on a time dependent grade two fluid model in two dimensions we prove the existence of a weak solution of a time dependent grade two fluid model in a plane lipschitz domain and uniqueness of the solution in a convex polygon the method of proof is constructive and can be adapted to the numerical analysis of finite element schemes for solving this problem numerically

the camassa holm equation as a geodesic flow on the diffeomorphism group misiolek j geom phys 24 203 208 1998 has shown that the camassa holm equation is a geodesic flow on the bott virasoro group in this paper it is shown that the camassa holm equation for the case 0 is the geodesic spray of the weak riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating the h1 inner product over the entire group this paper uses the right trivialization technique to rigorously verify that the euler poincare theory for lie groups can be applied to diffeomorphism groups the observation made in this paper has led to physically meaningful generalizations of the ch equation to higher dimensional manifolds

finite difference solutions of a non linear schrodinger equation abstract this paper describes a finite difference method to approximate a schrodinger equation with a power non linearity a special case of this equation is currently used to model the propagation of a laser beam in a plasma the main feature of the method we present is that it satisfies a discrete analogue of some important conservation laws of the equations we present numerical results which show in particular the propagation and the formation of solitions in the one dimensional case

strong solution for a stochastic model of two dimensional second grade fluids existence uniqueness and asymptotic behavior abstract we investigate a stochastic evolution equation for the motion of a second grade fluid filling a bounded domain of r 2 global existence and uniqueness of strong probabilistic solution is established in contrast to previous results on this model we show that the sequence of galerkin approximation converges in mean square to the exact strong probabilistic solution of the problem we also give two results on the long time behavior of the solution mainly we prove that the strong solution of our stochastic model converges exponentially in mean square to the stationary solution of the time independent second grade fluids equations if the deterministic part of the external force does not depend on time if the deterministic forcing term explicitly depends on time then the strong probabilistic solution decays exponentially in mean square

time domain model reduction of time delay systems via orthogonal polynomial expansions abstract this paper investigates model reduction of time delay systems in the time domain based on the expansion of systems over orthogonal polynomials time delay systems are first expanded over generalized laguerre polynomials the nice properties of generalized laguerre polynomials lead to a direct system expansion and a sylvester equation with special structures which enables an efficient calculation of laguerre coefficients of systems projection methods are then adopted to generate reduced models and we show that a desired number of laguerre coefficients are preserved by reduced models further we extend the proposed method to general orthogonal polynomials where the relationship between taylor expansion and orthogonal polynomial expansion is examined to achieve the expansion of time delay systems systems with multiple delays and delays appearing in the derivative of the states are also discussed two numerical examples are simulated to showcase the efficiency of our approach

on the existence and uniqueness of solutions to stochastic three dimensional lagrangian averaged navier stokes equations we prove the existence and uniqueness of solutions for a stochastic version of the threedimensional lagrangian averaged navier stokes equation in a bounded domain to this end we previously prove some existence and uniqueness results for an abstract stochastic equation and justify that our model falls within this framework

exponential mixing for stochastic model of two dimensional second grade fluids abstract in this paper we establish the exponential mixing property of stochastic model for the incompressible second grade fluids the general criterion established by odasso 2008 plays an important role

on a leray α model of turbulence in this paper we introduce and study a new model for three dimensional turbulence the leray model this model is inspired by the lagrangian averaged navier stokes model of turbulence also known navier stokes model or the viscous camassa holm equations as in the case of the lagrangian averaged navier stokes model the leray model compares successfully with empirical data from turbulent channel and pipe flows for a wide range of reynolds numbers we establish here an upper bound for the dimension of the global attractor the number of degrees of freedom of the leray model of the order of l l d 12 7 where l is the size of the domain and l d is the dissipation length scale this upper bound is much smaller than what one would expect for three dimensional models i e l l d 3 this remarkable result suggests that the leray model has a great potential to become a good sub grid scale large eddy simulation model of turbulence we support this observation by studying analytically and computationally the energy spectrum and show that in addition to the usual k 5 3 kolmogorov power law the inertial range has a steeper power law spectrum for wavenumbers larger than 1 finally we propose a prandtl like boundary layer model induced by the leray model and show a very good agreement of this model with empirical data for turbulent boundary layers

strong solutions for a stochastic model of two dimensional second grade fluids driven by levy noise abstract in this paper we consider stochastic second grade fluids driven by levy noise on a bounded domain of r 2 global existence and uniqueness of strong probabilistic solutions are established which improves the existing corresponding results for martingale solutions in literature

geometry and curvature of diffeomorphism groups withh1metric and mean hydrodynamics holm marsden and ratiu adv in math 137 1998 1 81 derived a new model for the mean motion of an ideal fluid in euclidean space given by the equationv t u t v t 2 u t t u t grad p t where div u 0 andv 1 2 u in this model the momentumvis transported by the velocityu with the effect that nonlinear interaction between modes corresponding to length scales smaller than is negligible we generalize this equation to the setting of ann dimensional compact riemannian manifold the resulting equation is the euler poincare equation associated with the geodesic flow of theh1right invariant metric on ds the group of volume preserving hilbert diffeomorphisms of classhs we prove that the geodesic spray is continuously differentiable fromtds m intottds m so that a standard picard iteration argument proves existence and uniqueness on a finite time interval our goal in this paper is to establish the foundations for lagrangian stability analysis following arnold ann inst grenoble16 1966 319 361 to do so we use submanifold geometry and prove that the weak curvature tensor of the right invarianth1metric on ds is a bounded trilinear map in thehstopology from which it follows that solutions to jacobi s equation exist using such solutions we are able to study the infinitesimal stability behavior of geodesics

well posedness of stochastic partial differential equations with lyapunov condition abstract in this paper we show the existence and uniqueness of strong solutions for a large class of spde where the coefficients satisfy the local monotonicity and lyapunov condition one sided linear growth condition moreover some new invariance result and stronger regularity estimate are also established for the solutions as examples the main result is applied to stochastic tamed 3d navier stokes equations stochastic generalized curve shortening flow singular stochastic p laplace equations stochastic fast diffusion equations stochastic burgers type equations and stochastic reaction diffusion equations

the camassa holm equations and turbulence abstract in this paper we will survey our results on the camassa holm equations and their relation to turbulence as discussed in s chen c foias d d holm e olson e s titi s wynne the camassa holm equations as a closure model for turbulent channel and pipe flow phys rev lett 81 1998 5338 s chen c foias d d holm e olson e s titi s wynne a connection between the camassa holm equations and turbulent flows in channels and pipes phys fluids in press in particular we will provide a more detailed mathematical treatment of those equations for pipe flows which yield accurate predictions of turbulent flow profiles for very large reynolds numbers there are many facts connecting the camassa holm equations to turbulent fluid flows the dimension of the attractor agrees with the heuristic argument based on the kolmogorov statistical theory of turbulence the statistical properties of the energy spectrum agree in numerical simulation with the kolmogorov power law furthermore comparison of mean flow profiles for turbulent flow in channels and pipes given by experimental and numerical data show acceptable agreement with the profile of the corresponding solution of the camassa holm equations

krylov subspace methods for reduced order modeling in circuit simulation the simulation of electronic circuits involves the numerical solution of very large scale sparse in general nonlinear systems of differential algebraic equations often the size of these systems can be reduced considerably by replacing the equations corresponding to linear subcircuits by approximate models of much smaller state space dimension in this paper we describe the use of krylov subspace methods for generating such reduced order models of linear subcircuits particular emphasis is on reduced order modeling techniques that preserve the passivity of linear rlc subcircuits

regularity of the global attractor and finite dimensional behavior for the second grade fluid equations abstract this paper is devoted to the large time behavior and especially to the regularity of the global attractor of the second grade fluid equations in the two dimensional torus we first recall that for any size of the material coefficient 0 these equations are globally well posed and admit a compact global attractor a in h 3 t 2 2 we prove that for any 0 there exists 0 such that a belongs to h 3 t 2 2 if the forcing term is in h 1 t 2 2 we also show that this attractor is contained in any sobolev space h 3 m t 2 2 provided that is small enough and the forcing term is regular enough these arguments lead also to a new proof of the existence of the compact global attractor a furthermore we prove that on a the second grade fluid system can be reduced to a finite dimensional system of ordinary differential equations with an infinite delay moreover the existence of a finite number of determining modes for the equations of the second grade fluid is established

exponential mixing for stochastic pdes the non additive case we establish a general criterion which ensures exponential mixing of parabolic stochastic partial differential equations spde driven by a non additive noise which is white in time and smooth in space we apply this criterion on two representative examples 2d navier stokes ns equations and complex ginzburg landau cgl equation with a locally lipschitz noise due to the possible degeneracy of the noise doob theorem cannot be applied hence a coupling method is used in the spirit of kuksin and shirikyan j math pures appl 1 567 602 2002 and mattingly commun math phys 230 421 462 2002

large deviations and transitions between equilibria for stochastic landau lifshitz gilbert equation we study a stochastic landau lifshitz equation on a bounded interval and with finite dimensional noise we first show that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property next we prove the large deviations principle for the small noise asymptotic of solutions using the weak convergence method an essential ingredient of the proof is the compactness or weak to strong continuity of the solution map for a deterministic landau lifschitz equation when considered as a transformation of external fields we then apply this large deviations principle to show that small noise can cause magnetisation reversal we also show the importance of the shape anisotropy parameter for reducing the disturbance of the solution caused by small noise the problem is motivated by applications from ferromagnetic nanowires to the fabrication of magnetic memories

equivalence of laguerre based model order reduction and moment matching the equivalence between the classical moment matching and the laguerre based reduction approaches is shown in addition this equivalence is generalized to include a family of coefficients known as generalized markov parameters

decompositions of multiattribute utility functions based on convex dependence we describe a method of assessing von neumann morgenstern utility functions on a two attribute space and its extension to n attribute spaces first we introduce the concept of convex dependence between two attributes where we consider the change of shapes of conditional utility functions then we establish theorems which show how to decompose a two attribute utility function using the concept of convex dependence this concept covers a wide range of situations involving trade offs the convex decomposition includes as special cases keeney s additive multiplicative decompositions fishburn s bilateral decomposition and bell s decomposition under the interpolation independence moreover the convex decomposition is an exact grid model which was axiomatized by fishburn and farquhar finally we extend the convex decomposition theorem from two attributes to an arbitrary number of attributes

2d stochastic navier stokes equations driven by jump noise abstract in the paper we are studying the existence and uniqueness of the solution of an abstract nonlinear equation driven by a multiplicative noise of levy type our result is formulated in an abstract setting this type of equation covers the stochastic 2d navier stokes equations the 2d stochastic magneto hydrodynamic equations the 2d stochastic boussinesq model for the benard convection the 2d stochastic magnetic bernard problem the 3d stochastic leray model for the navier stokes equations and several stochastic shell models of turbulence

the anisotropic lagrangian averaged euler and navier stokes equations the purpose of this paper is twofold first we give a derivation of the lagrangian averaged euler lae and navier stokes lans equations this theory involves a spatial scale and the equations are designed to accurately capture the dynamics of the euler and navier stokes equations at length scales larger than while averaging the motion at scales smaller than the derivation involves an averaging procedure that combines ideas from both the material lagrangian and spatial eulerian viewpoints this framework allows the use of a variant of g i taylor s frozen turbulence hypothesis as the foundation for the model equations more precisely the derivation is based on the strong physical assumption that fluctutations are frozen into the mean flow in this article we use this hypothesis to derive the averaged lagrangian for the theory and all the terms up to and including order 2 are accounted for

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

existence of a martingale solution of the stochastic navier stokes equations in unbounded 2d and 3d domains abstract stochastic navier stokes equations in 2d and 3d possibly unbounded domains driven by a multiplicative gaussian noise are considered the noise term depends on the unknown velocity and its spatial derivatives the existence of a martingale solution is proved the construction of the solution is based on the classical faedo galerkin approximation the compactness method and the jakubowski version of the skorokhod theorem for nonmetric spaces moreover some compactness and tightness criteria in nonmetric spaces are proved compactness results are based on a certain generalization of the classical dubinsky theorem

incompressible euler as a limit of complex fluid models with navier boundary conditions in this article we study the limit 0 of solutions of the euler equations and the limit 0 of solutions of the second grade fluid equations in a bounded domain both in two and in three space dimensions we prove that solutions of the complex fluid models converge to solutions of the incompressible euler equations in a bounded domain with navier boundary conditions under the hypothesis that there exists a uniform time of existence for the approximations independent of and this additional hypothesis is not necessary in 2d where global existence is known and for axisymmetric flows without swirl for which we prove global existence our conclusion is strong convergence in l2 to a solution of the incompressible euler equations assuming smooth initial data

multiattribute risk linearity in applying multiattribute utility theory one typically assumes the condition of mutual utility independence and uses a multiattribute utility function that is additive or multiplicative this paper proposes a model in which this condition is not satisfied that is the risk attitude for one attribute depends on the amounts of the other attributes we introduce a condition on this dependence called multiattribute risk linearity that implies a logarithmic form for the multiattribute utility function and we show that a logarithmic utility function can be determined by assessment procedures that require the same number of indifference assessments as those for a multiplicative utility function

on the convergence rate of the euler α an inviscid second grade complex fluid model to the euler equations we study the convergence rate of the solutions of the incompressible euler an inviscid second grade complex fluid equations to the corresponding solutions of the euler equations as the regularization parameter approaches zero first we show the convergence in hs s n 2 1 in the whole space and that the smooth euler solutions exist at least as long as the corresponding solution of the euler equations next we estimate the convergence rate for two dimensional vortex patch with smooth boundaries

model order reduction methods for coupled systems in the time domain using laguerre polynomials in this paper based on laguerre polynomials we present new methods for model reduction of coupled systems in the time domain by appropriately selected projection matrices a reduced order system is produced to retain the topology structure of the original system meanwhile it preserves a desired number of laguerre coefficients of the system s output thereby providing good approximation accuracy we also study the two sided projection method in the time domain as well as the stability of reduced order systems two numerical examples are used to illustrate the efficiency of the proposed methods

finite degree utility independence when u is a von neumann morgenstern utility function on x y y is utility independent of x if u can be written as ux y fxgy ax with f positive this paper introduces a fundamental extension of utility independence that is based on induced indifference relations over gambles on one factor when the level of the other factor is fixed it is proved that y is degree n utility independent of x if and only if u can be written as ux y ftxgty fnxgny ax and cannot be written in a similar way with fewer than n products of single factor functions a similar theorem holds when the roles of y and x are interchanged it follows that if y is degree n utility independent of x then x is degree m utility independent of y for some m n 1 n n 1 it is then shown that u can be represented in terms of n conditional utility functions on y m conditional utility functions on x and at most n 1m 1 scaling constants

stochastic equations in hilbert space with application to navier stokes equations in any dimension abstract we give an existence theorem for an abstract nonlinear stochastic evolution equation in a hilbert space the result is applicable to the stochastic navier stokes equation in any dimension with a nonlinear noise term

convergence of the 2d euler α to euler equations in the dirichlet case indifference to boundary layers abstract in this article we consider the euler system as a regularization of the incompressible euler equations in a smooth two dimensional bounded domain for the limiting euler system we consider the usual non penetration boundary condition while for the euler regularization we use velocity vanishing at the boundary we also assume that the initial velocities for the euler system approximate in a suitable sense as the regularization parameter 0 the initial velocity for the limiting euler system for small values of this situation leads to a boundary layer which is the main concern of this work our main result is that under appropriate regularity assumptions and despite the presence of this boundary layer the solutions of the euler system converge as 0 to the corresponding solution of the euler equations in l 2 in space uniformly in time we also present an example involving parallel flows in order to illustrate the indifference to the boundary layer of the 0 limit which underlies our work

grade two fluids on non smooth domain driven by multiplicative noise existence uniqueness and regularity the author presented a lecture on the subject of this paper at the workshop on numerics for stochastic partial differential equations and their applications held at the johann radon institute for computational and applied mathematics ricam linz austria during the period 04 16 december 2016

on time domain simulation of lossless transmission lines with nonlinear terminations a time domain approach is presented to solve nonlinear circuits with lossless transmission lines mathematically the circuits are described by a special kind of nonlinear differential algebraic equations daes with multiple constant delays in order to directly compute these delay systems in time domain decoupling by waveform relaxation wr is applied to the systems for the relaxation based method we provide a new convergence proof numerical experiments are given to illustrate the novel approach

stochastic generalized magnetohydrodynamics equations with not regular multiplicative noise well posedness and invariant measure abstract in this paper we study a stochastic magnetohydrodynamics mhd system with fractional diffusion and resistivity 0 in r d d 2 3 our main goal is to identify the conditions on under which we can prove the existence of a martingale solution the pathwise uniqueness of solution and the existence of invariant measure when the noises are multiplicative and take values in functional space bigger than the space of square integrable functions roughly speaking we prove that if 1 0 and the driving noises take values in h then the stochastic system has at least a weak martingale solution we also establish the pathwise uniqueness of solution whenever d 2 finally under the latter condition and under the addition of linear damping to the equations we are able to establish the existence of an invariant measure

a semi discrete scheme for the stochastic nonlinear schrodinger equation we study the convergence of a semi discretized version of a numerical scheme for a stochastic nonlinear schrodinger equation the nonlinear term is a power law and the noise is multiplicative with a stratonovich product our scheme is implicit in the deterministic part of the equation as is usual for conservative equations we also use an implicit discretization of the noise which is better suited to stratonovich products we consider a subcritical nonlinearity so that the energy can be used to obtain an a priori estimate however in the semi discrete case no ito formula is available and we have to use a discrete form of this tool also in the course of the proof we need to introduce a cut off of the diffusion coefficient which allows to treat the nonlinearity then we prove convergence by a compactness argument due to the presence of noise and to the implicit discretization of the noise this is rather complicated and technical we finally obtain convergence of the discrete solutions in various topologies

martingale and stationary solutions for stochastic navier stokes equations we prove the existence of martingale solutions and of stationary solutions of stochastic navier stokes equations under very general hypotheses on the diffusion term the stationary martingale solutions yield the existence of invariant measures when the transition semigroup is well defined the results are obtained by a new method of compactness

an efficient hybrid reduction method for time delay systems using hermite expansions abstractthis paper presents an efficient model reduction method for time delay systems in the time domain we expand the systems under a hermite polynomial basis and show that hermite coefficients of the expansion are determined by a linear equation thus can be calculated efficiently such linear relationship is well taken in the projection methods of model reduction and reduced models are generated to preserve a desired number of hermite coefficients in the time domain in contrast to other existing techniques which aim at approximating the transfer function of time delay systems in the frequency domain we also exploit two sided projections for time delay systems leading to a hybrid reduction method which generates reduced models sharing the nice properties both in the time and frequency domains two numerical examples illustrate the feasibility and effectiveness of the approach