on flows associated to sobolev vector fields in wiener spaces an approach a la diperna lions in this paper we extend the diperna lions theory of flows associated to sobolev vector fields to the case of cameron martin valued vector fields in wiener spaces e having a sobolev regularity the proof is based on the analysis of the continuity equation in e and on uniform gaussian commutator estimates in finite dimensional spaces

equations differentielles sur l espace de wiener et formules de cameron martin non lineaires proprietes algebriques des approximations sur r n une formule de perturbation des flots et convergence des approximations formule de changement de variable sur x et equation differentielle

convergence of numerical approximations to non linear continuity equations with rough force fields we prove quantitative regularity estimates for the solutions to non linear continuity equations and their discretized numerical approximations on cartesian grids when advected by a rough force field this allows us to not only recover the known optimal regularity for linear transport equations but also to obtain the convergence of a wide range of numerical schemes our proof is based on novel commutator estimates quantifying and extending to the non linear case the classical commutator approach of the theory of renormalized solutions

compactness for nonlinear continuity equations abstract we prove compactness and hence existence for solutions to a class of nonlinear transport equations the corresponding models combine the features of linear transport equations and scalar conservation laws we introduce a new method which gives quantitative compactness estimates compatible with both frameworks

collocation methods for index 1 daes with a singularity of the first kind we study the convergence behavior of collocation schemes applied to approximate solutions of bvps in linear index 1 daes which exhibit a critical point at the left boundary such a critical point of the dae causes a singularity within the inherent ode system we focus our attention on the case when the inherent ode system is singular with a singularity of the first kind apply polynomial collocation to the original dae system and consider different choices of the collocation points such as equidistant gaussian or radau points we show that for a well posed boundary value problem for daes having a sufficiently smooth solution the global error of the collocation scheme converges with the order o h s where s is the number of collocation points superconvergence cannot be expected in general due to the singularity not even for the differential components of the solution the theoretical results are illustrated by numerical experiments

on continuity equations in infinite dimensions with non gaussian reference measure let be a gaussian measure on a locally convex space and h be the corresponding cameron martin space it has been recently shown by l ambrosio and a figalli that the linear rst order pde

codes on the klein quartic ideals and decoding corresp a sequence of codes with particular symmetries and with large rates compared to their minimal distances is constructed over the field gf 2 3 in the sequence there is for instance a code of length 21 and dimension 10 with minimal distance 9 and a code of length 21 and dimension 16 with minimal distance 3 the codes are constructed from algebraic geometry using the dictionary between coding theory and algebraic curves over finite fields established by goppa the curve used in the present work is the klein quartic this curve has the maximal number of rational points over gf 2 3 allowed by serre s improvement of the hasse weil bound which together with the low genus accounts for the good code parameters the klein quartic has the frobenius group g of order 21 acting as a group of automorphisms which accounts for the particular symmetries of the codes in fact the codes are given alternative descriptions as left ideals in the group algebra gf 2 3 g this description allows for easy decoding for instance in the case of the single error correcting code of length 21 and dimension 16 with minimal distance 3 decoding is obtained by multiplication with an idempotent in the group algebra

analysis and numerical approximation of singular boundary value problems with the p laplacian in fluid mechanics this paper studies a generalization of the cahn hilliard continuum model for multi phase fluids where the classical laplacian has been replaced by a degenerate one i e the so called p laplacian the solution s asymptotic behavior is analyzed at two singular points namely at the origin and at infinity an efficient technique for treating such singular boundary value problems is presented and results of numerical integration are discussed and compared with earlier computed data

on the space of bv functions and a related stochastic calculus in infinite dimensions functions of bounded variation bv functions are defined on an abstract wiener space e h in a way similar to that in finite dimensions some characterizations are given which justify describing a bv function as a function in l log l 1 2 with the first order derivative being an h valued measure it is also shown that the space of bv functions is obtained by a natural extension of the sobolev space d1 1 moreover some stochastic formulae related to bv functions are investigated

analysis of a new error estimate for collocation methods applied to singular boundary value problems we discuss an a posteriori error estimate for the numerical solution of boundary value problems for nonlinear systems of ordinary differential equations with a singularity of the first kind the estimate for the global error of an approximation obtained by collocation with piecewise polynomial functions is based on the defect correction principle we prove that for collocation methods which are not superconvergent the error estimate is asymptotically correct as an essential prerequisite we derive convergence results for collocation methods applied to nonlinear singular problems

unicite de solutions d equations differentielles sur l espace de wiener soit x l espace de wiener et a un champ de vecteurs sur x on definit la notion de solution de l equation differentielle du dt 1 u t u 0 x on demontre un theoreme d unicite

vector bundles and codes on the hermitian curve the construction of algebraic geometry ag codes can be seen as a distinctly geometric process and yet decoding procedures tend to rely on algebraic ideas that have no direct geometric interpretation recently however trygve johnsen observed that decoding can be viewed in abstract terms of a class of vector bundles on the underlying curve the present paper describes these objects at a concrete computational level for the hermitian codes comega d mpinfin defined over fq 2 q a power of 2 the construction of explicit representations of the vector bundles by transition matrices involves finding functions on the curve that satisfy a certain property in their power series expansions around pinfin computing the image of the corresponding global sections under serre duality and finding a suitable open cover of the curve the cover enables any rational point to be expressed as a line bundle by a simple kind of transition function a special case is considered in which these functions can be realized as ratios of linear forms

critical non sobolev regularity for continuity equations with rough velocity fields abstract we present a divergence free vector field in the sobolev space h 1 such that the flow associated to the field does not belong to any sobolev space the vector field is deterministic but constructed as the realization of a random field combining simple elements this construction illustrates the optimality of recent quantitative regularity estimates as it gives a straightforward example of a well posed flow which has nevertheless only very weak regularity

numerical solution of integro differential equations arising from singular boundary value problems abstract we consider the numerical solution of a singular boundary value problem on the half line for a second order nonlinear ordinary differential equation due to the fact that the nonlinear differential equation has a singularity at the origin and the boundary value problem is posed on an unbounded domain the proposed approaches are complex and require a considerable computational effort in the present paper we describe an alternative approach based on the reduction of the original problem to an integro differential equation though the problem is posed on the half line we just need to approximate the solution on a finite interval by analyzing the behavior of the numerical approximation on this interval we identify the solution that satisfies the prescribed boundary condition although the numerical algorithm is much simpler than the ones proposed before it provides accurate approximations we illustrate the proposed methods with some numerical examples

bv functions and distorted ornstein uhlenbeck processes over the abstract wiener space let e h be an abstract wiener space and h be the class of functions l1 e satisfying the ray hamza condition in every direction l e for h the closure e f of the symmetric form e u v e u z v z h z dz u v fc1b is a quasi regular dirichlet form on l2 f d f supp yielding an associated diffusion m xt pz on f called a distorted ornstein uhlenbeck process a function on e is called a bv function bv e in notation if p 1 lp e and v supg fc1b e g h z 1 e g z z dz is finite for h bv e there exist a positive finite measure d on f and a weakly measurable function f h such that z h 1 d a e and f g z z dz f g z z h d dz g fc1b e further the sample path of m admits an expression as a sum of e valued cafs xt x0 wt 12 t0 xs ds 12 t0 xs dl d s where wt is an e valued brownian motion and l d t is a pcaf of m with revuz measure d a measurable set e is called caccioppoli if i bv e in this case the support of the measure di is concentrated in and the above equations reduce to the gauss formula and the skorohod equation for the modified reflecting ornstein uhlenbeck process respectively a related coarea formula is also presented

lagrangian flows and the one dimensional peano phenomenon for odes we consider the one dimensional ordinary differential equation with a vector field which is merely continuous and nonnegative and satisfies a condition on the amount of zeros although it is classically known that this problem lacks uniqueness of classical trajectories we show that there is uniqueness for the so called regular lagrangian flow by now usual notion of flow in nonsmooth situations as well as uniqueness of distributional solutions for the associated continuity equation the proof relies on a space reparametrization argument around the zeros of the vector field

bv functions in abstract wiener spaces abstract functions of bounded variation in an abstract wiener space i e an infinite dimensional banach space endowed with a gaussian measure and a related differential structure have been introduced by m fukushima and m hino using dirichlet forms and their properties have been studied with tools from analysis and stochastics in this paper we reformulate in an integral geometric vein and with purely analytical tools the definition and the main properties of bv functions and investigate further properties

sobolev functions on infinite dimensional domains abstract we study extensions of sobolev and bv functions on infinite dimensional domains along with some positive results we present a negative solution of the long standing problem of existence of sobolev extensions of functions in gaussian sobolev spaces from a convex domain to the whole space

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 and uniqueness of martingale solutions for sdes with rough or degenerate coefficients abstract in this paper we extend recent results on the existence and uniqueness of solutions of odes with non smooth vector fields to the case of martingale solutions in the stroock varadhan sense of sdes with non smooth coefficients in the first part we develop a general theory which roughly speaking allows to deduce existence uniqueness and stability of martingale solutions for l d almost every initial condition x whenever existence and uniqueness is known at the pde level in the l setting and conversely if existence and uniqueness of martingale solutions is known for l d a e initial condition then existence and uniqueness for the pde holds in the second part of the paper we consider situations where on the one hand no pointwise uniqueness result for the martingale problem is known and on the other hand well posedness for the fokker planck equation can be proved thus the theory developed in the first part of the paper is applicable in particular we will study the fokker planck equation in two somehow extreme situations in the first one assuming uniform ellipticity of the diffusion coefficients and lipschitz regularity in time we are able to prove existence and uniqueness in the l 2 setting in the second one we consider an additive noise and assuming the drift b to have bv regularity and allowing the diffusion matrix a to be degenerate also identically 0 we prove existence and uniqueness in the l setting therefore in these two situations our theory yields existence uniqueness and stability results for martingale solutions

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

lusin type approximation of sobolev by lipschitz functions in gaussian and rcd k spaces abstract we establish new approximation results in the sense of lusin of sobolev functions by lipschitz ones in some classes of non doubling metric measure structures our proof technique relies upon estimates for heat semigroups and applies to gaussian and rcd k spaces as a consequence we obtain quantitative stability for regular lagrangian flows in gaussian settings

stochastic lagrangian particle approach to fractal navier stokes equations in this article we study the fractal navier stokes equations by using the stochastic lagrangian particle path approach in constantin and iyer comm pure appl math lxi 330 345 2008 more precisely a stochastic representation for the fractal navier stokes equations is given in terms of stochastic differential equations driven by levy processes based on this representation a self contained proof for the existence of a local unique solution for the fractal navier stokes equation with initial data in mathbb w 1 p is provided and in the case of two dimensions or large viscosity the existence of global solutions is also obtained in order to obtain the global existence in any dimensions for large viscosity the gradient estimates for levy processes with time dependent and discontinuous drifts are proved

lagrangian flows driven by bv fields in wiener spaces we establish the renormalization property for essentially bounded solutions of the continuity equation associated to bounded variation bv fields in wiener spaces with values in the associated cameron martin space thus obtaining by standard arguments new uniqueness and stability results for correspondent lagrangian flows

decoding algebraic geometric codes up to the designed minimum distance a simple decoding procedure for algebraic geometric codes c sub omega d g is presented this decoding procedure is a generalization of peterson s decoding procedure for the bch codes it can be used to correct any d 1 2 or fewer errors with complexity o n sup 3 where d is the designed minimum distance of the algebraic geometric code and n is the codelength

lagrangian solutions to the 2d euler system with l1 vorticity and infinite energy we consider solutions to the two dimensional incompressible euler system with only integrable vorticity thus with possibly locally infinite energy with such regularity we use the recently developed theory of lagrangian flows associated to vector fields with gradient given by a singular integral in order to define lagrangian solutions for which the vorticity is transported by the flow we prove strong stability of these solutions via strong convergence of the flow under the only assumption of l1 weak convergence of the initial vorticity the existence of lagrangian solutions to the euler system follows for arbitrary l1 vorticity relations with previously known notions of solutions are established

on the decoding of algebraic geometric codes this paper provides a survey of the existing literature on the decoding of algebraic geometric codes definitions theorems and cross references will be given we show what has been done discuss what still has to be done and pose some open problems

asymptotically correct error estimation for collocation methods applied to singular boundary value problems we discuss an a posteriori error estimate for collocation methods applied to boundary value problems in ordinary differential equations with a singularity of the first kind as an extension of previous results we show the asymptotical correctness of our error estimate for the most general class of singular problems where the coefficient matrix is allowed to have eigenvalues with positive real parts this requires a new representation of the global error for the numerical solution obtained by piecewise polynomial collocation when applied to our problem class

the divergence of banach space valued random variables on wiener space the domain of definition of the divergence operator on an abstract wiener space w h is extended to include w valued and open image in new window valued integrands the main properties and characterizations of this extension are derived and it is shown that in some sense the added elements in s extended domain have divergence zero these results are then applied to the analysis of quasiinvariant flows induced by w valued vector fields and among other results it turns out that these divergence free vector fields are responsible for generating measure preserving flows

estimates and regularity results for the diperna lions flow in this paper we derive new simple estimates for ordinary differential equations with sobolev coefficients these estimates not only allow to recover some old and recent results in a simple direct way but they also have some new interesting corollaries

density profile equation with p laplacian analysis and numerical simulation analytical properties of a nonlinear singular second order boundary value problem in ordinary differential equations posed on an unbounded domain for the density profile of the formation of microscopic bubbles in a nonhomogeneous fluid are discussed especially sufficient conditions for the existence and uniqueness of solutions are derived two approximation methods are presented for the numerical solution of the problem one of them utilizes the open domain matlab code bvpsuite the results of numerical simulations are presented and discussed

a note on hermitian codes over gf q sup 2 properties of codes defined by goppa s algebraic geometric method using hermitian curves are discussed generator and parity check matrices for such codes are derived explicitly results concerning the minimum distance and the weight distribution of the codes are provided

analytical numerical investigation of bubble type solutions of nonlinear singular problems in this work we are concerned about a singular boundary value problem for a second order nonlinear ordinary differential equation arising in hydrodynamics and nonlinear field theory when centrally symmetric bubble type solutions are sought we are interested on solutions of this equation which are strictly increasing on the positive semi axis and have finite limits at zero and infinity necessary conditions for the existence of such solutions are obtained in the form of a restriction on the equation parameters the asymptotic behavior of certain solutions of this equation is analyzed near the two singularities when r 0 and r where the considered boundary conditions define one parameter families of solutions based on the analytic study an efficient numerical method is proposed to compute approximately the needed solutions of the above problem some results of the numerical experiments are displayed and their physical interpretation is discussed

fast decoding of codes from algebraic plane curves improvement to an earlier decoding algorithm for codes from algebraic geometry is presented for codes from an arbitrary regular plane curve the authors correct up to d 2 m sup 2 8 m 4 9 8 errors where d is the designed distance of the code and m is the degree of the curve the complexity of finding the error locator is o n sup 7 3 where n is the length of the code for codes from hermitian curves the complexity of finding the error values given the error locator is o n sup 2 and the same complexity can be obtained in the general case if only d 2 m sup 2 2 errors are corrected