global well posedness of incompressible inhomogeneous fluid systems with bounded density or non lipschitz velocity in this paper we first prove the global existence of weak solutions to the d dimensional incompressible inhomogeneous navier stokes equations with initial data a 0 in l infty mathbb r d u 0 u 0 h u 0 d in dot b 1 frac d p p r mathbb r d which satisfy mu a 0 l infty u 0 h dot b 1 frac d p p r rm exp c r mu 2r u 0 d dot b 1 frac d p p r 2r leqq c 0 mu for some positive constants c 0 c r and 1 p d 1 r the regularity of the initial velocity is critical to the scaling of this system and is general enough to generate non lipschitz velocity fields furthermore with additional regularity assumptions on the initial velocity or on the initial density we can also prove the uniqueness of such a solution we should mention that the classical maximal l p l q regularity theorem for the heat kernel plays an essential role in this context

regularity of renormalized solutions in the boltzmann equation with long range interactions it is well established that renormalized solutions of the boltzmann equation enjoy some kind of regularity or at least compactness in the velocity variable when the angular collision kernel is nonintegrable however obtaining explicit estimates in convenient and natural functional settings proves rather difficult in this work we derive a velocity smoothness estimate from the a priori control of the renormalized dissipation as a direct consequence of our result we show that in the presence of long range interactions any renormalized solution f t x v to the boltzmann equation satisfies locally for every and for some s 0 depending on p we also provide an application of this new estimate to the hydrodynamic limit of the boltzmann equation without cutoff 2012 wiley periodicals inc

on the decay and stability to global solutions of the 3 d inhomogeneous navier stokes equations in this chapter we consider the large time decay and stability to any given global smooth solution of the three dimensional inhomogeneous incompressible navier stokes equations

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

time domain model order reduction using general orthogonal polynomials for k power bilinear systems abstractin this paper we propose a model order reduction mor method based on general orthogonal polynomials for k power bilinear systems in the time domain constructing proper projection matrices by solving a series of linear equations a reduced k power bilinear system is produced which preserves the original coupled structure it can match several expansion coefficients of the original output then the error bound of our algorithm is also investigated moreover the stability of the reduced system is discussed as well finally two numerical examples are provided to illustrate the effectiveness of our algorithm

chebyshev legendre super spectral viscosity method for nonlinear conservation laws in this paper a super spectral viscosity method using the chebyshev differential operator of high order d s sqrt 1 x 2 opx s is developed for nonlinear conservation laws the boundary conditions are treated by a penalty method compared with the second order spectral viscosity method the super one is much weaker while still guaranteeing the convergence of the bounded solution of the chebyshev galerkin chebyshev collocation or legendre galerkin approximations to nonlinear conservation laws which is proved by compensated compactness arguments

convergence analysis of the chebyshev legendre spectral method for a class of fredholm fractional integro differential equations abstract in this paper we propose and analyze a spectral chebyshev legendre approximation for fractional order integro differential equations of fredholm type the fractional derivative is described in the caputo sense our proposed method is illustrated by considering some examples whose exact solutions are available we prove that the error of the approximate solution decays exponentially in l 2 norm

solutions of navier stokes maxwell systems in large energy spaces large weak solutions to navier stokes maxwell systems are not known to exist in their corresponding energy space in full generality here we mainly focus on the three dimensional setting of a classical incompressible navier stokes maxwell system and in an effort to build solutions in the largest possible functional spaces prove that global solutions exist under the assumption that the initial velocity and electromagnetic fields have finite energy and that the initial electromagnetic field is small in dot h s left mathbb r 3 right with s in left frac 12 frac 32 right we also apply our method to improve known results in two dimensions by providing uniform estimates as the speed of light tends to infinity the method of proof relies on refined energy estimates and a gr onwall like argument along with a new maximal estimate on the heat flow in besov spaces the latter parabolic estimate allows us to bypass the use of the so called chemin lerner spaces altogether which is crucial and could be of independent interest

remarks on infinite energy solutions of nonlinear wave equations abstract we first consider the existence of a solution of the critical semilinear wave equation in besov space which extends the results in p germain global infinite energy solutions of critical semilinear wave equation revista matematica iberoamericana 24 2 2008 463 497 to general dimensions next we derive the existence and uniqueness of global solutions for a semilinear wave equation in marcinkiewicz space

an extension of the spectral tau method for numerical solution of multi order fractional differential equations with convergence analysis the main purpose of this paper is to provide an efficient numerical approach for the fractional differential equations fdes based on a spectral tau method an extension of the operational approach of the tau method with the orthogonal polynomial bases is proposed to convert fdes to its matrix vector multiplication representation the fractional derivatives are described in the caputo sense the spectral rate of convergence for the proposed method is established in the l 2 norm we tested our procedure on several examples and observed that the obtained numerical results confirm the theoretical prediction of the exponential rate of convergence

second order spectral differentiation matrices the second order spectral differentiation matrix is required when spectral methods are used to solve second order differential equations for methods based on polynomials the precise form of this matrix will depend on the method used in each case however the eigenvalue with largest magnitude grows as o n 4 which has implications for the stability when explicit methods are used to solve time dependent problems this paper investigates the relationship between the different matrices and shows that they are related by a rank two update furthermore it is shown that the matrix can be chosen so that the magnitude of the largest eigenvalue grows as o n 2 rather than o n 4 and the properties of the underlying polynomials in this case are discussed

the large time behavior of solutions in the critical l p framework for compressible viscous and heat conductive gas flows in this paper we are concerned with non isentropic navier stokes equations governing compressible viscous and heat conductive gases we formulate an additional regularity assumption of low frequencies in rd d 3 and then establish the sharp time weighted inequality which allows us to get the time decay estimates of global strong solutions in the lp critical besov spaces precisely it is shown that if the initial data belong to some besov space b2 s1 with s1 1 d2 s0 s0 2dp d2 then the lp norm of the critical global solutions admits the time decay t s12 d2 12 1p in particular t d2p if s1 s0 which coincides with that of the heat kernel in the lp framework in comparison with the work of danchin and xu j math fluid mech 20 1641 1665 2018 the low frequency assumption is improved such that the regularity s1 belongs to the whole range 1 d2 s0

the stability of pseudospectral chebyshev methods abstract the stability of pseudospectral chebyshev methods is demonstrated for parabolic and hyperbolic problems with variable coefficients the choice of collocation points is discussed numerical examples are given for the case of variable coefficient hyperbolic equations author

lyapunov equations energy functionals and model order reduction of bilinear and stochastic systems we discuss the relation of a certain type of generalized lyapunov equations to gramians of stochastic and bilinear systems together with the corresponding energy functionals while gramians and energy functionals of stochastic linear systems show a strong correspondence to the analogous objects for deterministic linear systems the relation of gramians and energy functionals for bilinear systems is less obvious we discuss results from the literature for the latter problem and provide new characterizations of input and output energies of bilinear systems in terms of algebraic gramians satisfying generalized lyapunov equations in any of the considered cases the definition of algebraic gramians allows us to compute balancing transformations and implies model reduction methods analogous to balanced truncation for linear deterministic systems we illustrate the performance of these model reduction methods by showing numerical experiments for different bilinear systems

h2 optimal model order reduction by two sided technique on grassmann manifold via the cross gramian of bilinear systems abstractin this paper we discuss the optimal h2 model order reduction mor problem for bilinear systems the h2 optimal mor problem of bilinear systems is considered as the minimisation problem on grassmann manifold which is stored as a quotient space of the noncompact stifiel manifold grassmann manifold whose tangent space is endowed with a riemannian metric is a riemannian manifold in its tangent space equipped with the riemannian metric we derive the negative gradients of the cost function i e the steepest descent direction of the cost function after that the formulas of geodesic on grassmann manifold are given then we perform linear searches along geodesics to obtain the optimal solutions thereby a two sided mor iterative algorithm is proposed to construct an order reduced bilinear system which is used to simulate the output and input responses of the original bilinear system numerical examples demonstrate the effectiveness of our algorithm

computation of system balancing transformations and other applications of simultaneous diagonalization algorithms an algorithm is presented in this paper for computing state space balancing transformations directly from a state space realization the algorithm requires no squaring up or unnecessary matrix products various algorithmic aspects are discussed in detail a key feature of the algorithm is the determination of a contragredient transformation through computing the singular value decomposition of a certain product of matrices without explicitly forming the product other contragredient transformation applications are also described it is further shown that a similar approach may be taken involving the generalized singular value decomposition to the classical simultaneous diagonalization problem these svd based simultaneous diagonalization algorithms provide a computational alternative to existing methods for solving certain classes of symmetric positive definite generalized eigenvalue problems

model reduction by moment matching for linear and nonlinear systems the model reduction problem for single input single output linear and nonlinear systems is addressed using the notion of moment a re visitation of the linear theory allows to obtain novel results for linear systems and to develop a nonlinear enhancement of the notion of moment this in turn is used to pose and solve the model reduction problem by moment matching for nonlinear systems to develop a notion of frequency response for nonlinear systems and to solve model reduction problems in the presence of constraints on the reduced model connections between the proposed results projection methods the covariance extension problem and interpolation theory are presented finally the theory is illustrated by means of simple worked out examples and case studies

a low frequency assumption for optimal time decay estimates to the compressible navier stokes equations the global existence issue in critical spaces for compressible navier stokes equations was addressed by danchin invent math 141 579 614 2000 and then developed by charve and danchin arch rational mech anal 198 233 271 2010 chen et al commun pure appl math 63 1173 1224 2010 and haspot arch rational mech anal 202 427 460 2011 in more general lp setting the main aim of this paper is to exhibit more precisely time decay estimates of solutions constructed in the critical regularity framework to the best of our knowledge the low frequency assumption usually plays a key role in the large time asymptotics of solutions which was firstly observed by matsumura and nishida j math kyoto univ 20 67 104 1980 in the l 1 mathbb r d space we now claim a new low frequency assumption for barotropic compressible navier stokes equations which may be of interest in the mathematical analysis of viscous fluids precisely if the initial density and velocity additionally belong to some besov space dot b sigma 1 2 infty mathbb r d with the regularity sigma 1 in 1 d 2 2d p d 2 then a sharp time weighted inequality including enough time decay information can be available where optimal decay exponents for the high frequencies are exhibited the proof mainly depends on some non standard besov product estimates as a by product those optimal time decay rates of lq lr type are also captured in the critical framework

global well posedness for the 3d navier sokes equations with a large component of vorticity abstract in this paper we give a new approach to improve the leray s result concerning the cauchy problem to the 3d navier stokes equations in particular global well posedness with a large component of the initial vorticity is obtained our idea is considering the vorticity equations and using some suitable function spaces

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

incompressible limit of oldroyd b fluids in the whole space abstract this paper is dedicated to the study of compressible oldroyd b model in the framework of critical spaces we first prove the local well posedness for the compressible oldroyd b model with initial data 0 1 u 0 0 b 2 1 d 2 b 2 1 d 2 1 d b 2 1 d 2 d d the global well posedness is also established provided the initial data and coupling constant are sufficiently small on the basis of this we also consider the incompressible limit problem for ill prepared initial data we prove that as the mach number tends to zero the global solution to the compressible oldroyd b fluids converges to the solution to the corresponding incompressible model in some function spaces in the meanwhile the converge rates are obtained in some sense to fulfill our proof we give a new commutator estimate

a new frequency weighted balanced truncation method and an error bound in this paper a new frequency weighted model reduction method with an a priori error bound is proposed the method is a generalization of enns technique 1984 and yields stable models even when both input and output weightings are included the proposed method is compared with other existing methods using numerical examples

well posedness of the viscous boussinesq system in besov spaces of negative regular index s 1 this paper is concerned with well posedness of the natural convection in a viscous incompressible fluid we first prove that the n dimensional boussinesq system is well posed for initial data u0 0 either belonging to b 1 1 b 1 1 bp r 1 or to b 1 1 bp 1 with 1 r n2 0 and then we prove that this system is well posed for initial data belonging to b 1 1 b 1 1 bn2 1 1 bn2 1 1 well posedness bp rs lt bp rs

large global solutions to the navier stokes equations slowly varying in one direction in to previous papers by the authors classes of initial data to the three dimensional incompressible navier stokes equations were presented generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large the aim of this article is to provide new examples of arbitrarily large initial data giving rise to global solutions in the whole space contrary to the previous examples the initial data has no particular oscillatory properties but varies slowly in one direction the proof uses the special structure of the nonlinear term of the equation

spectral methods on arbitrary grids stable and spectrally accurate numerical methods are constructed on arbitrary grids for partial differential equations these new methods are equivalent to conventional spectral methods but do not rely on specific grid distributions specifically we show how to implement legendre galerkin legendre collocation and laguerre galerkin methodology on arbitrary grids

preconditioning on element interfaces for the p version finite element method and spectral element method in the preconditioned iterative method based on nonoverlapping domain decompositions the preconditioner is usually of block diagonal type to calculate its action on a residual vector it is required to solve a linear system of algebraic equations with the coefficient matrix being the schur complement block corresponding to the interface to reduce the computational cost the schur complement blocks in the preconditioners can be replaced with other simpler matrices which are called the interface preconditioners in this paper we discuss various interface preconditioners for the p version finite element method and spectral element method in r2 and r3 emphases are placed on 1 the lower and upper bounds of the condition numbers for the preconditioned schur complement blocks using the interface preconditioners which characterize the enlargement of the overall condition numbers and 2 the algorithms to efficiently implement the interface preconditioners numerical results are presented for the condition numbers using various interface preconditioners

global stability of large solutions to the 3d navier stokes equations we prove the stability of mildly decaying global strong solutions to the navier stokes equations in three space dimensions combined with previous results on the global existence of large solutions with various symmetries this gives the first global existence theorem for large solutions with approximately symmetric initial data the stability of unforced 2d flow under 3d perturbations is also obtained

an efficient chebyshev algorithm for the solution of optimal control problems in this paper we derive an efficient chebyshev algorithm for solving optimal control problems the chebyshev expansions are employed to approximate both the control and the state functions the discretizing process and the related techniques are unique compared to existing methods the optimal control problems are transformed into the resulting mathematical programming problems theoretical analysis is given to support the method further numerical examples and comparisons are presented to illustrate the efficiency of the method

global strong solutions for equations related to the incompressible viscoelastic fluids with a class of large initial data abstract we investigate the global strong solutions for a system related to the incompressible viscoelastic fluids of oldroyd b type with a class of large initial data e 0 e 0 i j n n v 0 v 01 v 0 n we obtain the global existence and uniqueness of a solution provided that e 0 h v 0 h are sufficiently small in b 2 1 n 2 1 b 2 1 n 2 b 2 1 n 2 1 while e 0 n and v 0 n are large where e 0 h e 0 i j n 1 n v 0 h v 01 v 0 n 1 and e 0 n e 0 n 1 e 0 n n

global well posedness in spatially critical besov space for the boltzmann equation the unique global strong solution in the chemin lerner type space to the cauchy problem on the boltzmann equation for hard potentials is constructed in a perturbation framework such a solution space is of critical regularity with respect to the spatial variable and it can capture the intrinsic properties of the boltzmann equation for the proof of global well posedness we develop some new estimates on the nonlinear collision term through the littlewood paley theory

existence of global strong solutions in critical spaces for barotropic viscous fluids this paper is dedicated to the study of viscous compressible barotropic fluids in dimension n 2 we address the question of the global existence of strong solutions for initial data close to a constant state having critical besov regularity first this article shows the recent results of charve and danchin arch ration mech anal 198 1 233 271 2010 and chen et al commun pure appl math 63 1173 1224 2010 with a new proof our result relies on a new a priori estimate for the velocity that we derive via the intermediary of the effective velocity which allows us to cancel out the coupling between the density and the velocity as in haspot well posedness in critical spaces for barotropic viscous fluids 2009 second we improve the results of charve and danchin 2010 and chen et al 2010 by adding as in charve and danchin 2010 some regularity on the initial data in low frequencies in this case we obtain global strong solutions for a class of large initial data which rely on the results of hoff arch rational mech anal 139 303 354 1997 hoff commun pure appl math 55 11 1365 1407 2002 and hoff j math fluid mech 7 3 315 338 2005 and those of charve and danchin 2010 and chen et al 2010 we conclude by generalizing these results for general viscosity coefficients

global solutions to the 3d incompressible nematic liquid crystal system abstract in this paper we consider the global well posedness of the cauchy problem of the 3d incompressible nematic liquid crystal system with initial data in the critical besov space b 2 1 1 2 r 3 b 2 1 3 2 r 3 in particular we prove that there exists a positive constant c 0 such that the nematic liquid crystal system has a unique global solution with initial data u 0 d 0 u 0 h u 0 3 d 0 which satisfies 1 1 d 0 d 0 b 2 1 3 2 1 u 0 h b 2 1 1 2 exp c 0 2 u 0 3 b 2 1 1 2 1 2 c 0 for some c 0 sufficiently small where d 0 is a constant vector with d 0 1 here and are two positive viscosity constants

canonical forms for bilinear systems abstract a method is presented which enables the transformation of the state equations of bilinear systems to canonical forms it is based on the properties of reachability and observability a distinct feature is that the derived canonical forms are unique with respect to their structure and determined by a minimum of non zero parameters

on the global wellposedness to the 3 d incompressible anisotropic navier stokes equations corresponding to the wellposedness result 2 for the classical 3 d navier stokes equations ns with initial data in the scaling invariant besov space mathcal b 1 frac3p p infty here we consider a similar problem for the 3 d anisotropic navier stokes equations ans where the vertical viscosity is zero in order to do so we first introduce the besov sobolev type spaces mathcal b frac12 frac12 4 and mathcal b frac12 frac12 4 t then with initial data in the scaling invariant space mathcal b frac12 frac12 4 we prove the global wellposedness for ans provided the norm of initial data is small enough compared to the horizontal viscosity in particular this result implies the global wellposedness of ans with high oscillatory initial data 1 2

optimal decay rates on the asymptotics of orthogonal polynomial expansions for functions of limited regularities in this paper new and optimal asymptotics on the decay of the coefficients for functions of limited regularity expanded in terms of jacobi and gegenbauer polynomial series are presented for a class of functions with interior singularities the decay of the coefficient is of the same asymptotic order for arbitrary alpha beta 1 which confirms that the decay of the coefficients in the jacobi polynomial series without normalization is a factor of sqrt n slower compared with the chebyshev expansion while for functions with boundary singularities the decay depends on alpha and beta with alpha beta 1 for gegenbauer expansion it is related to the parameter lambda whatever f with interior or boundary singularities all of these asymptotic analysis are optimal moreover under the optimal asymptotic analysis it derives that the truncated spectral expansions with some specific parameters can achieve the optimal convergence rates i e the same as the best polynomial approximation in the sense of absolute maximum error norm numerical examples illustrate the perfect coincidence with the estimates

on the stability in weak topology of the set of global solutions to the navier stokes equations let x be a suitable function space and let mathcal g subset x be the set of divergence free vector fields generating a global smooth solution to the incompressible homogeneous three dimensional navier stokes equations we prove that a sequence of divergence free vector fields converging in the sense of distributions to an element of mathcal g belongs to mathcal g if n is large enough provided the convergence holds anisotropically in frequency space typically this excludes self similar type convergence anisotropy appears as an important qualitative feature in the analysis of the navier stokes equations it is also shown that initial data which do not belong to mathcal g hence which produce a solution blowing up in finite time cannot have a strong anisotropy in their frequency support

stability to the global large solutions of 3 d navier stokes equations in this chapter we consider the stability of global large solutions to the 3 d classical isotropic navier stokes equations in the anisotropic sobolev spaces

global large solutions to 3 d inhomogeneous navier stokes system with one slow variable under a nonlinear smallness condition on the isotropic critical besov norm to the fluctuation of the initial density and the critical anisotropic besov norm of the horizontal components of the initial velocity which have to be exponentially small compared with the critical anisotropic besov norm to the third component of the initial velocity we investigate the global wellposedness of 3 d inhomogeneous incompressible navier stokes equations 1 2 in the critical besov spaces the novelty of this results is that the isotropic space structure to the inhomogeneity of the initial density function is consistent with the propagation of anisotropic regularity for the velocity field in the second part we apply the same idea to prove the global wellposedness of 1 2 with some large data which are slowly varying in one direction

a fast simple and stable chebyshev legendre transform using an asymptotic formula a fast simple and numerically stable transform for converting between legendre and chebyshev coefficients of a degree n polynomial in mathcal o n log n 2 log log n operations is derived the fundamental idea of the algorithm is to rewrite a well known asymptotic formula for legendre polynomials of large degree as a weighted linear combination of chebyshev polynomials which can then be evaluated by using the discrete cosine transform numerical results are provided to demonstrate the efficiency and numerical stability since the algorithm evaluates a legendre expansion at an n 1 chebyshev grid as an intermediate step it also provides a fast transform between legendre coefficients and values on a chebyshev grid

the chebyshev legendre method implementing legendre methods on chebyshev points a new collocation method for the numerical solution of partial differential equations is presented this method uses the chebyshev collocation points but because of the way the boundary conditions are implemented it has all the advantages of the legendre methods in particular l 2 estimates can be easily obtained for hyperbolic and parabolic problems

a rational spectral method for the kdv equation on the half line a petrov galerkin method using orthogonal rational functions is proposed for the korteweg de vries kdv equation on the half line with initial boundary values the nonlinear term and the right hand side term are treated by chebyshev rational interpolation explicitly and the linear terms are computed with the galerkin method implicitly such an approach is applicable using fast algorithms numerical results are presented for problems with both exponentially and algebraically decaying solutions respectively highlighting the performance of the proposed method

frequency interval model reduction of bilinear systems in this technical note a new method for model reduction of bilinear systems is presented the method is developed in particular for many applications in which one is interested to approximate a system in a given frequency interval to this end new generalized frequency interval gramians are introduced for bilinear systems it is shown that these gramians are the solutions to the so called frequency interval generalized lyapunov equations the conditions for these equations to be solvable are derived and an algorithm is proposed to solve such equations iteratively the method is further illustrated with the help of an illustrative example the numerical results show that the method is more accurate than its previous counterpart which is based on the ordinary gramians

domain decomposition method for the h p version finite element method domain decomposition method for the h p version of the finite element method in two and three dimensions are discussed using the framework of additive schwarz method various iterative methods are described with their condition numbers estimated further to reduce the cost for solving the sub problems on element interfaces different inexact interface solvers are proposed the effects on the overall condition number as well as their efficient implementation are analyzed

on the wellposedness of three dimensional inhomogeneous navier stokes equations in the critical spaces we prove the local wellposedness of three dimensional incompressible inhomogeneous navier stokes equations with initial data in the critical besov spaces without assumptions of small density variation furthermore if the initial velocity field is small enough in the critical besov space dot b 1 2 2 1 mathbb r 3 this system has a unique global solution

a survey of model reduction by balanced truncation and some new results balanced truncation is one of the most common model reduction schemes in this note we present a survey of balancing related model reduction methods and their corresponding error norms and also introduce some new results five balancing methods are studied 1 lyapunov balancing 2 stochastic balancing 3 bounded real balancing 4 positive real balancing and 5 frequency weighted balancing for positive real balancing we introduce a multiplicative type error bound moreover for a certain subclass of positive real systems a modified positive real balancing scheme with an absolute error bound is proposed we also develop a new frequency weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians two numerical examples are illustrated to verify the efficiency of the proposed methods

the optimal upper and lower bounds of convergence rates for the 3d navier stokes equations under large initial perturbation abstract this paper is concerned with the optimal algebraic convergence rates for leray weak solutions of the 3d navier stokes equations in morrey space it is shown that if the global leray weak solution u x t of the 3d navier stokes equations satisfies u l r 0 m p q r 3 2 r 3 p 2 3 2 p p q 2 then even for the large initial perturbation every weak solution v x t of the perturbed navier stokes equations converges algebraically to u x t with the optimal upper and lower bounds c 1 1 t 2 v t u t l 2 c 2 1 t 2 for large t 1 2 5 2 the findings are mainly based on the developed fourier splitting methods and iterative process

on the role of the besov spaces for the solutions of the generalized burgers equation in homogeneous sobolev spaces abstract the generalized burgers equation tu xxu xuk 1 0 with initial data u0 in homogeneous sobolev spaces are investigated the starting point of this work is the construction of solutions in l 0 b 1 k for small size initial data in b 1 k if in addition the initial data belongs to l p s then the obtained solution is actually in l 0 l p s

sobolev estimates for solutions of the transport equation and ode flows associated to non lipschitz drifts it is known after jabin j differ equ 260 5 4739 4757 2016 and alberti et al ann pde 5 1 9 2019 that ode flows and solutions of the transport equation associated to sobolev vector fields do not propagate sobolev regularity even of fractional order in this paper we improve the result at clop and jylha j differ equ 266 8 4544 4567 2019 and show that some kind of propagation of sobolev regularity happens as soon as the gradient of the drift is exponentially integrable we provide sharp sobolev estimates and new examples as an application of our main theorem we generalize a regularity result for the 2d euler equation obtained by bahouri and chemin in bahouri and chemin arch ration mech anal 127 2 159 181 1994

efficient spectral sparse grid methods and applications to high dimensional elliptic problems we develop in this paper some efficient algorithms which are essential to implementations of spectral methods on the sparse grid by smolyak s construction based on a nested quadrature more precisely we develop a fast algorithm for the discrete transform between the values at the sparse grid and the coefficients of expansion in a hierarchical basis and by using the aforementioned fast transform we construct two very efficient sparse spectral galerkin methods for a model elliptic equation in particular the chebyshev legendre galerkin method leads to a sparse matrix with a much lower number of nonzero elements than that of low order sparse grid methods based on finite elements or wavelets and can be efficiently solved by a suitable sparse solver ample numerical results are presented to demonstrate the efficiency and accuracy of our algorithms

zero mach number limit for compressible flows with periodic boundary conditions we address the question of convergence to the incompressible navier stokes equations for slightly compressible viscous flows with ill prepared initial data and periodic boundary conditions the case of the whole space has been studied in an earlier paper by the author the functional setting is very close to the one used by h fujita and t kato for incompressible flows for arbitrarily large initial data we show that the compressible flow with small mach number exists as long as the incompressible one does in particular it exists globally if the corresponding incompressible solution exists for all time we further state a convergence result for the slightly compressible solution filtered by the group of acoustics introduction an engineer would hardly make a distinction between incom pressible flows and slightly compressible flows indeed incompressible navier stokes equations are usually regarded as an appropriate model for describing water in the present paper we study whether such an approximation is relevant from a mathematical viewpoint in the case of periodic boundary conditions the case of the whole space has been treated in 8 after some convenient rescaling see for instance the appendix of 16 or the introduction of 21 the equations for a slightly compressible fluid read

a new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations a new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested this method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions stability and convergence results are proven for the chebyshev approximation of linear scalar hyperbolic equations the eigenvalues of this method applied to parabolic equations are shown to be real and negative

local in time results for local and non local capillary navier stokes systems with large data abstract in this article we study three capillary compressible models the classical local navier stokes korteweg system and two non local models for large initial data bounded away from zero and with a reference pressure state which is not necessarily stable p can be non positive we prove that these systems have a unique local in time solution and we study the convergence rate of the solutions of the non local models towards the local korteweg model the results are given for constant viscous coefficients and we explain how to extend them for density dependant coefficients

a legendre petrov galerkin and chebyshev collocation method for third order differential equations a legendre petrov galerkin lpg method for the third order differential equation is developed by choosing appropriate base functions the method can be implemented efficiently also this new approach enables us to derive an optimal rate of convergence in l2 norm the method is applied to some nonlinear problems such as the korteweg de vries kdv equation with the chebyshev collocation treatment for the nonlinear term it is a legendre petrov galerkin and chebyshev collocation lpg cc method numerical experiments are given to confirm the theoretical result

a stable penalty method for the compressible navier stokes equations i open boundary conditions the purpose of this paper is to present asymptotically stable open boundary conditions for the numerical approximation of the compressible navier stokes equations in three spatial dimensions the treatment uses the conservation form of the navier stokes equations and utilizes linearization and localization at the boundaries based on these variables the proposed boundary conditions are applied through a penalty procedure thus ensuring correct behavior of the scheme as the reynolds number tends to infinity the versatility of this method is demonstrated for the problem of a compressible flow past a circular cylinder

global well posedness for compressible viscoelastic fluids near equilibrium in this paper we prove local well posedness for compressible viscoelastic fluids of the oldroyd model under the assumption that the initial density is bounded away from zero and global well posedness near equilibrium the proof of global well posedness relies on some intrinsic properties of viscoelastic fluids and on a uniform estimate for a linearized hyperbolic parabolic system with convection terms

a schur method for balanced truncation model reduction it is shown that a not necessarily balanced state space realization of the moore reduced model can be computed directly without balancing via projections defined in terms of arbitrary bases for the left and right eigenspaces associated with the large eigenvalues of the product pq of the reachability and controllability grammians two specific methods for computing these bases are proposed one based on the ordered schur decomposition of pq and the other based on the cholesky factors of p and q the algorithms perform reliably even for nonminimal models

global solutions to the 3 d incompressible inhomogeneous navier stokes system in this paper we consider the global well posedness of the 3 d incompressible inhomogeneous navier stokes equations with initial data in the critical besov spaces a0 bq 13q r3 u0 u0h u03 bp 1 1 3p r3 for p q satisfying 1 q p 6 and 1q 1p 13 more precisely we prove that there exist two positive constants c0 c0 such that if a0 bq 13q u0h bp 1 1 3p exp c0 u03 bp 1 1 3p2 2 c0 then 1 3 has a unique global solution a l r bq 13q r3 u l r bp 1 1 3p r3 l1 r bp 11 3p r3 in particular this result implies the global well posedness result in abidi and paicu 2007 2 for the inhomogeneous navier stokes system with small initial data

global solutions to the 3 d incompressible anisotropic navier stokes system in the critical spaces in this paper we consider the global wellposedness of the 3 d incompressible anisotropic navier stokes equations with initial data in the critical besov sobolev type spaces mathcal b and mathcal b frac12 frac12 4 see definitions 1 1 and 1 2 below in particular we proved that there exists a positive constant c such that ans has a unique global solution with initial data u 0 u 0 h u 0 3 which satisfies u 0 h mathcal b exp bigl frac c nu 4 u 0 3 mathcal b 4 bigr leq c 0 nu or u 0 h mathcal b frac12 frac12 4 exp bigl frac c nu 4 u 0 3 mathcal b frac12 frac12 4 4 bigr leq c 0 nu for some c 0 sufficiently small to overcome the difficulty that gronwall s inequality can not be applied in the framework of chemin lerner type spaces widetilde l p t mathcal b we introduced here sort of weighted chemin lerner type spaces widetilde l 2 t f mathcal b for some apropriate l 1 function f t

an unconstrained h 2 model order reduction optimisation algorithm based on the stiefel manifold for bilinear systems abstractin this paper the optimal h 2 model order reduction mor problem for bilinear systems is explored the orthogonality constraint of the cost function generated by the h 2 mor error makes i

approximate solution of the sturm liouville problems with legendre galerkin chebyshev collocation method abstract in this paper we use the legendre galerkin chebyshev collocation method lgcc to approximate the eigenvalues of a regular sturm liouville problem with three kinds of boundary conditions unlike the classical chebyshev collocation method this method preserves the symmetry of the problem numerical results show that the method is of high accuracy

erratum to global wellposed problem for the 3 d incompressible anisotropic navier stokes equations in an anisotropic space in this paper we consider a global wellposed problem for the 3 d incompressible anisotropic navier stokes equations ans we prove the global wellposedness for ans provided the initial horizontal data are sufficient small in the scaling invariant besov sobolev type space b 0 frac 1 2 in particular the result implies the global wellposedness of ans with large initial vertical velocity

on the stability of global solutions to the 3d boussinesq system abstract in this paper we investigate the stability to any given global smooth solutions of the 3d boussinesq system assuming that u is a global solution called reference solution later with initial data 0 u 0 we prove that a small perturbation of 0 u 0 still generates a global smooth solution to the boussinesq system and this solution keeps close to the reference solution

largest well posed spaces for the general diffusion system with nonlocal interactions abstract the authors derive a general diffusion gd system with nonlocal interactions of special structure via energetic variational approach and observe that there exist two critical values of s i e s 1 2 1 for the nonlocal interactions where s 1 2 reflects how strong nonlocal property we have and s 1 affects the linearization and choice of initial data spaces the authors also establish the global existence and uniqueness of mild solution

a new dual petrov galerkin method for third and higher odd order differential equations application to the kdv equation a new dual petrov galerkin method is proposed analyzed and implemented for third and higher odd order equations using a spectral discretization the key idea is to use trial functions satisfying the underlying boundary conditions of the differential equations and test functions satisfying the dual boundary conditions the method leads to linear systems which are sparse for problems with constant coefficients and well conditioned for problems with variable coefficients our theoretical analysis and numerical results indicate that the proposed method is extremely accurate and efficient and most suitable for the study of complex dynamics of higher odd order equations

optimal error estimates of the chebyshev legendre spectral method for solving the generalized burgers equation in this paper the chebyshev legendre collocation method is applied to the generalized burgers equation optimal error estimate of the method is proved for the problem with the dirichlet boundary conditions also a legendre galerkin chebyshev collocation method is given for the generalized burgers equation the scheme is basically formulated in the legendre spectral form but with the nonlinear term being treated by the chebyshev collocation method so that the scheme can be implemented at chebyshev gauss lobatto points efficiently optimal order convergence is also obtained through coupling estimates in the l2 norm and the h1 norm

on h2 model reduction of bilinear systems the h 2 model reduction problem for continuous time bilinear systems is studied in this paper by defining the h 2 norm of bilinear systems in terms of the state space matrices the h 2 model reduction error is computed via the reachability or observability gramian necessary conditions for the reduced order bilinear models to be h 2 optimal are given the gradient flow approach is used to obtain the solution of the h 2 model reduction problem the formulation allows certain properties of the original models to be preserved in the reduced order models the model reduction procedure developed can also be applied to finite dimensional linear time invariant systems a numerical example is employed to illustrate the effectiveness of the proposed method

a triangular spectral element method using fully tensorial rational basis functions a rational approximation in a triangle is proposed and analyzed in this paper the rational basis functions in the triangle are obtained from the polynomials in the reference square through a collapsed coordinate transform optimal error estimates for the l 2 and h 1 0 orthogonal projections are derived with upper bounds expressed in the original coordinates in the triangle it is shown that the rational approximation is as accurate as the polynomial approximation in the triangle illustrative numerical results which are in agreement with the theoretical estimates are presented

gevrey regularity of mild solutions to the parabolic elliptic system of drift diffusion type in critical besov spaces abstract this paper is devoted to studying analyticity of mild solutions to the parabolic elliptic system of drift diffusion type with small initial data in critical besov spaces more precisely by using multilinear singular integrals and fourier localization argument we show that global in time mild solutions are gevrey regular i e e t 1 v e t 1 w l t b p r 2 n p r n l t 1 b p r n p r n for all 1 p 2 n and 1 r where 1 is the fourier multiplier whose symbol is given by 1 i 1 n i as a corollary we obtain decay estimates of mild solutions in critical besov spaces without using cumbersome recursive estimation of higher order derivatives

on gramians and balanced truncation of discrete time bilinear systems this paper obtains a balanced truncation model reduction method for discrete time bilinear systems after deriving the sufficient conditions for the discrete time bilinear systems to lead to bounded outputs for some bounded inputs we study the properties of the reachability and observability gramians the gramians can be computed by solving two generalized lyapunov equations by balancing the reachability and observability gramians a model reduction method is obtained some propositions of the reduced order models are given a numerical example is given to illustrate the effectiveness of the proposed method

a stable penalty method for the compressible navier stokes equations iii multidimensional domain decomposition schemes this paper concluding the trilogy develops schemes for the stable solution of wave dominated unsteady problems in general three dimensional domains the schemes utilize a spectral approximation in each subdomain and asymptotic stability of the semidiscrete schemes is established the complex computational domains are constructed by using nonoverlapping quadrilaterals in the two dimensional case and hexahedrals in the three dimensional space to illustrate the ideas underlying the multidomain method a stable scheme for the solution of the three dimensional linear advection diffusion equation in general curvilinear coordinates is developed the analysis suggests a novel yet simple stable treatment of geometric singularities like edges and vertices the theoretical results are supported by a two dimensional implementation of the scheme the main part of the paper is devoted to the development of a spectral multidomain scheme for the compressible navier stokes equations on conservation form and a unified approach for dealing with the open boundaries and subdomain boundaries is presented well posedness and asymptotic stability of the semidiscrete scheme is established in a general curvilinear volume with special attention given to a hexahedral domain the treatment includes a stable procedure for dealing with boundary conditions at a solid wall the efficacy of the scheme for the compressible navier stokes equations is illustrated by obtaining solutions to subsonic and supersonic boundary layer flows with various types of boundary conditions the results are found to agree with the solution of the compressible boundary layer equations