation laws with artificial viscosity terms added the system is assumed to admit a quadratic entropy which is consistent with the viscosity matrix and the initial data is assumed to be close to a constant in l2 l in particular our result applies to the equations of compressible fluid flow in two and three space variables

on blowup of classical solutions to the compressible navier stokes equations in this paper we study the finite time blow up of smooth solutions to the compressible navier stokes system when the initial data contain vacuums we prove that any classical solutions of viscous compressible fluids without heat conduction will blow up in finite time as long as the initial data has an isolated mass group see definition 2 2 the results hold regardless of either the size of the initial data or the far fields being vacuum or not this improves the blowup results of xin comm pure appl math 51 229 240 1998 by removing the crucial assumptions that the initial density has compact support and the smooth solution has finite total energy furthermore the analysis here also yields that any classical solutions of viscous compressible fluids without heat conduction in bounded domains or periodic domains will blow up in finite time if the initial data have an isolated mass group satisfying some suitable conditions

numerical investigation of cavitation in multidimensional compressible flows the compressible navier stokes equations for an ideal polytropic gas are considered in r n n 2 3 the question of possible vacuum formation an open theoretical problem is investigated numerically using highly accurate computational methods the flow is assumed to be symmetric about the origin with a purely radial velocity field the numerical results indicate that there are weak solutions to the navier stokes system in two and three space dimensions which display formation of vacuum when the initial data are discontinuous and sufficiently large the initial density is constant while the initial velocity field is symmetric points radially away from the origin and belongs to h s loc for all s n 2 in addition in the one dimensional case the numerical solutions are in agreement with known theoretical results

serrin type blowup criterion for full compressible navier stokes system the authors establish a serrin type blowup criterion for the cauchy problem of the three dimensional full compressible navier stokes system which states that a strong or smooth solution exists globally provided that the velocity satisfies serrin s condition and that the temporal integral of the maximum norm of the divergence of the velocity is bounded in particular this criterion extends the well known serrin s blowup criterion for the three dimensional incompressible navier stokes equations to the three dimensional full compressible system and is just the same as that of the barotropic case

frequency analysis of a uniform ring perturbed by point masses and springs abstract perturbation expansions of solutions for a uniform thin linear elastic ring perturbed by point masses and radial massless springs are developed the perturbation locations divide the ring into uniform segments so a variational formulation is used to determine the boundary conditions that must be satisfied between adjoining segments the motion of each segment can be represented as a weighted sum of the eigenfunctions for the uniform thin ring so when the boundary conditions are enforced the resulting algebraic relations are expanded as a function of the perturbation parameter the perturbation mass normalized by the ring mass a series of algebraic problems are sequentially solved to yield perturbation expansions for the modal frequencies and eigenmodes single mass dual mass and mass spring case studies are considered the perturbation results show excellent agreement with finite element analysis of a thin ring for mass perturbations up to 15 of the nominal ring mass the results are also compared to rayleigh ritz analysis

existence and uniqueness of solutions for some hyperbolic systems of conservation laws we study the cauchy problem for systems of conservation laws which belong to the temple class the compensated compactness theory is used to prove existence of solutions some uniqueness results are established by means of holmgren s principle

existence results for viscous polytropic fluids with vacuum abstract we consider the full navier stokes equations for viscous polytropic fluids with nonnegative thermal conductivity we prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition the initial density need not be positive and may vanish in an open set moreover our results hold for both bounded and unbounded domains

symmetric nonbarotropic flows with large data and forces we prove the global existence of weak solutions of the navier stokes equations of compressible nonbarotropic flow in three space dimensions with initial data and external forces which are large discontinuous and spherically or cylindrically symmetric the analysis allows for the possibility that a vacuum state emerges at the origin or axis of symmetry and the equations hold in the sense of distributions in the set where the density is positive in addition the mass and momentum equations hold weakly in the entire space time domain but with a nonstandard interpretation of the viscosity terms as distributions solutions are obtained as limits of solutions in annular regions between two balls or cylinders and the analysis allows for the possibility that energy is absorbed into the origin or axis and is lost in the limit as the inner radius goes to zero

on a parabolic hyperbolic chemotaxis system with discontinuous data well posedness stability and regularity abstract the global dynamics and regularity of parabolic hyperbolic systems is an interesting topic in pdes due to the coupling of competing dissipation and hyperbolic effects this paper is concerned with the cauchy problem of a parabolic hyperbolic system derived from a chemotaxis model describing the dynamics of the initiation of tumor angiogenesis it is shown that as time tends to infinity the cauchy problem with large amplitude discontinuous data admit global weak solutions which converge to a constant state resp a viscous shock wave if the asymptotic states of initial values at far field are equal resp unequal our results improve the previous results where initial value was required to be continuous and have small amplitude numerical simulations are performed to verify our analytical results illustrate the possible regularity of solutions and speculate the minimal regularity of initial data required to obtain the smooth classical solutions of the concerned parabolic hyperbolic system

solutions a variations bornees pour certains systemes hyperboliques de lois de conservation abstract blake temple trans amer math soc 280 1983 781 795 has described the hyperbolic systems of two conservation laws whose shock and rarefaction curves coincide in this note we prove the global existence of weak solutions for such systems with any bounded variation initial condition the proof is based upon standard numerical schemes as well as upon parabolic regularization the key is that the total variation of the riemann invariants is decreasing in time at least in the case of the initial condition with compact support we prove by using the glimm scheme that the system is decoupling in two conservation laws in one unknown in finite time

existence of global weak solutions for 3d degenerate compressible navier stokes equations in this paper we prove the existence of global weak solutions for 3d compressible navier stokes equations with degenerate viscosity the method is based on the bresch and desjardins commun math phys 238 211 223 2003 entropy conservation the main contribution of this paper is to derive the mellet and vasseur commun partial differ equ 32 431 452 2007 type inequality for weak solutions even if it is not verified by the first level of approximation this provides existence of global solutions in time for the compressible barotropic navier stokes equations the result holds for any gamma 1 in two dimensional space and for 1 gamma 3 in three dimensional space in both case with large initial data possibly vanishing on the vacuum this solves an open problem proposed by lions mathematical topics in fluid mechanics vol 2 compressible models 1998

dynamics of singularity surfaces for compressible viscous flows in two space dimensions we prove the global existence of solutions of the navier stokes equations of compressible barotropic flow in two space dimensions which exhibit convecting singularity curves the fluid density and velocity gradient have jump discontinuities across these curves exactly as predicted by the rankine hugoniot conditions and these jump discontinuities decay exponentially in time more rapidly for smaller viscosities the singularity curves remain c1 despite the fact that the velocity fields which convect them are not continuously differentiable 2002 wiley periodicals inc

global well posedness of the cauchy problem of two dimensional compressible navier stokes equations in weighted spaces abstract in this paper we study the global well posedness of classical solution to 2d cauchy problem of the compressible navier stokes equations with large initial data and vacuum it is proved that if the shear viscosity is a positive constant and the bulk viscosity is the power function of the density that is with 3 then the 2d cauchy problem of the compressible navier stokes equations on the whole space r 2 admits a unique global classical solution u which may contain vacuums in an open set of r 2 note that the initial data can be arbitrarily large to contain vacuum states various weighted estimates of the density and velocity are obtained in this paper and these self contained estimates reflect the fact that the weighted density and weighted velocity propagate along with the flow

cauchy problem for the vlasov poisson boltzmann system the dynamics of dilute electrons can be modelled by the fundamental vlasov poisson boltzmann system which describes mutual interactions of the electrons through collisions in the self consistent electric field in this paper it is shown that any smooth perturbation of a given global maxwellian leads to a unique global in time classical solution when either the mean free path is small or the background charge density is large moreover the solution converges to the global maxwellian when time tends to infinity the analysis combines the techniques used in the study of conservation laws with the decomposition of the boltzmann equation introduced in 17 19 by obtaining new entropy estimates for this physical model

nonlinear stability of strong traveling waves for the singular keller segel system with large perturbations abstract this paper is concerned with the nonlinear stability of traveling wave solutions for a conserved system of parabolic equations derived from a singular chemotaxis model describing the initiation of tumor angiogenesis when the initial datum is a continuous small perturbation with zero integral from the spatially shifted traveling wave the asymptotic stability of the large amplitude strong traveling waves has been established in a series of works 29 34 35 by the second author with his collaborators in this paper we shall show that similar stability results indeed hold true for large and discontinuous initial data i e the initial perturbation from the traveling wave could be discontinuous and has large oscillations such as riemann data with large jumps to the best of our knowledge this paper provides a first result on the asymptotic stability of large amplitude traveling waves with large initial perturbation for a system of conservation laws although similar results have been available for the scalar equations cf 8 42 we also extend existing results to the initial data with lower regularity

systems of conservation laws with invariant submanifolds systems of conservation laws with coinciding shock and rarefaction curves arise in the study of oil reservoir simulation multicomponent chromatogra phy as well as in the study of nonlinear motion in elastic strings here we characterize this phenomenon by deriving necessary and sufficient conditions on the geometry of a wave curve in order that the shock wave curve coincide with its associated rarefaction wave curve for a system of conservation laws this coinci dence is the one dimensional case of a submanifold of the state variables being invariant for the system of equations and the necessary and sufficient conditions are derived for invariant submanifolds of arbitrary dimension in the case of 2 x 2 systems we derive explicit formulas for the class of flux functions that give rise to the coupled nonlinear conservation laws for which the shock and rarefaction wave curves coincide introduction systems of conservation laws which have coinciding shock and rarefaction curves arise in the study of oil reservoir simulation nonlinear wave motion in elastic strings as well as in multicomponent chromatography 1 4 5 6 9 11 12 these systems have many interesting features the riemann problem for these equations can be explicitly solved in the large and wave interactions have a simplified structure even in the presence of a nonconvex flux function for this reason these systems represent some of the few examples for which the cauchy problem has been solved for arbitrary data of bounded variation also hyperbolic degeneracies appear in each of these systems in the present paper we are concerned with characterizing the phenomenon of coinciding shock and rarefaction curves this phenomenon turns out to be a special case of the general phenomenon of a submanifold of r being invariant for weak solutions of a given system of n conservation laws invariant manifolds of higher dimension also appear in physical conservation equations in which the conserved quantities represent concentrations i e when the concentrations c of any of the species vanish the system must reduce to a lower order system that expresses the conservation of the remaining species thus c 0 defines a manifold which is invariant in the sense that solutions that start on the manifold remain there for all time moreover such an invariant manifold of dimension n 1 is the boundary of an dimensional invariant region in the sense of chueh conley and smoller 14 in the case of

global strong solutions to the 3d full compressible navier stokes equations with density temperature dependent viscosities in bounded domains abstract we consider the three dimensional full compressible navier stokes system with density temperature dependent viscosities in smooth bounded domains for the case when the velocity u and absolute temperature admit the dirichlet boundary condition the strong solutions exist globally in time provided that u 0 l 2 2 0 l 2 2 is suitably small through some time weighted a priori estimates the main difficulties caused by the density temperature dependent viscosities and the bounded domain are overcome moreover the time uniform upper bounds for the l p norm of the gradient of the density are obtained which is of independent interest for compressible fluids when initial vacuum is allowed

a beale kato majda criterion for three dimensional compressible viscous heat conductive flows we prove a blow up criterion in terms of the upper bound of 1 for a strong solution to three dimensional compressible viscous heat conductive flows the main ingredient of the proof is an a priori estimate for a quantity independently introduced in haspot regularity of weak solutions of the compressible isentropic navier stokes equation arxiv 1001 1581 2010 and sun et al j math pure appl 95 36 47 2011 whose divergence can be viewed as the effective viscous flux

existence of global strong solutions in critical spaces for barotropic viscous fluids this paper is dedicated to the study of viscous compressible barotropic fluids in dimension n 2 we address the question of the global existence of strong solutions for initial data close to a constant state having critical besov regularity first this article shows the recent results of charve and danchin arch ration mech anal 198 1 233 271 2010 and chen et al commun pure appl math 63 1173 1224 2010 with a new proof our result relies on a new a priori estimate for the velocity that we derive via the intermediary of the effective velocity which allows us to cancel out the coupling between the density and the velocity as in haspot well posedness in critical spaces for barotropic viscous fluids 2009 second we improve the results of charve and danchin 2010 and chen et al 2010 by adding as in charve and danchin 2010 some regularity on the initial data in low frequencies in this case we obtain global strong solutions for a class of large initial data which rely on the results of hoff arch rational mech anal 139 303 354 1997 hoff commun pure appl math 55 11 1365 1407 2002 and hoff j math fluid mech 7 3 315 338 2005 and those of charve and danchin 2010 and chen et al 2010 we conclude by generalizing these results for general viscosity coefficients

a multi fluid compressible system as the limit of weak solutions of the isentropic compressible navier stokes equations this paper mainly concerns the mathematical justification of a viscous compressible multi fluid model linked to the baer nunziato model used by engineers see for instance ishii thermo fluid dynamic theory of two phase flow eyrolles paris 1975 under a stratification assumption more precisely we show that some approximate finite energy weak solutions of the isentropic compressible navier stokes equations converge on a short time interval to the strong solution of this viscous compressible multi fluid model provided the initial density sequence is uniformly bounded with corresponding young measures which are linear convex combinations of m dirac measures to the authors knowledge this provides in the multidimensional in space case a first positive answer to an open question see hillairet j math fluid mech 9 343 376 2007 with a stratification assumption the proof is based on the weak solutions constructed by desjardins commun partial differ equ 22 5 6 977 1008 1997 and on the existence and uniqueness of a local strong solution for the multi fluid model established by hillairet assuming initial density to be far from vacuum in a first step adapting the ideas from hoff and santos arch ration mech anal 188 509 543 2008 we prove that the sequence of weak solutions built by desjardins has extra regularity linked to the divergence of the velocity without any relation assumption between and coupled with the uniform bound of the density property this allows us to use appropriate defect measures and their nice properties introduced and proved by hillairet aspects interactifs de la m ecanique des fluides phd thesis ens lyon 2005 in order to prove that the young measure associated to the weak limit is the convex combination of m dirac measures finally under a non degeneracy assumption of this combination stratification assumption this provides a multi fluid system using a weak strong uniqueness argument we prove that this convex combination is the one corresponding to the strong solution of the multi fluid model built by hillairet if initial data are equal we will briefly discuss this assumption to complete the paper we also present a blow up criterion for this multi fluid system following huang et al in serrin type criterion for the three dimensional viscous compressible flows arxiv 2010

a multidimensional model for the combustion of compressible fluids we consider a multidimensional model for the combustion of compressible reacting fluids the flow is governed by the navier stokes in eulerian coordinates and the chemical reaction is irreversible and is governed by the arrhenius kinetics the existence of globally defined weak solutions is established by using weak convergence methods compactness and interpolation arguments in the spirit of feireisl 16 and p l lions 24

stability of godunov s method for a class of 2 2 systems of conservation laws on demontre la stabilite et la convergence du schema de godunov pour une classe particuliere de systemes 2 2 non lineaires de lois de conservation

global well posedness of compressible navier stokes equations for some classes of large initial data we prove the global well posedness of three dimensional compressible navier stokes equations for some classes of large initial data which may have large oscillation for the density and large energy for the velocity the proof uses the special structure of the system especially the effective viscous flux

well posedness in critical spaces for the system of compressible navier stokes in larger spaces this paper is dedicated to the study of viscous compressible barotropic fluids in dimension n geq2 we address the question of well posedness for it large data having critical besov regularity our result improves the analysis of r danchin in cite dw of chen et al in cite c2 and of the author in cite h1 h2 inasmuch as we may take initial density in b nn p 1 with 1 leq p infty our result relies on a new a priori estimate for the velocity where we introduce a new unknown called textit effective velocity to weaken one the coupling between the density and the velocity in particular for the first time we obtain uniqueness without imposing hypothesis on the gradient of the density

entropy bounded solutions to the one dimensional compressible navier stokes equations with zero heat conduction and far field vacuum abstract the entropy is one of the fundamental states of a fluid and in the viscous case the equation that it satisfies is highly singular in the region close to the vacuum in spite of its importance in the gas dynamics the mathematical analyses on the behavior of the entropy near the vacuum region were rarely carried out in particular in the presence of vacuum either at the far field or at some isolated interior points it was unknown whether the entropy remains its boundedness the results obtained in this paper indicate that the ideal gases retain their uniform boundedness of the entropy locally or globally in time if the vacuum occurs at the far field only and the density decays slowly enough at the far field precisely we consider the cauchy problem to the one dimensional full compressible navier stokes equations without heat conduction and establish the local and global existence and uniqueness of entropy bounded solutions in the presence of vacuum at the far field only it is also shown that different from the case that with compactly supported initial density the compressible navier stokes equations with slowly decaying initial density can propagate the regularities in inhomogeneous sobolev spaces

asymptotic stability of riemann solutions for a class of multidimensional systems of conservation laws with viscosity we prove the asymptotic stability of two state nonplanar riemann solutions for a class of multidimensional hyperbolic systems of conservation laws when the initial data are perturbed and viscosity is added the class considered here is those systems whose flux functions in different directions share a common complete system of riemann invariants the level surfaces of which are hyperplanes in particular we obtain the uniqueness of the self similar l entropy solution of the two state nonplanar riemann problem the asymptotic stability to which the main result refers is in the sense of the convergence as t in lloc1 of the space of directions x t that is the solution u t x of the perturbed problem satisfies u t t r as t in lloc1 n where r is the self similar entropy solution of the corresponding two state nonplanar riemann problem

asymptotic behavior of compressible p th power newtonian fluid with large initial data abstract in this paper we study a large time behavior of the global spherically or cylindrically symmetric solutions in h 1 for the p th power compressible newtonian fluid in multi dimension with large initial data precisely if the initial data is spherically symmetric the space dimension can be taken not less than two on the other hand if the initial data is cylindrically symmetric the smallness of initial data is not needed the main concern of the present paper is to investigate the exponential decay rate of a solution toward the stationary solution as time goes to infinity we get the uniform positive lower and upper bounds of the density by using different methods

global existence for 1d compressible isentropic navier stokes equations with large initial data we prove the global existence of weak solutions of the cauchy problem for the navier stokes equations of compressible isentropic flow of a polytropic gas in one space dimension the initial velocity and density are assumed to be in l2 and l2 n bv respectively modulo additive constants in particular no smallness assumptions are made about the intial data in addition we prove a result concerning the asymptotic decay of discontinuities in the solution when the adiabatic constant exceeds 3 2

global solutions of the navier stokes equations for multidimensional compressible flow with discontinuous initial data abstract we prove the global existence of weak solutions of the navier stokes equations for compressible isothermal flow in two and three space dimensions when the initial density is close to a constant in l 2 and l and the initial velocity is small in l 2 and bounded in l 2 n in two dimensions the l 2 norms must be weighted slightly a great deal of qualitative information about the solution is obtained for example we show that the velocity and vorticity are relatively smooth in positive time as is the effective viscous flux f which is the divergence of the velocity minus a certain multiple of the pressure we find that f plays a crucial role in the entire analysis particularly in closing the required energy estimates understanding rates of regularization near the initial layer and most important obtaining time independent pointwise bounds for the density

on the statistics of natural frequency splitting for rings with random mass imperfections abstract this paper considers the statistical distribution of natural frequency splits for an initially perfect ring with different types of random mass imperfection the analysis used to derive analytical expressions for the natural frequency splits is based on a rayleigh ritz approach in which it is assumed that the mode shapes of the imperfect rings are identical to those of a perfect ring the types of random mass imperfection investigated are i random harmonic variations in the mass per unit length around the circumference of the ring ii the attachment of random point masses at random locations on the ring and iii the attachment of random point masses at uniformly spaced positions on the ring for case i it is found that the frequency splits always have a rayleigh distribution for case ii an expression for the statistical distribution is deduced which tends to a rayleigh distribution as the number of attached masses increases for case iii it is found that the frequency split distribution is dependent upon the mode considered and the number of attached masses and that in some situations the frequency splits have a half gaussian i e non rayleigh distribution

the in plane vibration of thin rings with in plane profile variations part ii application to nominally circular rings abstract geometric profile variations always exist in nominally circular rings due to limitations in the manufacturing processes such profile variations are known to lead to frequency splitting between pairs of modes which are degenerate in a perfect ring in this paper the effects of circumferential profile variations on the in plane vibration characteristics of such rings are studied using a numerical method the inner and outer ring surfaces are described in a very general way by fourier series and the rayleigh ritz method is used to obtain the natural frequencies and mode shapes results are presented for a number of example cases which include single and multiple harmonic variations in profile the relationship between the patterns of frequency splitting and the harmonic content of the ring profile is investigated and the most important causes of frequency splitting are identified

asymptotic stability of riemann solutions in bgk approximations to certain multidimensional systems of conservation laws abstract we prove the asymptotic stability of nonplanar two states riemann solutions in bgk approximations of a class of multidimensional systems of conservation laws the latter consists of systems whose flux functions in different directions share a common complete system of riemann invariants the level surfaces of which are hyperplanes the asymptotic stability to which the main result refers is in the sense of the convergence as t in l loc 1 of the space of directions x t that is the solution z t x of the perturbed cauchy problem for the corresponding bgk system satisfies x z t t d r as t in l loc 1 r n where r is the self similar entropy solution of the two states nonplanar riemann problem for the system of conservation laws

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

blow up criterions of strong solutions to 3d compressible navier stokes equations with vacuum abstract in the paper we establish a blow up criterion in terms of the integrability of the density for strong solutions to the cauchy problem of compressible isentropic navier stokes equations in r 3 with vacuum under the assumptions on the coefficients of viscosity 29 3 this extends the corresponding results in huang et al 2011 sun et al 2011 20 36 where a blow up criterion in terms of the upper bound of the density was obtained under the condition 7 as a byproduct the restriction 7 in fan et al 2010 sun et al 2011 12 37 is relaxed to 29 3 for the full compressible navier stokes equations by giving a new proof of lemma 3 1 besides we get a blow up criterion in terms of the upper bound of the density and the temperature for strong solutions to the cauchy problem of the full compressible navier stokes equations in r 3 the appearance of vacuum could be allowed this extends the corresponding results in sun et al 2011 37 where a blow up criterion in terms of the upper bound of 1 was obtained without vacuum the effective viscous flux plays a very important role in the proofs

lagrangian structure for two dimensional non barotropic compressible fluids abstract we study the lagrangian structure for weak solutions of two dimensional navier stokes equations for a non barotropic compressible fluid i e we show the uniqueness of particle trajectories for two dimensional compressible fluids including the energy equation our result extends partially the result obtained for barotropic fluids by hoff and santos 10

convergence rate of the solution toward boundary layer solution for initial boundary value problem of the 2 d viscous conservation laws abstract in this paper we study the large time behavior of the solution to the initial boundary value problem for 2 d viscous conservation laws in the space x bt the global existence and the asymptotic stability of a stationary solution are proved by kawashima et al 1 here we investigate the convergence rate of solution toward the boundary layer solution with the non degenerate case where f u b h 2 sobolev space and via the weighted energy method we draw the conclusion that the solution converges to the corresponding boundary layer solution with algebraic or exponential rate in time under the assumption that the initial perturbation decays with algebraic or exponential in the spatial direction

global solutions to a model for exothermically reacting compressible flows with large discontinuous initial data we prove the global existence of solutions of the navier stokes equations describing the dynamic combustion of a compressible exothermically reacting fluid and we study the large time behavior of solutions giving necessary and sufficient conditions for complete combustion in certain cases the adiabatic constants and specific heats of the burned product and unburned reactant fluids may differ and the initial data may be large and discontinuous

the in plane vibration of thin rings with in plane profile variations part i general background and theoretical formulation abstract this paper presents a methodology for the analysis of the free in plane vibration of thin rings with profile variations in the circumferential direction the methodology is suitable for any thin ring which is bounded by closed curves which are single valued functions of circumferential position the inner and outer profiles are expressed as fourier series thus allowing any profile to be approximated with any degree of accuracy an iterative numerical procedure for determining the true middle surface and the corresponding thickness at each cross section around the circumference is established a reduced plane stress form of novozhilov s thin shell theory is used to model the deformation mechanics of the ring the eigenvalue problem is then formulated using the rayleigh ritz method in conjunction with a harmonic series description of the displacements general expressions are presented for the corresponding mass and stiffness matrices a companion paper presents a comprehensive set of results which illustrates application of the theory

large time behavior of entropy solutions of conservation laws abstract we are concerned with the large time behavior of discontinuous entropy solutions for hyperbolic systems of conservation laws we present two analytical approaches and explore their applications to the asymptotic problems for discontinuous entropy solutions these approaches allow the solutions of arbitrarily large oscillation without a priori assumption on the ways from which the solutions come the relation between the large time behavior of entropy solutions and the uniqueness of riemann solutions leads to an extensive study of the uniqueness problem we use a direct method to show the large time behavior of large l solutions for a class of m m systems including a model in multicomponent chromatography we employ the uniqueness of riemann solutions and the convergence of self similar scaling sequence of solutions to show the asymptotic behavior of large bv solutions for the 3 3 system of euler equations in thermoelasticity these results indicate that the riemann solution is the unique attractor of large discontinuous entropy solutions whose initial data are l l 1 or bv l 1 perturbation of the riemann data for these systems these approaches also work for proving the large time behavior of approximate solutions to hyperbolic conservation laws

a simple theory for the analysis and correction of frequency splitting in slightly imperfect rings abstract the flexural radial vibration of circular rings with imperfections represented by small attached masses and springs is considered the concept of an equivalent imperfection mass is introduced this is used to quantify the frequency splitting which occurs in slightly imperfect axisymmetric bodies the concept is used to estimate the trimming mass which must be added to or removed from the ring and at which point in order to reduce or eliminate the frequency split in a given pair of modes the method can be applied to any axisymmetric body for which the mode shapes can be measured or computed

construction of solutions for compressible isentropic navier stokes equations in one space dimension with nonsmooth initial data we prove the global existence of weak solutions for the cauchy problem for the navier stokes equations for one dimensional isentropic flow when the initial velocity is in l 2 and the initial density is in l 2 bv solutions are obtained as limits of approximations obtained by building heuristic jump conditions into a semi discrete difference scheme this allows for a rather simple analysis in which pointwise control is achieved through piecewise h 1 and total variation estimates

universal attractor in h4 for the nonlinear one dimensional compressible navier stokes equations abstract this paper is concerned with the regularity exponential stability of solutions and existence of universal attractor in h 4 for a nonlinear one dimensional compressible navier stokes equations describing a motion of heat conductive viscous real gas in a bounded domain 0 1 some new ideas and more delicated estimates are introduced to prove these results

lagrangean structure and propagation of singularities in multidimensional compressible flow we study the propagation of singularities in solutions of the navier stokes equations of compressible barotropic fluid flow in two and three space dimensions the solutions considered are in a fairly broad regularity class for which initial densities are nonnegative and essentially bounded initial energies are small and initial velocities are in certain fractional sobolev spaces we show that if the initial density is bounded below away from zero in an open set v then each point of v determines a unique integral curve of the velocity field and that this system of integral curves defines a locally bi holder homeomorphism of v onto its image at each positive time this lagrangean structure is then applied to show that if the initial density has a limit at a point of such a set v from a given side of a continuous hypersurface in v then at each later time both the density and the divergence of the velocity have limits at the transported point from the corresponding side of the transported hypersurface which is also a continuous manifold if the limits from both sides exist then the rankine hugoniot conditions hold in a strict pointwise sense showing that the jump in the divergence of the velocity is proportional to the jump in the pressure this leads to a derivation of an explicit representation for the strength of the jump in the logarithm of the density from which it follows that discontinuities persist for all time convecting along fluid particle paths and in the case that the pressure is strictly increasing in density having strengths which decay exponentially in time

decay of entropy solutions of nonlinear conservation laws we are concerned with the asymptotic behavior of entropy solutions of nonlinear conservation laws the main objective of this paper is to present an analytical approach and to explore its applications to studying the large time behavior of periodic entropy solutions of hyperbolic conservation laws the asymptotic decay of periodic solutions of nonlinear hyperbolic conservation laws is an important nonlinear phenomenon it is observed that the genuine nonlinearity of equations forces the waves of each characteristic family to interact vigorously and to cancel each other the insightful analysis of glimm lax gl for scalar equations and 2 2 systems has indicated that the resultant mutual cancellation of interacting shock and rarefaction waves of the same family induces the decay of periodic solutions such a result was first shown by lax l1 in 1957 for one dimensional convex scalar conservation laws dafermos d1 applying his uniform processes proved the decay result for the case that the set of inflection points of the flux does not have an accumulation point the glimm lax theory gl indicates that for 2 2 strictly hyperbolic and genuinely nonlinear systems any periodic glimm solution decays like o 1 t this result was proved by using the approximate characteristic method in the glimm difference solutions provided that the oscillation of the corresponding initial data is small recently using the method of generalized characteristics dafermos d3 showed that any periodic solution with local bounded variation and small oscillation for 2 2 systems decays asymptotically with the same detailed structure pas found by lax l1 for the

asymptotic behavior of solutions to a model for the flow of a reacting fluid we give a complete analysis of solutions of a model for the flow of a multispecies reacting fluid occupying a thin cylinder whose walls may be semipermeable with respect to some or all of the chemical species we prove the global existence of solutions and establish a number of time independent a priori bounds sufficient to determine the corresponding time asymptotic steady state we then derive necessary conditions and sufficient conditions ensuring that this steady state reflects complete combustion that is that at least one of the reactant species is depleted

compressible navier stokes equations with zero heat conductivity abstract we study the large time behavior of solutions of the compressible navier stokes equations without heat conductivity acoustic waves are dissipative due to the viscosity entropy waves are nondissipative due to lack of heat conductivity thus the system is of composite type the simplest prototype of this class of nonlinear partial differential equations we study the nonlinear interactions of waves using a combination of energy method weighted energy method and pointwise estimates

frequency split and vibration localization in imperfect rings abstract the dynamics of linearly elastic imperfect rings vibrating in their own plane is considered in this paper imperfections are modeled as perturbations of the uniform linear mass density and bending stiffness of a perfect ring a perturbation expansion and a spectral representation are employed and a variational formulation of the vibration problem is obtained a linear theory is deduced by retaining only the leading order terms in the variational formulation the linear theory yields simple closed form expressions for the eigenfrequencies and the modal shapes which are accurate when the imperfections are sufficiently small an enhanced nonlinear theory is also derived which is accurate even when the ring imperfections are not small in this case an iterative solution procedure is developed the proposed theories are validated by considering some case study problems and using the ritz rayleigh solution as a benchmark finally the linear theory is applied to the frequency trimming problem of an imperfect ring a simple closed form expression for the trimming masses is presented valid for trimming any selected number of eigenmodes

systems of partial differential equations of mixed hyperbolic parabolic type abstract we prove the global existence and uniqueness of solutions of certain mixed hyperbolic parabolic systems of partial differential equations in one space dimension with initial data that is assumed to be pointwise bounded with possibly large oscillation and with small total energy the systems we consider are general enough to include the navier stokes equations of compressible flow the equations of compressible mhd models of chemical combustion and others in particular the application of our results to the mhd system gives an existence result which is new