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

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

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

the classification and representation of single traveling wave solutions to the generalized fornberg whitham equation abstract by the complete discrimination system for polynomial method we give the classification and representation of single traveling wave solutions to the generalized fornberg whitham equation with n 1 and n 2

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

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

on computing of arbitrary positive integer powers for one type of odd order symmetric circulant matrices ii this article is an extension of the work j rimas on computing of arbitrary positive integer powers for one type of symmetric odd order circulant matrices i appl math comput in press in which the general expression of the lth power email protected n for one type of symmetric odd order circulant matrices is given in this new paper we present the complete derivation of this general expression expressions of eigenvectors and jordan s form of the matrix and of the transforming matrix and its inverse are given too

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

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

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

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

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

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

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

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

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

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

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