low mach number limit of full navier stokes equations in a 3d bounded domain abstract this paper studies the low mach number limit of the full compressible navier stokes equations in a three dimensional bounded domain where the velocity field and the temperature satisfy the slip boundary conditions and the neumann boundary condition respectively the uniform estimates in the mach number for the strong solutions are derived in a short time interval provided that the initial density and temperature are close to the constant states and satisfy the bounded derivative conditions thus the solutions of the full compressible navier stokes equations converge to the one of the isentropic incompressible navier stokes equations as the mach number vanishes

three dimensional vibrations of thick circular and annular plates abstract the ritz method is applied in a three dimensional 3 d analysis to obtain accurate frequencies for thick circular and annular plates the method is formulated in a manner which allows one to have any combination of free or fixed plate boundaries admissible functions for the three displacement components are chosen as trigonometric functions in the circumferential co ordinate and algebraic polynomials in the radial and axial co ordinates upper bound convergence of the non dimensional frequencies to at least four significant figures is demonstrated comparisons of results are made with ones obtained by others using 2 d mindlin thick plate theory and with other 3 d solutions extensive and accurate four significant figure frequencies are presented for completely free circular plates having thickness to diameter ratios of 0 2 0 3 0 4 and 0 5 for poisson s ratios 0 0 3 and 0 499 frequencies are also given for thick annular plates having a thickness to outer diameter of 0 2 inside to outside diameter ratios of 0 1 0 5 and 0 9 and 0 3 all 3 d modes are included in the analyses e g flexural thickness shear in plane stretching and torsional the circular and annular plate frequency data given is exact to at least four digits thus being benchmark data against which results from 2 d thick plate theories or other approximate methods e g finite elements may be compared

on the flexural vibration of cylinders under axial loads numerical and experimental study abstract the flexural vibration of a homogeneous isotropic linearly elastic cylinder of any aspect ratio is analysed in this paper natural frequencies of a cylinder under uniformly distributed axial loads acting on its bases are calculated numerically by the ritz method with terms of power series in the coordinate directions as approximating functions the effect of axial loads on the flexural vibration cannot be described by applying infinitesimal strain theory therefore geometrically nonlinear strain displacement relations with second order terms are considered here the natural frequencies of free free clamped clamped and sliding sliding cylinders subjected to axial loads are calculated using the proposed three dimensional ritz approach and are compared with those obtained with the finite element method and the bernoulli euler theory different experiments with cylinders axially compressed by a hydraulic press are carried out and the experimental results for the lowest flexural frequency are compared with the numerical results an approach based on the ritz formulation is proposed for the flexural vibration of a cylinder between the platens of the press with constraints varying with the intensity of the compression the results show that for low compressions the cylinder behaves similarly to a sliding sliding cylinder whereas for high compressions the cylinder vibrates as a clamped clamped one

weakly nonlinear dissipative approximations of hyperbolic parabolic systems with entropy hyperbolic parabolic systems have spatially homogenous stationary states when the dissipation is weak one can derive weakly nonlinear dissipative approximations that govern perturbations of these constant states these approximations are quadratically nonlinear when the original system has an entropy the approximation is formally dissipative in a natural hilbert space we show that when the approximation is strictly dissipative it has global weak solutions for all initial data in that hilbert space we also prove a weak strong uniqueness theorem for it in addition we give a kawashima type criterion for this approximation to be strictly dissipative we apply the theory to the compressible navier stokes system

incompressible limit of a compressible micropolar fluid model with general initial data abstract in this paper we consider the incompressible limit of a compressible micropolar fluid model with general initial data in the whole space r 3 it is proved that as the mach number goes to zero the weak solutions of the compressible micropolar fluids converge to the strong solution of the ideal incompressible micropolar fluids as long as the latter exists moreover the rate of convergence is also obtained

an analytical numerical and experimental study of the axisymmetric vibrations of a short cylinder this paper presents a study of standing axisymmetric waves in homogeneous isotropic linear and elastic cylinders whose length and diameter are of the same order of magnitude it is analytically demonstrated that only materials with a poisson s ratio equal to zero can vibrate in pure radial and longitudinal axisymmetric modes and hence the universal frequencies independent of poisson s ratio and the corresponding universal aspect ratios can be accurately calculated the ritz method is applied to numerically determine the natural frequencies and the mode shapes of finite cylinders vibration of a cylinder in the neighbourhood of the universal point is analysed both free and forced vibrations are experimentally detected by using a laser speckle interferometer that allows the determination of the natural frequencies and the mode shapes from the numerical calculations it is confirmed that for materials with poisson s ratio not equal to zero no crossings occur between the curves of non dimensional frequency versus aspect ratio neither for antisymmetric modes themselves nor for symmetric modes there is a good agreement between the analytical solutions the numerical results and the experimental measurements

on the normal modes of free vibration of inhomogeneous and anisotropic elastic objects the hamilton s principle approach to the calculation of vibrational modes of elastic objects with free boundaries is exploited to compute the resonance frequencies of a variety of anisotropic elastic objects including spheres hemispheres spheroids ellipsoids cylinders eggs shells bells sandwiches parallelepipeds cones pyramids prisms tetrahedra octahedra and potatoes the paramount feature of this calculation which distinguishes it from previous ones is the choice of products of powers of the cartesian coordinates as a basis for expansion of the displacement in a truncated complete set enabling one to analytically evaluate the required matrix elements for these systems because these basis functions are products of powers of x y and z this scheme is called the xyz algorithm the xyz algorithm allows a general anisotropic elastic tensor with any position dependence and any shape with arbitrary density variation a number of plots of resonance spectra of families of elastic objects are

small entire functions with extremely fast growth abstract we prove in this note that given 0 1 2 there exists a linear manifold m of entire functions satisfying that m is dense in the space of all entire functions such that lim z exp z f j z 0 on any plane strip for every f m and for every derivation index j moreover the growth index of each nonnull function of m is infinite with respect to any prefixed sequence of nonconstant entire functions

compressible navier stokes system large solutions and incompressible limit abstract here we prove the existence of global in time regular solutions to the two dimensional compressible navier stokes equations supplemented with arbitrary large initial velocity v 0 and almost constant density 0 for large volume bulk viscosity the result is generalized to the higher dimensional case under the additional assumption that the strong solution of the classical incompressible navier stokes equations supplemented with the divergence free projection of v 0 is global the systems are examined in r d with d 2 in the critical b 2 1 s besov spaces framework

the low mach number limit for the full navier stokes fourier system we study the low mach number asymptotic limit for solutions to the full navier stokes fourier system supplemented with ill prepared data and considered on an arbitrary time interval convergencetowards the incompressible navier stokes equations is shown

lineability criteria with applications lineability is a property enjoyed by some subsets within a vector space x a subset a of x is called lineable whenever a contains except for zero an infinite dimensional vector subspace if additionally x is endowed with richer structures then the more stringent notions of dense lineability maximal dense lineability and spaceability arise naturally in this paper several lineability criteria are provided and applied to specific topological vector spaces mainly function spaces sometimes such criteria furnish unified proofs of a number of scattered results in the related literature families of strict order integrable functions hypercyclic vectors non extendable holomorphic mappings riemann non lebesgue integrable functions sequences not satisfying the lebesgue dominated convergence theorem nowhere analytic functions bounded variation functions entire functions with fast growth and peano curves among others are analyzed from the point of view of lineability

layered incompressible fluid flow equations in the limit of low mach number and strong stratification abstract we investigate the low mach number regimes of the complete navier stokes fourier system which describes an evolution of a viscous heat conducting gas with a heat source term interpreted as a radiation cooling in models of atmospheric flows this investigation is performed in the context of weak solutions on an arbitrary large time interval and for the ill prepared initial data the physically expected limiting equations as the mach and froude numbers tend to zero at the same rate are layered incompressible flow equations with strong stratification we give a rigorous mathematical proof of this result

low mach number limit of viscous polytropic fluid flows abstract this paper studies the singular limit of the non isentropic navier stokes equations with zero thermal coefficient in a two dimensional bounded domain as the mach number goes to zero a uniform existence result is obtained in a time interval independent of the mach number provided that the initial data satisfy the bounded derivative conditions that is the time derivatives up to order two are bounded initially and navier s slip boundary condition is imposed

low mach number limit of the full navier stokes equations the low mach number limit for classical solutions of the full navier stokes equations is here studied the combined effects of large temperature variations and thermal conduction are taken into account in particular we consider general initial data the equations lead to a singular problem depending on a small scaling parameter whose linearized system is not uniformly well posed yet it is proved that solutions exist and they are uniformly bounded for a time interval which is independent of the mach number ma 0 1 the reynolds number re 1 and the peclet number pe 1 based on uniform estimates in sobolev spaces and using a theorem of g metivier s schochet 30 we next prove that the penalized terms converge strongly to zero this allows us to rigorously justify at least in the whole space case the well known computations given in the introduction of p l lions book 26

a lot of counterexamples to liouville s theorem abstract we prove in this paper that given 0 1 2 there exists a linear manifoldmof entire functions satisfying thatmis dense in the space of all entire functions and in addition limz exp z f j z 0 on any plane strip for everyf mand for every derivation indexj moreover it is shown the existence of an entire function with infinite growth index satisfying the latter property

symmetrization of ritz approximation functions for vibrational analysis of trigonal cylinders in the ritz method of calculating vibrational normal modes a set of finite series approximation functions provides a matrix eigenvalue equation for the coefficients in the series and the resonant frequency the matrix problem usually can be block diagonalized by grouping the functions into subsets according to their properties under the symmetry operations that are common to the specimen geometry and crystal class this task is addressed in this study for the case of cylindrical specimens of crystals belonging to one of the higher trigonal crystal classes the existence of doubly degenerate resonant modes significantly complicates the analysis group theoretical projection operators are employed to extract from series approximation functions in cylindrical coordinates the terms that transform according to each irreducible representation of the point group this provides a complete symmetry based block diagonalization and categorization of the modal symmetries off diagonal projection operators are used to provide relations between the displacement patterns of degenerate modes the method of analysis is presented in detail to assist in its application to other geometries crystal structures and or forms of ritz approximation functions

global existence in critical spaces for flows of compressible viscous and heat conductive gases we are concerned with global existence and uniqueness of strong solutions for a general model of viscous and heat conductive gases the initial data are supposed to be close to a stable equilibrium with constant density and temperature using uniform estimates for the linearized system with a convection term we get global well posedness in a functional setting invariant with respect to the scaling of the associated equations in space dimension n 3 we also show a smoothing effect on the velocity and the temperature and a decay on the difference between the density and the constant reference state these results extend a previous paper devoted to the barotropic case see 5

optimization of the ritz method to calculate axisymmetric natural vibration frequencies of cylinders a straightforward accurate technique is presented for calculating natural frequencies for axisymmetric vibration of cylinders of any aspect ratio the ritz variational method is applied by using algebraic polynomials in the form of power series in the axial and radial coordinates the adequate choice of the maximum exponents for each coordinate is important factor in order to achieve better accuracy compatible with an admissible time of calculation a computational model is developed that provides the optimum selection of exponents by following an automatic iterative process this leads to a precise calculation of frequencies in a minimum time of calculation the method is tested by comparing numerical and experimental results and a good agreement is obtained the common hypothesis on the number of terms in the radial and axial coordinates is not sufficiently justified as is proved

the zero mach limit of compressible flows we prove that compressible navier stokes flows in two and three space dimensions converge to incompressible navier stokes flows in the limit as the mach number tends to zero no smallness restrictions are imposed on the external force the initial velocity or the time interval we assume instead that the incompressible flow exists and is reasonably smooth on a given time interval and prove that compressible flows with compatible initial data converge uniformly on that time interval our analysis shows that the essential mechanism in this process is a hyperbolic effect which becomes stronger with smaller mach number and which ultimately drives the density to a constant

the free vibrations of inhomogeneous elastic cylinders and spheres abstract the variational statement governing equations and corresponding ritz approximations are derived in cartesian cylindrical and spherical coordinates for the evaluation of the natural frequencies of free vibrations of elastic cylinders and spheres the formulation can account for orthotropic material symmetry and can be applied to either solid or hollow geometries the approximating functions selected for the displacement components depend on the geometry and coordinate system used to describe the problem and are a combination of power series fourier series and spherical harmonics representative examples are given in the various coordinate systems for both the cylinder and the sphere in general excellent agreement is found with results obtained by other methods

zero mach number limit of the compressible primitive equations well prepared initial data author s liu xin titi edriss s abstract this work concerns the zero mach number limit of the compressible primitive equations the primitive equations with the incompressibility condition are identified as the limiting equations the convergence with well prepared initial data i e initial data without acoustic oscillations is rigorously justified and the convergence rate is shown to be of order mathcal o varepsilon as varepsilon rightarrow 0 where varepsilon represents the mach number as a byproduct we construct a class of global solutions to the compressible primitive equations which are close to the incompressible flows

linear subsets of nonlinear sets in topological vector spaces for the last decade there has been a generalized trend in mathematics on the search for large algebraic structures linear spaces closed subspaces or infinitely generated algebras composed of mathematical objects enjoying certain special properties this trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis operator theory summability theory polynomials in banach spaces hypercyclicity and chaos and general functional analysis this expository paper is devoted to providing an account on the advances and on the state of the art of this trend nowadays known as lineability and spaceability

the incompressible limit for compressible mhd equations in lp type critical spaces abstract in this paper the global well posedness and the low mach number limit for compressible viscous magnetohydrodynamic equations in the critical l p type besov space are considered more precisely we will show that in the isentropic case the solution of compressible magnetohydrodynamic equations will converge to that of incompressible magnetohydrodynamic equations when the mach number tends to zero in the critical l p framework moreover the convergence rate will be obtained

axisymmetric free vibrations of finite anisotropic cylinders abstract a technique is presented for obtaining estimates for the natural frequencies of axisymmetric free vibration for anisotropic cylinders of finite length the weak forms of the governing differential equations are solved by using the ritz method with terms of power series in the co ordinate directions as the approximating functions the dimensions of the cylinder are arbitrary and a wide range of geometries can be considered the solution technique is applied to the example problems of an isotropic cylinder and a cylinder composed of a material with hexagonal symmetry the method gives natural frequencies and mode shapes that are in excellent agreement with existing solutions the condition of traction free faces of the cylinder is also accurately satisfied

time periodic solution to a two phase model with magnetic field in a periodic domain abstract in the paper we consider a two phase model with magnetic field and a time periodic external force in a periodic domain in r n the existence of the time periodic solutions to the model is obtained by the topological degree theory and parabolic regularization method

continuum based free vibration of circular trigonal and isotropic plates the free vibration behavior of completely unrestrained elastic circular plates with trigonal and isotropic material symmetry is studied with an approach involving approximate continuum solutions to the three dimensional theory of linear elasticity of primary interest are 1 the influence of trigonal material symmetry on the modes of free vibration and 2 the accuracy of thin plate theory relative to the more exact three dimensional theory resonant frequencies are calculated from the weak form of the equations of motion for the plate through the use of approximation functions and the ritz method formulated in cylindrical coordinates this approach enables the resulting eigenvalue problem to be split through group theoretical symmetry analysis representative examples are given and quantitative limits are discussed

zero mach number limit for compressible flows with periodic boundary conditions we address the question of convergence to the incompressible navier stokes equations for slightly compressible viscous flows with ill prepared initial data and periodic boundary conditions the case of the whole space has been studied in an earlier paper by the author the functional setting is very close to the one used by h fujita and t kato for incompressible flows for arbitrarily large initial data we show that the compressible flow with small mach number exists as long as the incompressible one does in particular it exists globally if the corresponding incompressible solution exists for all time we further state a convergence result for the slightly compressible solution filtered by the group of acoustics introduction an engineer would hardly make a distinction between incom pressible flows and slightly compressible flows indeed incompressible navier stokes equations are usually regarded as an appropriate model for describing water in the present paper we study whether such an approximation is relevant from a mathematical viewpoint in the case of periodic boundary conditions the case of the whole space has been treated in 8 after some convenient rescaling see for instance the appendix of 16 or the introduction of 21 the equations for a slightly compressible fluid read

traction free vibrations of finite trigonal elastic cylinders the unrestrained traction free vibrations of finite elastic cylinders with trigonal material symmetry are studied using two approaches based on the ritz method which formulate the weak form of the equations of motion in cylindrical and rectangular coordinates elements of group theory are used to divide approximation functions into orthogonal subsets thus reducing the size of the computational problem and classifying the general symmetries of the vibrational modes results for the special case of an isotropic cylinder are presented and compared with values published by other researchers for the isotropic case the relative accuracy of the formulations in cylindrical and rectangular coordinates can be evaluated because exact analytical solutions are known for the torsional modes the calculation in cylindrical coordinates is found to be more accurate for a given number of terms in the series approximation functions for a representative trigonal material langatate calculations of the resonant frequencies and the sensitivity of the frequencies on each of the elastic constants are presented the dependence on geometry ratio of length to diameter is briefly explored the special case of a transversely isotropic cylinder with the elastic stiffness c14 equal to zero is also considered

low mach number limit for the non isentropic navier stokes equations abstract we study the low mach number limit of the local in time solutions to the compressible navier stokes equations with zero heat conductivity coefficient as the mach number tends to zero a uniform existence result for the one dimensional initial boundary value problem is proved provided that the initial data are well prepared in the sense that the temporal derivatives up to order two are bounded initially

incompressible limit of global strong solutions to 3 d barotropic navier stokes equations with well prepared initial data and navier s slip boundary conditions abstract this paper studies the incompressible limit of global strong solutions to the three dimensional barotropic navier stokes equations associated with navier s slip boundary conditions provided that the initial data are sufficiently small and well prepared in the sense that the time derivatives up to first order of solutions are bounded initially the main idea is to derive a differential inequality with decay so that the estimates are bounded uniformly both in the mach number 0 0 for some 0 0 and the time t 0

low mach number limit of viscous compressible flows in the whole space this paper is devoted to the low mach number limit of weak solutions to the compressible navier stokes equations for isentropic fluids in the whole space r d d 2 or 3 this problem was investigated by p l lions and n masmoudi we present here a different approach based upon strichartz s estimates for the linear wave equation in the inviscid case which improves the convergence result and simplifies the proof we prove that the velocity field is strongly compact and converges to a global weak solution of the incompressible navier stokes equations

small entire functions with infinite growth index abstract in this paper we prove that given 0 there exists a dense linear manifold m of entire functions such that formula for every f m and l straight line and with infinite growth index for all non null functions of m moreover every non null function of m has exactly 2 2 1 julia directions and if l is a straight line that does not contain a julia line then for every f m formula and for j 1 f j is bounded and integrable with respect to the length measure on l and lf j 0

forced vibrations of solid elastic cylinders abstract a method is presented to determine the vibration response of a solid elastic isotropic cylinder with arbitrary length to radius ratio when subject to arbitrary distribution of axisymmetric excitation on its surfaces in this method the axial and radial components of displacement are expressed as a sum of two infinite series each term in both the series is an exact solution to the governing equations of motion and has a coefficient that is used to satisfy boundary conditions one series contains bessel functions that form a complete set in the radial direction and the other contains trigonometric functions that form a complete set in the axial direction the components of stress are also expressed in terms of complete sets of functions by using the expression for displacement the coefficients in the series are determined by using the orthogonal properties of the functions to satisfy the boundary conditions in a mean square error sense numerical results are presented to illustrate the broadband responses of cylinders to uniform and concentrated loads on the flat and curved surfaces they are in good agreement with results obtained using atila a commercial finite element software

global existence in critical spaces for compressible navier stokes equations we investigate global strong solutions for isentropic compressible fluids with initial data close to a stable equilibrium we obtain the existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations more precisely the initial velocity has the same critical regularity index as for the incompressible homogeneous navier stokes equations and one more derivative is needed for the density we point out a smoothing effect on the velocity and a l 1 decay on the difference between the density and the constant reference state the proof lies on uniform estimates for a mixed hyperbolic parabolic linear system with a convection term

incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions this paper is concerned with the incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions it is rigorously shown that the weak solutions of the compressible magnetohydrodynamic equations converge to the strong solution of the viscous or inviscid incompressible magnetohydrodynamic equations as long as the latter exists both for the well prepared initial data and general initial data furthermore the convergence rates are also obtained in the case of the well prepared initial data

the incompressible limit in l p type critical spaces this paper aims at justifying the low mach number convergence to the incompressible navier stokes equations for viscous compressible flows in the ill prepared data case the fluid domain is either the whole space or the torus a number of works have been dedicated to this classical issue all of them being to our knowledge related to l 2 spaces and to energy type arguments in the present paper we investigate the low mach number convergence in the l p type critical regularity framework more precisely in the barotropic case the divergence free part of the initial velocity field just has to be bounded in the critical besov space dot b d p 1 p r cap dot b 1 infty 1 for some suitable p r in 2 4 times 1 infty we still require l 2 type bounds on the low frequencies of the potential part of the velocity and on the density though an assumption which seems to be unavoidable in the ill prepared data framework because of acoustic waves in the last part of the paper our results are extended to the full navier stokes system for heat conducting fluids

incompressible limits of the navier stokes equations for all time abstract we study the asymptotic behaviors of the regular solutions to the compressible navier stokes equations for well prepared initial data for all time as the mach number tends to zero by deriving a differential inequality with certain decay property the estimates obtained in this paper are uniform both in time and mach number

orthogonality and normalization of torsional modes of vibration of solid elastic spheres in this paper the work of lamb on the free torsional vibrations of spheres or vibrations of the first class has been extended in particular the conditions for the orthogonality of the normal modes have been established these conditions have allowed the normalized displacements to be presented as a function of the radial distance from the centre of the sphere three dimensional diagrams have been devised which show the displacements of the surface of the sphere these modal diagrams also show the nodes and nodal lines