vanishing viscosity with short wave long wave interactions for multi d scalar conservation laws we consider a system coupling a multidimensional semilinear schrodinger equation and a multidimensional nonlinear scalar conservation law with viscosity which is motivated by a model of short wave long wave interaction introduced by benney 1977 we prove the global existence and uniqueness of the solution of the cauchy problem for this system we also prove the convergence of the whole sequence of solutions when the viscosity e and the interaction parameter approach zero so that o e1 2 we also indicate how to extend these results to more general systems which couple multidimensional semilinear systems of schrodinger equations with multidimensional nonlinear systems of scalar conservation laws mildly coupled

one dimensional transport equations with discontinuous coefficients we consider one dimensional linear transport equations with bounded but possibly discontinuous coefficient a the cauchy problem is studied from two different points of view in the first case we assume that a is piecewise continuous we give an existence result and a precise description of the solutions on the lines of discontinuity in the second case we assume that a satisfies a one sided lipschitz condition we give existence uniqueness and general stability results for backward lipschitz solutions and forward measure solutions by using a duality method we prove that the flux associated to these measure solutions is a product by some canonical representative of a key words linear transport equations discontinuous coefficients weak stability duality product of a measure by a discontinuous function nonnegative solutions 1991 mathematics subject classification primary 35f10 35b35 34a12 to appear in nonlinear analysis tma departement de mathematiques et applications umr cnrs 8553 ecole normale superieure et cnrs 45 rue d ulm 75230 paris cedex 05 france francois bouchut ens fr mathematiques applications et physique mathematique d orleans umr cnrs 6628 universite d orleans 45067 orleans cedex 2 france james math cnrs fr

entropy solutions for stochastic porous media equations we provide an entropy formulation for porous medium type equations with a stochastic non linear spatially inhomogeneous forcing well posedness and l1 contraction is obtained in the class of entropy solutions our scope allows for porous medium operators u m 1u for all m 1 and holder continuous diffusion nonlinearity with exponent 1 2

measure valued solutions to initial boundary value problems for certain systems of conservation laws existence and dynamics a framework for studying initial boundary value problems for systems of conservation laws in what concerns to the existence of measure valued solutions and their asymptotic behavior is developed here with the helpful introduction of a class of flux maps which allow a rather complete treatment of these questions including systems of practical importance as those arising in multiphase flow in porous media the systems of this class may in general admit umbilic points submanifolds where genuine nonlinearity fails as well as elliptic regions we prove the existence of measure valued solutions by using the vanishing viscosity method and also finite difference schemes the main result about the dynamics of the measure valued solutions is that for certain special boundary values given by constant states the time averages of these m v solutions converge weakly to the dirac measure concentrated at those states for a e space variable the rate of convergence of the time averages of the expected values can be estimated by properties of the flux maps only

the small dispersion limit of the korteweg de vries equation iii in parts i and ii we have derived explicit formulas for the distribution limit u of the solution of the kdv equation as the coefficient of uxxx tends to zero this formula contains n parameters 1 n whose values as well as whose number depends on x and t in section 4 we have shown that for t tb n 1 and the value of was determined in section 5 we have shown that the parameters i satisfy a nonlinear system of partial differential equations in part iii section 6 we show that for t large n 3 and we determine the asymptotic behavior of 1 2 3 and of u and u2 for t large the explicit formulas show that u and u2 are o t 1 and o t 2 respectively see formulas 6 2 and 6 24 in section 7 we study initial data whose value tends to zero as x and to 1 as x if one accepts some plausible guesses about the behavior of solutions with such initial data we derive an explicit formula for the solution and determine the large scale asymptotic behavior of the solution the function s is expressible in terms of complete elliptic integrals a similar formula is derived for u2 in section 8 we indicate how to extend the treatment of this series of papers to multihumped but still negative initial data

renormalized entropy solutions of scalar conservation laws with boundary condition we study an initial boundary value problem for a scalar conservation law u t div phi u f on a bounded domain existence and uniqueness of a renormalized entropy solution is established for general l 1 data phi is an element ofc r r n

on the representation of operators in bases of compactly supported wavelets this paper describes exact and explicit representations of the differential operators d n dx n n 1 2 cdots in orthonormal bases of compactly supported wavelets as well as the representations of the hilbert transform and fractional derivatives the method of computing these representations is directly applicable to multidimensional convolution operators also sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring o n log n operations for computing the wavelet coefficients of all n circulant shifts of a vector of the length n 2 n is constructed as an example of an application of this algorithm it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector see g beylkin r r coifman and v rokhlin comm pure appl math 44 1991 pp 141 183 may be reduced from o n to o log 2 n significant entries

quasiconvex elastodynamics weak strong uniqueness for measure valued solutions a weak strong uniqueness result is proved for measure valued solutions to the system of conservation laws arising in elastodynamics the main novelty brought forward by the present work is that the underlying stored energy function of the material is assumed strongly quasiconvex the proof employs tools from the calculus of variations to establish general convexity type bounds on quasiconvex functions and recasts them in order to adapt the relative entropy method to quasiconvex elastodynamics

generalized schultz iterative methods for the computation of outer inverses abstract we consider a general matrix iterative method of the type x k 1 x k p a x k for computing an outer inverse a r g n g 2 for given matrices a c m n and g c n m such that a r g n g c m here p x is an arbitrary polynomial of degree d the convergence of the method is proven under certain necessary conditions and the characterization of all methods having order r is given the obtained results provide a direct generalization of all known iterative methods of the same type moreover we introduce one new method and show a procedure how to improve the convergence order of existing methods this procedure is demonstrated on one concrete method and the improvement is confirmed by numerical examples

quasiconvexity and relaxation of nonconvex problems in the calculus of variations keywords quasiconvexity relaxation reference caa article 1982 002doi 10 1016 0022 1236 82 90046 5view record in web of science record created on 2008 11 25 modified on 2017 05 12

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

oscillations and concentrations in weak solutions of the incompressible fluid equations the authors introduce a new concept of measure valued solution for the 3 d incompressible euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations one application of the concepts developed here is the following important result a sequence of leray hopf weak solutions of the navier stokes equations converges in the high reynolds number limit to a measure valued solution of 3 d euler defined for all positive times the authors present several explicit examples of solution sequences for 3 d incompressible euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations an extensions of the concept of young measure is developed to incorporate these complex phenomena in the measure valued solutions constructed here

optimal factoring of fir filters new insights suggest that the most efficient fir digital filters can be created by using a scaled sequence of stages each representing a factor of the filter s transfer function a crucial capability for building such filters concerns finding the best fir filter factors then carefully scaling and sequencing them the efficiency of the resulting structure depends heavily upon obtaining such optimal factors we offer an algorithm to find scale and sequence optimally factored fir filters

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

on the design of highly accurate and efficient iir and fir filters we describe a systematic method for designing highly accurate and efficient infinite impulse response iir and finite impulse response fir filters given their specifications in our approach we first meet the specifications by constructing an iir filter with possibly a large number of poles we then construct for any given accuracy an optimal iir version of such filter with a minimal number of poles finally also for any given accuracy we convert the iir filter to an efficient fir filter cascade either serial or parallel since in this fir approximation the non causal part of the iir filter only introduces an additional delay as a function of the desired accuracy our iir construction does not have to enforce causality thus we obtain a simple method for constructing linear phase filters if the specifications so require all of these procedures are accomplished via robust fast algorithms we provide several illustrative examples of our method

generalized matrix inversion is not harder than matrix multiplication starting from the strassen method for rapid matrix multiplication and inversion as well as from the recursive cholesky factorization algorithm we introduced a completely block recursive algorithm for generalized cholesky factorization of a given symmetric positive semi definite matrix a r n x n we used the strassen method for matrix inversion together with the recursive generalized cholesky factorization method and established an algorithm for computing generalized 2 3 and 2 4 inverses introduced algorithms are not harder than the matrix matrix multiplication

multi dimensional scalar balance laws with discontinuous flux abstract we consider scalar balance laws with a dissipative source term the flux function may be discontinuous with respect to both the space variable x and the unknown quantity u we formulate the definition of entropy weak solutions and provide existence and uniqueness to the considered problem the problem is formulated in the framework of multi valued mappings the notion of entropy measure valued solutions is used to prove the so called contraction principle and comparison principle

well posedness of the transport equation by stochastic perturbation we consider the linear transport equation with a globally holder continuous and bounded vector field with an integrability condition on the divergence while uniqueness may fail for the deterministic pde we prove that a multiplicative stochastic perturbation of brownian type is enough to render the equation well posed this seems to be the first explicit example of a pde of fluid dynamics that becomes well posed under the influence of a multiplicative noise the key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of ito tanaka type

the caratheodory fejer method for recursive digital filter design a new technique for rational digital filter design is presented which is based on results in complex function theory due to takagi krein and others starting from a truncated or windowed impulse response the method computes the unique optimum rational chebyshev approximation with a prescribed number of stable poles both phase and magnitude are matched deleting the noncausal unstable part of the chebyshev approximation yields a stable approximation of specified order m n which is close to optimal in the chebyshev sense no iteration is involved except in the determination of an eigenvalue and eigenvector of the hankel matrix of impulse response coefficients in this paper the algorithm is specified and practical examples are discussed

a new convergence proof for finite volume schemes using the kinetic formulation of conservation laws we give a new convergence proof for finite volume schemes approximating scalar conservation laws the main ingredients of the proof are the kinetic formulation of scalar conservation laws a discrete entropy inequality and the velocity averaging technique

on nonlinear stochastic balance laws we are concerned with multidimensional stochastic balance laws we identify a class of nonlinear balance laws for which uniform spatial bv bound for vanishing viscosity approximations can be achieved moreover we establish temporal equicontinuity in l1 of the approximations uniformly in the viscosity coefficient using these estimates we supply a multidimensional existence theory of stochastic entropy solutions in addition we establish an error estimate for the stochastic viscosity method as well as an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux and random source functions various further generalizations of the results are discussed

well posedness theory for degenerate parabolic equations on riemannian manifolds abstract we consider the degenerate parabolic equation t u div f x u div div a x u x m t 0 on a smooth compact d dimensional riemannian manifold m g here for each u r x f x u is a vector field and x a x u is a 1 1 tensor field on m such that u a x u t x m is non decreasing with respect to u the fact that the notion of divergence appearing in the equation depends on the metric g requires revisiting the standard entropy admissibility concept we derive it under an additional geometry compatibility condition and as a corollary we introduce the kinetic formulation of the equation on the manifold using this concept we prove well posedness of the corresponding cauchy problem

the zero dispersion limit of the korteweg de vries equation for initial potentials with non trivial reflection coefficient the inverse scattering method is used to determine the distribution limit as 0 of the solution u x t of the initial value problem ut 6uux 2uxxx 0 u x 0 v x where v x is a positive bump which decays sufficiently fast as x x the case v x 0 has been solved by peter d lax and c david levermore 8 9 10 the computation of the distribution limit of u x t as 0 is reduced to a quadratic maximization problem which is then solved

computing the moore penrose inverse using its error bounds abstract a new iterative scheme for the computation of the moore penrose generalized inverse of an arbitrary rectangular or singular complex matrix is proposed the method uses appropriate error bounds and is applicable without restrictions on the rank of the matrix but it requires that the rank of the matrix is known in advance or computed beforehand the method computes a sequence of monotonic inclusion interval matrices which contain the moore penrose generalized inverse and converge to it successive interval matrices are constructed by using previous approximations generated from the hyperpower iterative method of an arbitrary order and appropriate error bounds of the moore penrose inverse a convergence theorem of the introduced method is established numerical examples involving randomly generated matrices are presented to demonstrate the efficacy of the proposed approach the main property of our method is that the successive interval matrices are not defined using principles of interval arithmetic but using accurately defined error bounds of the moore penrose inverse

on the strong pre compactness property for entropy solutions of a degenerate elliptic equation with discontinuous flux abstract under some non degeneracy condition we show that sequences of entropy solutions of a semi linear elliptic equation are strongly pre compact in the general case of a caratheodory flux vector the proofs are based on localization principles for h measures corresponding to sequences of measure valued functions

stochastic scalar conservation laws we introduce a notion of stochastic entropic solution a la kruzkov but with ito s calculus replacing deterministic calculus this results in a rich family of stochastic inequalities defining what we mean by a solution a uniqueness theory is then developed following a stochastic generalization of l1 contraction estimate an existence theory is also developed by adapting compensated compactness arguments to stochastic setting we use approximating models of vanishing viscosity solution type for the construction while the uniqueness result applies to any spatial dimensions the existence result in the absence of special structural assumptions is restricted to one spatial dimension only

finite volume relaxation schemes for multidimensional conservation laws we consider finite volume relaxation schemes for multidimensional scalar conservation laws these schemes are constructed by appropriate discretization of a relaxation system and it is shown to converge to the entropy solution of the conservation law with a rate of h 1 4 in l 0 t l loc 1 r d

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

continuity equations and ode flows with non smooth velocity in this paper we review many aspects of the well posedness theory for the cauchy problem for the continuity and transport equations and for the ordinary differential equation ode in this framework we deal with velocity fields that are not smooth but enjoy suitable weak differentiability assumptions we first explore the connection between the partial differential equation pde and the ode in a very general non smooth setting then we address the renormalization property for the pde and prove that such a property holds for sobolev velocity fields and for bounded variation velocity fields finally we present an approach to the ode theory based on quantitative estimates

convergence of a residual based artificial viscosity finite element method we present a residual based artificial viscosity finite element method to solve conservation laws the galerkin approximation is stabilized by only residual based artificial viscosity without any least squares supg or streamline diffusion terms we prove convergence of the method applied to a scalar conservation law in two space dimensions toward an unique entropy solution for implicit time stepping schemes

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

weak asymptotic decay via a relaxed invariance principle for a wave equation with nonlinear non monotone damping this paper considers the problem of asymptotic decay as t of solutions of the wave equation u tt u a x u t u t x a bounded open connected set in n n 1 with smooth boundary u 0 on the nonlinear function is not assumed to be globally lipschitz continuous 0 y 2 y n 1 0 y 1 y 1 y n 1 0 for all y n 1 is not assumed to be monotone in y 1 under additional restrictions on the kernel of conditions are given which imply that u u t converges to 0 0 weakly in h h 1 0 l 2 as t the work generalises earlier results of dafermos and haraux where strong decay in h as t was obtained in the case y 1 y n 1 q y 1 q monotone on

stochastic integration in umd banach spaces in this paper we construct a theory of stochastic integration of processes with values in l h e where h is a separable hilbert space and e is a umd banach space i e a space in which martingale differences are unconditional the integrator is an h cylindrical brownian motion our approach is based on a two sided l p decoupling inequality for umd spaces due to garling which is combined with the theory of stochastic integration of l h e valued functions introduced recently by two of the authors we obtain various characterizations of the stochastic integral and prove versions of the ito isometry the burkholder davis gundy inequalities and the representation theorem for brownian martingales

kinetic decomposition for singularly perturbed higher order partial differential equations abstract in this paper we consider singularly perturbed higher order partial differential equations we establish the condition under which the approximate solutions converge in a strong topology to the entropy solution of a scalar conservation laws using methodology developed in hwang and tzavaras comm partial differential equations 27 2002 1229 first we obtain the approximate transport equation for the given dispersive equations then using the averaging lemma we obtain the convergence

path dependent convex conservation laws abstract for convex scalar conservation laws in 1 d driven by a rough path z t in the sense of lions perthame and souganidis in 32 we show that it is possible to replace z t by a piecewise linear path and still obtain the same solution at a given time this result is connected to the spatial regularity of solutions we show that solutions are spatially lipschitz continuous for an a priori set of times depending on the path and the initial data fine properties of the map z u for a prescribed time are studied we provide a detailed description of the properties of the rough path z t that influence the solution this description is extracted by a factorization of the solution operator at time in a companion paper 24 we make use of the observations herein to construct computationally efficient numerical methods

formulation and convergence of the finite volume method for conservation laws on spacetimes with boundary we study nonlinear hyperbolic conservation laws posed on a differential n 1 manifold with boundary referred to as a spacetime and defined from a prescribed flux field of n forms depending on a parameter the unknown variable a class of equations proposed by lefloch and okutmustur far east j math sci 31 49 83 2008 our main result is a proof of the convergence of the finite volume method for weak solutions satisfying suitable entropy inequalities a main difference with previous work is that we allow for slices with a boundary and in addition introduce a new formulation of the finite volume method involving the notion of total flux functions under a natural global hyperbolicity condition on the flux field and the spacetime and by assuming that the spacetime admits a foliation by compact slices with boundary we establish an existence and uniqueness theory for the initial and boundary value problem and we prove a contraction property in a geometrically natural l 1 type distance

statistical solutions and onsager s conjecture abstract we prove a version of onsager s conjecture on the conservation of energy for the incompressible euler equations in the context of statistical solutions as introduced recently by fjordholm et al 2017 as a byproduct we also obtain an alternative proof for the conservative direction of onsager s conjecture for weak solutions under a weaker besov type regularity assumption than previously known

scalar conservation laws on constant and time dependent riemannian manifolds abstract in this paper we establish well posedness for scalar conservation laws on closed manifolds m endowed with a constant or a time dependent riemannian metric for initial values in l m in particular we show the existence and uniqueness of entropy solutions as well as the l 1 contraction property and a comparison principle for these solutions throughout the paper the flux function is allowed to depend on time and to have non vanishing divergence furthermore we derive estimates of the total variation of the solution for initial values in bv m and we give in the case of a time independent metric a simple geometric characterisation of flux functions that give rise to total variation diminishing estimates

scalar conservation laws with stochastic forcing we show that the cauchy problem for a randomly forced periodic multi dimensional scalar first order conservation law with additive or multiplicative noise is well posed it admits a unique solution characterized by a kinetic formulation of the problem which is the limit of the solution of the stochastic parabolic approximation

the nehari shuffle fir q filter design with guaranteed error bounds an approach to the problem of designing a finite impulse response filter of specified length q which approximates in uniform frequency l sub infinity norm a given desired possibly infinite impulse response causal stable filter transfer function is presented an algorithm independent lower bound on the achievable approximation error is derived and an approximation method that involves the solution of a fixed number of all pass nehari extension problems and is therefore called the nehari shuffle is presented upper and lower bounds on the approximation error are derived for the algorithm examples indicate that the method closely approaches the derived global lower bound the method is compared with the preuss complex remez exchange algorithm in some examples

a wise method for designing iir filters the problem of designing optimal digital iir filters with frequency responses approximating arbitrarily chosen complex functions is considered the real valued coefficients of the filter s transfer function are obtained by numerical minimization of carefully formulated cost which is referred here to as the weighted integral of the squared error wise criterion the wise criterion linearly combines the wls criterion that is used in the weighted least squares approach toward filter design and some time domain components the wls part of wise enforces the quality of the frequency response of the designed filter while the time domain part of the wise criterion restricts the positions of the filter s poles to the interior of an origin centred circle with arbitrary radius this allows one not only to achieve stability of the filter but also to maintain some safety margins a great advantage of the proposed approach is that it does not impose any constraints on the optimization problem and the optimal filter can be sought using off the shelf optimization procedures the power of the proposed approach is illustrated with filter design examples that compare favorably with results published in research literature

on stochastic conservation laws and malliavin calculus abstract for stochastic conservation laws driven by a semilinear noise term we propose a generalization of the kru kov entropy condition by allowing the kru kov constants to be malliavin differentiable random variables existence and uniqueness results are provided our approach sheds some new light on the stochastic entropy conditions put forth by feng and nualart 17 and bauzet vallet and wittbold 2 and in our view simplifies some of the proofs

convergence of approximations to stochastic scalar conservation laws we develop a general framework for the analysis of approximations to stochastic scalar conservation laws our aim is to prove under minimal consistency properties and bounds that such approximations are converging to the solution to a stochastic scalar conservation law the weak probabilistic convergence mode is convergence in law the most natural in this context we use also a kinetic formulation and martingale methods our result is applied to the convergence of the finite volume method in the companion paper 15

erratum to existence and strong pre compactness properties for entropy solutions of a first order quasilinear equation with discontinuous flux sequences of entropy solutions of a non degenerate first order quasilinear equation are shown to be strongly pre compact in the general case of a caratheodory flux vector existence of the weak and entropy solution to the cauchy problem for such an evolutionary equation is also established the proofs are based on the general localization principle for h measures corresponding to sequences of measure valued functions