hamiltonian perspective on compartmental reaction diffusion networks inspired by the recent developments in modeling and analysis of reaction networks we provide a geometric formulation of the reversible reaction networks under the influence of diffusion using the graph knowledge of the underlying reaction network the obtained reaction diffusion system is a distributed parameter port hamiltonian system on a compact spatial domain motivated by the need for computer based design we offer a spatially consistent discretization of the pde system and in a systematic manner recover a compartmental ode model on a simplicial triangulation of the spatial domain exploring the properties of a balanced weighted laplacian matrix of the reaction network and the laplacian of the simplicial complex we characterize the space of equilibrium points and provide a simple stability analysis on the state space modulo the space of equilibrium points the paper rules out the possibility of the persistence of spatial patterns for the compartmental balanced reaction diffusion networks

evolutionary subgradient inclusions with nonlinear weakly continuous operators and applications abstract in this paper we consider the first order evolutionary inclusions with nonlinear weakly continuous operators and a multivalued term which involves the clarke subgradient of a locally lipschitz function first we provide a surjectivity result for stationary inclusion with weakly weakly upper semicontinuous multifunction then we use this result to prove the existence of solutions to the rothe sequence and the evolutionary subgradient inclusion finally we apply our results to the non stationary navier stokes equation with nonmonotone and multivalued frictional boundary conditions

global strong solutions of two dimensional navier stokes equations with nonlinear slip boundary conditions abstract the two dimensional time dependent navier stokes equations with nonlinear slip boundary conditions are investigated in this paper since the nonlinear slip boundary conditions of this type include the subdifferential property the weak variational formulation is the variational inequality the existence uniqueness and regularity of global weak solutions are shown using the regularized method moreover the continuous dependence property of the weak solution for given initial data and the behavior of the global weak solution as t are established

on the vanishing viscosity limit for the 3d navier stokes equations with a slip boundary condition where and below and denote the div and curl operators respectively n is the outward normal and is the unit tangential vector of the investigation of vanishing viscosity limit of solutions of the navier stokes equations both in the two and three spacial dimensional cases is a classical issue there are two related questions arising from here one is how to describe the inviscid limiting behavior of the navier stokes equation and the other is that does the euler equation can be approximated by the navier stokes equations in the case that the solution to the this research is supported in part by zheng ge ru foundation and hong kong rgc earmarked research grants cuhk 4028 04p cuhk 4040 02p and cuhk 4279 00p

explicit simplicial discretization of distributed parameter port hamiltonian systems simplicial dirac structures as finite analogues of the canonical stokes dirac structure capturing the topological laws of the system are defined on simplicial manifolds in terms of primal and dual cochains related by the coboundary operators these finite dimensional dirac structures offer a framework for the formulation of standard input output finite dimensional port hamiltonian systems that emulate the behavior of distributed parameter port hamiltonian systems this paper elaborates on the matrix representations of simplicial dirac structures and the resulting port hamiltonian systems on simplicial manifolds employing these representations we consider the existence of structural invariants and demonstrate how they pertain to the energy shaping of port hamiltonian systems on simplicial manifolds

the analysis of stationary viscous incompressible flow through a rotating radial blade machine existence of a weak solution abstract the paper deals with the analysis of the mathematical model of the two dimensional stationary viscous incompressible flow through a rotating radial blade machine the flow is described and studied in the rotating frame the paper provides the classical and weak formulation of the corresponding boundary value problem the boundary condition on the outflow is the so called natural boundary condition with the nonlinear term proposed by bruneau and fabrie 1996 1 and also modified by a term arising from the rotation of the machine the existence of a weak solution is proved

study of coating flow by the finite element method abstract silliman s analysis of slot coating is extended to accommodate film flows with highly bent menisci as in slide and curtain coating by combining polar and cartesian coordinate parametrizations of meniscus shape the nonlinear algebraic equations from the subdomain galerkin weighted residual method and finite element basis functions are solved by newton s method which is perfected for the free boundary problem convergence behavior is examined slot coating results reveal slow flow zones and show that at high capillary number and low metering rate the downstream meniscus can no longer attach to the slot comparison of coyne and elrod s approximate solutions with the finite element solutions shows that their assumption of a parabolic velocity profile leads to reasonable approximations of meniscus shapes

on existence and uniqueness of solutions to a boussinesq system with nonlinear and mixed boundary conditions abstract we study a boussinesq system in a bounded domain with an outlet boundary portion where fluid can leave or re enter on this boundary part we consider a do nothing condition for the fluid flow and a new artificial condition for the heat transfer that couples nonlinearly the fluid velocity and temperature the latter can be further adjusted if convective or conductive phenomena are dominant we prove existence and in some cases uniqueness of weak solutions to stationary and evolutionary problems by a fixed point strategy under suitable assumptions on the data a variety of numerical tests shows the improved performance of the new artificial condition with respect to other standard choices in the literature

topology optimization of fluid structure interaction problems in poroelasticity abstract this paper presents a method for applying topology optimization to fluid structure interaction problems in saturated poroelastic media the method relies on a multiple scale method applied to periodic media the resulting model couples the stokes flow in the pores of the structure with the deformation of the elastic skeleton through a macroscopic darcy type flow law the method allows to impose pressure loads for static problems through a one way coupling while transient problems are fully coupled modeling the interaction between fluid and solid the material distribution is determined by topology optimization in order to optimize the performance of a shock absorber and test the pressure loading capabilities and optimization of an internally pressurized lid

finite element exterior calculus from hodge theory to numerical stability this article reports on the confluence of two streams of research one emanating from the fields of numerical analysis and scientific computation the other from topology and geometry in it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de rham cohomology and hodge theory are key tools for exploring the well posedness of the continuous problem the discretization methods we consider are finite element methods in which a variational or weak formulation of the pde problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace after a brief introduction to finite element methods we develop an abstract hilbert space framework for analyzing the stability and convergence of such discretizations in this framework the differential complex is represented by a complex of hilbert spaces and stability is obtained by transferring hodgetheoretic structures that ensure well posedness of the continuous problem from the continuous level to the discrete we show stable discretization is achieved if the finite element spaces satisfy two hypotheses they can be arranged into a subcomplex of this hilbert complex and there exists a bounded cochain projection from that complex to the subcomplex in the next part of the paper we consider the most canonical example of the abstract theory in which the hilbert complex is the de rham complex of a domain in euclidean space we use the koszul complex to construct two families of finite element differential forms show that these can be arranged in subcomplexes of the de rham complex in numerous ways and for each construct a bounded cochain projection the abstract theory therefore applies to give the stability and convergence of finite element approximations of the hodge laplacian other applications are considered as well especially the elasticity complex and its application to the equations of elasticity background material is included to make the presentation self contained for a variety of readers

pseudoholomorphic strips in symplectizations ii fredholm theory and transversality this paper is part of a larger program the investigation of the chord problem in three dimensional contact geometry the main tool will be pseudoholomorphic strips in the symplectisation of a three dimensional contact manifold with two totally real submanifolds l0 l1 as boundary conditions the submanifolds l0 and l1 do not intersect transversally in this paper we will develop a nonlinear fredholm theory that guarantees the existence of a family of embedded pseudoholomorphic strips near a given one with suitable properties c 2003 wiley periodicals inc

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

numerical methods for the stokes and navier stokes equations driven by threshold slip boundary conditions abstract in this article we discuss the numerical solution of the stokes and navier stokes equations completed by nonlinear slip boundary conditions of friction type in two and three dimensions to solve the stokes system we first reduce the related variational inequality into a saddle point point problem for a well chosen augmented lagrangian to solve this saddle point problem we suggest an alternating direction method of multiplier together with finite element approximations the solution of the navier stokes system combines finite element approximations time discretization by operator splitting and augmented lagrangian method numerical experiment results for two and three dimensional flow confirm the interest of these approaches

on coupled navier stokes and energy equations in exterior like domains we investigate a class of nonlinear evolution systems modeling time dependent flows of incompressible viscous and heat conducting fluids with temperature dependent transport coefficients in three dimensional exterior like domains we prove a local existence theorem for the fully coupled parabolic system with a source term involving the square of the velocity gradient and a combination of dirichlet and artificial boundary conditions

on the regularity of flows with ladyzhenskaya shear dependent viscosity and slip or non slip boundary conditions navier stokes equations with shear dependent viscosity under the classical nonslip boundary condition were introduced and studied in the 1960s by o a ladyzhenskaya and in the case of gradient dependent viscosity by j l lions a particular case is the well known smagorinsky turbulence model this is nowadays a central subject of investigation on the other hand boundary conditions of slip type seems to be more realistic in some situations in particular in numerical applications they are a main research subject the existence of weak solutions u to the above problems with slip or nonslip type boundary conditions is well known in many cases however regularity up to the boundary still presents many open questions in what follows we present some regularity results in the stationary case for weak solutions to this kind of problem see theorem 3 1 and theorem 3 2 the evolution problem is studied in a forthcoming paper 6 a cornerstone in our proof is the classical nirenberg translation method see 38

on the navier stokes flows for heat conducting fluids with mixed boundary conditions abstract we consider a coupled model for steady flows of viscous incompressible heat conducting fluids with temperature dependent material coefficients in a fixed three dimensional open cylindrical channel we introduce the banach spaces x and y to be the space of possible solutions of this problem and the space of its data respectively we show that the corresponding operator of the problem acting between x and y is frechet differentiable applying the local diffeomorphism theorem we get the local solvability results for a variational formulation

variational inequality for the rotating navier stokes equations with subdifferential boundary conditions a steady viscous incompressible fluid through a rotating channel satisfies the rotating navier stokes equations subdifferential boundary conditions are imposed a variational inequality is derived for this system furthermore under some assumptions the weak solution of the variational inequality exists and is unique

korn s inequalities and their applications in continuum mechanics korn s inequalities have played a central role in the development of linear elasticity not only in connection with the basic theoretical issues such as existence and uniqueness but also in a variety of applications the korn inequalities and other related inequalities for integrals of quadratic functionals also arise in the analysis of viscous incompressible fluid flow the dimensionless optimal constants appearing in these inequalities the korn constants depend only on the shape of the domains of concern information on the geometric dependence of these constants is essential in applications in this review article we summarize the major results on korn s inequalities for bounded domains in two and three dimensions with emphasis on results concerning the korn constants some applications in continuum mechanics are also described

topology optimization for three phase materials distribution in a dissipative expansion chamber by unified multiphase modeling approach the sound attenuation performance of a dissipative expansion chamber is a combinational result of reflective and dissipative effects although it is well known that the performance can be substantially improved by altering the distribution of constituent materials the process of finding an optimal distribution of the materials still remains a challenge this work proposes a new design method for interior space of a dissipative expansion chamber by using topology optimization method different from the existing topology optimizations for expansion chamber designs based on simplified material modeling the present design deals with fully modeled multiphase constituent materials such as acoustic poroelastic and elastic one difficulties in the optimization formulation for the multiphase material distribution arise from extremely different acoustic behavior of the materials and the use of various governing equations for different phase materials to systematically vary the attributes of the chamber interior space a unified multiphase modeling approach that allows continuous variations between the three phase materials within the same implementation is employed with an elaborately derived penalty parameters of material interpolation functions various design examples are successfully solved for wide frequency bands and the optimal configurations clearly demonstrate the importance of using specific configurations tuned to different target frequencies

stokes system with local coulomb s slip boundary conditions analysis of discretized models and implementation abstract the theoretical part of the paper analyzes discretized stokes systems with local coulomb s slip boundary conditions solutions to discrete models are defined by means of fixed points of an appropriate mapping we prove the existence of a fixed point establish conditions guaranteeing its uniqueness and examine how they depend on the discretization parameter h and the slip coefficient the second part of the paper is devoted to computational aspects numerical experiments are presented

slip with friction and penetration with resistance boundary conditions for the navier stokes equations numerical tests and aspects of the implementation we consider slip with friction and penetration with resistance boundary conditions in the steady state navier stokes equations this paper describes some aspects of the implementation of these boundary conditions for finite element discretizations numerical tests on two and three dimensional channel flows across a step using the slip with friction boundary condition study the influence of the friction parameter on the position of the reattachment point and the reattachment line of the recirculating vortex respectively

penalty finite element method for stokes problem with nonlinear slip boundary conditions abstract the penalty finite element method for stokes problem with nonlinear slip boundary conditions based on the finite element subspace v h m h which satisfies the discrete inf sup condition is investigated in this paper since this class of nonlinear slip boundary conditions include the subdifferential property the weak variational formulation associated with stokes problem is variational inequality under some regularity assumptions we obtain the optimal h 1 and l 2 error estimates between u and u h and between u and u h e where the error orders are e h for h 1 error and e h 2 for l 2 error

two dimensional poroelastic acoustical foam shape design for absorption coefficient maximization by topology optimization method optimal shape design of a two dimensional poroelastic acoustical foam is formulated as a topology optimization problem for a poroelastic acoustical system consisting of an air region and a poroelastic foam region two different physical regions are continuously changed in an iterative design process to automatically account for the moving interfaces between two regions we propose a new unified model to analyze the whole poroelastic acoustical foam system with one set of governing equations biot s equations are modified with a material property interpolation from a topology optimization method with the unified analysis model we carry out two dimensional optimal shape design of a poroelastic acoustical foam by a gradient based topology optimization setting the specific objective is the maximization of the absorption coefficient in low and middle ranges of frequencies with different amounts of a poroelastic material the performances of the obtained shapes are compared with those of well known wedge s

certifying spatially uniform behavior in reaction diffusion pde and compartmental ode systems we present a condition that guarantees spatial uniformity for the asymptotic behavior of the solutions of a reaction diffusion pde with neumann boundary conditions this condition makes use of the jacobian matrix of the reaction terms and the second neumann eigenvalue of the laplacian operator on the given spatial domain and eliminates the global lipschitz assumptions commonly used in mathematical biology literature we then derive numerical procedures that employ linear matrix inequalities to certify this condition and illustrate these procedures on models of several biochemical reaction networks finally we present an analog of this pde result for the synchronization of a network of identical ode models coupled by diffusion terms from a systems biology perspective the main contribution of the paper is to blend analytical and numerical tools from nonlinear systems and control theory to derive a relaxed and verifiable condition for spatial uniformity of biological processes

two level variational multiscale finite element methods for navier stokes type variational inequality problem in this paper we present two level variational multiscale finite element method based on two local gauss integrations for navier stokes equations with friction boundary conditions which are of the form of navier stokes type variational inequality of the second kind we solve navier stokes type variational inequality problem on the coarse mesh and solve linearized navier stokes type variational inequality problem corresponding to newton iteration on the fine mesh the error estimates in h 1 norm for velocity and l 2 norm for pressure are derived meanwhile uzawa iteration schemes are constructed to solve the subproblems in this two level method finally the numerical results are displayed to support the theoretical analysis

existence of the solution to stationary navier stokes equations with nonlinear slip boundary conditions abstract the stationary navier stokes equations with nonlinear slip boundary conditions are investigated in this paper because the boundary conditions include the subdifferential property on the part boundary the variational formulation of this problem is the variational inequality problem of the second kind with navier stokes operator the main purpose of the paper is to study the existence of the weak solution and the strong solution to this variational inequality problem in terms of the yosida s regularity method

3d unsteady navier stokes problem with memory and subdifferential boundary condition abstract we consider a mathematical model which describes the motion of a 3d unsteady fluid flow governed by the navier stokes system and subjected to mixed boundary conditions with a given velocity on one part of the boundary and nonlinear slip conditions with a memory term reminiscent of coulomb s friction law on the other part we establish first some regularity properties and estimates for a simplified model then we prove the existence of a solution to our problem by using a successive approximation technique and compactness arguments based on helly s theorem for the velocity field

a passivity based approach to stability of spatially distributed systems with a cyclic interconnection structure a class of distributed systems with a cyclic interconnection structure is considered these systems arise in several biochemical applications and they can undergo diffusion driven instability which leads to a formation of spatially heterogeneous patterns in this paper a class of cyclic systems in which addition of diffusion does not have a destabilizing effect is identified for these systems global stability results hold if the ldquosecantrdquo criterion is satisfied in the linear case it is shown that the secant condition is necessary and sufficient for the existence of a decoupled quadratic lyapunov function which extends a recent diagonal stability result to partial differential equations for reaction diffusion equations with nondecreasing coupling nonlinearities global asymptotic stability of the origin is established all of the derived results remain true for both linear and nonlinear positive diffusion terms similar results are shown for compartmental systems

two grid mfeas for the incompressible stokes type variational inequality with damping abstract in this paper we study two grid mixed finite element algorithms mfeas for the stationary incompressible stokes equations with damping and nonlinear slip boundary conditions we set up the weak formulation and obtain the well posedness then employing the mixed finite element method some optimal error estimates between the exact solution and the approximation solution are derived two kinds of two grid mfeas based on the stokes correction and newton type correction are established some error estimates for two grid mfeas are obtained several numerical results are presented to confirm our theoretical findings

pressure projection stabilized finite element method for stokes problem with nonlinear slip boundary conditions in this paper we consider the pressure projection stabilized finite element method for the stokes problem with nonlinear slip boundary conditions whose variational formulation is the variational inequality problem of the second kind with the stokes operator the h 1 and l 2 error estimates for the velocity and the l 2 error estimate for the pressure are obtained finally the numerical results are displayed to verify the theoretical analysis

a coherent analysis of stokes flows under boundary conditions of friction type this paper is concerned with the stationary and nonstationary flow of viscous incompressible fluid under boundary conditions of friction type which are certain nonlinear boundary conditions similar to the so called signorini boundary condition in elasticity we assume that the flow is governed by the linear stokes equation while the boundary condition is nonlinear from the methodological viewpoint the analysis is carried out in a coherent way starting from study of the related boundary value problems for the stationary flow by means of the theory of variational inequalities and getting to wellposedness of the initial boundary value problems for the nonstationary flow by means of the nonlinear semigroup theory from the viewpoint of applications we mention original motivations and include some new generalizations like the cases of anisotropic friction and inhomogeneous boundary value

adaptive discontinuous galerkin methods for solving an incompressible stokes flow problem with slip boundary condition of frictional type abstract we consider the numerical solution of a variational inequality arising in the study of an incompressible stokes flow with a nonlinear slip boundary condition of frictional type we study several discontinuous galerkin dg schemes for solving the problem and derive reliable and efficient residual type a posteriori error estimators an adaptive dg method is constructed based on the error estimators to solve the problem some numerical examples are presented to illustrate the efficiency of the adaptive dg method

on a strong solution of the non stationary navier stokes equations under slip or leak boundary conditions of friction type abstract strong solutions of the non stationary navier stokes equations under non linearized slip or leak boundary conditions are investigated we show that the problems are formulated by a variational inequality of parabolic type to which uniqueness is established using galerkin s method and deriving a priori estimates we prove global and local existence for 2d and 3d slip problems respectively for leak problems under no leak assumption at t 0 we prove local existence in 2d and 3d cases compatibility conditions for initial states play a significant role in the estimates

semi discrete stabilized finite element methods for navier stokes equations with nonlinear slip boundary conditions based on regularization procedure based on the pressure projection stabilized methods the semi discrete finite element approximation to the time dependent navier stokes equations with nonlinear slip boundary conditions is considered in this paper because this class of boundary condition includes the subdifferential property then the variational formulation is the navier stokes type variational inequality problem using the regularization procedure we obtain a regularized problem and give the error estimate between the solutions of the variational inequality problem and the regularized problem with respect to the regularized parameter varepsilon which means that the solution of the regularized problem converges to the solution of the navier stokes type variational inequality problem as the parameter varepsilon longrightarrow 0 moreover some regularized estimates about the solution of the regularized problem are also derived under some assumptions about the physical data the pressure projection stabilized finite element methods are used to the regularized problem and some optimal error estimates of the finite element approximation solutions are derived

distributed algorithms for reaching consensus on general functions this paper presents analysis and design results for distributed consensus algorithms in multi agent networks we consider continuous consensus functions of the initial state of the network agents under mild smoothness assumptions we obtain necessary and sufficient conditions characterizing any algorithm that asymptotically achieves consensus this characterization is the building block to obtain various design results for networks with weighted directed interconnection topologies we first identify a class of smooth functions for which one can synthesize in a systematic way distributed algorithms that achieve consensus we apply this result to the family of weighted power mean functions and characterize the exponential convergence properties of the resulting algorithms we establish the validity of these results for scenarios with switching interconnection topologies finally we conclude with two discontinuous distributed algorithms that achieve respectively max and min consensus in finite time

correction to structure and stability of certain chemical networks and applications to the kinetic proofreading model of t cell receptor signal transduction deals with the theory of structure stability robustness and stabilization for an appealing class of nonlinear systems which arises in the analysis of chemical networks the results given here extend but are also heavily based upon certain previous work by feinberg 1977 1979 1987 1995 and horn and jackson 1972 of which a self contained and streamlined exposition is included the theoretical conclusions are illustrated through an application to the kinetic proofreading model proposed by mckeithan 1995 for t cell receptor signal transduction

a steady flow through a plane cascade of profiles with an arbitrarily large inflow_the mathematical model existence of a weak solution the paper deals with a mathematical model of a viscous stationary incompressible flow through a cascade of profiles the problem for the navier stokes system is formulated in a domain corresponding to one spatial period of the cascade we consider several types of boundary conditions on various parts of the boundary the inflow the artificial lower and upper boundaries the profile the outflow we solve the problem with an arbitrarily large inflow into the turbine we formulate the artificial boundary condition on the outflow the modification of the so called do nothing condition which enables us to prove the existence of a weak solution for any inflow

some properties on the surfaces of vector fields and its application to the stokes and navier stokes problems with mixed boundary conditions abstract in this paper we are concerned with the stationary and non stationary stokes and navier stokes problems with mixed boundary conditions involving velocity pressure rotation stress and normal derivative of velocity together relying on the relations among strain rotation normal derivative of velocity and shape of boundary surface we get variational formulations of the stokes and navier stokes problems with the mixed boundary conditions also we study existence and uniqueness of solutions to the problems

numerical analysis of a characteristic stabilized finite element method for the time dependent navier stokes equations with nonlinear slip boundary conditions abstract based on a characteristic method this work is concerned with a finite element approximation to the time dependent navier stokes equations with nonlinear slip boundary conditions since this slip boundary condition of friction type contains a subdifferential property its continuous variational problem is formulated as an inequality which can turn into an equality problem by using a powerful regularized method then a fully discrete characteristic scheme under the stabilized lower order finite element pairs is proposed for the equality problem optimal error estimates for velocity and pressure are derived under the corresponding l 2 h 1 norms finally a smooth problem test is reported to demonstrate the theoretically predicted convergence order and the expected slip phenomena and the simulation of a bifurcated blood flow model is displayed to illustrate the efficiency of the proposed method

regularity of solutions to a non homogeneous boundary value problem for general stokes systems in r n we give a simple and very complete proof of the existence of a strong h2 solution to the non homogeneous problem 1 1 under the non homogeneous boundary conditions 1 6 here we consider the half space case rn n 3 see theorem 1 2 this regularity result was previously obtained by solonnikov and scadilov in reference 33 for the classical stokes system 0 g x 0 in the 3 d homogenous case a 0 b 0 and a suitable open subset of r3

steady flow of generalized newtonian fluid with multivalued rheology and nonmonotone friction law abstract we study the stationary incompressible flow of a generalized newtonian fluid described by a nonlinear multivalued maximal monotone constitutive law and a multivalued nonmonotone frictional boundary condition we provide results on the existence and uniqueness of a solution to the variational form of the problem when the multivalued laws are of a subdifferential form we prove the existence of a solution to a variational hemivariational inequality for the flow s velocity field

the method of moving asymptotes a new method for structural optimization a new method for non linear programming in general and structural optimization in particular is presented in each step of the iterative process a strictly convex approximating subproblem is generated and solved the generation of these subproblems is controlled by so called moving asymptotes which may both stabilize and speed up the convergence of the general process

a projection method based on self adaptive rules for stokes equations with nonlinear slip boundary conditions abstract in this work we propose and study a self adaptive projection method for the numerical solution of stokes equations under slip boundary conditions we introduce the projection operator to formulate the slip constraint as a fixed point equation to solve this problem we suggest a projection method by reformulating the nonlinear slip boundary conditions into an iterative form to make the method more efficient we find a self adaptive rule that uses iterative functions to adjust the penalty parameter automatically we show the convergence of the method in function space and give its application in detail finally the numerical results are given to support our theoretical results

finite element approximation of incompressible navier stokes equations with slip boundary condition we consider mixed finite element approximations of the stationary incompressible navier stokes equations with slip boundary condition simultaneously approximating the velocity pressure and normal stress component the stability of the schemes is achieved by adding suitable consistent penalty terms corresponding to the normal stress component and to the pressure a new method of proving the stability of the discretizations allows us to obtain optimal error estimates for the velocity pressure and normal stress component in natural norms without using duality arguments and without imposing uniformity conditions on the finite element partition the schemes can easily be implemented into existing finite element codes for the navier stokes equations with standard dirichlet boundary conditions

on existence regularity and uniqueness of thermally coupled incompressible flows in a system of three dimensional pipes abstract we study an initial boundary value problem for time dependent flows of heat conducting viscous incompressible fluids in a system of three dimensional pipes on a time interval 0 t we are motivated by the bounded domain approach with do nothing boundary conditions in terms of the velocity pressure and temperature of the fluid such flows are described by a coupled parabolic system with strong nonlinearities and including the natural boundary conditions for the velocity and temperature of the fluid on the part of the boundary where the fluid is supposed to leave the channel the present analysis is devoted to the proof of the existence regularity and uniqueness of the solution for the problem described above