the steady motion of a navier stokes liquid around a rigid body let mathcal r be a body moving by prescribed rigid motion in a navier stokes liquid mathcal l that fills the whole space and is subject to given boundary conditions and body force under the assumptions that with respect to a frame mathcal f attached to mathcal r these data are time independent and that their magnitude is not too large we show the existence of one and only one corresponding steady motion of mathcal l with respect to mathcal f such that the velocity field at the generic point x in space decays like x 1 these solutions are physically reasonable in the sense of finn 10 in particular they are unique and satisfy the energy equation among other things this result is relevant in engineering applications involving orientation of particles in viscous liquid 14

existence of suitable weak solutions to the navier stokes equations in time varying domains abstract we consider suitable weak solutions to the navier stokes equations in time varying domains we develop schauder theory for the approximate stokes equations in time varying domains whose solutions satisfy a uniform localized energy estimate including boundary existence of suitable weak solutions in time varying domains follows from compactness in lebesgue and sobolev spaces

polygonal figures and their form spaces ii angle permutations in equilateral polygons abstract in a closed nonregular equilateral polygonal curve angles cannot be permuted arbitrarily without opening the curve here we investigate the possible angle permutations that leave a polygonal curve closed to this end we introduce angle sequences i e symbol sequences describing the number and arrangement of like angle values in classes of forms we find that there exist universal angle sequences giving closed polygons for arbitrary angle values provided that the angle sum is kept constant at one of certain permissible values the universal angle sequences are periodic they consist of at least two identical series of symbols within a series the symbols can be permuted freely on condition that in all other series the very same permutation is applied all these correlated permutations when applied to the angles of a corresponding polygonal curve leave this curve closed polygons belonging to a given angle sequence form class constitute a submanifold of the manifold m i of equilateral polygons within the form space these submanifolds structure m i in a hierarchical manner so that the form class concept gives a valuable complementary classification scheme for the forms of m i furthermore we find that polygons with constant angle composition the term composition is used here in its chemical sense show a wide variety of possible shapes from quasi circular to wormlike and even self intersecting ones corresponding to different degrees of segregation of like angles within the polygonal curve

nonuniqueness for the heat flow of harmonic maps on the disk we prove that for suitable initial data the heat flow of harmonic maps from the disk to the sphere admits infinitely many solutions characterised by backward bubbling at some arbitrarily large time all having uniformly bounded energy

convergence analysis of a penalization method for the three dimensional motion of a rigid body in an incompressible viscous fluid we present and analyze a penalization method which extends the the method of ph angot c h bruneau and p fabrie numer math 81 1999 pp 487 520 to the case of a rigid body moving freely in an incompressible fluid the fluid solid system is viewed as a single variable density flow the interface is captured by a color function satisfying a transport equation the solid velocity is computed by averaging at every time the flow velocity in the solid phase this velocity is used to penalize the flow velocity at the fluid solid interface and to move the interface numerical illustrations are provided to illustrate our convergence result a discussion of our result in the light of existing existence results is also given

on the global existence and convergence to steady state of navier stokes flow past an obstacle that is started from rest a new approach to the existence theory for the navier stokes equations using a technique of kato 15 further developed in combination with estimates for oseen s equation by kobayashi shibata 17 and shibata 24 has made possible the solution of a long standing open problem often referred to as finn s starting problem this paper provides the solution

spectral analysis of operators with the two point bohr spectrum in this paper the representation theory of locally compact abelian groups is used to introduce linear operators with the two point bohr spectrum and to study their spectral properties in particular the structure of the inverse operators is considered and the applications to the theory of difference operators are discussed

global weak solutions for the two dimensional motion of several rigid bodies in an incompressible viscous fluid we consider the two dimensional motion of several non homogeneous rigid bodies immersed in an incompressible non homogeneous viscous fluid the fluid and the rigid bodies are contained in a fixed open bounded set of r 2 the motion of the fluid is governed by the navier stokes equations for incompressible fluids and the standard conservation laws of linear and angular momentum rule the dynamics of the rigid bodies the time variation of the fluid domain due to the motion of the rigid bodies is not known a priori so we deal with a free boundary value problem the main novelty here is the demonstration of the global existence of weak solutions for this problem more precisely the global character of the solutions we obtain is due to the fact that we do not need any assumption concerning the lack of collisions between several rigid bodies or between a rigid body and the boundary we give estimates of the velocity of the bodies when their mutual distance or the distance to the boundary tends to zero

point vortex dynamics as zero radius limit of the motion of a rigid body in an irrotational fluid the vortex point system is usually considered as an idealized model where the vorticity of an ideal incompressible two dimensional fluid is concentrated in a finite number of moving points in the case of a single vortex in an otherwise irrotational ideal fluid occupying a bounded and simply connected two dimensional domain the motion is given by the so called kirchhoff routh velocity which depends only on the domain the main result of this paper establishes that this dynamics can also be obtained as the limit of the motion of a rigid body immersed in such a fluid when the body shrinks to a pointwise massless particle with fixed circulation the rigid body is assumed to be only accelerated by the force exerted by the fluid pressure on its boundary the fluid velocity and pressure being given by the incompressible euler equations with vanishing vorticity the circulation of the fluid velocity around the particle is conserved as time proceeds according to kelvin s theorem and gives the strength of the limit point vortex in order to understand the solid dynamics one has to evaluate the pressure field on the boundary of the solid that is where the singularity is concentrated at the limit moreover the newton equations driving the particle s dynamics involve a singular perturbation problem in time whereas the fluid state may be seen as solving an auxiliary problem the fluid velocity can indeed be recovered by an elliptic type problem where time appears only as a parameter since the analysis involves singular perturbation problem both in space and in time and that nothing excludes a priori some sharp energy transfer from the fluid toward the rigid immersed particle it is crucial to recast the solid dynamics under normal forms this form makes appear the gyroscopic structure of these singularities and allows one to obtain uniform estimates on the dynamics thanks to some energy type quantities modulated by the limit dynamics and therefore to describe the transition of the dynamics in the limit in order to get such normal forms we first establish that the newton equations can be seen as a geodesic equation with a metric associated with the well known added inertia phenomenon under an applied force similar to the lorentz force which can be seen as an extension of the celebrated kutta joukowski lift force then in the limit surprising relations between leading and subprincipal orders of various terms and the modulation variables show up and allow us to establish a normal form with a more gyroscopic structure as a byproduct of our analysis we also prove that in the different regime where the body shrinks with a fixed mass the limit equation is a second order differential equation involving a kutta joukowski type lift force which extends the result of glass o lacave c sueur f on the motion of a small body immersed in a two dimensional incompressible perfect fluid bull soc math france volume 142 fascicule 3 489 536 2014 to the case where the domain occupied by the solid fluid system is bounded

an almost periodic noncommutative wiener s lemma we develop a theory of almost periodic elements in banach algebras and present an abstract version of a noncommutative wiener s lemma the theory can be used for example to derive some of the recently obtained results in time frequency analysis such as the spectral properties of the finite linear combinations of time frequency shifts

deformations isosymmetric manifolds and higher dimensional form space symmetries for point ensembles polygonal forms under o 2 symmetry i two and three points abstract the analysis of polygonal forms and their form transitions using normal deformations 1 has been extended to a global analysis of form spaces for polygonal forms specified by n points in a plane coincidences of the points are explicitly allowed and the origin of the form space is taken to correspond to a coincidence of all n points this choice is natural since for every form there exists a totally symmetric deformation leading to exactly this configuration the form space f for n points in a plane is a 2n dimensional euclidean space whose points represent all possible n vertex polygons simple and self intersecting ones by fixing the centres of gravity of the forms in the origin of the x y plane the dimension of the form space reduces to 2n 2 to give a reduced form space f within the form spaces isosymmetric manifolds geometric loci of all forms having the same symmetry g are determined these manifolds inscribed in a form space f define the corresponding symmetry space s the symmetry g s of the reduced symmetry space for three points has been determined in part the four dimensional symmetry group 21 04 has been found to be a subgroup of the full noncrystallographic symmetry of s for some cases the relation g f g s has been exemplified

weak strong uniqueness for the compressible fluid rigid body interaction abstract in this work we study the coupled system of partial and ordinary differential equations describing the interaction between a compressible isentropic viscous fluid and a rigid body moving freely inside the fluid in particular the position and velocity of the rigid body in the fluid are unknown and the motion of the rigid body is driven by the normal stress forces of the fluid acting on the boundary of the body we prove that the strong solution which is known to exist under certain smallness assumptions is unique in the class of weak solutions to the problem the proof relies on a correct definition of the relative energy to use this tool we then have to introduce a change of coordinates to transform the strong solution to the domain of the weak solution in order to use it as a test function in the relative energy inequality estimating all arising terms we prove that the weak solution has to coincide with the transformed strong solution and finally that the transformation has to be in fact an identity

slanted matrices banach frames and sampling in this paper we present a rare combination of ab stract results on the spectral properties of slanted matrices and some of their very specific applications to frame theory and sam pling problems we show that for a large class of slanted matrices boundedness below of the corresponding operator in p for some p implies boundedness below in p for all p we use the established result to enrich our understanding of banach frames and obtain new results for irregular sampling problems we also present a version of a non commutative wiener s lemma for slanted matri ces

smooth solutions for motion of a rigid body of general form in an incompressible perfect fluid in this paper we consider the interactions between a rigid body of general form and the incompressible perfect fluid surrounding it local well posedness in the space c 0 t hs is obtained for the fluid rigid body system

existence of weak solutions up to collision for viscous fluid solid systems with slip we study in this paper the movement of a rigid solid inside an incompressible navier stokes flow within a bounded domain we consider the case where slip is allowed at the fluid solid interface through a navier condition taking into account slip at the interface is very natural within this model as classical no slip conditions lead to unrealistic collisional behavior between the solid and the domain boundary we prove for this model existence of weak solutions of leray type up to collision in three dimensions the key point is that due to the slip condition the velocity field is discontinuous across the fluid solid interface this prevents from obtaining global h1 bounds on the velocity which makes many aspects of the theory of weak solutions for dirichlet conditions unadapted

on a model for mixture flows derivation dissipation and stability properties we propose a new model describing mixture flows the non linear system couples heterogeneous navier stokes equations with a constraint on the mean volume velocity of the flow the pde system is obtained from a more microscopic viewpoint involving a vlasov like equation describing the disperse phase through a certain hydrodynamic limit the model has remarkable dissipation properties inherited from the structure of the fluid kinetic description based on these properties together with additional estimates that can be obtained in the one dimension framework we establish the stability of weak solutions

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

inertial motions of a rigid body with a cavity filled with a viscous liquid we study inertial motions of the coupled system mathscr s constituted by a rigid body containing a cavity entirely filled with a viscous liquid we show that for arbitrary initial data having only finite kinetic energy every corresponding weak solution a la leray hopf converges as time goes to infinity to a uniform rotation unless two central moments of inertia of mathscr s coincide and are strictly greater than the third one this corroborates a famous conjecture of n ye zhukovskii in several physically relevant cases moreover we show that in a known range of initial data this rotation may only occur along the central axis of inertia of mathscr s with the larger moment of inertia we also provide necessary and sufficient conditions for the rigorous nonlinear stability of permanent rotations which improve and or generalize results previously given by other authors under different types of approximation finally we present results obtained by a targeted numerical simulation that on the one hand complement the analytical findings whereas on the other hand point out new features that the analysis is yet not able to catch and as such lay the foundation for interesting and challenging future investigation

control at a distance of the motion of a rigid body immersed in a two dimensional viscous incompressible fluid abstract we consider the motion of a rigid body immersed in a two dimensional viscous incompressible fluid with navier slip with friction conditions at the solid boundary the fluid solid system occupies the whole plane we prove the small time global exact controllability of the position and velocity of the solid when the control takes the form of a distributed force supported in a compact subset with nonvoid interior of the fluid domain away from the body the strategy relies on the introduction of a small parameter we consider fast and strong amplitude controls for which the navier stokes rigid body system behaves like a perturbation of the euler rigid body system by the means of a multi scale asymptotic expansion we construct a controlled solution to the navier stokes rigid body system thanks to some controlled solutions to euler rigid body type systems and by using that the influence of the boundary layer on the solid motion turns out to be sufficiently small

isosymmetric manifolds in form spaces and the normal deformations of polygonal forms abstract form variations are described in an appropriately constructed form space f typically an r n where every point of f represents a different form regarding the symmetries of the forms f can be divided into disjunct isosymmetric manifolds i e points lines surfaces and volumes whose points correspond to forms with equal symmetries these manifolds are derived from a symmetry analysis of possible deformations of the forms this analysis is comparable to the construction of symmetry coordinates in a normal coordinate analysis of molecules and results in normal modes of deformation normal deformations of these forms from the symmetry species of a normal deformation the symmetry of the resulting form can be inferred transformation of the form space coordinates into normal coordinates the differentials of which are the normal deformations facilitates the description of the high dimensional form spaces and can be made the basis of an easy symmetry diagnosis of forms furthermore the problem of an ascent in symmetry by deformation is discussed

a simple 1d model of inviscid fluid solid interaction we analyze a one dimensional fluid particle interaction model composed by the burgers equation for the fluid velocity and an ordinary differential equation which governs the particle movement the coupling is achieved through a friction term one of the novelties is to consider entropy weak solutions involving shock waves the difficulty is the interaction between these shock waves and the particle we prove that the riemann problem with arbitrary data always admits a solution which is explicitly constructed besides two asymptotic behaviors are described the long time behavior and the behavior for large friction coefficients

polygonal figures and their form spaces i the structure of isodiastemic manifolds manifolds of equilateral polygons abstract the geometric structure of isodiastemic manifolds i e manifolds on which the equilaterality of polygons is preserved within a form space f spanned by n 2 consecutive vertex angles i internal coordinates of plane n gons is reported for n 5 and n 6 the curved isodiastemic manifolds are almost everywhere locally n 3 dimensional exceptions are singular points linear forms for even n where the local dimension is n 2 the isodiastemic manifolds for n 5 and n 6 are subdivided into five n 3 dimensional submanifolds of nondegenerate forms forms without coincident points each comprising forms with the same angle sum called a form family the submanifolds are bounded by n 4 dimensional dividing manifolds of degenerate forms which are structured hierarchically according to the types of degeneracy of their forms for n 6 the boundary polyhedra for two submanifolds are described in detail the manifold of spearhead shaped forms is equivalent to an octahedron whereas the hexagon manifold is equivalent to a special icosihexahedron 1

a fuzzy symmetry concept for forms with imperfect symmetries we develop a new symmetry concept for forms whose symmetries are only approximate the concept is based on fuzzy set theory the crisp set s g of objects having a certain symmetry g is replaced by a fuzzy set s g with a membership function 0 1 corresponding to the degree of fulfilment of the symmetry requirements of g degree of symmetry the degree of symmetry s g of a given form s relative to a symmetry group g is evaluated on the basis of a harmonic analysis of the corresponding form function r depending on the symmetry of the form certain coefficients of the fourier series vanish the extinction rules for the fourier coefficients are deduced from the symmetries of the individual cosine terms cos we define the degree of symmetry as the relative weight of all terms which are compatible with g thus s is a continous measure of symmetry it may quantitatively characterise continuous transitions of forms between different symmetries and also the degree of membership of s in s g depending on the values of s we propose a gradation of fuzzy symmetries actual apparent approximate distant hidden missing symmetry

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

existence of solutions for the equations modeling the motion of rigid bodies in an ideal fluid in this paper we study the motion of rigid bodies in a perfect incompressible fluid the rigid fluid system fils a bounded domain inr 3 adapting the strategy from bourguignon and brezis 1 we use the stream lines of the fluid and we eliminate the pressure by solving a neumann problem in this way the system is reduced to an ordinary dierential equation on a closed infinite dimensional manifold using this formulation we prove the local in time existence and uniqueness of strong solutions notation throughout this paper denotes an open bounded and connected subset of r 3 and s0 is a closed set with nonempty interior and with smooth boundary such that s0 we denote as usual by so3 r the special orthogonal group on r 3 we will often use functions defined from a time interval to r 3 or to so3 r these functions will be denoted using bold characters such as h 0 t r 3 or r 0 t so3 r the same kind of notation will be used for three other time dependent vector fields k and which will be defined in the sequel the five time dependent fields mentioned above will define the state z of the fluid solid system a vector from r 3 or a matrix from so3 r will be denoted by h or by r respectively the transposed of a matrix will be denoted by so that the column vector of components a and b is denoted either a

application of finite element method in aeroelasticity the subject of this paper is the numerical simulation of the interaction of two dimensional incompressible viscous flow and a vibrating airfoil which can rotate around the elastic axis and oscillate in the vertical direction the numerical simulation consists of the finite element approximation of the navier stokes equations coupled with the system of ordinary differential equations describing the airfoil motion the arbitrary lagrangian eulerian ale formulation of the navier stokes equations stabilization the finite element discretization and coupling of both models is discussed moreover the reynolds averaged navier stokes rans system of equations together with the spallart almaras turbulence model is also discussed the computational results of aeroelastic calculations are presented and compared with the nastran code solutions

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

uniqueness results for weak solutions of two dimensional fluid solid systems in this paper we consider two systems modelling the evolution of a rigid body in an incompressible fluid in a bounded domain of the plane the first system corresponds to an inviscid fluid driven by the euler equation whereas the other one corresponds to a viscous fluid driven by the navier stokes system in both cases we investigate the uniqueness of weak solutions a la yudovich for the euler case a la leray for the navier stokes case as long as no collision occurs

regularity issues in the problem of fluid structure interaction we investigate the evolution of rigid bodies in a viscous incompressible fluid the flow is governed by the 2d navier stokes equations set in a bounded domain with dirichlet boundary conditions the boundaries of the solids and the domain have holder regularity c1 0 1 first we show the existence and uniqueness of strong solutions up to the collision a key ingredient is a bmo bound on the velocity gradient which substitutes to the standard h2 estimate for smoother domains then we study the asymptotic behaviour of one c1 body falling over a flat surface we show that a collision is possible in finite time if and only if 1 2