remark on the stability of the large stationary solutions to the navier stokes equations under the general flux condition abstract consider stationary weak solutions of the navier stokes equations in a bounded domain in r 3 under the nonhomogeneous boundary condition we give a new approach for the stability of the stationary flow in the l 2 framework furthermore we give some examples of stable solutions which may be large in l 3 or w 1 3 2

existence uniqueness and l q estimates for the stokes problem in an exterior domain as is well known the study of the properties of solutions to the stokes system plays an essential role in the mathematical theory of viscous fluid flows governed by the navier stokes equations it is not art overstatement to say that existence uniqueness and a priori estimates in sobolev spaces for the stokes system are fundamental in every basic question related to the navier stokes equations such as existence regularity asymptotic behavior in time etc see e g ladyzhenskaya 1969 solonni ov 1977 and te am 1977 so far however the stokes system has been fully analyzed only for bounded domains see sobolevskn 1960 ladvznenskaya 1959 solonnu ov 1960 vorovlch yot oowcr 1961 cattabrma 1961 in particular in the paper of cattabriga a complete treatment is given in the sobotev spaces w l 1 1 q c for space dimension n 3 cattabriga s results essentially state that if 2 is a bounded domain in r a of class c m m max l 2 2 l 1 then given f e wt q y2 3 gc wt l q o q e wl 2 1 q q 8o a 1 q there exists one and only one distributional solution v c w1 z q q 3 p e wt l q o to the stokes system

genetic algorithm approach on multi criteria minimum spanning tree problem minimum spanning tree mst problem is of high importance in network optimization the multi criteria mst mc mst is a more realistic representation of the practical problem in the real world but it is difficult for the traditional network optimization technique to deal with in this paper a genetic algorithm ga approach is developed to deal with this problem without neglecting its network topology the proposed method adopts the prufer number as the tree encoding and applies the multiple criteria decision making mcdm technique and nondominated sorting technique to make the ga search give out all pareto optimal solutions either focused on the region near the ideal point or distributed all along the pareto frontier compared with the enumeration method of pareto optimal solution the numerical analysis shows the efficiency and effectiveness of the ga approach on the mc mst problem

on uniqueness for the navier stokes system in 3d bounded lipschitz domains in this paper we prove uniqueness of solutions of the navier stokes system in cb 0 t l3 3 l 0 t l3 2 where is a bounded lipschitz domain in r3

bounded imaginary powers and h calculus of the stokes operator in two dimensional exterior domains the present contribution deals with the stokes operator aq on lq 1 q where is an exterior domain in 2 of class c2 it is proved that aq admits a bounded h calculus this implies the existence of bounded imaginary powers of aq which has several important applications so far this property was only known for exterior domains in n n 3 in particular this shows that aq has maximal regularity on lq for the proof the resolvent aq 1 has to be analyzed for and 0 for large this is done using an approximate resolvent based on the results of 3 which were obtained by applying the calculus of pseudodifferential boundary value problems for small we analyze the representation of the resolvent developed in 11 by a potential theoretical method

singular initial data and uniform global bounds for the hyper viscous navier stokes equation with periodic boundary conditions abstract in the hyper viscous navier stokes equations of incompressible flow the operator a is replaced by a a b aa ba for real numbers a b with 1 and b 0 we treat here the case a 0 and equip a and hence a a b with periodic boundary conditions over a rectangular solid r n for initial data in l p with n 2p 1 2 we establish local existence and uniqueness of strong solutions generalizing a result of giga miyakawa for 1 and b 0 specializing to the case p 2 which holds a particular physical relevance in terms of the total energy of the system it is somewhat interesting to note that the condition n 4 1 2 is sufficient also to establish global existence of these unique regular solutions and uniform higher order bounds for the borderline case n 4 1 2 we generalize standard existing for n 3 folklore results and use energy techniques and gronwall s inequality to obtain first a time dependent h bound and then convert to a time independent global exponential h bound this is to be expected given that uniform bounds already exist for n 2 1 6 pp 78 79 and the folklore bounds already suggest that the n 4 1 2 cases for n 3 should behave as well as the n 2 case what is slightly less expected is that the n 3 cases are easier to prove and give better bounds e g the uniform bound for n 3 depends on the square of the data in the exponential rather than the fourth power for n 2 more significantly for n 4 1 2 we use our own entirely semigroup techniques to obtain uniform global bounds which bootstrap directly from the uniform l2 estimate and are algebraic in terms of the uniform l2 bounds on the initial and forcing data the integer powers on the square of the data increase without bound as n 4 1 2 thus anticipating the exponential bound in the borderline case n 4 1 2 we prove our results for the case a 1 and b 0 the general case with a 0 and b 0 can be recovered by using norm equivalence we note that the hyperviscous navier stokes equations have both physical and numerical application

behavior of derivatives of eigenvalues and eigenvectors in curve veering and mode localization and their relation to close eigenvalues the problem to measure the phenomena of eigenvalue curve veering and mode localization is addressed in this paper the second derivative of an eigenvalue and the first derivative of an eigenvector are taken as the measures numerically showing curve veering and mode localization based on the measurement close eigenvalues as a key factor for the occurrence of the phenomena are defined two eigenvalues are considered to be close if their difference is small enough to cause the occurrence of the phenomena the curve veering and mode localization can be noted by comparison of the derivatives with a critical value and hence the associated eigenvalues are close weakly coupled springs are given as an example

global solutions to semilinear parabolic systems for small data abstract we consider semilinear parabolic systems u t au f u g u 0 u 0 a being the generator of an analytic semigroup with spectrum a in the right half plane for appropriate small data g u 0 global solutions are obtained if 0 belongs to the resolvent set and local existence is proved if 0 a the inverse function theorem is used in a class of functions which are holder continuous with respect to time t the asymptotic behaviour of the solution is given too as a first application we prove the existence of global respectively local solutions to the navier stokes equations in l spaces in bounded respectively unbounded domains second we consider the case where a is an elliptic operator of order 2 m and f has bounded growth third a quasilinear example where the nonlinearity involves derivatives of the same order as a is studied then an example of a nonautonomous nonlinearity is given and finally the convergence of solutions of the time dependent system to solutions of the stationary semilinear elliptic one is derived for 0 a

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

on energy inequality smoothness and large time behavior in l 2 for weak solutions of the navier stokes equations in exterior domains soit un domaine exterieur de r n n 3 4 a frontiere lisse sur le cylindre d espace temps 0 on considere le probleme de navier stokes non stationnaire u t u u u p f dans 0 u 0 dans 0 u 0 sur 0 u x 0 a x avec des donnees prescrites f et a

convergence of euler stokes splitting of the navier stokes equations we consider approximation by partial time steps of a smooth solution of the navier stokes equations in a smooth domain in two or three space dimensions with no slip boundary condition for small k 0 we alternate the solution for time k of the inviscid euler equations with tangential boundary condition and the solution of the linear stokes equations for time k with the no slip condition imposed we show that this approximation remains bounded in h2 p and is accurate to order k in lp for p the principal difficulty is that the initial state for each stokes step has tangential velocity at the boundary generated during the euler step and thus does not satisfy the boundary condition for the stokes step the validity of such a fractional step method or splitting is an underlying principle for some computational methods 1994 john wiley sons inc

existence and asymptotic behavior for a convection problem abstract we prove global existence and exponential decay of solutions for a system which arise in thermal convection flow for sufficiently small initial data these results improve previous ones in funkcial ekvac 34 1991 449 further we investigate the behavior of solutions for arbitrarily large initial data in particular we show that the length of the interval on which we have existence and exponential decay is inverse proportional to the size of the initial data

on stability of the spatially inhomogeneous navier stokes boussinesq system with general nonlinearity this paper considers l 2 asymptotic stability of the spatially inhomogeneous navier stokes boussinesq system with general nonlinearity including both power nonlinear terms and convective terms we construct a local in time strong solution of the system by applying semigroup theory on hilbert spaces and fractional powers of the stokes laplace operator it is also shown that under some assumptions on an energy inequality the system has a unique global in time strong solution when the initial datum is sufficiently small furthermore we investigate the asymptotic stability of the global in time strong solution by using an energy inequality maximal l p in time regularity for hilbert space valued functions and fractional powers of linear operators in a solenoidal l 2 space we introduce new methods for showing the asymptotic stability by applying an energy inequality and maximal l p in time regularity for hilbert space valued functions our approach in this paper can be applied to show the asymptotic stability of energy solutions for various incompressible viscous fluid systems and the stability of small stationary solutions whose structure is not clear

inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry let r cn n be a nontrivial involution i e r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a let s be one of the following subsets of cn n i r symmetric matrices ii hermitian r symmetric matrices iii r skew symmetric matrices iv hermitian r skew symmetric matrices let z cn m with rank z m and diag 1 m the inverse eigenproblem consists of finding z such that the set s z a s az z is nonempty and to find the general form of a s z in all cases we use the special spectral properties of s to essentially characterize the set of admissible pairs z and the special structure of the members of s to obtain the general solution of the inverse eigenproblem given an arbitrary b s the approximation problem consists of finding the unique matrix ab s z that best approximates b in the frobenius norm it is not necessary to assume that r r in connection with the inverse eigenproblem for r symmetric or r skew symmetric matrices however we impose this additional assumption in connection with the inverse eigenproblem for hermitian r symmetric or r skew symmetric matrices and in connection with the approximation problem for i iv

the hopf bifurcation with symmetry for the navier stokes equations in l p ω n with application to plane poiseuille flow the existence of periodic solutions of the navier stokes equations in function spaces based upon lp nis proved the paper has three parts a a proof of the existence of strong solutions of the evolution equation with initial data in a solenoidal subspace of lp n b the evolution equation is restricted to a space of time periodic functions and a fredholm integral equation on this space is formed the lyapunov schmidt method is applied to prove the existence of bifurcating time periodic solutions in the presence of symmetry c the theory is applied to the bifurcation of periodic solutions from planar poiseuille flow in the presence of symmetry so 2 x o 2 x s1 yielding new results for this classic problem the o 2 invariance is in the spanwise direction with the periodicity in time and in the streamwise direction we find that generically there is a bifurcation to both oblique travelling waves and to travelling waves that are stationary in the spanwise direction there are however points of degeneracy on the neutral surface a numerical method is used to identify these points and an analysis in the neighborhood of the degenerate points yields more complex periodic solutions as well as branches of quasi periodic solutions

mode localization phenomena in large space structures the possibility of localization or confinement of vibratory modes in large space structures is investigated theoretically and numerically these structures belong to a class of periodic structures that have recently been shown to be sensitive to periodicity b reaking disorder or imperfections when localization occurs the modal amplitude of a global mode becomes confined to a local region of the structure with serious implications for the control problem the results of this study indicate that mode localization is most likely to occur in structures consisting of a large number of weakly coupled substructures certain large space structures with high modal densities fall in this category and it is therefore important to include the effect of structural imperfections and disorder when designing control systems for shape or directional control of such structures k kh kw ku k k m m m n n p qi qt t u v w uf 8fj e 6

a pair of simultaneous linear matrix equations a1xb1 c1 a2xb2 c2 and a matrix programming problem necessary and sufficient conditions are obtained for a pair of matrix equations a1xb1 c1 a2xb2 c2 on a general field f to have a common solution along with the expression for a general common solution when certain conditions hold common solutions of minimum rank are described a matrix programming problem is solved en route

compact gravity inversion we present a new criterion for the inversion of gravity data the principle employed is to minimize the volume of the causative body which is equivalent to maximizing its compactness the anomalous density distribution is obtained using an iterative technique which is numerically stable and rapidly convergent the principle can also be adapted to include modeling of gravity anomalies by single density sources the advantage of this approach is that desirable geologic characteristics are automatically incorporated into the model with a minimum of subjective judgments on the part of the interpreter the treatment of noise in the data fits naturally into the formulation of the inversion procedure the method is illustrated by the inversion of noise free and noisy data generated from a two dimensional model consisting of a regular array of identical rectangular blocks whose densities can be individually specified in every case the algorithm successfully recovers the correct density distribution from the dat

new thoughts on old results of r t seeley in this paper we show that the lp realization of a vector valued elliptic boundary value problem j admits a bounded calculus on lp g e 1 p provided the top order coefficients of are holder continuous here g denotes a domain in n with compact c2m boundary and e a banach space of class our proof is based on an abstract perturbation result for operators admitting bounded calculus and kernel estimates for the solution of j

mode localization in multispan beams the finite element method is used to study localization as a function of timoshenko beam effects beam end conditions span length mass and stiffness imperfection viscous damping axial force transverse support and rotational coupling stiffness and modeling resolution three configurations are studied starting with two separate 2 span models and culminating in a 10 span configuration that resembles lattice type large space structures

stochastic volterra equations in banach spaces and stochastic partial differential equation abstract in this paper we study the existence uniqueness and large deviation estimate for stochastic volterra integral equations with singular kernels in 2 smooth banach spaces then we apply them to a large class of semilinear stochastic partial differential equations spde and obtain the existence of unique maximal strong solutions in the sense of sde and pde under local lipschitz conditions moreover stochastic navier stokes equations are also investigated

eigensolutions sensitivity for nonsymmetric matrices with repeated eigenvalues a perturbation algorithm is developed for evaluating eigenvalue and eigenvector directional derivatives of nonsymmetric defective matrices some properties are recalled in particular chains of generalized right and left eigenvectors and their orthogonal properties are defined small perturbations of the matrix are then considered an asymptotic expansion of the eigensolutions of the perturbed problem is obtained in terms of noninteger powers of the perturbation parameter particular attention is devoted to the eigenvectors of the perturbed system and to the strong coupling that occurs between the chains

frequency coalescence and mode localization phenomena a geometric theory physical systems the natural frequencies of which depend parametrically on certain quantities may exhibit the phenomenon of frequency coalescence when two or more natural frequencies become equal or the phenomenon of avoided crossings when two or more frequencies approach rapidly and then avoid crossing each other in the process the natural modes undergo significant variations it is shown that both phenomena can be described in a unified manner from the singularities of the mapping from the complex parameter plane into the complex frequency plane in particular the formation of a saddle point in the frequency plane and a branch point in the parameter space is the basis for explaining the properties associated with these phenomena the question of sensitivity of the system response near such singularities is subsequently examined it is shown that the development of a branch point singularity in the complex parameter plane can cause large or even infinite sensitivity of the system about this point depending on the distance of the branch point from the real axis in the parameter space it is further shown that mode localization is also associated with the appearance of a branch point near the real axis of the complex parameter plane which causes the large sensitivity that characterizes the phenomenon the theory is applied to examples of conservative and dissipative systems undergoing free vibrations

adaptive learning 3d gravity inversion for salt body imaging we have developed an iterative scheme for inverting gravity data produced by salt bodies with density contrasts relative to the sediments varying from positive to negative crossing in this way the nil zone our inversion method estimates a 3d density contrast distribution through a piecewise constant function defined on a user specified grid of cells it consists of two nested iterative loops the outer loop uses an adaptive learning strategy that starts with a coarse grid of cells a set of first guess geometric elements axes and points and the corresponding assigned density contrasts from the second iteration on this strategy refines the grid and automatically creates a new set of geometric elements points only and associated density contrasts each geometric element operates as the first guess skeletal outline of a section of the salt body to be imaged the inner loop estimates the 3d density contrast distribution for the grid of cells and for the set of geometric elements defined in the oute

three dimensional gravity inversion using graph theory to delineate the skeleton of homogeneous sources abstractwe developed a new strategy based on graph theory concepts to invert gravity data using an ensemble of simple point masses our method consisted of a genetic algorithm with elitism to generate a set of possible solutions each estimate was associated to a graph to solve the minimum spanning tree mst problem to produce unique and stable estimates we restricted the position of the point masses by minimizing the statistical variance of the distances of an mst jointly with the data misfit function during the iterations of the genetic algorithm hence the 3d spatial distribution of the point masses identified the skeleton of homogeneous gravity sources in addition our method also gave an estimation of the anomalous mass of the source so together with the anomalous mass the skeleton could aid other 3d methods with promising geometric a priori parameters several tests with different values of regularizing parameter were made to bespeak this new regularizing strategy the inversion results ap

a singular perturbation analysis of eigenvalue veering and modal sensitivity in perturbed linear periodic systems abstract an investigation into the eigenvalue loci veering and modal sensitivity is presented for mistuned structural systems examples from both the weakly coupled uniaxial component systems and the cyclic symmetric systems are considered the analysis is based on singular perturbation techniques it is shown that uniform asymptotic expansions for the eigenvalues and eigenvectors can be constructed in terms of the mistuning parameters and that these solutions are in excellent agreement with the exact solutions the modal sensitivity function and the eigenvector rotations are then used to show the singular behavior of the eigenvectors in the neighborhood of a singular point rapid changes in the sensitivity function may signify large relative differences in the amplitudes of forced vibratory response between the tuned and the mistuned states

on a larger class of stable solutions to the navier stokes equations in exterior domains where w w x w1 x wn x and x denote the unknown velocity vector and the unknown pressure of the fluid at point x respectively while f f x f i j x i j 1 n is the given tensor with div f n j 1 f 1 j xj nj 1 f n j xj denoting the external force in our previous paper 23 we established the theory of existence uniqueness and regularity for solutions to n s0 in the class w ln

peristaltic transport of a heat conducting viscous fluid as an application of abstract differential equations and semigroup of operators abstract a recent abstract result of rankin trans amer math soc to appear is applied to the peristaltic transport of a heat conducting fluid in a flexible tube the oberbeck boussinesq equations are used as the governing equations a pressure drop over a wavelength of the tube is prescribed to ensure uniqueness and newton s cooling law for the temperature is imposed on the boundary with the notion of a weak solution introduced local existence uniqueness regularity and continuation of solutions are considered in particular w r 2 regularity of the solutions is established

asymptotic stability of large solutions with large perturbation to the navier stokes equations abstract consider weak solutions w of the navier stokes equations in serrin s class w l 0 l q for 2 3 q 1 with 3 q where is a general unbounded domain in r 3 we shall show that although the initial and external disturbances from w are large every perturbed flow v with the energy inequality converges asymptotically to w as v t w t l 2 0 v t w t l 2 o t 1 2 as t

existence of optimal controls for viscous flow problems a class of optimal control problems in viscous flow is studied main result is the existence theorem for optimal control three typical flow control problems are formulated within this general class

semi group theory for the stokes operator with navier type boundary conditions on l p spaces in this article we consider the stokes problem with navier type boundary conditions on a domain omega not necessarily simply connected since under these conditions the stokes problem has a non trivial kernel we also study the solutions lying in the orthogonal of that kernel we prove the analyticity of several semigroups generated by the stokes operator considered in different functional spaces we obtain strong weak and very weak solutions for the time dependent stokes problem with the navier type boundary condition under different hypotheses on the initial data u0 and external force f then we study the fractional and pure imaginary powers of several operators related with our stokes operators using the fractional powers we prove maximal regularity results for the homogeneous stokes problem on the other hand using the boundedness of the pure imaginary powers we deduce maximal l p l q regularity for the inhomogeneous stokes problem

3 d inversion of gravity data we present two methods for inverting surface gravity data to recover a 3 d distribution of density contrast in the first method we transform the gravity data into pseudomagnetic data via poisson s relation and carry out the inversion using a 3 d magnetic inversion algorithm in the second we invert the gravity data directly to recover a minimum structure model in both approaches the earth is modeled by using a large number of rectangular cells of constant density and the final density distribution is obtained by minimizing a model objective function subject to fitting the observed data the model objective function has the flexibility to incorporate prior information and thus the constructed model not only fits the data but also agrees with additional geophysical and geological constraints we apply a depth weighting in the objective function to counteract the natural decay of the kernels so that the inversion yields depth information applications of the algorithms to synthetic and field data produce density models representative of true structures our results have shown that the inversion of gravity data with a properly designed objective function can yield geologically meaningful information

multi parameter perturbation methods for the eigensolution sensitivity analysis of nearly resonant non defective multi degree of freedom systems the dynamic behavior of structural systems may be strongly characterized by the occurrence of multiple internal resonances for particular combinations of the mechanical parameters the linear models governing these resonant or nearly resonant systems tend to exhibit high sensitivity of the eigenvalues and eigenvectors to small parameter modifications this pathological condition is recognized as a source of relevant phenomena such as frequency veering and mode localization or hybridization the paper presents the generalization of uniformly valid perturbation methods to perform eigensolution sensitivity analyses in multi degree of freedom hamiltonian systems with a generic number of close eigenvalues the leading idea is to treat systematically nearly resonant systems as multi parameter perturbations of a perfectly resonant non defective though a priori unknown reference system given a single nearly resonant system a multi parameter perturbation method is presented to achieve a two fold objective first identify a close resonant system suited to serve as a starting point for sensitivity analyses inverse problem second to approximate asymptotically the eigensolution of all the nearly resonant systems which may arise from its generic perturbation direct problem the conditions of existence and uniqueness of the inverse problem solution are discussed the direct problem solution is analyzed with a focus on the eigensolution sensitivity to parameter perturbations with different physical meanings such as a slight geometric disorder or weak elastic coupling in periodic structures finally the procedure is verified on a prototypical structural system describing the section dynamics of a suspended bridge

traveltime inversion for 2 d anomaly structures this work presents a traveltime inversion method that uses parametric functions to represent 2 d anomaly structures these functions are described by a small set of unknown parameters which in turn are obtained after solving a highly nonlinear optimization problem via simulated annealing sa the procedure favors neither smooth nor high contrasting anomalies and keeps the number of unknowns very small so as to make the problem tractable using sa yet the strategy allows one to accommodate a large class of velocity models results indicate that this new approach typically yields better images than a standard linearized inversion based on a cell parameterization scheme

a common solution to a pair of linear matrix equations over a principal ideal domain abstract a necessary and sufficient condition for the existence of a common solution to a pair of linear matrix equations over a principal ideal domain is obtained the equations are of the type a i b i xc i for i 1 2 the solvability condition is that the equations each have a solution and a bilateral linear matrix equation made up of the matrices a i b i and c i has a solution

solutions in lebesgue spaces of the navier stokes equations with dynamic boundary conditions this paper although self contained is a continuation of the work done in 8 where the motion of a viscous incompressible fluid is considered in conjunction with the rotation of a rigid body which is immersed in the fluid the resulting mathematical model is a navier stokes problem with dynamic boundary conditions in 8 a unique l 2 3 solution is constructed under certain regularity assumptions on the initial states in this paper we consider the navier stokes problem with dynamic boundary conditions in the lebesgue spaces l r 3 3

l2 decay for the navier stokes flow in halfspaces on etudie le comportement asymptotique dans l 2 des solutions faibles des equations de navier stokes dans des demi espaces

l q theory of the kelvin voigt equations in bounded domains abstract we investigate the well posedness of the kelvin voigt equations in spaces based on l q 1 q where is a bounded domain of r d d 2

the solution of nonlinear inverse problems and the levenberg marquardt method although the levenberg marquardt damped least squares method is an extremely powerful tool for the iterative solution of nonlinear problems its theoretical basis has not been described adequately in the literature this is unfortunate because levenberg and marquardt approached the solution of nonlinear problems in different ways and presented results that go far beyond the simple equation that characterizes the method the idea of damping the solution was introduced by levenberg who also showed that it is possible to do that while at the same time reducing the value of a function that must be minimized iteratively this result is not obvious although it is taken for granted moreover levenberg derived a solution more general than the one currently used marquardt started with the current equation and showed that it interpolates between the ordinary least squares method and the steepest descent method in this tutorial the two papers are combined into a unified presentation which will help the reader gain a better understanding of what happens when solving nonlinear problems because the damped least squares and steepest descent methods are intimately related the latter is also discussed in particular in its relation to the gradient when the inversion parameters have the same dimensions and units the direction of steepest descent is equal to the direction of minus the gradient in other cases it is necessary to introduce a metric i e a definition of distance in the parameter space to establish a relation between the two directions although neither levenberg nor marquardt discussed these matters their results imply the introduction of a metric some of the concepts presented here are illustrated with the inversion of synthetic gravity data corresponding to a buried sphere of unknown radius and depth finally the work done by early researchers that rediscovered the damped least squares method is put into a historical context

on the initial boundary value problem for a bingham fluid in a three dimensional domain the initial boundary value problem associated with the motion of a bingham fluid is considered the existence and uniqueness of strong solution is proved under a certain assumption on the data it is also shown that the solution exists globally in time when the data are small and that the solution converges to a periodic solution if the external force is time periodic o introduction the purpose of this paper is to establish the existence of strong solutions to a variational inequality which describes the motion of a bingham fluid in a bounded three dimensional domain a bingham fluid is a rigid viscoplastic fluid which is governed by a special constitutive law such that it moves like a rigid body if a certain function of the stresses does not reach the yield limit and it behaves like a viscous fluid when the yield limit is reached since the motion is governed by two entirely different stress strain relations depending on the state of stresses the conservation of momentum is expressed in terms of a variational inequality so that one can avoid the difficulty of separating the fluid zone and the 1 rlglcl zone the initial boundary value problem we shall study is formulated as 0 l it uj u a u w u b u u w j w j u f w a in 0 t for each test function w such that v w 0 in q and w 0 on aq 0 2 v u 0 in q x 0 t 0 3 u 0 on aq x 0 t 0 4 8 x 0 vo x in q here q is a bounded domain in r3 with smooth boundary aq a x t denotes the velocity of the fluid and f x t stands for external force we assume that the density the yield limit and the viscosity are positive constants in particular the received by the editors august 15 1986 1980 mathematics subject classijication 1985 revision primary 35q20 35b10 35b40 35b65 secondary 76a05 76d99 this research was supported by the air force office of scientific research under grant afosr86 0085 and by nsf grant dms 8521848 r 1987 american mathematical society 0002 9947 87 1 00 25 per page

inverse eigenproblem for r symmetric matrices and their approximation let email protected c n x n be a nontrivial involution i e r r 1 i n we say that email protected c n x n is r symmetric if rgr g the set of all nxnr symmetric matrices is denoted by gsc n x n in this paper we first give the solvability condition for the following inverse eigenproblem iep given a set of vectors x i i 1 m in c n and a set of complex numbers l i i 1 m find a matrix email protected gsc n x n such that x i i 1 m and l i i 1 m are respectively the eigenvalues and eigenvectors of a we then consider the following approximation problem given an nxn matrix email protected find email protected email protected s e such that email protected email protected email protected min a s email protected email protected email protected where s e is the solution set of iep and email protected email protected is the frobenius norm we provide an explicit formula for the best approximation solution email protected by means of the canonical correlation decomposition

incompressible inhomogeneous fluids in bounded domains of r3 with bounded density abstract in this paper we study the incompressible inhomogeneous navier stokes equations in bounded domains of r 3 involving bounded density functions 1 a based on the corresponding theory of besov spaces on domains we first obtain the global existence of weak solutions u with initial data a 0 l u 0 b q s 1 3 q for 1 q 3 1 s furthermore with additional regularity assumptions on the initial velocity we also prove the uniqueness of such a solution it is a generalization of a result established by huang et al 2013 20 for the whole space r 3

very weak solutions and large uniqueness classes of stationary navier stokes equations in bounded domains of r2 extending the notion of very weak solutions developed recently in the three dimensional case to bounded domains r2 we obtain a new class of unique solutions u in lq q 2 to the stationary navier stokes system u u u p f divu k u g with data f k g of low regularity as a main consequence we obtain a new uniqueness class also for classical weak or strong solutions indeed such a solution is unique if its lq norm is sufficiently small or the data satisfy the uniqueness condition of a very weak solution

a class of solutions to stationary stokes and navier stokes equations with boundary data in w 1 q q we develop a theory for a general class of very weak solutions to stationary stokes and navier stokes equations in a bounded domain with boundary of class c2 1 corresponding to boundary data in the distribution space w 1 q q 1 q these solutions exist and are unique for small data in the nonlinear case in their class of existence and satisfy a correponding estimate in terms of the data moreover they become regular if the data are regular to our knowledge the only existence result for solutions attaining such boundary data is due to giga 16 proposition 2 2 for the stokes case however the methods and the approach used in the present paper are different than giga s and cover more general issues including the nonlinear navier stokes equations and the precise way in which the boundary data are attained by the solutions we also introduce in the last section a further generalization of the solution class

discontinuous galerkin approximations for an optimal control problem of three dimensional navier stokes voigt equations we analyze a fully discrete scheme based on the discontinuous in time galerkin approach which is combined with conforming finite element subspaces in space for the distributed optimal control problem of the three dimensional navier stokes voigt equations with a quadratic objective functional and box control constraints the space time error estimates of order o sqrt tau h where tau and h are respectively the time and space discretization parameters are proved for the difference between the locally optimal controls and their discrete approximations

stochastic two dimensional euler equations the existence of a martingale solution to 2 dimensional stochastic euler equations is proved the constructed solution is a limit as the viscosity converges to zero of a sequence of solutions to modified navier stokes equations

decay estimates of gradient of a generalized oseen evolution operator arising from time dependent rigid motions in exterior domains let us consider the motion of a viscous incompressible fluid past a rotating rigid body in three dimensions where the translational and angular velocities of the body are prescribed but time dependent in a reference frame attached to the body we have the navier stokes system with the drift and one half of the coriolis terms in a fixed exterior domain the existence of the evolution operator t t s in the space l q generated by the linearized non autonomous system was proved by hansel and rhandi j reine angew math 694 1 26 2014 and the large time behavior of t t s f in l r for t s rightarrow infty was then developed by hishida math ann 372 915 949 2018 when f is taken from l q with q leqq r the contribution of the present paper concerns such l q l r decay estimates of nabla t t s with optimal rates which must be useful for the study of stability attainability of the navier stokes flow in several physically relevant situations our main theorem completely recovers the l q l r estimates for the autonomous case stokes and oseen semigroups those semigroups with rotating effect in three dimensional exterior domains which were established by hishida and shibata arch ration mech anal 193 339 421 2009 iwashita math ann 285 265 288 1989 kobayashi and shibata math ann 310 1 45 1998 maremonti and solonnikov ann sc norm super pisa 24 395 449 1997 and shibata in amann arendt hieber neubrander nicaise von below eds functional analysis and evolution equations the gunter lumer volume birkhauser basel pp 595 611 2008

on stokes operators with variable viscosity in bounded and unbounded domains we consider a generalization of the stokes resolvent equation where the constant viscosity is replaced by a general given positive function such a system arises in many situations as linearized system when the viscosity of an incompressible viscous fluid depends on some other quantities we prove that an associated stokes like operator generates an analytic semi group and admits a bounded h calculus which implies the maximal l q regularity of the corresponding parabolic evolution equation the analysis is done for a large class of unbounded domains with w 2 frac1r r boundary for some r d with r q q in particular the existence of an l q helmholtz projection is assumed

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

erratum to perturbation and interpolation theorems for the h calculus with applications to differential operators we prove comparison theorems for the h calculus that allow to transfer the property of having a bounded h calculus from one sectorial operator to another the basic technical ingredient are suitable square function estimates these comparison results provide a new approach to perturbation theorems for the h calculus in a variety of situations suitable for applications our square function estimates also give rise to a new interpolation method the rademacher interpolation we show that a bounded h calculus is characterized by interpolation of the domains of fractional powers with respect to rademacher interpolation this leads to comparison and perturbation results for operators defined in interpolation scales such as the l p scale we apply the results to give new proofs on the h calculus for elliptic differential operators including schrodinger operators and perturbed boundary conditions as new results we prove that elliptic boundary value problems with bounded uniformly coefficients have a bounded h calculus in certain sobolev spaces and that the stokes operator on bounded domains with c 1 1 has a bounded h calculus in the helmholtz scale l p p 1

l q l r estimates for solutions of the nonstationary stokes equations in an exterior domain and the navier stokes initial value problems inl q spaces

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

mode localization and eigenvalue loci veering phenomena in disordered structures an investigation of the effects of disorder on the modes of vibration of nearly periodic structures is presented it is shown that in structures with close eigenvalues small structural irregularities result in both strong localization of the mode shapes and abrupt veering away or mutual repulsion of the loci of the eigenvalues when these are plotted against a parameter representing the disorder in the system perturbation methods for the eigenvalue problem are applied to predict the occurrence of strong localization and eigenvalue loci veering which are shown to be two manifestations of the same phenomenon also perturbation methods that handle the dramatic effects of small disorder are developed to analyze eigenvalue loci veering and strong localization two representative disordered nearly periodic structures are studied a mistuned assembly of coupled oscillators and a multi span beam with irregular spacing of the supports

gravity and magnetic inversion with minimization of a specific functional the nonuniqueness of gravity or magnetic data inversion is well known in order to remove ambiguity some authors have sought solutions minimizing a functional describing geometrical or physical properties last and kubik 1983 in particular developed a method explaining the observed anomaly by structures of minimum volume in this method the domain where anomalous sources are searched is divided into elementary prisms of a constant density or susceptibility contrast each elementary contrast is allowed to vary individually thus a contrast distribution is computed the search for this kind of solution leads in general to geologically more appropriate bodies but exceptions do occur in this paper the technique is broadened to include the search for solutions minimizing the moment of inertia with respect to the center of gravity or with respect to a given dip line passing through it the resulting structures are both deeper and more compact precisely as is required in specific cases theoretical and

eigensolutions of perturbed nearly defective matrices abstract perturbation methods for approximating eigenvalues and eigenvectors of perturbed nearly defective mistuned systems have been discussed it is shown that the mistuning causes taylor expansions to be not uniformly valid even in small intervals of the perturbation parameter subsequently rendering them useless for practical purposes the problem is overcome by starting the perturbation expansion from an exactly defective tuned system close to the mistuned one asymptotic expansions are then obtained in terms of fractional powers of two parameters the modification and the mistuning parameter however as the tuned system is unknown an inverse problem has to be solved in order to determine it an algorithm valid for the particular but frequent case of several couples of nearly coincident eigenvalues has been detailed for this problem an outline of a more general case is also given illustrative examples are presented

comments on curve veering in eigenvalue problems abstract the dependence of eigenvalues on a system parameter is frequently illustrated by a family of loci when two loci approach each other they often cross or abruptly diverge the latter case called curve veering has been observed in approximate solutions associated with discretized models the influence of discretization in producing curve veering has raised doubt on the validity of many approximate solutions the existence of curve veering in continuous models is illustrated by presenting the exact solution of an elementary eigenvalue problem veering is then examined in a general eigenvalue problem criteria are established to distinguish veerings from crossings in both continuous and discretized models the application of the criteria is illustrated by examples

mode localization and frequency veering in a non conservative mechanical system with dissimilar components abstract a singular perturbation method is applied in carrying out an investigation on the connection between frequency veering and mode localization phenomena in a two degree of freedom damped linear oscillator the components of the system examined are dissimilar and as a result the modes are localized for most of the combinations of the parameters however the natural frequencies become close leading to curve veering only for some special combinations of the parameters a typical mathematical consequence is that regular perturbation expansions of the natural frequencies in terms of a single parameter break down in the vicinity of curve veering in the present study damping effects are included and this breakdown is shown to happen for the natural frequencies as well as the damping ratios of the system examined this difficulty is eventually overcome by applying the method of matched asymptotic expansions accurate uniformly valid expressions are derived for both the frequencies of free oscillation and the corresponding damping ratios

on the mathrm l p theory of the navier stokes equations on three dimensional bounded lipschitz domains on a bounded lipschitz domain omega subset mathbb r d d ge 3 we continue the study of shen arch ration mech anal 205 2 395 424 2012 and of kunstmann and weis j evol equ 387 409 2016 of the stokes operator on mathrm l p sigma omega we employ their results in order to determine the domain of the square root of the stokes operator as the space mathrm w 1 p 0 sigma omega for frac 1 p frac 1 2 frac 1 2d varepsilon and some varepsilon 0 this characterization provides gradient estimates as well as mathrm l p mathrm l q mapping properties of the corresponding semigroup in the three dimensional case this provides a means to show the existence of solutions to the navier stokes equations in the critical space mathrm l infty 0 infty mathrm l 3 sigma omega whenever the initial velocity is small in the mathrm l 3 norm finally we present a different approach to the mathrm l p theory of the navier stokes equations by employing the maximal regularity proven by kunstmann and weis j evol equ 387 409 2016

global well posedness of the two dimensional exterior navier stokes equations for non decaying data we prove global well posedness of the two dimensional exterior navier stokes equations for bounded initial data with a finite dirichlet integral subject to the non slip boundary condition as an application we construct global solutions for asymptotically constant initial data and arbitrary large reynolds numbers

h calculus for the stokes operator on lq spaces it is proved that the stokes operator on a bounded domain an exterior domain or a perturbed half space admits a bounded h calculus on lq if q 1

inhomogeneous navier stokes equations in the half space with only bounded density abstract in this paper we establish the global existence and uniqueness of solutions to the inhomogeneous navier stokes system in the half space the initial density only has to be bounded and close enough to a positive constant the initial velocity belongs to some critical besov space and the l norm of the inhomogeneity plus the critical norm to the horizontal components of the initial velocity has to be very small compared to the exponential of the norm to the vertical component of the initial velocity with a little bit more regularity for the initial velocity those solutions are proved to be unique in the last section of the paper our results are partially extended to the bounded domain case

an existence theorem for the navier stokes flow in the exterior of a rotating obstacle we consider the three dimensional navier stokes initial value problem in the exterior of a rotating obstacle it is proved that a unique solution exists locally in time if the initial velocity possesses the regularity l 1 2 this regularity assumption is the same as that in the famous paper of fujita kato an essential step for the proof is the deduction of a certain smoothing property together with estimates near t 0 of the semigroup which is not an analytic one generated by the operator in the space l 2 where stands for the angular velocity of the rotating obstacle and p denotes the projection associated with the helmholtz decomposition

linear inversion of gravity data for 3 d density distributions we have developed an improved levenburg marquart technique to rapidly invert bouguer gravity data for a 3 d density distribution as a source of the observed field this technique is designed to replace tedious forward modeling with an automatic solver that determines density models constrained by geologic information supplied by the user where such information is not available objective models are generated the technique estimates the density distribution within the source volume using a least squares inverse solution that is obtained iteratively by singular value decomposition using orthogonal decomposition of matrices with sequential householder transformations the source volume is subdivided into a series of right rectangular prisms of specified size but of unknown density this discretization allows the construction of a system of linear equations relating the observed gravity field to the unknown density distribution convergence of the solution to the system is tightly controlled by a damping pa