the effects of rounding errors in the nodes on barycentric interpolation we analyze the effects of rounding errors in the nodes on polynomial barycentric interpolation these errors are particularly relevant for the first barycentric formula with the chebyshev points of the second kind here we propose a method for reducing them

global strong solution for the korteweg system in dimension n 2 this work is devoted to prove the existence of global strong solution in dimension n 2 for a general isothermal model of capillary fluids derived by j e dunn and j serrin 1985 see 16 which can be used as a phase transition model in a first part we prove the existence of strong solution in finite time for large initial data with a precise bound by below on the life span t of existence of the solution this one depends on the norm of the initial data 0 v 0 the second part consists in proving the existence of global strong solution with particular choice on the capillary coefficient 2 and on the viscosity tensor which corresponds to the shallow water case 2 div du to do this we derivate different energy estimate on the density and the effective velocity v which ensures that the strong solution can be extended beyond t the main difficulty consists in controlling the vacuum or in other words to estimate the l norm of 1 the proof relies mostly on a method introduced by de giorgi 14 see also ladyzhenskaya et al in 30 for the parabolic case to obtain regularity results for elliptic equations with discontinuous diffusion coefficients and a suitable bootstrap argument

accurate computation of divided differences of the exponential function abstract the traditional recurrence for the computation of exponential divided differences along with a new method based on the properties of the exponential function are studied in detail in this paper our results show that it is possible to combine these two methods to compute exponential divided differences accurately a hybrid algorithm is presented for which our error bound grows quite slowly with the order of the divided difference author

global solutions of the super critical 2d quasi geostrophic equation in besov spaces abstract in this paper we study the super critical 2d dissipative quasi geostrophic equation we obtain some regularization effects allowing us to prove a global well posedness result for small initial data lying in critical besov spaces constructed over lebesgue spaces l p with p 1 local results for arbitrary initial data are also given

strong solution for korteweg system in bmo 1 rn with initial density in l in this paper we investigate the question of the existence of strong solution in finite time for the korteweg system for small initial data provided that the initial momentum 0 u 0 belongs to bmo 1 t r n for t 0 and the initial density 0 is in l r n with n 1 and far away from the vacuum this result extends the so called koch tataru theorem for the korteweg system it is also interesting to observe that any initial shock on the density is instantaneously regularized inasmuch as the density becomes lipschitz for any t with t 0 we also prove the existence of global strong solution for initial data 0 1 0 u 0 b n 2 1 2 r n b n 2 2 r n l r n b n 2 1 2 r n n this result allows in particular to extend the notion of oseen solution corresponding to particular solution of the incompressible navier stokes system in dimension n 2 to the korteweg system provided that the vorticity of the momentum 0 u 0 is a dirac mass 0 with sufficiently small ihowever unlike the navier stokes equations the property of self similarity is not conserved for the korteweg system since there is no invariance by scaling because the term of pressure

a numerical variational method for extracting 3d singularities abstract a finite element procedure is developed for calculating the order and mode of singularities at 3d vertices in anisotropic linear elastic solids and composites it is an extension of the method of bazant and estenssoro although it differs from the latter in some essential aspects it is based on a variational principle derived from the statement of virtual work on the surface of the unit sphere surrounding the singularity the sphere is divided into six node spherical triangles the singularities of the spherical coordinates at the poles are avoided by coordinate transformations the three matrices of the quadratic eigenvalue problem are explicitly evaluated and used to advantage real and complex eigenvalues and eigenvectors arc calculated by inverse treppen iteration with a relatively small mesh accuracy is to two decimal places the method is capable of solving any 3d and any 2d vertex singularity problem with reasonable accuracy and without any assumptions with regard to the behavior of displacements in the neighborhood of line singularities it is quite robust and stable several examples and applications to practical problems are given a procedure to handle the incompressible case is discussed

a priori estimates for the 3d quasi geostrophic system abstract the present article is devoted to the 3d dissipative quasi geostrophic system qg this system can be obtained as a limit model of the primitive equations in the asymptotics of strong rotation and stratification and involves a non radial non local homogeneous pseudo differential operator of order 2 denoted by and whose semigroup kernel reaches negative values after a refined study of the non local part of we prove a priori estimates in the general l p setting for the 3d qg model the main difficulty of this article is to study this highly non classical operator and the commutator of with a lagrangian change of variable an important application of these a priori estimates providing bound from below to the lifespan of the solutions of the primitive equations for ill prepared blowing up initial data can be found in a companion paper

fast and stable contour integration for high order divided differences via elliptic functions in this paper we will present a new method for evaluating high order divided differences for certain classes of analytic possibly operator values functions this is a classical problem in numerical mathematics but also arises in new applications such as e g the use of generalized convolution quadrature to solve retarded potential integral equations the functions which we will consider are allowed to grow exponentially to the left complex half plane and the interpolation points are scattered in a large real interval our approach is based on the representation of divided differences as contour integral and we will employ a subtle parameterization of the contour in combination with a quadrature approximation by the trapezoidal rule

the large time behavior of solutions in the critical l p framework for compressible viscous and heat conductive gas flows in this paper we are concerned with non isentropic navier stokes equations governing compressible viscous and heat conductive gases we formulate an additional regularity assumption of low frequencies in rd d 3 and then establish the sharp time weighted inequality which allows us to get the time decay estimates of global strong solutions in the lp critical besov spaces precisely it is shown that if the initial data belong to some besov space b2 s1 with s1 1 d2 s0 s0 2dp d2 then the lp norm of the critical global solutions admits the time decay t s12 d2 12 1p in particular t d2p if s1 s0 which coincides with that of the heat kernel in the lp framework in comparison with the work of danchin and xu j math fluid mech 20 1641 1665 2018 the low frequency assumption is improved such that the regularity s1 belongs to the whole range 1 d2 s0

the stability of barycentric interpolation at the chebyshev points of the second kind we present a new analysis of the stability of the first and second barycentric formulae for interpolation at the chebyshev points of the second kind our theory shows that the second formula is more stable than previously thought and our experiments confirm its stability in practice we also extend our current understanding regarding the accuracy problems of the first barycentric formula

on the numerical stability of newton s formula for lagrange interpolation abstract it was previously noticed in the literature that the computation of newton s formula can be very sensitive to the ordering of the input data and it may be numerically unstable even for interpolation at the chebyshev nodes nevertheless to the best of our knowledge the effects of rounding errors in the computation of newton s formula have not been properly quantized yet in this note we make a full stability analysis of the propagation of rounding errors in the computation of newton s formula for lagrange interpolation we derive sharp upper bounds for the numerical error in the computations of three algorithms for computing polynomial interpolants in newton form that can be used to understand the instabilities mentioned above we throw light on and give a practical meaning to the condition number defined in carnicer et al 2019 by showing that it is the right thermometer for measuring the effects of rounding errors in the computation of newton s formula

the nonrelativistic limit in radiation hydrodynamics i weak entropy solutions for a model problem abstract this paper is concerned with a model system for radiation hydrodynamics in multiple space dimensions the system depends singularly on the light speed c and consists of a scalar nonlinear balance law coupled via an integral type source term to a family of radiation transport equations we first show existence of entropy solutions to cauchy problems of the model system in the framework of functions of bounded variation this is done by using difference schemes and discrete ordinates then we establish strong convergence of the entropy solutions indexed with c as c goes to infinity the limit function satisfies a scalar integro differential equation

analysis of a three dimensional dissimilar material joint with one real singularity using a conservative integral abstract in the present study a conservative integral based on the betti reciprocal principle is formulated to determine the intensity of singularity at a vertex of the interface in three dimensional dissimilar material joints with one real singularity eigenanalysis formulated using a three dimensional finite element method fem is used to calculate the order of stress singularity angular functions of displacements and stresses models with various element sizes and various integral areas are used to investigate the effect of the integration area on the accuracy of the results the results are compared with those obtained from the boundary element method bem using a curve fitting technique to calculate the intensity of singularity in addition models of various lengths and various material combinations are used to investigate the stress singularity characteristics in three dimensional dissimilar material joints the results of the present study indicate that the conservative integral can be used to determine the intensity of singularity in three dimensional bi material joints the accuracy of the results can be improved by mesh refinement finally the relationships among the intensity of singularity the order of stress singularity and the model geometry are discussed

incompressible limit of oldroyd b fluids in the whole space abstract this paper is dedicated to the study of compressible oldroyd b model in the framework of critical spaces we first prove the local well posedness for the compressible oldroyd b model with initial data 0 1 u 0 0 b 2 1 d 2 b 2 1 d 2 1 d b 2 1 d 2 d d the global well posedness is also established provided the initial data and coupling constant are sufficiently small on the basis of this we also consider the incompressible limit problem for ill prepared initial data we prove that as the mach number tends to zero the global solution to the compressible oldroyd b fluids converges to the solution to the corresponding incompressible model in some function spaces in the meanwhile the converge rates are obtained in some sense to fulfill our proof we give a new commutator estimate

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

efficient computation of order and mode of corner singularities in 3d elasticity a general numerical procedure is presented for the efficient computation of corner singularities which appear in the case of non smooth domains in three dimensional linear elasticity for obtaining the order and mode of singularity a neighbourhood of the singular point is considered with only local boundary conditions the weak formulation of the problem is approximated by a galerkin petrov finite element method a quadratic eigenvalue problem p q 2r u 0 is obtained with explicitly analytically defined matrices p q r moreover the three matrices are found to have optimal structure so that p r are symmetric and q is skew symmetric which can serve as an advantage in the following solution process on this foundation a powerful iterative solution technique based on the arnoldi method is submitted for not too large systems this technique needs only one direct factorization of the banded matrix p for finding all eigenvalues in the interval e 0 5 1 0 no eigenpairs can be lost as well as the corresponding eigenvectors which is a great improvement in comparison with the normally used determinant method for large systems a variant of the algorithm with an incomplete factorization of p is implemented to avoid the appearance of too much fill in to illustrate the effectiveness of the present method several new numerical results are presented in general they show the dependence of the singular exponent on different geometrical parameters and the material properties copyright 2001 john wiley sons ltd

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

global well posedness in spatially critical besov space for the boltzmann equation the unique global strong solution in the chemin lerner type space to the cauchy problem on the boltzmann equation for hard potentials is constructed in a perturbation framework such a solution space is of critical regularity with respect to the spatial variable and it can capture the intrinsic properties of the boltzmann equation for the proof of global well posedness we develop some new estimates on the nonlinear collision term through the littlewood paley theory

global solutions of a high dimensional system for korteweg materials the global existence of solutions to a high dimensional system of korteweg materials is established when the initial data are small the complete model was proposed by dunn and serrin and the local existence of the solution was obtained in an earlier paper by the authors

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

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

relative entropy in multi phase models of 1d elastodynamics convergence of a non local to a local model abstract in this paper we study a local and a non local regularization of the system of nonlinear elastodynamics with a non convex energy we show that solutions of the non local model converge to those of the local model in a certain regime the arguments are based on the relative entropy framework and provide an example of how local and non local regularizations may compensate for non convexity of the energy and enable the use of the relative entropy stability theory even if the energy is not quasi or poly convex

polynomial interpolation lagrange versus newton we show that the lagrangian form of the interpolating polynomial can be calculated with the same number of arithmetic operations as the newtonian form

stress singularity analysis in three dimensional bonded structure abstract stress analysis for joints between two isotropic materials is performed using three dimensional boundary element method with rongved s fundamental solution rongved l 1955 force interior to one of two joined semi infinite solids in second midwestern conf on solid mechanics pp 1 13 which satisfies boundary conditions at the interface the stress distribution around the vertex of the joint is precisely examined and the order of stress singularity obtained in three dimensional analysis is compared with that in two dimensional analysis it is shown that the order of the stress singularity along the stress singularity line which is the intersection of the interface with the free surface varies from the value in plane strain to that in plane stress as the opposite side free surface is approached the order of stress singularity in the spherical coordinates centered at the vertex is next investigated when singularity lines with a singularity order meet with each other at the vertex the order of stress singularity is larger than that for plane strain and plane stress when the vertex angle between two side free surfaces is varied the order of stress singularity is also altered in particular a bound at which singularity disappears at the plane for dundurs composite parameters is examined for a cubic dissimilar material it is demonstrated that the bound varies little in comparison with the one obtained from the two dimensional theory

antisymmetrical 3 d stress field near the crack front of a thin elastic plate abstract based on detailed three dimensional finite clement analysis the near tip field of a thin elastic plate remotely subjected to mode ii antisymmctrical loading is investigated the computed results show the transition to a three dimensional state to occur near the radial distance from the crack tip of 1 5 times the plate thickness in the close vicinity of the crack front the asymptotic stress field is characterized by a combination of plane strain mode ii and anti plane mode iii singular 1 r fields the domain of such fields extends approximately 0 5 of the thickness in the radial direction the mixed mode stress intensity factors along the crack front are determined from 3 d conservation integrals and unlike the case of symmetrical loading their magnitudes are greater towards the free surface near the intersection of the crack front and free surface the stress field converges to the antisymmetrical corner singularity solution the asymptotic form of stress near the intersection under general loading conditions is presented along with the corner stress intensity factors

a boundary value method for the singular behavior of bimaterial systems under inplane loading abstract based on the governing equations of linear elasticity this paper develops a novel boundary value method to study the singular behavior of elastic stress fields at the corners of bimaterial wedges and junctions by using the eigenfunction expansion technique the resulted one dimensional differential system which consists of the reduced equilibrium equations and boundary conditions just relates to an angular coordinate in the polar coordinate system implementing discretization of this differential system by the finite cloud method we readily derive the so called generalized eigenproblem in the singular eigenvalue the performance of the methodology is subsequently verified through the well known crack and interface crack problems demonstrating high accuracy and quick convergence characteristics in addition a selected set of practically useful models is numerically analyzed to examine the angular variations of the displacement and stress fields and the influences of wedge side boundary conditions to singular behavior are also studied

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

global small solutions to an mhd type system the three dimensional case in this paper we consider the global well posedness of a three dimensional incompressible mhd type system with smooth initial data that is close to some nontrivial steady state it is a coupled system between the navier stokes equations and a free transport equation with a universal nonlinear coupling structure the main difficulty of the proof lies in exploring the dissipative mechanism of the system due to the fact that there is a free transport equation of in the coupled equations and only the horizontal derivatives of is dissipative with respect to time to achieve this we first employ anisotropic littlewood paley analysis to establish the key l1 lip 3 estimate to the third component of the velocity field then we prove the global well posedness to this system by the energy method which depends crucially on the divergence free condition of the velocity field 2014 wiley periodicals inc

on a lagrangian method for the convergence from a non local to a local korteweg capillary fluid model in the present article we are interested in further investigations for the barotropic compressible navier stokes system endowed with a non local capillarity we studied in 7 thanks to an accurate study of the associated linear system using a lagrangian change of coordinates we provide more precise energy estimates in terms of hybrid besov spaces naturally depending on a threshold frequency which is determined in function of the physical parameter distinguishing the low and the high regimes it allows us in particular to prove the convergence of the solutions from the non local to the local korteweg system another mathematical interest of this article is the study of the effect of the lagrangian change on the non local capillary term

travelling waves for the gross pitaevskii equation ii the purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the gross pitaevskii equation in dimensions two and three our arguments based on minimization under constraints yield a full branch of solutions and extend earlier results see 3 4 8 where only a part of the branch was built in dimension three we also show that there are no travelling wave solutions of small energy

on the influence of gravity on density dependent incompressible periodic fluids abstract the present work is devoted to the analysis of density dependent incompressible fluids in a 3d torus when the froude number e goes to zero we consider the very general case where the initial data do not have a zero horizontal average where we only have smoothing effect on the velocity but not on the density and where we can have resonant phenomena on the domain we explicitly determine the limit system when e 0 and prove its global wellposedness finally we prove that for large initial data the density dependent incompressible fluid system is globally wellposed provided that e is small enough

a global existence result for the compressible navier stokes equations in the critical l p framework the present paper is dedicated to the global well posedness issue for the barotropic compressible navier stokes system in the whole space mathbb r d with d 2 we aim at extending the work by danchin inventiones mathematicae 141 3 579 614 2000 to a critical framework which is not related to the energy space for small perturbations of a stable equilibrium state in the sense of suitable lp type besov norms we establish the global existence as a consequence like for incompressible flows one may exhibit a class of large highly oscillating initial velocity fields for which global existence and uniqueness holds true in passing we obtain new estimates for the linearized and the paralinearized systems which may be of interest for future works on compressible flows

numerical solution of navier stokes korteweg systems by local discontinuous galerkin methods in multiple space dimensions compressible liquid vapor flow with phase transitions can be described by systems of navier stokes korteweg type they extend the navier stokes equations by nonlinear higher grade terms which take the form of either differential or nonlocal integral operators a numerical approximation method on the basis of the local discontinuous galerkin method in multiple space dimensions is suggested for isothermal flows it relies on a specific discretization of a non conservative formulation to enhance the performance of the overall scheme two techniques are used i local spatial adaptivity based on gradient indicators for the density and ii parallelism based on domain decomposition the paper concludes with numerical experiments in two and three space dimensions they show the reliability and efficiency of the proposed approach as well as they demonstrate the applicability of the models for several important phase transition phenomena

application of the finite element iterative method to the eigenvalue problem of a crack between dissimilar media the finite element iterative method is applied to the eigenvalue problem of a crack between dissimilar media in this case the transfer matrix is non symmetric which leads to complex eigenvalues the singularity obtained agrees with the analytical results of r 1 2 ie the method of evaluating the eigenfunctions is general and can be applied to more complex cases of material and geometry which are frequently encountered in composite materials a powerful method for evaluating stress intensities in dissimilar media is given in the appendix the method is also reduced for homogeneous media to give stress intensity factors for modes i ii and iii

local in time results for local and non local capillary navier stokes systems with large data abstract in this article we study three capillary compressible models the classical local navier stokes korteweg system and two non local models for large initial data bounded away from zero and with a reference pressure state which is not necessarily stable p can be non positive we prove that these systems have a unique local in time solution and we study the convergence rate of the solutions of the non local models towards the local korteweg model the results are given for constant viscous coefficients and we explain how to extend them for density dependant coefficients

existence of global strong solution for korteweg system with large infinite energy initial data this work is devoted to the study of the initial boundary value problem for a general isothermal model of capillary fluids derived by j e dunn and j serrin 1985 see cite fds which can be used as a phase transition model we will prove the existence of local and global under a condition of smallness on the initial data strong solutions with discontinuous initial density where ln rho 0 belongs to the besov space b n 2 infty r n in passing our result allow to consider initial data with discontinuous interfaces up our knowledge in the literature the results of existence of strong solutions require generally continuous initial density it implies also that we can work with initial data space of infinite energy our result relies on the fact that the density can be written as the sum of the solution rho l associated to linear system and a remainder density bar rho which is more regular than rho l by taking into account the regularizing effects induced on the bilinear convection term the main difficulty concerns the proof of new estimate of maximum principle type for the linear system associated to the korteweg system the proof is based on a characterization of the besov space in terms of the semi group associated to this linear system let also mention that we prove the existence of global strong solution with a smallness hypothesis which is subcritical in terms of the scaling of the equations it allows us to exhibit a family of large energy initial data for the scaling of the equations providing global strong solution in particular for the first time up our knowledge we show the existence of global strong solution for some large energy initial data when n 2 we finish this paper by introducing the notion of quasi solutions for the korteweg s system a tool which has been developed in the framework of the compressible navier stokes equations cite arxiv arxiv1 hal cras1 cras2 which enables us to improve the previous result and by obtaining the existence of global strong solution with large initial velocity in b n 1 2 infty in particular when n 2 it implies the existence of global strong solution with large energy data as a corollary we get global existence and uniqueness for highly compressiblekorteweg system when n geq2 it means that for any large initial data under an irrotational condition on the initial velocity we have the existence of global strong solution provided that the pressure is sufficiently highly compressible

3 d elastic stress singularity at polyhedral corner points abstract the three dimensional stress singularity at the top of an arbitrary polyhedral corner is considered based on the boundary integral equations the problem is reduced by the mellin transform to a system of certain one dimensional integral equations the orders of stress singularity are spectral points of the integral operators while angular distribution and intensity factors are found as residues at those points numerical results are obtained by means of the galerkin discretization scheme using expansions in terms of orthogonal polynomials with the proper weights some of the results illustrating the orders dependence on the elastic properties and corner geometry for a wedge shaped punch and a crack for an elastic trihedron and for a surface breaking crack are given

equations d ondes quasilineaires et estimations de strichartz dans cet article notre but est de resoudre des equations d ondes quasilineaires pour des donnees initiales moins regulieres que ce qu imposent les methodes d energie ceci impose de demontrer des estimees de type strichartz pour des operateurs d ondes a coefficients seulement lipschitziens

global well posedness for compressible viscoelastic fluids near equilibrium in this paper we prove local well posedness for compressible viscoelastic fluids of the oldroyd model under the assumption that the initial density is bounded away from zero and global well posedness near equilibrium the proof of global well posedness relies on some intrinsic properties of viscoelastic fluids and on a uniform estimate for a linearized hyperbolic parabolic system with convection terms

global strong solution for the korteweg system with quantum pressure in dimension n ge 2 this work is devoted to prove the existence of global strong solution in dimension n ge 2 for a isothermal model of capillary fluids derived by dunn and serrin see arch ration mech anal 88 2 95 133 1985 which can be used as a phase transition model we will restrict us to the case of the so called compressible navier stokes system with quantum pressure which corresponds to the korteweg system with capillary coefficient such that kappa rho frac kappa 1 rho with kappa 1 0 and rho the density in a first part we prove the existence of strong solution in finite time for large initial data in critical besov spaces with precise estimate on the life span t the second part consists in proving the existence of global strong solution with large initial data for a specific choice on the capillary coefficient kappa 1 mu 2 with mu the viscosity coefficient to do this we derive different energy estimates on the density and the effective velocity v which allow us to extend the strong solution beyond t the main difficulty consists in estimating the l infty norm of frac 1 rho the proof relies mostly on a method introduced by de giorgi mem accad sci torino cl sci fis mat nat 3 3 25 43 1957 see also ladyzhenskaya et al linear and quasilinear equations of parabolic type ams translations providence 1968 for the parabolic case to obtain regularity results for elliptic equations with discontinuous diffusion coefficients and a suitable bootstrap argument

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

on the global well posedness of the euler boussinesq system with fractional dissipation abstract we study the global well posedness of the euler boussinesq system with the term dissipation d on the temperature equation we prove that for 1 the coupled system has a global unique solution for initial data with critical regularities

corner stress singularities in an fgm thin plate abstract describing the behaviors of stress singularities correctly is essential for obtaining accurate numerical solutions of complicated problems with stress singularities this analysis derives asymptotic solutions for functionally graded material fgm thin plates with geometrically induced stress singularities the classical thin plate theory is used to establish the equilibrium equations for fgm thin plates it is assumed that the young s modulus varies along the thickness and poisson s ratio is constant the eigenfunction expansion method is employed to the equilibrium equations in terms of displacement components for an asymptotic analysis in the vicinity of a sharp corner the characteristic equations for determining the stress singularity order at the corner vertex and the corresponding corner functions are explicitly given for different combinations of boundary conditions along the radial edges forming the sharp corner the non homogeneous elasticity properties are present only in the characteristic equations corresponding to boundary conditions involving simple support finally the effects of material non homogeneity following a power law on the stress singularity orders are thoroughly examined by showing the minimum real values of the roots of the characteristic equations varying with the material properties and vertex angle

non existence of classical solutions with finite energy to the cauchy problem of the compressible navier stokes equations the well posedness of classical solutions with finite energy to the compressible navier stokes equations cns subject to arbitrarily large and smooth initial data is a challenging problem in the case when the fluid density is away from vacuum strictly positive this problem was first solved for the cns in either one dimension for general smooth initial data or multi dimension for smooth initial data near some equilibrium state that is small perturbation antontsev et al in boundary value problems in mechanics of nonhomogeneous fluids north holland publishing co amsterdam 1990 kazhikhov in sibirsk mat zh 23 60 64 1982 kazhikhov et al in prikl mat meh 41 282 291 1977 matsumura and nishida in proc jpn acad ser a math sci 55 337 342 1979 j math kyoto univ 20 67 104 1980 commun math phys 89 445 464 1983 in the case that the flow density may contain a vacuum the density can be zero at some space time point it seems to be a rather subtle problem to deal with the well posedness problem for cns the local well posedness of classical solutions containing a vacuum was shown in homogeneous sobolev space without the information of velocity in l2 norm for general regular initial data with some compatibility conditions being satisfied initially cho et al in j math pures appl 9 83 243 275 2004 cho and kim in j differ equ 228 377 411 2006 manuscr math 120 91 129 2006 choe and kim in j differ equ 190 504 523 2003 and the global existence of a classical solution in the same space is established under the additional assumption of small total initial energy but possible large oscillations huang et al in commun pure appl math 65 549 585 2012 however it was shown that any classical solutions to the compressible navier stokes equations in finite energy inhomogeneous sobolev space cannot exist globally in time since it may blow up in finite time provided that the density is compactly supported xin in commun pure appl math 51 229 240 1998 in this paper we investigate the well posedess of classical solutions to the cauchy problem of navier stokes equations and prove that the classical solution with finite energy does not exist in the inhomogeneous sobolev space for any short time under some natural assumptions on initial data near the vacuum this implies in particular that the homogeneous sobolev space is as crucial as studying the well posedness for the cauchy problem of compressible navier stokes equations in the presence of a vacuum at far fields even locally in time