global well posedness of incompressible inhomogeneous fluid systems with bounded density or non lipschitz velocity in this paper we first prove the global existence of weak solutions to the d dimensional incompressible inhomogeneous navier stokes equations with initial data a 0 in l infty mathbb r d u 0 u 0 h u 0 d in dot b 1 frac d p p r mathbb r d which satisfy mu a 0 l infty u 0 h dot b 1 frac d p p r rm exp c r mu 2r u 0 d dot b 1 frac d p p r 2r leqq c 0 mu for some positive constants c 0 c r and 1 p d 1 r the regularity of the initial velocity is critical to the scaling of this system and is general enough to generate non lipschitz velocity fields furthermore with additional regularity assumptions on the initial velocity or on the initial density we can also prove the uniqueness of such a solution we should mention that the classical maximal l p l q regularity theorem for the heat kernel plays an essential role in this context

strong solutions to the cauchy problem of two dimensional incompressible fluid models of korteweg type abstract this paper studies the local existence of strong solutions to the cauchy problem of the incompressible fluid models of korteweg type with vacuum as far field density the corresponding 3d problem has been solved by tan and wang 2010 21 notice that the technique used by tan and wang fails treating the 2d case because the l p norm p 2 of the velocity u cannot be controlled in terms only of u and u here in the present paper we will use the framework of weighted approximation estimates introduced by liang 2015 14 for navier stokes equations to obtain the local existence of strong solutions provided the initial density does not decay very slowly at infinity with the compact support case included

on the decay and stability to global solutions of the 3 d inhomogeneous navier stokes equations in this chapter we consider the large time decay and stability to any given global smooth solution of the three dimensional inhomogeneous incompressible navier stokes equations

nonuniqueness of nematic liquid crystal flows in dimension three abstract for suitable initial and boundary data we construct infinitely many weak solutions to the nematic liquid crystal flows in dimension three these solutions are in the axisymmetric class with bounded energy and backward bubbling at a large time

weak solution to the steady compressible flow of nematic liquid crystals abstract in this study three dimensional equations for the steady compressible flow of nematic liquid crystals are considered in a bounded domain for the adiabatic exponent 1 we establish the existence of a weak solution by three level approximation and adapting the weak convergence method

local existence and uniqueness of strong solutions to the navier stokes equations with nonnegative density abstract in this paper we consider the initial boundary value problem to the nonhomogeneous incompressible navier stokes equations local strong solutions are established for any initial data 0 u 0 w 1 l h 0 1 with 1 and if 2 then the strong solution is unique the initial density is allowed to be nonnegative and in particular the initial vacuum is allowed the assumption on the initial data is weaker than the previous widely used one that 0 u 0 h 1 l h 0 1 h 2 and no compatibility condition is required

local strong solution and blow up criterion for the 2d nonhomogeneous incompressible fluids abstract this paper concerns the cauchy problem for nonhomogeneous incompressible navier stokes equations in the two dimensional 2d space by means of the weighted initial density we obtain the existence and uniqueness of local strong solution with vacuum as a far field state moreover the solution is shown to be globally defined in time as long as the l norm of a weighted density keeps bounded

global strong solution to the 2d density dependent liquid crystal flows with vacuum abstract we establish the global existence and uniqueness of strong solutions to the 2d density dependent liquid crystal flows with vacuum in a bounded smooth domain

an improved a priori error analysis for finite element approximations of signorini s problem the present paper is concerned with the unilateral contact model in linear elastostatics the so called signorini problem our results can also be applied to the scalar signorini problem a standard continuous linear finite element approximation is first chosen to approach the two dimensional problem we develop a new error analysis in the h 1 norm using estimates on poincare constants with respect to the size of the areas of the noncontact sets in particular we do not assume any additional hypothesis on the finiteness of the set of transition points between contact and noncontact this approach allows us to establish better error bounds under sole h tau assumptions on the solution if 3 2 tau 2 we improve the existing rate by a factor h tau 3 2 2 and if tau 2 the existing rate h 3 4 is improved by a new rate of h sqrt vert ln h vert using the same finite element spaces as previously we then consider another discrete approximation of the nonlinear contact condit

a lagrangian approach for the incompressible navier stokes equations with variable density we investigate the cauchy problem for the inhomogeneous navier stokes equations in the whole n dimensional space under some smallness assumption on the data we show the existence of global in time unique solutions in a critical functional framework the initial density is required to belong to the multiplier space of input amssym dot b n p 1 p 1 bbb r n in particular piecewise constant initial densities are admissible data provided the jump at the interface is small enough and generate global unique solutions with piecewise constant densities using lagrangian coordinates is the key to our results as it enables us to solve the system by means of the basic contraction mapping theorem as a consequence conditions for uniqueness are the same as for existence 2012 wiley periodicals inc

a primal dual active set algorithm for three dimensional contact problems with coulomb friction in this paper efficient algorithms for contact problems with tresca and coulomb friction in three dimensions are presented and analyzed the numerical approximation is based on mortar methods for nonconforming meshes with dual lagrange multipliers using a nonsmooth complementarity function for the three dimensional friction conditions a primal dual active set algorithm is derived the method determines active contact and friction nodes and at the same time resolves the additional nonlinearity originating from sliding nodes no regularization and no penalization are applied and superlinear convergence can be observed locally in combination with a multigrid method it defines a robust and fast strategy for contact problems with tresca or coulomb friction the efficiency and flexibility of the method is illustrated by several numerical examples

nonparabolic dissipative systems modeling the flow of liquid crystals we study a simplified system which retains most of the interesting mathematical properties of the original ericksen leslie equations for the flow of liquid crystals this is a coupled nonparabolic dissipative dynamic system we derive several energy laws which enable us to prove the global existence of the weak solutions and the classical solutions we also discuss uniqueness and some stability properties of the system 1995 john wiley sons inc

inhomogeneous dirichlet boundary condition in the a posteriori error control of the obstacle problem abstract we study a posteriori error control of finite element approximation of the elliptic obstacle problem with nonhomogeneous dirichlet boundary condition the results in the article are two fold firstly we address the influence of the inhomogeneous dirichlet boundary condition in residual based a posteriori error control of the elliptic obstacle problem secondly by rewriting the obstacle problem in an equivalent form we derive a posteriori error bounds which are in simpler form and efficient to accomplish this we construct and use a post processed solution u h of the discrete solution u h which satisfies the exact boundary conditions sharply although the discrete solution u h may not satisfy we propose two post processing methods and analyze them namely the harmonic extension and a linear extension the theoretical results are illustrated by the numerical results

convergence and superconvergence analysis of a nonconforming finite element method for solving the signorini problem abstract in this paper we present the carey nonconforming finite element approximation of the variational inequality resulting from the signorini problem firstly we show that if the displacement field is of h 2 regularity the optimal convergence rate of o h can be obtained with respect to the energy norm secondly if stronger but reasonable h 5 2 regularity is available the superconvergence rate of o h 3 2 can be derived through the interpolated postprocessing technique finally numerical experiments are given which are consistent with our theoretical analysis

global strong solutions for the incompressible nematic liquid crystal flows with density dependent viscosity coefficient abstract this paper is concerned with an initial boundary value problem of the incompressible nematic liquid crystal flows with density dependent viscosity in a smooth bounded domain r 3 the global well posedness of strong solutions with large oscillations is established in vacuum provided u 0 l 2 d 0 l 2 is suitably small with arbitrary large initial density which extended the local strong solution by gao et al 12 to be a global one

finite element method for stokes equations under leak boundary condition of friction type stokes equations are considered with the leak boundary condition of friction type we propose a discrete variational inequality based on one of the p1 p1 stabilized p1b p1 and p2 p1 finite elements we show that a suitable numerical integration approximation for the nonlinear term preserves in a discrete sense some properties known in continuous problems those properties are essential to justify our error estimate and numerical implementation in a consistent way numerical examples are presented to confirm the theoretical results

global strong solutions to incompressible ericksen leslie system in r3 abstract in this paper we consider the cauchy problem to the ericksen leslie system of liquid crystals in r 3 global well posedness of strong solutions are obtained under the condition that the product of u 0 2 d 0 2 and u 0 2 2 d 0 2 is suitably small this result can be viewed as a supplement to the local existence and blow up criteria recently established in hong et al 2014

global solution to the incompressible flow of liquid crystals abstract the initial boundary value problem for the three dimensional incompressible flow of liquid crystals is considered in a bounded smooth domain the existence and uniqueness is established for both the local strong solution with large initial data and the global strong solution with small data it is also proved that when the strong solution exists a weak solution must be equal to the unique strong solution with the same data

error estimates of the weakly over penalized symmetric interior penalty method for two variational inequalities in this paper we apply the weakly over penalized symmetric interior penalty method to solve some variational inequalities which include the signorini problem and the obstacle problem optimal a priori error estimates in energy norm are derived some numerical tests are provided to confirm our theoretical analysis

regularity and existence of global solutions to the ericksen leslie system in mathbb r 2 in this paper we first establish the regularity theorem for suitable weak solutions to the ericksen leslie system in mathbb r 2 building on such a regularity we then establish the existence of a global weak solution to the ericksen leslie system in mathbb r 2 for any initial data in the energy space under the physical constraints on the leslie coefficients ensuring the dissipation of energy of the system which is smooth away from at most finitely many times this extends earlier works by lin et al arch ration mech anal 197 297 336 2010 on a simplified nematic liquid crystal flow to the general ericksen leslie system

global weak solutions to the equations of compressible flow of nematic liquid crystals in two dimensions we consider weak solutions to a simplified ericksen leslie system of two dimensional compressible flow of nematic liquid crystals an initial boundary value problem is first studied in a bounded domain by developing new techniques and estimates to overcome the difficulties induced by the supercritical nonlinearity nabla mathbf d 2 mathbf d in the equations of angular momentum on the direction field and adapting the standard three level approximation scheme and the weak convergence arguments for the compressible navier stokes equations we establish the global existence of weak solutions under a restriction imposed on the initial energy including the case of small initial energy then the cauchy problem with large initial data is investigated and we prove the global existence of large weak solutions by using the domain expansion technique and the rigidity theorem provided that the second component of initial data of the direction field satisfies some geometric angle condition

the local average contact lac method abstract the local average contact lac method allows the handling of nonmatching meshes in an easy way by averaging locally the interpenetration between the contacting bodies in this paper we consider the frictionless unilateral contact problem and several numerical experiments involving two and three dimensional bodies discretized with various linear and quadratic finite elements we also present a convergence analysis of the method in the geometrical nonconforming case in which the boundary points of the candidate contact areas do not coincide

global existence of solutions of the liquid crystal flow for the oseen frank model in r2 abstract in the first part of this paper we establish the global existence of solutions of the liquid crystal gradient flow for the well known oseen frank model the liquid crystal flow is a prototype of equations from the ericksen leslie system in the hydrodynamic theory and generalizes the heat flow for harmonic maps into the 2 sphere the ericksen leslie system is a system of the navier stokes equations coupled with the liquid crystal flow in the second part of this paper we also prove the global existence of solutions of the ericksen leslie system for a general oseen frank model in r 2

unique solvability for the density dependent navier stokes equations abstract in this paper we consider the incompressible navier stokes equations with a density dependent viscosity in a bounded domain of r n n 2 3 we prove the local existence of unique strong solutions for all initial data satisfying a natural compatibility condition this condition is also necessary for a very general initial data moreover we provide a blow up criterion for the regularity of the strong solution for these results the initial density need not be strictly positive it may vanish in an open subset of

global solutions to the 3d incompressible nematic liquid crystal system abstract in this paper we consider the global well posedness of the cauchy problem of the 3d incompressible nematic liquid crystal system with initial data in the critical besov space b 2 1 1 2 r 3 b 2 1 3 2 r 3 in particular we prove that there exists a positive constant c 0 such that the nematic liquid crystal system has a unique global solution with initial data u 0 d 0 u 0 h u 0 3 d 0 which satisfies 1 1 d 0 d 0 b 2 1 3 2 1 u 0 h b 2 1 1 2 exp c 0 2 u 0 3 b 2 1 1 2 1 2 c 0 for some c 0 sufficiently small where d 0 is a constant vector with d 0 1 here and are two positive viscosity constants

a point to segment contact formulation for isogeometric nurbs based finite elements formulation of isogeometric finite elements has received a great deal of attention in the recent past the present study deals with the treatment of problems in structural mechanics including large elastic deformations and contact one decisive difference between isogeometric finite elements based on nurbs functions and standard finite elements fe based on lagrange polynomials is higher inter element continuity this is a promising characteristics to model contact problems as all issues associated with kinks between elements are naturally avoided particularly in cases with large sliding this has the potential to be an attractive feature we present a bilateral isogeometric collocation contact formulation for geometrically non linear two dimensional problems using greville and botella points to collocate the contact integrals different methods to obtain accurate and physically meaningful stress distributions are investigated and compared the results show that the higher inter element continuity in context of non smooth problems may imply undesired effects in the numerical solution such as unphysical stress oscillations similar effects have been observed in standard p fem numerical experiments indicate that these oscillations may be avoided if the basis functions of contact and non contact zones are separated by knot relocation and knot repetition

hybrid finite element methods for the signorini problem we study three mixed linear finite element methods for the numerical simulation of the two dimensional signorini problem applying falk s lemma and saddle point theory to the resulting discrete mixed variational inequality allows us to state the convergence rate of each of them two of these finite elements provide optimal results under reasonable regularity assumptions on the signorini solution and the numerical investigation shows that the third method also provides optimal accuracy

global solution to the 3 d density dependent incompressible flow of liquid crystals abstract in this paper we consider the global wellposedness of the 3 d density dependent incompressible flow of liquid crystals in the critical besov spaces we first get the local solution for this system without assumptions of small density variation and then extend this local solution to be global under the smallness assumptions on initial density initial orientation field and the horizontal components of the initial velocity field this result implies the global well posedness of this system with large initial vertical velocity component

optimal convergence for discrete variational inequalities modelling signorini contact in 2d and 3d without additional assumptions on the unknown contact set the basic h 1 finite element error estimate of order h with only h 2 regularity on the solution has not been yet established for the simplest two dimensional signorini problem approximated by a discrete variational inequality or the equivalent mixed method and linear finite elements to obtain an optimal error bound in this basic case and also when considering more general cases e g the three dimensional problem quadratic finite elements additional assumptions on the exact solution in particular on the unknown contact set see z belhachmi and f ben belgacem math comp 72 2003 pp 83 104 s hueber and b wohlmuth siam j numer anal 43 2005 pp 156 173 b wohlmuth a popp m gee and w wall comput mech 49 2012 pp 735 747 had to be used in this paper we consider finite element approximations of the two dimensional and three dimensional signorini problems with linear and quadratic finite elements in the analysis we remove all the additional assumptions and pr

global regularity to the 3d incompressible nematic liquid crystal flows with vacuum abstract this paper is concerned with the 3d incompressible nematic liquid crystal equations in smooth bounded domains where the velocity u and macroscopic molecular orientation d admit the dirichlet and neumann boundary condition respectively global in time strong solution is proved to exist when the basic energy is suitably small the initial vacuum is allowed

on the global wellposedness to the 3 d incompressible anisotropic navier stokes equations corresponding to the wellposedness result 2 for the classical 3 d navier stokes equations ns with initial data in the scaling invariant besov space mathcal b 1 frac3p p infty here we consider a similar problem for the 3 d anisotropic navier stokes equations ans where the vertical viscosity is zero in order to do so we first introduce the besov sobolev type spaces mathcal b frac12 frac12 4 and mathcal b frac12 frac12 4 t then with initial data in the scaling invariant space mathcal b frac12 frac12 4 we prove the global wellposedness for ans provided the norm of initial data is small enough compared to the horizontal viscosity in particular this result implies the global wellposedness of ans with high oscillatory initial data 1 2

global regularity of 2d density patches for inhomogeneous navier stokes this paper is about lions open problem on density patches lions in mathematical topics in fluid mechanics vol 1 volume 3 of oxford lecture series in mathematics and its applications clarendon press oxford university press new york 1996 whether or not inhomogeneous incompressible navier stokes equations preserve the initial regularity of the free boundary given by density patches using classical sobolev spaces for the velocity we first establish the propagation of c 1 gamma regularity with 0 gamma 1 in the case of positive density furthermore we go beyond this to show the persistence of a geometrical quantity such as the curvature in addition we obtain a proof for c 2 gamma regularity

global strong solution for 3d viscous incompressible heat conducting navier stokes flows with non negative density abstract we are concerned with an initial boundary value problem for the nonhomogeneous heat conducting navier stokes flows with non negative density first of all we show that for the initial density allowing vacuum the strong solution exists globally if the velocity satisfies the serrin s condition then under some smallness condition we prove that there is a unique global strong solution to the 3d viscous nonhomogeneous heat conducting navier stokes flows our method relies upon the delicate energy estimates and regularity properties of stokes system and elliptic equation

a posteriori error estimator and error control for contact problems in this paper we consider two error estimators for one body contact problems the first error estimator is defined in terms of h div conforming stress approximations and equilibrated fluxes while the second is a standard edge based residual error estimator without any modification with respect to the contact we show reliability and efficiency for both estimators more over the error is bounded by the first estimator with a constant one plus a higher order data oscillation term plus a term arising from the contact that is shown numerically to be of higher order the second estimator is used in a control based afem refinement strategy and the decay of the error in the energy is shown several numerical tests demonstrate the performance of both estimators

fujita kato theorem for the 3 d inhomogeneous navier stokes equations abstract in this paper we prove the global existence and uniqueness of the solution with discontinuous density for the 3 d inhomogeneous navier stokes equations if the initial data 0 u 0 l r 3 h s r 3 with s 1 2 satisfies 0 c 0 0 c 0 u 0 h 1 2 e for some small e 0 depending only on c 0 c 0 furthermore we introduce the dual method to show that if u 0 l p r 3 for p 6 5 2 the velocity satisfies the decay estimate k u t l 2 c 1 t k 2 p for t 1 k 0 1 with p 3 2 1 p 1 2

an error estimate for the signorini problem with coulomb friction approximated by finite elements the present paper is concerned with the unilateral contact model and the coulomb friction law in linear elastostatics we consider a mixed formulation in which the unknowns are the displacement field and the normal and tangential constraints on the contact area the chosen finite element method involves continuous elements of degree one and continuous piecewise affine multipliers on the contact zone a convenient discrete contact and friction condition is introduced in order to perform a convergence study we finally obtain a first a priori error estimate under the assumptions ensuring the uniqueness of the solution to the continuous problem

equilibration techniques for solving contact problems with coulomb friction abstract in this paper we consider residual and equilibrated error indicators for contact problems with coulomb friction the contact problem is handled within the abstract framework of saddle point problems more precisely the non penetration constraint and the friction law is realized as a variationally consistent weak formulation in terms of a localized dual lagrange multiplier space thus from the displacement we can easily compute in a local post process the lagrange multiplier which acts as a neumann condition on the possible contact zone having computed the discrete lagrange multiplier we can apply standard error estimators by replacing the unknown neumann data by its approximation as it is shown in 1 this results in an error estimator for a one sided contact problem without friction here we consider more general situations and discuss two additional contact terms which measure the non conformity of the discrete lagrange multiplier numerical results in two and three dimensions illustrate the flexibility of the approach and show the influence of the material parameters on the adaptive refinement process

global solution to the 3d inhomogeneous nematic liquid crystal flows with variable density abstract in this paper we investigate the global existence and uniqueness of solution to the 3d inhomogeneous incompressible nematic liquid crystal flows with variable density in the framework of besov spaces it is proved that there exists a global and unique solution to the nematic liquid crystal flows if the initial data 0 1 u 0 n 0 e 3 m b p 1 3 p 1 r 3 b p 1 3 p 1 r 3 b p 1 3 p r 3 with 1 p 6 and satisfies 0 1 m b p 1 3 p 1 u 0 b p 1 3 p 1 n 0 e 3 b p 1 3 p c for some small c 0 depending only on p here m b p 1 3 p 1 r 3 is the multiplier space of besov space b p 1 3 p r 3 using the lagrangian approach in danchin and mucha 2012 2013 7 8 is the key to our results

stability to the global large solutions of 3 d navier stokes equations in this chapter we consider the stability of global large solutions to the 3 d classical isotropic navier stokes equations in the anisotropic sobolev spaces

a monotonic method for the numerical solution of some free boundary value problems this work presents an efficient monotonic algorithm for the numerical solution of the obstacle problem and the signorini problems when they are discretized either by the finite element method or by the finite volume method the convergence of this algorithm applied to the discrete problem is proven in both cases

numerical implementation of two nonconforming finite element methods for unilateral contact we consider the finite element approximation of the unilateral contact problem between elastic bodies we are interested in a practical problem which often occurs in finite element computations concerning two independently discretized bodies in unilateral contact it follows that the nodes of both bodies located on the contact surface do not fit together we present two different approaches in order to define unilateral contact on nonmatching meshes the first is an extension of the mortar finite element method to variational inequalities that defines the contact in a global way on the contrary the second one expresses local node on segment contact conditions in both cases the theoretical approximation properties are given then we implement and compare the two methods

on the wellposedness of three dimensional inhomogeneous navier stokes equations in the critical spaces we prove the local wellposedness of three dimensional incompressible inhomogeneous navier stokes equations with initial data in the critical besov spaces without assumptions of small density variation furthermore if the initial velocity field is small enough in the critical besov space dot b 1 2 2 1 mathbb r 3 this system has a unique global solution

liquid crystal flows in two dimensions this paper is concerned with a simplified hydrodynamic equation proposed by ericksen and leslie modeling the flow of nematic liquid crystals in dimension two we establish both interior and boundary regularity theorems for such a flow under smallness conditions as a consequence we establish the existence of global in time weak solutions on a bounded smooth domain in mathbb r 2 which are smooth everywhere with possible exceptions of finitely many singular times

numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element methods this contribution deals with the finite element approximation of the variational inequalities coming from the signorini problem and the unilateral contact between two elastic bodies the numerical analysis we carry out improves former results and states optimal convergence rates under reasonable regularity hypotheses for both conforming and nonconforming methods

coupling finite element and spectral methods first results a poisson equation on a rectangular domain is solved by coupling two methods the domain is divided in two squares a finite element approximation is used on the first square and a spectral discretization is used on the second one two kinds of matching conditions on the interface are presented and compared in both cases error estimates are proved

on critical cases of sobolev s inequalities abstract we present a new form of the trudinger type inequality which shows an explicit dependence moreover we give an alternative proof of the brezis gallouet wainger inequality

a non steady system with friction boundary conditions for flow of heat conducting incompressible viscous fluids abstract in this paper we are concerned with a non steady system for motion of incompressible viscous newtonian heat conducting fluids under mixed boundary conditions the boundary conditions for fluid may include tresca slip leak and one sided leak conditions based on total stress velocity total pressure rotation total stress together and the conditions for temperature may include dirichlet neumann and robin conditions together relying on the relations among strain rotation normal derivative of velocity and shape of boundary surface we get variational formulation the formulation consists of a variational inequality for velocity due to the boundary conditions of friction type and a variational equation for temperature we first study existence estimation and relative compactness of solutions to an approximate equation by passing to limits as the parameter for approximation goes to zero we get existence of a solution with defect measure for temperature

on the abstract cauchy problem of parabolic type in spaces of continuous functions on etudie l existence l unicite et les proprietes de regularite des solutions du probleme de cauchy u t au r f t o t t u o u 0 ou a d a ce e est un operateur lineaire clos dans l espace de banach e

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

global weak solution and blow up criterion of the general ericksen leslie system for nematic liquid crystal flows abstract in this paper we investigate the three dimensional general ericksen leslie e l system with ginzburg landau type approximation modeling nematic liquid crystal flows first by overcoming the difficulties from lack of maximum principle for the director equation and high order nonlinearities for the stress tensor we prove existence of global in time weak solutions under physically meaningful boundary conditions and suitable assumptions on the leslie coefficients which ensures that the total energy of the e l system is dissipated moreover for the e l system with periodic boundary conditions we prove the local well posedness of classical solutions under the so called parodi s relation and establish a blow up criterion in terms of the temporal integral of both the maximum norm of the curl of the velocity field and the maximum norm of the gradient of the liquid crystal director field

local well posedness of strong solutions to density dependent liquid crystal system abstract in this paper we study the cauchy problem to the density dependent liquid crystal system in r 3 we establish the local existence and uniqueness of strong solutions to this system in order to overcome the difficulties caused by the high order coupling terms a biharmonic regularization of the system as an auxiliary system is introduced and we make full use of the intrinsic cancellation properties between the high order coupling terms

global strong and weak solutions to inhomogeneous nematic liquid crystal flow in two dimensions abstract we consider the strong and weak solutions to the cauchy problem of the inhomogeneous incompressible nematic liquid crystal equations in two dimensions we first establish the local existence and uniqueness of strong solutions by using the standard domain expanding method and then extend such a local strong solution to be a global one provided the initial density is away from vacuum and the initial direction field satisfies some geometric structure the size of the initial data can be large based on such global existence results by using the compactness argument we obtain the global existence of weak solutions with nonnegative initial density

strong solutions to the density dependent incompressible nematic liquid crystal flows abstract in this paper we investigate the density dependent incompressible nematic liquid crystal flows in n dimensional n 2 or 3 bounded domain the local existence and uniqueness of strong solutions are obtained when the viscosity coefficient of fluid depends on density furthermore one establishes blowup criterions for the regularity of the strong solutions in dimensions two and three respectively in particular we build a blowup criterion just in terms of the gradient of density if the initial direction field satisfies some geometric configuration for these results the initial density need not be strictly positive

mixed finite element methods for smooth domain formulation of crack problems the discretization by finite element methods of a new variational formulation of crack problems is considered the new formulation called the smooth domain method is derived for crack problems in the case of an elastic membrane inequality type boundary conditions are prescribed at the crack faces the resulting model takes the form of a unilateral contact problem on the crack we study and implement various mixed finite element methods for the numerical approximation of the model a priori error estimates are derived and results of computations are provided the convergence rates obtained from the numerical simulations are in agreement with the theoretical estimates

existence of solutions for the ericksen leslie system we study the general ericksen leslie system which describes the flow of liquid crystal materials the dissipation property of the system is established and is used to prove the global existence of weak solutions we also study the existence of classical solutions and the asymptotic stability of the solutions

global strong solution to the 2d nonhomogeneous incompressible mhd system abstract in this paper we first prove the unique global strong solution with vacuum to the two dimensional nonhomogeneous incompressible mhd system as long as the initial data satisfies some compatibility condition as a corollary the global existence of strong solution with vacuum to the 2d nonhomogeneous incompressible navier stokes equations is also established our main result improves all the previous results where the initial density need to be strictly positive the key idea is to use some critical sobolev inequality of logarithmic type which is originally due to brezis and wainger 1980 7

a nitsche based method for unilateral contact problems numerical analysis we introduce a nitsche based formulation for the finite element discretization of the unilateral contact problem in linear elasticity it features a weak treatment of the non linear contact conditions through a consistent penalty term without any additional assumption on the contact set we can prove theoretically its fully optimal convergence rate in the h1 norm for linear finite elements in two dimensions which is o h 1 2 when the solution lies in h 3 2 0 1 2 an interest of the formulation is that conversely to lagrange multiplier based methods no other unknown is introduced and no discrete inf sup condition needs to be satisfied

global solutions to the 3 d incompressible inhomogeneous navier stokes system in this paper we consider the global well posedness of the 3 d incompressible inhomogeneous navier stokes equations with initial data in the critical besov spaces a0 bq 13q r3 u0 u0h u03 bp 1 1 3p r3 for p q satisfying 1 q p 6 and 1q 1p 13 more precisely we prove that there exist two positive constants c0 c0 such that if a0 bq 13q u0h bp 1 1 3p exp c0 u03 bp 1 1 3p2 2 c0 then 1 3 has a unique global solution a l r bq 13q r3 u l r bp 1 1 3p r3 l1 r bp 11 3p r3 in particular this result implies the global well posedness result in abidi and paicu 2007 2 for the inhomogeneous navier stokes system with small initial data

global solutions to the 3 d incompressible anisotropic navier stokes system in the critical spaces in this paper we consider the global wellposedness of the 3 d incompressible anisotropic navier stokes equations with initial data in the critical besov sobolev type spaces mathcal b and mathcal b frac12 frac12 4 see definitions 1 1 and 1 2 below in particular we proved that there exists a positive constant c such that ans has a unique global solution with initial data u 0 u 0 h u 0 3 which satisfies u 0 h mathcal b exp bigl frac c nu 4 u 0 3 mathcal b 4 bigr leq c 0 nu or u 0 h mathcal b frac12 frac12 4 exp bigl frac c nu 4 u 0 3 mathcal b frac12 frac12 4 4 bigr leq c 0 nu for some c 0 sufficiently small to overcome the difficulty that gronwall s inequality can not be applied in the framework of chemin lerner type spaces widetilde l p t mathcal b we introduced here sort of weighted chemin lerner type spaces widetilde l 2 t f mathcal b for some apropriate l 1 function f t

on the well posedness of 2 d incompressible navier stokes equations with variable viscosity in critical spaces abstract in this paper we first prove the local well posedness of the 2 d incompressible navier stokes equations with variable viscosity in critical besov spaces with negative regularity indices without smallness assumption on the variation of the density the key is to prove for p 1 4 and a b p 1 2 p r 2 that the solution mapping h a f to the 2 d elliptic equation div 1 a div f is bounded on b p 1 2 p 1 r 2 more precisely we prove that b p 1 2 p 1 c 1 a b p 1 2 p 2 f b p 1 2 p 1 the proof of the uniqueness of solution to 1 2 relies on a lagrangian approach 15 16 17 when the viscosity coefficient is a positive constant we prove that 1 2 is globally well posed

erratum to global wellposed problem for the 3 d incompressible anisotropic navier stokes equations in an anisotropic space in this paper we consider a global wellposed problem for the 3 d incompressible anisotropic navier stokes equations ans we prove the global wellposedness for ans provided the initial horizontal data are sufficient small in the scaling invariant besov sobolev type space b 0 frac 1 2 in particular the result implies the global wellposedness of ans with large initial vertical velocity

a stabilized lagrange multiplier method for the finite element approximation of contact problems in elastostatics in this work we consider a stabilized lagrange or kuhn tucker multiplier method in order to approximate the unilateral contact model in linear elastostatics the particularity of the method is that no discrete inf sup condition is needed in the convergence analysis we propose three approximations of the contact conditions well adapted to this method and we study the convergence of the discrete solutions several numerical examples in two and three space dimensions illustrate the theoretical results and show the capabilities of the method

an optimal a priori error estimate for nonlinear multibody contact problems nonconforming domain decomposition methods provide a powerful tool for the numerical approximation of partial differential equations for the discretization of a nonlinear multibody contact problem we use linear mortar finite elements based on dual lagrange multipliers under some regularity assumptions on the solution an optimal convergence order of h 0 5 nu 0 nu leq 0 5 can be established in two dimensions 2d and three dimensions 3d compared with a standard linear saddle point formulation two additional terms which provide a measure for the nonconformity and the nonlinearity of the approach have to be taken in account numerical examples illustrating the performance of the nonconforming method and confirming our theoretical result are presented

quadratic finite element approximation of the signorini problem applying high order finite elements to unilateral contact variational inequalities may provide more accurate computed solutions compared with linear finite elements up to now there was no significant progress in the mathematical study of their performances the main question is involved with the modeling of the nonpenetration signorini condition on the discrete solution along the contact region in this work we describe two nonconforming quadratic finite element approximations of the poisson signorini problem responding to the crucial practical concern of easy implementation and we present the numerical analysis of their efficiency by means of falk s lemma we prove optimal and quasi optimal convergence rates according to the regularity of the exact solution

uniform global strong solutions of the 2d density dependent incompressible magnetic benard problem in a bounded domain abstract in this paper we use the bootstrap argument to prove the uniform in k global strong solutions of the 2d density dependent incompressible magnetic benard problem in a bounded domain here k is the heat conductivity coefficient

global regularity and uniqueness of weak solution for the 2 d liquid crystal flows abstract we prove the global existence and regularity of weak solution for the 2 d liquid crystal flows with the large initial velocity the uniqueness of weak solution is also proved by using the littlewood paley analysis

a posteriori error estimations of residual type for signorini s problem this paper presents an a posteriori error analysis for the linear finite element approximation of the signorini problem in two space dimensions a posteriori estimations of residual type are defined and upper and lower bounds of the discretization error are obtained we perform several numerical experiments in order to compare the convergence of the terms in the error estimator with the discretization error

incompressible flows with piecewise constant density we investigate the incompressible navier stokes equations with variable density the aim is to prove existence and uniqueness results in the case of discontinuous initial density in dimension n 2 3 assuming only that the initial density is bounded and bounded away from zero and that the initial velocity is smooth enough we get the local in time existence of unique solutions uniqueness holds in any dimension and for a wider class of velocity fields in particular all those results are true for piecewise constant densities with arbitrarily large jumps global results are established in dimension two if the density is close enough to a positive constant and in n dimensions if in addition the initial velocity is small the lagrangian formulation for describing the flow plays a key role in the analysis that is proposed in the present paper

superconvergence of finite element method for the signorini problem in this paper we study the superconvergence of the frictionless signorini problem when approximated by bilinear finite elements by virtue of the information on the contact zone we can derive a superconvergence rate of o h 3 2 under a proper regularity assumption finally a numerical test is given to verify our result

two results on solution uniqueness and multiplicity for the linear elastic friction problem with normal compliance abstract this study is concerned with the frictional contact problem governed by the normal compliance law in linear elasticity the paper presents two contributions dealing with the stationary problem we first obtain improved bounds ensuring the uniqueness of a solution second we exhibit examples in which infinitely many solutions to the problem exist

global well posedness for the incompressible navier stokes equations with density dependent viscosity coefficient abstract this paper is concerned with an initial boundary value problem of the incompressible navier stokes equations with density dependent viscosity in a smooth bounded domain r 3 the global well posedness of strong solutions with large oscillations is established in both non vacuum and vacuum cases provided the initial velocity is suitably small in certain sense it is worth pointing out that there isn t any smallness condition on the density and its gradient