weak solutions for an inviscid two phase flow model in physical vacuum abstract we investigate a free boundary problem for the inviscid two phase flow model with moving physical vacuum boundary which is a degenerate and characteristic hyperbolic system of conservation laws based on a suitable transformation a special degenerate parabolic approximation and delicate uniform energy estimates we prove the existence and uniqueness of the local weak solutions this can be regarded as a generalization of the results in coutand and shkoller 2011 5 from single phase gas model to two phase gas liquid model

a 3d non newtonian fluid structure interaction model for blood flow in arteries the mathematical modelling and numerical simulation of the human cardiovascular system is playing nowadays an important role in the comprehension of the genesis and development of cardiovascular diseases in this paper we deal with two problems of 3d modelling and simulation in this field which are very often neglected in the literature on the one hand blood flow in arteries is characterized by travelling pressure waves due to the interaction of blood with the vessel wall on the other hand blood exhibits non newtonian properties like shear thinning viscoelasticity and thixotropy the present work is concerned with the coupling of a generalized newtonian fluid accounting for the shear thinning behaviour of blood with an elastic structure describing the vessel wall to capture the pulse wave due to the interaction between blood and the vessel wall we provide an energy estimate for the coupling and compare the numerical results with those obtained with an equivalent fluid structure interaction model using a newtonian fluid

zero surface tension limit of viscous surface waves we consider the free boundary problem for a layer of viscous incompressible fluid in a uniform gravitational field lying above a rigid bottom and below the atmosphere for the semi small initial data we prove the zero surface tension limit of the problem within a local time interval the unique local strong solution with surface tension is constructed as the limit of a sequence of approximate solutions to a special parabolic regularization for the small initial data we prove the global in time zero surface tension limit of the problem

observer based feedback boundary stabilization of the navier stokes equations abstract this paper aims at designing an observer based feedback law which locally stabilizes the solution to the two dimensional navier stokes equations with mixed boundary conditions we consider a finite number of controls acting on a portion of the boundary through robin boundary conditions and construct a linear luenberger observer based on the point observations of the linearized navier stokes equations the sensor location for the point observations is determined by the response of feedback functional gains we prove that the nonlinear system coupled with the observer through the feedback law is locally exponentially stable numerical experiments based on a taylor hood finite element method are presented to illustrate the design for different reynolds numbers

a multilevel technique for the approximate solution of operator lyapunov and algebraic riccati equations we consider multigrid or more appropriately multilevel techniques for the numerical solution of operator lyapunov and algebraic riccati equations the riccati equation which is quadratic plays an essential role in the solution of linear quadratic optimal control problems the linear lyapunov equation is important in the stability theory for linear systems and its solution is the primary step in the newton kleinman nk algorithm for the solution of algebraic riccati equations both equations are operator equations when the underlying linear system is infinite dimensional in this case finite dimensional discretization is required however as the level of discretization increases the convergence rate of the standard iterative techniques for solving high order matrix lyapunov and riccati equations decreases to deal with this multileveling is introduced into the iterative nk method for solving the algebraic riccati equation and smith s method for solving matrix lyapunov equations theoretical result

the immersed boundary lattice boltzmann method for solving fluid particles interaction problems a new computational method the immersed boundary lattice boltzmann method is presented this method is a combination and utilizes the most desirable features of the lattice boltzmann and the immersed boundary methods the method uses a regular eulerian grid for the flow domain and a lagrangian grid to follow particles that are contained in the flow field the rigid body conditions for the fluid and the particles are enforced by a penalty method which assumes that the particle boundary is deformable with a high stiffness constant the velocity field of the fluid and particles is solved by adding a force density term into the lattice boltzmann equation this novel method preserves the advantages of lbm in tracking a group of particles and at the same time provides an alternative and better approach to treating the solid fluid boundary conditions the method also solves the problems of fluctuation of the forces and velocities on the particles when the bounce back boundary conditions are applied this method enables one to simulate problems with particle deformation and fluid structure deformation its results are validated by comparison with results from other methods

well posedness in smooth function spaces for moving boundary 1 d compressible euler equations in physical vacuum we prove well posedness for the 3 d compressible euler equations with moving physical vacuum boundary with an equation of state given by the so called gamma gas law for gamma 1 the physical vacuum singularity requires the sound speed c to go to zero as the square root of the distance to the moving boundary and thus creates a degenerate and characteristic hyperbolic free boundary system wherein the density vanishes on the free boundary the uniform kreiss lopatinskii condition is violated and manifest derivative loss ensues nevertheless we are able to establish the existence of unique solutions to this system on a short time interval which are smooth in sobolev spaces all the way to the moving boundary and our estimates have no derivative loss with respect to initial data our proof is founded on an approximation of the euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term chosen to preserve as much of the geometric structure of the euler equations as possible we first construct solutions to this degenerate parabolic regularization using a new higher order hardy type inequality we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter solutions to the compressible euler equations are found in the limit as the artificial viscosity tends to zero our regular solutions can be viewed as degenerate viscosity solutions out methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws

a limit theorem for moments in space of the increments of brownian local time we prove a limit theorem for moments in space of the increments of brownian local time as special cases for the second and third moments previous results by chen et al ann prob 38 2010 396 438 and rosen stoch dyn 11 2011 5 48 which were later reproven by hu and nualart electron commun probab 15 2010 396 410 and rosen in seminaire de probabilites xliii 2011 95 104 springer are included furthermore a conjecture of rosen for the fourth moment is settled in comparison to the previous methods of proof we follow a fundamentally different approach by exclusively working in the space variable of the brownian local time which allows to give a unified argument for arbitrary orders the main ingredients are perkins semimartingale decomposition the kailath segall identity and an asymptotic ray knight theorem by pitman and yor

a priori estimates for the free boundary 3d compressible euler equations in physical vacuum we prove a priori estimates for the three dimensional compressible euler equations with moving physical vacuum boundary with an equation of state given by p c for 1 the vacuum condition necessitates the vanishing of the pressure and hence density on the dynamic boundary which creates a degenerate and characteristic hyperbolic free boundary system to which standard methods of symmetrizable hyperbolic equations cannot be applied

open and traction boundary conditions for the incompressible navier stokes equations we present numerical schemes for the incompressible navier stokes equations nse with open and traction boundary conditions we use pressure poisson equation ppe formulation and propose new boundary conditions for the pressure on the open or traction boundaries after replacing the divergence free constraint by this pressure poisson equation we obtain an unconstrained nse for stokes equation with open boundary condition on a simple domain we prove unconditional stability of a first order semi implicit scheme where the pressure is treated explicitly and hence is decoupled from the computation of velocity using either boundary condition the schemes for the full nse that treat both convection and pressure terms explicitly work well with various spatial discretizations including spectral collocation and c 0 finite elements moreover when reynolds number is of o 1 and when the first order semi implicit time stepping is used time step size of o 1 is allowed in benchmark computations for the full nse besides standard stability and accuracy check various numerical results including flow over a backward facing step flow past a cylinder and flow in a bifurcated tube are reported numerically we have observed that using ppe formulation enables us to use the velocity pressure pairs that do not satisfy the standard inf sup compatibility condition our results extend that of johnston and liu h johnston j g liu accurate stable and efficient navier stokes solvers based on explicit treatment of the pressure term j comp phys 199 1 2004 221 259 which deals with no slip boundary conditions only

improvements on open and traction boundary conditions for navier stokes time splitting methods we present in this paper a numerical scheme for incompressible navier stokes equations with open and traction boundary conditions in the framework of pressure correction methods a new way to enforce this type of boundary condition is proposed and provides higher pressure and velocity convergence rates in space and time than found in the present state of the art we illustrate this result by computing some numerical and physical tests in particular we establish reference solutions of a laminar flow in a geometry where a bifurcation takes place and of the unsteady flow around a square cylinder

analysis of differential equations modelling the reactive flow through a deformable system of cells a system of model equations coupling fluid flow deformation of solid structure and chemical reactions is formulated starting from processes in biological tissue the main aim of this paper is to analyse this non standard system where the elasticity modules are functionals of a concentration and the diffusion coefficients of the chemical substances are functions of their concentrations a new approach and new methods are required and adapted to these nonlinearities and the transmission conditions on the interface solid fluid strong solutions for the initial and boundary value problem are constructed under suitable regularity assumptions on the data and stability estimates of the solutions with respect to the initial and boundary values are proved these estimates imply uniqueness directly the approach of the paper can be used in more general problems modeling reactive flow and transport and its interaction with elastic cell structures in a forthcoming paper the approach of this paper is used for getting the upscaled system modeling reactive flow through biological tissue on the macroscopic scale starting from a system on the cell level

a coupled momentum method for modeling blood flow in three dimensional deformable arteries abstract blood velocity and pressure fields in large arteries are greatly influenced by the deformability of the vessel moreover wave propagation phenomena in the cardiovascular system can only be described considering wall deformability since blood is usually described as an incompressible fluid however computational methods for simulating blood flow in three dimensional models of arteries have either considered a rigid wall assumption for the vessel or significantly simplified or reduced geometries computing blood flow in deformable domains using standard techniques like the ale method remains a formidable problem for large realistic anatomic and physiologic models of the cardiovascular system we have developed a new method to simulate blood flow in three dimensional deformable models of arteries the method couples the equations of the deformation of the vessel wall at the variational level as a boundary condition for the fluid domain we consider a strong coupling of the degrees of freedom of the fluid and the solid domains and a linear membrane model enhanced with transverse shear for the vessel wall the effect of the vessel wall boundary is therefore added in a monolithic way to the fluid equations resulting in a remarkably robust scheme we present here the mathematical formulation of the method and discuss issues related to the fluid solid coupling membrane formulation time integration method and boundary and initial conditions implementation is discussed and results with simple geometries as well as large subject specific models are presented

a clt for the l2 modulus of continuity of brownian local time let l x t x t r 1 x r 1 denote the local time of brownian motion and l x t 2 dx let n 0 1 be independent of t for each fixed t l x h t l t x 2 dx 4ht h 3 2 l 64 3 1 2 t as h 0 equivalently l x 1 t l t x 2 dx 4t h 3 2 l 64 3 1 2 1 as t

rational rates of uniform decay for strong solutions to a fluid structure pde system abstract in this work we investigate the uniform stability properties of solutions to a well established partial differential equation pde model for a fluid structure interaction the pde system under consideration comprises a stokes flow which evolves within a three dimensional cavity moreover a kirchhoff plate equation is invoked to describe the displacements along a fixed portion say of the cavity wall contact between the respective fluid and structure dynamics occurs on the boundary interface the main result in the paper is as follows the solutions to the composite pde system corresponding to smooth initial data decay at the rate of o 1 t our method of proof hinges upon the appropriate invocation of a relatively recent resolvent criterion for polynomial decays of c 0 semigroups while the characterization provided by said criterion originates in the context of operator theory and functional analysis the work entailed here is wholly within the realm of pde

heat wave interaction in 2 3 dimensions optimal rational decay rate abstract in this paper we consider a simplified version of a fluid structure pde model which has been of longstanding interest within the mathematical and biological sciences in it a n dimensional heat equation replaces the original stokes system so as to ultimately have a vector valued heat equation and vector valued wave equation compose the coupled pde system under study the coupling between the two disparate pde components occurs across a boundary interface as such the entire pde dynamics manifests features of both hyperbolicity and parabolicity for this heat structure system our main result of uniform stability is as follows given smooth initial data i e data in the domain of the underlying semigroup generator of the coupled pde system the corresponding solutions decay at the rate of o t 1 this establishes the long conjectured optimal rate the problem of obtaining sharp rational decay rates for the heat wave pde under present consideration has been a much considered problem with the modus operandi of earlier efforts taking place within the time domain by contrast we adopt here a frequency domain approach which is based upon a recent resolvent criterion and which was initiated in our prior effort wherein we obtained the rate of decay o 1 t the present optimal improvement o t 1 is achieved by employing an additional tool in our analysis a microlocal analysis argument to estimate a critical term involving two problematic boundary traces

asymptotic laws of planar brownian motion on presente des resultats recents sur la distribution asymptotique des nombres de passages et d enroulements

an example in linear quadratic optimal control we construct a simple example of a quadratic optimal control problem for an infinite dimensional linear system based on a shift semigroup this system has an unbounded control operator the cost is quadratic in the input and the state and the weighting operators are bounded despite its extreme simplicity this example has all the unexpected features discovered recently by o staffans and also by m weiss and g weiss more precisely in the formula linking the optimal feedback operator to the optimal cost operator as well as in the riccati equation the weighting operator of the input has to be replaced by another operator which can be derived from the spectral factorization of the popov function

stabilization of explicit coupling in fluid structure interaction involving fluid incompressibility abstract in this work we propose a stabilized explicit coupling scheme for fluid structure interaction problems involving a viscous incompressible fluid the coupled discrete formulation is based on nitsche s method with a time penalty term giving l 2 control on the fluid pressure variations at the interface for a linear model problem we prove that the scheme is stable in the energy norm irrespectively of the so called added mass effect namely the fluid structure density ratio and the geometry of the domain numerical experiments in the linear and non linear case show that optimal time accuracy can be obtained by performing one defect correction iteration

optimal order multigrid methods for solving exterior boundary value problems the coupling of boundary elements and finite elements combines the advantage of boundary elements for treating domains extended to infinity and that of finite elements in treating the nonhomogeneity of equations and the complexity of domains in the case of the laplacian by taking a circle or a sphere as the artificial coupling boundary it is shown that the corresponding boundary integral equation can be solved without any cost and the coupled system is reduced to a simple finite element system two multigrid methods are proposed to solve this finite element linear system both methods are of optimal order and can be used to solve such finite element equations as efficiently as to solve those arising from interior boundary value problems numerical experiments are included to show the efficiency and advantages of the methods an apparent significance of the methods is that the boundary elements appear neither in the discretization nor in the coding

unsteady interaction of a viscous fluid with an elastic shell modeled by full von karman equations abstract we study well posedness and asymptotic dynamics of a coupled system consisting of linearized 3d navier stokes equations in a bounded domain and a classical nonlinear full von karman shallow shell equation that accounts for both transversal and lateral displacements on a flexible part of the boundary we also take into account rotational inertia of filaments of the shell our main result shows that the problem generates a semiflow in an appropriate phase space the regularity provided by viscous dissipation in the fluid allows us to consider simultaneously both cases of presence inertia in the lateral displacements and its absence our second result states the existence of a compact global attractor for this semiflow in the case of presence of rotational damping in the transversal component and a particular structure of external forces

interface feedback control stabilization of a nonlinear fluid structure interaction we consider a model of fluid structure interaction in a bounded domain rn n 2 where is comprised of two open adjacent sub domains occupied respectively by the solid and the fluid this leads to a study of the navier stokes equation coupled on the boundary with the dynamic system of elasticity we shall consider models where the elastic body exhibits small but rapid oscillations these are established models arising in engineering applications when the structure is immersed in a viscous flow of liquid questions related to the stability of finite energy solutions are of paramount interest it was shown in lasiecka and lu 2011 14 that all data of finite energy produce solutions whose energy converges strongly to zero the cited result holds under partial flatness geometric condition whose role is to control the effects of the pressure in the ns equation related conditions has been used in avalos and triggiani 2008 23 for the analysis of the linear model the goal of the present work is to study uniform stability of all finite energy solutions corresponding to nonlinear interaction this particular question of interest in its own rights is also a necessary preliminary step for the analysis of optimal control strategies arising in infinite horizon control problems associated with the structure it is shown in this paper that a stress type feedback control applied on the interface of the structure produces solutions whose energy is exponentially stable

an eulerian description of fluids containing visco elastic particles equations governing the flow of fluid containing visco hyperelastic particles are developed in an eulerian framework the novel feature introduced here is to write an evolution equation for the strain it is envisioned that this will simplify numerical codes which typically compute the strain on lagrangian meshes moving through eulerian meshes existence results for the flow of linear visco hyperelastic particles in a newtonian fluid are established using a galerkin scheme

energy methods for quasilinear hyperbolic initial boundary value problems applications to elastodynamics the aim of this paper is to show how energy methods can be used to establish local in time existence of smooth solutions to certain initial boundary value problems for systems of quasilinear hyperbolic partial differential equations our particular objective is to treat the equations of nonlinear elastodynamics

weak solvability of interior transmission problems via mixed finite elements and dirichlet to neumann mappings we study the weak solvability of an interior linear nonlinear transmission problem arising in steady heat transfer and potential theory for the variational formulation we use a dirichlet to neumann mapping on the interface which is obtained from the application of the boundary integral method to the linear domain and we utilize a mixed finite element method in the nonlinear region existence and uniqueness of solution for the continuous formulation are provided and general approximation results for a fully discrete galerkin method are derived in particular a compatibility condition between the mesh sizes involved is deduced in order to conclude the solvability and stability of this galerkin scheme

well posedness in smooth function spaces for the moving boundary three dimensional compressible euler equations in physical vacuum we prove well posedness for the three dimensional compressible euler equations with moving physical vacuum boundary with an equation of state given by p c for 1 the physical vacuum singularity requires the sound speed c to go to zero as the square root of the distance to the moving boundary and thus creates a degenerate and characteristic hyperbolic free boundary system wherein the density vanishes on the free boundary the uniform kreiss lopatinskii condition is violated and manifest derivative loss ensues nevertheless we are able to establish the existence of unique solutions to this system on a short time interval which are smooth in sobolev spaces all the way to the moving boundary and our estimates have no derivative loss with respect to initial data our proof is founded on an approximation of the euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term chosen to preserve as much of the geometric structure of the euler equations as possible we first construct solutions to this degenerate parabolic regularization using a higher order version of hardy s inequality we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter solutions to the compressible euler equations are found in the limit as the artificial viscosity tends to zero our regular solutions can be viewed as degenerate viscosity solutions our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws

a three dimensional computational method for blood flow in the heart 1 immersed elastic fibers in a viscous incompressible fluid abstract this paper describes the numerical solution of the 3 dimensional equations of motion of a viscous incompressible fluid that contains an immersed system of elastic fibers implementation details such as vectorization and the efficient use of external memory are discussed the method is applied to the damped vibrations of a fiber wound toroidal tube and empirical evidence of convergence is presented

on the coupling of boundary integral and mixed finite element methods we present a numerical method for solving an exterior dirichlet problem in the plane the technique consists in coupling boundary integral and mixed finite element methods an artificial boundary is introduced separating an interior region from an exterior one from an integral representation of the solution in the exterior domain we deduce two integral equations which relate the solution and its normal derivative over the artificial boundary these integral equations are incorporated into the so called mixed formulation of the problem in the interior region and a finite element method is used to approximate the resulting variational problem

an effective fluid structure interaction formulation for vascular dynamics by generalized robin conditions in this work we focus on the modeling and numerical simulation of the fluid structure interaction mechanism in vascular dynamics we first propose a simple membrane model to describe the deformation of the arterial wall which is derived from the koiter shell equations and is applicable to an arbitrary geometry secondly we consider a reformulation of the fluid structure problem in which the newly derived membrane model thanks to its simplicity is embedded into the fluid equations and will appear as a generalized robin boundary condition the original problem is then reduced to the solution of subsequent fluid equations defined on a moving domain and may be achieved with a fluid solver only we also derive a stability estimate for the resulting numerical scheme finally we propose new outflow absorbing boundary conditions which are easy to implement and allow us to reduce significantly the spurious pressure wave reflections that typically appear in artificially truncated computational domains we present several numerical results showing the effectiveness of the proposed approaches

goal oriented error estimation and adaptivity for fluid structure interaction using exact linearized adjoints we develop duality based a posteriori error estimates for functional outputs of solutions of steady fluid structure interaction problems the crucial complication in obtaining these estimates pertains to the derivation of the coupled dual exact linearized adjoint problem owing to the free boundary character of fluid structure interaction we present two approaches to derive the dual problem in the domain map linearization approach the fluid subproblem is first transformed to a fixed reference domain after which one essentially linearizes with respect to the domain transformation map in the shape linearization approach fluid unknowns are fixed in the current configuration and a very weak formulation of the fluid subproblem is then linearized using shape derivative techniques we show that the dual problems correspond to coupled fluid structure problems with nonstandard coupling conditions furthermore we present numerical experiments that demonstrate the consistency of the dual based error estimates and their usefulness in goal oriented adaptive mesh refinement

riccati equation arising in a boundary control problem with distributed parameters global existence is proved for the solution of a riccati differential equation connected with the synthesis of a boundary control problem governed by parabolic partial differential equations

a fully eulerian formulation for fluid structure interaction problems in this work we present a fully eulerian framework for fluid structure interaction fsi problems coupling the incompressible navier stokes equations with a hyperelastic solid the fully eulerian framework is a monolithic implicit variational formulation for the coupled problem in contrast to the well established arbitrary lagrangian eulerian ale coordinates the fully eulerian framework formulates both subproblems fluid and solid in eulerian coordinates this concept circumvents various difficulties connected to ale coordinates since no artificial domain mapping is used the formulation is an interface capturing method and uses an extension of the solid s deformation the initial point set to detect the interface location by construction very large deformation as well as topology changes like contact of the solid to the domain boundary or other solid parts are possible

steady flow of a navier stokes liquid past an elastic body we perform a mathematical analysis of the steady flow of a viscous liquid mathcal l past a three dimensional elastic body mathcal b we assume that mathcal l fills the whole space exterior to mathcal b and that its motion is governed by the navier stokes equations corresponding to non zero velocity at infinity v as for mathcal b we suppose that it is a st venant kirchhoff material held in equilibrium either by keeping an interior portion of it attached to a rigid body or by means of appropriate control body force and surface traction we treat the problem as a coupled steady state fluid structure problem with the surface of mathcal b as a free boundary our main goal is to show existence and uniqueness for the coupled system liquid body for sufficiently small v this goal is reached by a fixed point approach based upon a suitable reformulation of the navier stokes equation in the reference configuration along with appropriate a priori estimates of solutions to the corresponding oseen linearization and to the elasticity equations

fully eulerian fluid structure interaction for time dependent problems abstract a fully eulerian framework formulation for solving time dependent fluid structure interaction problems is proposed in this work although some preliminary work on this approach exists for stationary configurations it remains to validate nonstationary processes the formulation is stated in an implicit monolithic frame of reference a finite difference scheme is used for temporal discretization whereas the spatial discretization is based on a galerkin finite element scheme the nonlinear problem is solved with a newton method and with analytical evaluation of the jacobian matrix in contrast to interface tracking methods for example the arbitrary lagrangian eulerian approach the interface must be captured which is similar to the level set method consequently the interface is allowed to intersect mesh cells which is a crucial difficulty of this method where appropriate treatment must be suggested the proposed formulation is substantiated by three numerical tests in which the performance of fully eulerian fluid structure interaction is demonstrated

on the convergence of fixed point iterations for the moving geometry in a fluid structure interaction problem abstract in this paper a fluid structure interaction problem for incompressible newtonian fluids is studied we prove the convergence of an iterative process with respect to the computational domain geometry moving according to a viscoelastic deformation in our previous joint works on numerical approximation of similar problems we refer to this approach as the global iterative method 1 2 this iterative approach can be understood as a linearization of the so called geometric nonlinearity of the underlying model the proof of the convergence is based on the banach fixed point theorem where the contractivity of the corresponding mapping is shown using the continuous dependence of the weak solution on the given domain deformation this estimate is obtained by remapping the problem onto a fixed domain and using appropriate divergence free test functions involving the difference of two solutions

approximation of the algebraic riccati equation in the hilbert space of hilbert schmidt operators this paper deals with the problem of approximating the infinite dimensional algebraic riccati equation considered as an abstract equation in the hilbert space of hilbert schmidt operators two kin

coupling of finite and boundary element methods for an elastoplastic interface problem a class of transmission problems is considered in which a nonlinear variational problem in one domain is coupled with a linear elliptic problem in a second domain a typical example is a problem from three dimensional elasticity theory where an elastoplastic material is embedded into a linear elastic material the nonlinear problem is given in variational form with a strictly convex functional the linear elliptic problem is described by boundary integral equations on the coupling boundary the typical saddle point structure of such problems is analyzed galerkin approximations are studied which consist of a finite element approximation in the first domain coupled with a boundary element method on the coupling boundary the convergence of the galerkin approximation is based on the saddle point structure which is shown to hold for the exact as well as the discretized problems

well posedness of the muskat problem with h2 initial data abstract we study the dynamics of the interface between two incompressible fluids in a two dimensional porous medium whose flow is modeled by the muskat equations for the two phase muskat problem we establish global well posedness and decay to equilibrium for small h 2 perturbations of the rest state for the one phase muskat problem we prove local well posedness for h 2 initial data of arbitrary size finally we show that solutions to the muskat equations instantaneously become infinitely smooth

lattice boltzmann method for simulating the viscous flow in large distensible blood vessels a lattice boltzmann method for simulating the viscous flow in large distensible blood vessels is presented by introducing a boundary condition for elastic and moving boundaries the mass conservation for the boundary condition is tested in detail the viscous flow in elastic vessels is simulated with a pressure radius relationship similar to that of the pulmonary blood vessels the numerical results for steady flow agree with the analytical prediction to very high accuracy and the simulation results for pulsatile flow are comparable with those of the aortic flows observed experimentally the model is expected to find many applications for studying blood flows in large distensible arteries especially in those suffering from atherosclerosis stenosis aneurysm etc

on the coupled bem and fem for a nonlinear exterior dirichlet problem in r2 in this paper we apply the coupling of boundary integral and finite element methods to solve a nonlinear exterior dirichlet problem in the plane specifically the boundary value problem consists of a nonlinear second order elliptic equation in divergence form in a bounded inner region and the laplace equation in the corresponding unbounded exterior region in addition to appropriate boundary and transmission conditions the main feature of the coupling method utilized here consists in the reduction of the nonlinear exterior boundary value problem to an equivalent monotone operator equation we provide sufficient conditions for the coefficients of the nonlinear elliptic equation from which existence uniqueness and approximation results are established then we consider the case where the corresponding operator is strongly monotone and lipschitz continuous and derive asymptotic error estimates for a boundary finite element solution we prove the unique solvability of the discrete operator equations and based on a strang type abstract error estimate we show the strong convergence of the approximated solutions moreover under additional regularity assumptions on the solution of the continous operator equation the asymptotic rate of convergenceo h is obtained

a note on optimal regularity and regularizing effects of point mass coupling for a heat wave system abstract we consider a coupled 1d heat wave system which serves as a simplified fluid structure interaction problem the system is coupled in two different ways the first when the interface does not have mass and the second when the interface does have mass we prove an optimal regularity result in sobolev spaces for both cases the main idea behind the proof is to reduce the coupled problem to a single nonlocal equation on the interface by using neummann to diriclet operator furthermore we show that point mass coupling regularizes the problem and quantify this regularization in the sense of sobolev spaces

motion of an elastic solid inside an incompressible viscous fluid the motion of an elastic solid inside an incompressible viscous fluid is ubiquitous in nature mathematically such motion is described by a pde system that couples the parabolic and hyperbolic phases the latter inducing a loss of regularity which has left the basic question of existence open until now in this paper we prove the existence and uniqueness of such motions locally in time when the elastic solid is the linear kirchhoff elastic material the solution is found using a topological fixed point theorem that requires the analysis of a linear problem consisting of the coupling between the time dependent navier stokes equations set in lagrangian variables and the linear equations of elastodynamics for which we prove the existence of a unique weak solution we then establish the regularity of the weak solution this regularity is obtained in function spaces that scale in a hyperbolic fashion in both the fluid and solid phases our functional framework is optimal and provides the a priori estimates necessary for us to employ our fixed point procedure

wellposedness spectral analysis and asymptotic stability of a multilayered heat wave wave system abstract in this work we consider a multilayered heat wave system where a 3 d heat equation is coupled with a 3 d wave equation via a 2 d interface whose dynamics is described by a 2 d wave equation this system can be viewed as a simplification of a certain fluid structure interaction fsi pde model where the structure is of composite type namely it consists of a thin layer and a thick layer we associate the wellposedness of the system with a strongly continuous semigroup and establish its asymptotic decay our first result is semigroup well posedness for the fsi pde dynamics utilizing here a lumer phillips approach we show that the fluid structure system generates a c 0 semigroup on a chosen finite energy space of data as our second result we prove that the solution to the fsi dynamics generated by the c 0 semigroup tends asymptotically to the zero state for all initial data that is the semigroup of the fsi system is strongly stable for this stability work we analyze the spectrum of the generator a and show that the spectrum of a does not intersect the imaginary axis

riccati theory and singular estimates for a bolza control problem arising in linearized fluid structure interaction abstract we consider a bolza boundary control problem involving a fluid structure interaction model the aim of this paper is to develop an optimal feedback control synthesis based on riccati theory the model considered consists of a linearized navier stokes equation coupled on the interface with a dynamic wave equation the model incorporates convective terms resulting from the linearization of the navier stokes equation around equilibrium the existence of the optimal control and its feedback characterization via a solution to a riccati equation is established the main mathematical difficulty of the problem is caused by unbounded action of control forces which in turn result in riccati equations with unbounded coefficients and in singular behavior of the gain operator this class of problems has been recently studied via the so called singular estimate control systems secs theory which is based on the validity of the singular estimate se g avalos differential riccati equations for the active control of a problem in structural acoustic j optim theory appl 91 1996 695 728 i lasiecka mathematical control theory of coupled pde s in nsf cmbs lecture notes siam 2002 with unbounded controls i lasiecka a tuffaha riccati equations for the bolza problem arising in boundary point control problems governed by c0 semigroups satisfying a singular estimate j optim theory appl 136 2008 229 246 it is shown that the fluid structure interaction does satisfy the singular estimate se condition this is accomplished by showing that the maximal abstract parabolic regularity is transported via hidden hyperbolic regularity of the boundary traces on the interface thus the established singular estimate allows for the application of recently developed general theory which in turn implies well posedness of the feedback synthesis and of the associated riccati equation moreover the singularities in the optimal control and in the feedback operator at the terminal time are quantitatively described

fluid structure algorithms based on steklov poincare operators in this paper we review some classical algorithms for fluid structure interaction problems and we propose an alternative viewpoint mutuated from the domain decomposition theory this approach yields preconditioned richardson iterations on the steklov poincare nonlinear equation at the fluid structure interface

stable loosely coupled type algorithm for fluid structure interaction in blood flow we introduce a novel loosely coupled type algorithm for fluid structure interaction between blood flow and thin vascular walls this algorithm successfully deals with the difficulties associated with the added mass effect which is known to be the cause of numerical instabilities in fluid structure interaction problems involving fluid and structure of comparable densities our algorithm is based on a time discretization via operator splitting which is applied in a novel way to separate the fluid sub problem from the structure elastodynamics sub problem in contrast with traditional loosely coupled schemes no iterations are necessary between the fluid and structure sub problems this is due to the fact that our novel splitting strategy uses the added mass effect to stabilize rather than to destabilize the numerical algorithm this stabilizing effect is obtained by employing the kinematic lateral boundary condition to establish a tight link between the velocities of the fluid and of the structure in each sub problem the stability of the scheme is discussed on a simplified benchmark problem and we use energy arguments to show that the proposed scheme is unconditionally stable due to the crucial role played by the kinematic lateral boundary condition the proposed algorithm is named the kinematically coupled scheme

modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow fluid structure interaction describing wave propagation in arteries driven by the pulsatile blood flow is a complex problem whenever possible simplified models are called for one dimensional models are typically used in arterial sections that can be approximated by the cylindrical geometry allowing axially symmetric flows although a good first approximation to the underlying problem the one dimensional model suffers from several drawbacks the model is not closed an ad hoc velocity profile needs to be prescribed to obtain a closed system and the model equations are quasi linear hyperbolic oversimplifying the viscous fluid dissipation typically producing shock wave solutions not observed in healthy humans in this manuscript we derived a simple closed reduced model that accounts for the viscous fluid dissipation to the leading order the resulting fluid structure interaction system is of hyperbolic parabolic type arterial walls were modeled by a novel linearly viscoelastic cylindrical koiter shell model and the flow of blood by the incompressible viscous navier stokes equations kelvin voigt type viscoelasticity was used to capture the hysteresis behavior observed in the measurements of the arterial stress strain response using the a priori estimates obtained from an energy inequality together with the asymptotic analysis and ideas from homogenization theory for porous media flows we derived an effective model which is an 2 approximation to the three dimensional axially symmetric problem whereis the aspect ratio of the cylindrical arterial section our model shows two interesting features of the underlying problem bending rigidity often times neglected in the arterial wall models plays a nonnegligible role in the 2 approximation of the original problem and the viscous fluid dissipation imparts long term viscoelastic memory effects on the motion of the arterial walls this does not to the leading order influence the hysteresis behavior of arterial walls the resulting model although two dimensional is in the form that allows the use of one dimensional finite element method techniques producing fast numerical solutions we devised a version of the douglas rachford time splitting algorithm to solve the underlying hyperbolic parabolic problem the results of the numerical simulations were compared with the experimental flow measurements performed at the texas heart institute and with the data corresponding to the hysteresis of the human femoral artery and the canine abdominal aorta excellent agreement was observed

finite volume and weno scheme in one dimensional vascular system modelling a finite volume and weno model is developed validated and applied to a one dimensional numerical investigation of the vascular system suitable balance laws are spatially integrated using a finite volume weno approach time integration is performed using a five steps fourth order accurate runge kutta strong stability preserving scheme a detailed investigation of different sets of reflecting and non reflecting boundary conditions is performed to achieve a correct representation of limited portions of the vascular system moreover special attention is devoted to the numerical discretisation of the source term finally as an application of the whole model the effect of an abdominal aorta aneurysm on the blood flow is evaluated

long time behavior of a coupled heat wave system arising in fluid structure interaction this paper is devoted to the analysis of the long time behavior of a coupled wave heat system in which a wave and a heat equation evolve in two bounded domains with natural transmission conditions at a common interface these conditions couple in particular the heat unknown with the velocity of the wave solution this model may be viewed as a simplified version of linearized models which arise in fluid structure interaction first we show the strong asymptotic stability of solutions to this system then based on the construction of ray like solutions by means of geometric optics expansions and a careful analysis of the transfer of energy at the interface we show the lack of uniform decay in general domains further we obtain a polynomial decay result for smooth solutions of the system under a suitable geometric assumption which guarantees that the heat domain envelopes the wave domain finally in the absence of geometric conditions we show a logarithmic decay result for the same system but with simplified transmission conditions at the interface we also analyze the difficulty there is to extend this result to the more natural transmission conditions

feedback stabilization of a thermal fluid system with mixed boundary control we consider the problem of local exponential stabilization of the nonlinear boussinesq equations with control acting on portion of the boundary in particular given a steady state solution on an bounded and connected domain r 2 we show that a finite number of controls acting on a part of the boundary through neumann robin boundary conditions is sufficient to stabilize the full nonlinear equations in a neighborhood of this steady state solution dirichlet boundary conditions are imposed on the rest of the boundary we prove that a stabilizing feedback control law can be obtained by solving a linear quadratic regulator lqr problem for the linearized boussinesq equations numerical result are provided for a 2d problem to illustrate the ideas

a primal mixed formulation for exterior transmission problems in bf r 2 we combine a primal mixed finite element approach with a dirichlet to neumann mapping arising from the boundary integral equation method to study the weak solvability and galerkin approximations of a class of linear exterior transmission problems in potential theory our results are mainly based on the babuska brezzi theory for variational problems with constraints we establish the uniqueness of solution for the continuous and discrete formulations and show that finite element subspac es of lagrange type satisfy the discrete compatibility conditions in addition we provide the error analysis including polygonal approximations of the domain and prove strong convergence of the galerkin solutions moreover under additional regularity assumptions on the solution of the continuous formulation we obtain the asymptotic rate of convergence o h

strong solutions to a nonlinear fluid structure interaction system abstract in this paper we prove the existence of local in time smooth solutions to the nonlinear fluid structure interaction model first introduced in j l lions quelques methodes de resolution des problemes aux limites non lineaires dunod 1969 and considered in v barbu z grujic i lasiecka a tuffaha existence of the energy level weak solutions for a nonlinear fluid structure interaction model in fluids and waves in contemp math vol 440 amer math soc providence ri 2007 pp 55 82 v barbu z grujic i lasiecka a tuffaha smoothness of weak solutions to a nonlinear fluid structure interaction model indiana univ math j 57 3 2008 1173 1207 in particular the strong solutions here are obtained given initial datum for the navier stokes equation in the space h 1 and initial data for the wave equation w 0 and w 1 in the spaces h 2 e and h 1 e respectively

simulations of the whirling instability by the immersed boundary method when an elastic filament spins in a viscous incompressible fluid it may undergo a whirling instability as studied asymptotically by wolgemuth powers and goldstein phys rev lett 84 2000 pp 16 23 we use the immersed boundary ib method to study the interaction between the elastic filament and the surrounding viscous fluid as governed by the incompressible navier stokes equations this allows the study of the whirling motion when the shape of the filament is very different from the unperturbed straight state

well posedness of the free boundary problem in incompressible elastodynamics abstract in this paper we prove the local well posedness of the free boundary problem in incompressible elastodynamics under a natural stability condition which ensures that the evolution equation describing the free boundary is strictly hyperbolic our result gives a rigorous confirmation that the elasticity has a stabilizing effect on the rayleigh taylor instability

existence of a weak solution to a fluid elastic structure interaction problem with the navier slip boundary condition abstract we study a nonlinear moving boundary fluid structure interaction fsi problem between an incompressible viscous newtonian fluid modeled by the 2d navier stokes equations and an elastic structure modeled by the shell or plate equations the fluid and structure are coupled via the navier slip boundary condition and balance of contact forces at the fluid structure interface the slip boundary condition might be more realistic than the classical no slip boundary condition in situations e g when the structure is rough and in modeling fsi dynamics near or at a contact cardiovascular tissue and cell seeded tissue constructs which consist of grooves in tissue scaffolds that are lined with cells are examples of rough elastic interfaces interacting with an incompressible viscous fluid the problem of heart valve closure is an example of a fsi problem with a contact involving elastic interfaces we prove the existence of a weak solution to this class of problems by designing a constructive proof based on the time discretization via operator splitting this is the first existence result for fluid structure interaction problems involving elastic structures satisfying the navier slip boundary condition

solving the linear quadratic optimal control problem for infinite dimensional systems abstract calculation of the solutions to linear quadratic optimal control problems for infinite dimensional systems is considered the sequence of solutions to a sequence of approximating finite dimensional problems converges to the optimal control for the infinite dimensional system if certain assumptions such as uniform stabilizability are satisfied we use this result to calculate controllers for the infinite dimensional system that are arbitrarily close to optimal a one dimensional heat equation and a problem of acoustic noise control are used to illustrate the algorithm the numerical results are discussed

well posedness of the free surface incompressible euler equations with or without surface tension we provide a new method for treating free boundary problems in perfect fluids and prove local in time well posedness in sobolev spaces for the free surface incompressible 3d euler equations with or without surface tension for arbitrary initial data and without any irrotationality assumption on the fluid this is a free boundary problem for the motion of an incompressible perfect liquid in vacuum wherein the motion of the fluid interacts with the motion of the free surface at highest order

a projection semi implicit scheme for the coupling of an elastic structure with an incompressible fluid we address the numerical simulation of fluid structure systems involving an incompressible viscous fluid this issue is particularly difficult to face when the fluid added mass acting on the structure is strong as it happens in hemodynamics for example indeed several works have shown that in such situations implicit coupling seems to be necessary in order to avoid numerical instabilities although significant improvements have been achieved during the last years solving implicit coupling often exhibits a prohibitive computational cost in this work we introduce a semi implicit coupling scheme which remains stable for a reasonable range of the discretization parameters the first idea consists in treating implicitly the added mass effect whereas the other contributions geometrical non linearities viscous and convective effects are treated explicitly the second idea relies on the fact that this kind of explicit implicit splitting can be naturally performed using a chorin temam projection scheme in the fluid we prove conditional stability of the scheme for a fully discrete formulation several numerical experiments point out the efficiency of the present scheme compared to several implicit approaches

the interaction between quasilinear elastodynamics and the navier stokes equations the interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations coupled to one another along the moving material interface through the continuity of the velocity and traction vectors we prove the existence and uniqueness locally in time of strong solutions in sobolev spaces for quasilinear elastodynamics coupled to the incompressible navier stokes equations unlike our approach in 5 for the case of linear elastodynamics we cannot employ a fixed point argument on the nonlinear system itself and are instead forced to regularize it by a particular parabolic artificial viscosity term we proceed to show that with this specific regularization we obtain a time interval of existence which is independent of the artificial viscosity together with a priori estimates we identify the global solution in both phases as well as the interface motion as a weak limit in strong norms of our sequence of regularized problems

lp moduli of continuity of gaussian processes and local times of symmetric levy processes let x x t t e r be a real valued symmetric levy process with continuous local times l x t t x e r x r and characteristic function ee i x t e t let 2 0 x y 4 0 sin 2 x y 2 d if 2 0 h is concave and satisfies some additional very weak regularity conditions then for any p 1 and all t r limh 0 b a l x h t l x t 0 h p dx 2 p 2 e p b a l x t p 2 dx for all a b in the extended real line almost surely and also in l m m 1 here is a normal random variable with mean zero and variance one this result is obtained via the eisenbaum isomorphism theorem and depends on the related result for gaussian processes with stationary increments g x x e r 1 for which e g x g y 2 2 0 x y limh 0 b a g x h g x 0 h p dx e p b a for all a b e r 1 almost surely

coupling of mixed finite elements and boundary elements for a hyperelastic interface problem we apply the coupling of mixed finite elements and boundary integral methods using two integral equations on the coupling boundary to study the weak solvability of a nonlinear elliptic problem arising in elastostatics our procedure is based on both the usual brezzi s theory for linear constrained variational problems and some fundamental tools from nonlinear functional analysis we derive existence and uniqueness of solution for the continuous variational formulations and provide general approximation results for the corresponding galerkin schemes although we consider bounded domains the same analysis applies for the case in which the boundary element region is unbounded

variational formulations of transmission problems via fem bem and dtn mappings the purpose of this work is to review some of the main results on the weak solvability and galerkin approximations of interior and exterior transmission problems arising in potential theory and elastostatics both linear and nonlinear model problems are included in our presentation we consider variational formulations based on the combination of mixed finite elements with either boundary integral equations or dirichlet to neumann dtn mappings we provide existence and uniqueness of solution for the continuous formulations and give general approximation estimates for the associated galerkin schemes some open questions are also mentioned

a survey of one dimensional random polymers in the last decade there has been an enormous progress in the mathematical understanding of one dimensional polymer measures which are path measures that suppress self intersections we are currently in the situation that many interesting questions have either been answered or that essential new ideas are needed in this survey paper we discuss the most relevant results open questions and heuristics

minimizing drag in a moving boundary fluid elasticity interaction abstract our goal is to minimize the fluid vorticity in the case of an elastic body moving and deforming inside the fluid using a distributed control this translates into analyzing an optimal control problem subject to a moving boundary fluid structure interaction fsi the fsi is described by the coupling of navier stokes and wave equations the control is inherently a nonlinear control acting as feedback on the moving frame its action depends on the flow map of the domain which is itself defined through the dynamics of the problem a key ingredient in the optimal control problem is represented by the long time behavior of the forced dynamics which was an open problem in the field our main results include existence of solutions for all times with small distributed sources and small initial data as well as existence of optimal control for the problem of minimization of drag in the fluid

well posedness for the classical stefan problem and the zero surface tension limit we develop a framework for a unified treatment of well posedness for the stefan problem with or without surface tension in the absence of surface tension we establish well posedness in sobolev spaces for the classical stefan problem we introduce a new velocity variable which extends the velocity of the moving free boundary into the interior domain the equation satisfied by this velocity is used for the analysis in place of the heat equation satisfied by the temperature solutions to the classical stefan problem are then constructed as the limit of solutions to a carefully chosen sequence of approximations to the velocity equation in which the moving free boundary is regularized and the boundary condition is modified in a such a way as to preserve the basic nonlinear structure of the original problem with our methodology we simultaneously find the required stability condition for well posedness and obtain new estimates for the regularity of the moving free boundary finally we prove that solutions of the stefan problem with positive surface tension sigma converge to solutions of the classical stefan problem as sigma to 0

existence of a solution to a fluid multi layered structure interaction problem abstract we study a nonlinear unsteady moving boundary fluid structure interaction fsi problem in which the structure is composed of two layers a thick layer and a thin layer which serves as a fluid structure interface with mass the fluid flow which is driven by the time dependent dynamic pressure data is modeled by the navier stokes equations for an incompressible viscous fluid defined on a 2d cylinder the elastodynamics of the cylinder wall is governed by the 1d linear wave equation modeling the thin structural layer and by the 2d equations of linear elasticity modeling the thick structural layer we prove existence of a weak solution to this nonlinear fsi problem as long as the cylinder radius is greater than zero the spaces of weak solutions presented in this manuscript reveal a striking new feature the presence of a thin fluid structure interface with mass regularizes solutions of the coupled problem

on well posedness for a free boundary fluid structure model we address a fluid structure interaction model describing the motion of an elastic body immersed in an incompressible fluid we establish a priori estimates for the local existence of solutions for a class of initial data which also guarantees uniqueness

an initial and boundary value problem modeling of fish like swimming in this paper we consider an initial and boundary value problem that models the self propelled motion of solids in a bidimensional viscous incompressible fluid the self propelling mechanism consisting of appropriate deformations of the solids is a simplified model of the propulsion mechanism of fish like swimmers the governing equations consist of the navier stokes equations for the fluid coupled to newton s laws for the solids since we consider the case in which the fluid solid system fills a bounded domain we have to tackle a free boundary value problem the main theoretical result in the paper asserts the global existence and uniqueness up to possible contacts of strong solutions of this problem the second novel result of this work is the provision of a numerical method for the fluid solid system this method provides a simulation of the simultaneous displacement of several swimmers and is tested on several examples

existence of weak solutions for the motion of rigid bodies in a viscous fluid we study the evolution of a finite number of rigid bodies within a viscous incompressible fluid in a bounded domain of r d d 2 or 3 with dirichlet boundary conditions by introducing an appropriate weak formulation for the complete problem we prove existence of solutions for initial velocities in h 1 0 omega in the absence of collisions solutions exist for all time in dimension 2 whereas in dimension 3 the lifespan of solutions is infinite only for small enough data

the uncoupling of boundary integral and finite element methods for nonlinear boundary value problems abstract in this paper the uncoupling of boundary integral and finite element methods is applied to study the weak solvability of certain nonlinear exterior boundary value problems as a model we consider a nonlinear second order elliptic equation in divergence form in a bounded inner region of the plane which becomes the laplace equation in the corresponding unbounded exterior region we provide sufficient conditions for the nonlinear coefficients from which existence uniqueness and approximation results are established in particular nonlinear equations yielding both monotone and nonmonotone operators are analyzed

existence of a weak solution to a nonlinear fluid structure interaction problem modeling the flow of an incompressible viscous fluid in a cylinder with deformable walls we study a nonlinear unsteady moving boundary fluid structure interaction fsi problem arising in modeling blood flow through elastic and viscoelastic arteries the fluid flow which is driven by the time dependent pressure data is governed by two dimensional incompressible navier stokes equations while the elastodynamics of the cylindrical wall is modeled by the one dimensional cylindrical koiter shell model two cases are considered the linearly viscoelastic and the linearly elastic koiter shell the fluid and structure are fully coupled two way coupling via the kinematic and dynamic lateral boundary conditions describing continuity of velocity the no slip condition and the balance of contact forces at the fluid structure interface we prove the existence of weak solutions to the two fsi problems the viscoelastic and the elastic case as long as the cylinder radius is greater than zero the proof is based on a novel semi discrete operator splitting numerical scheme known as the kinematically coupled scheme introduced in guidoboni et al j comput phys 228 18 6916 6937 2009 to numerically solve the underlying fsi problems the backbone of the kinematically coupled scheme is the well known marchuk yanenko scheme also known as the lie splitting scheme we effectively prove convergence of that numerical scheme to a solution of the corresponding fsi problem

symmetric coupling of boundary elements and raviart thomas type mixed finite elements in elastostatics both mixed finite element methods and boundary integral methods are important tools in computational mechanics according to a good stress approximation recently even low order mixed methods of raviart thomas type became available for problems in elasticity since either methods are robust for critical poisson ratios it appears natural to couple the two methods as proposed in this paper the symmetric coupling changes the elliptic part of the bilinear form only hence the convergence analysis of mixed finite element methods is applicable to the coupled problem as well specifically we couple boundary elements with a family of mixed elements analyzed by stenberg the locking free implementation is performed via lagrange multipliers numerical examples are included