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

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

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

the fully compressible semi geostrophic system from meteorology the semi geostrophic equations are an approximation to the 3 d euler equations for an atmosphere where the effects of rotation dominate they are used by meteorologists to model the formation of fronts mathematically rigorous results were obtained by benamou and brenier and by cullen and gangbo in the incompressible case in this paper we extend the results of benamou and brenier to the fully compressible case and show the existence of weak solutions to a reformulation of these equations in so called dual variables the monge kantorovich theory appears as a material tool in our study

rank one property for derivatives of functions with bounded variation when u is a function of bounded variation of the open set r into r du is a measure on which takes values in the space rm n of all m n matrices such that du i j ui xj for every i j with 1 i m 1 j n and we denote by dsu the part of the measure du which is singular with respect to lebesgue measure we recall that the rank of a matrix a rm n is dimension of the space spanned by its columns in r or by its rows in r and in particular the rank of a is one if and only if there exist e r and r with e 6 0 6 0 so that a e i e ai j ei j for all i j cf remark 1 7 when is a measure which takes values in the space rm n and d d denotes the radon nikodym derivative of with respect to the total variation we say that has rank one when the rank of the matrix d d x is one for almost every x in this paper we prove we prove the following statement see theorem 4 5 corollary 4 6 and remark 4 9

on the existence and weak stability of solutions to the compressible semigeostrophic equations abstract in this article existence and weak stability of solutions to the compressible semigeostrophic equations are studied both in physical and in dual spaces we present a proof of the existence of a weak solution to these equations in dual space assuming initial data in l 1 r 3 extending previous results cullen and maroofi 2003 8 an extension to the concept of lagrangian solutions due to cullen and feldman 2006 6 is presented to the compressible semigeostrophic system and it is shown the existence of lagrangian solutions in physical space with initial potential vorticities in l 1 as well a weak stability result with respect to perturbations of the potential vorticity in l 1

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

a hybrid phase flow method for hamiltonian systems with discontinuous hamiltonians in this paper we propose a new phase flow method for hamiltonian systems with discontinuous hamiltonians in the original phase flow method introduced by ying and candes j comput phys 220 2006 pp 184 215 the phase map should be smooth to ensure the accuracy of the interpolation such an interpolation is inaccurate if the phase map is nonsmooth for example when the hamiltonian is discontinuous we modify the phase flow method using a discontinuous hamiltonian solver and establish the stability for piecewise constant potentials of such a solver this extends the applicability of the highly efficient phase flow method to singular hamiltonian systems with a mild increase of algorithm complexity such a particle method can be useful for the computation of high frequency waves through interfaces

sobolev differentiable stochastic flows for sdes with singular coefficients applications to the transport equation in this paper we establish the existence of a stochastic flow of sobolev diffeomorphisms r d x 7 s t x r d s t r for a stochastic differential equation sde of the form dxt b t xt dt dbt s t r xs x r d the above sde is driven by a bounded measurable drift coefficientb r r d r d and a d dimensional brownian motion b more specifically we show that the stochastic flow s t of the sde lives in the space l 2 w 1 p r d w for all s t and all p 1 where w 1 p r d w denotes a weighted sobolev space with weight w possessing a p th moment with respect to lebesgue measure on r d this result is counter intuitive since the dominant culture in stochastic and deterministic dynamical systems is that the flow inherits its spatial regularity from the driving vector fields the spatial regularity of the stochastic flow yields existence and uniqueness of a sobolev differentiable weak solution of the stratonovich stochastic transport equation dtu t x b t x du t x dt p d 1 ei du t x db i t 0 u 0 x u0 x where b is bounded and measurable u0 is c 1 b and ei d 1 a basis for r d it is well known that the deterministic counter part of the above equation does not in general have a solution using stochastic perturbations and our analysis of the above sde we establish a deterministic flow of sobolev diffeomorphisms for classical one dimensional deterministic ode s driven by discontinuous vector fields furthermore and as a corollary of the latter result we construct a sobolev stochastic flow of diffeomorphisms for one dimensional sde s driven by discontinuous diffusion coefficients

on the dirichlet problem for first order quasilinear equations on a manifold we study the dirichlet problem for a first order quasilinear equation on a smooth manifold with boundary existence and uniqueness of a generalized entropy solution are established the uniqueness is proved under some additional requirement on the field of coefficients it is shown that generally the uniqueness fails the non uniqueness occurs because of presence of the characteristics not outgoing from the boundary including closed ones the existence is proved in general case moreover we establish that among generalized entropy solutions laying in the ball u r there exist the unique maximal and minimal solutions to prove our results we use the kinetic formulation similar to one by c imbert and j vovelle introduction let m be a n dimensional c smooth compact manifold with boundary s m thus in a neighborhood of each point x0 m we can define the local coordinates x1 xn j x x u corresponding to the chart u j v where u is a neighborhood of the point x0 coordinate neighborhood v is an open subset of the half space x x1 xn r x1 0 x x2 xn rn 1 and j u v is a c diffeomorphism clearly boundary points of u and v must correspond each other under the diffeomorphism j j u s v denote by tm t m the tangent and cotangent bundles on m and by txm t xm the corresponding fibres of these bundles at a point x m any vector field a a x txm being a section of the tangent bundle tm acts on smooth functions f x c m as a first order differential operator represented in local coordinates x1 xn as a a x xi i e a f ai x xi

a variational approach for the 2 dimensional semi geostrophic shallow water equations existence of weak solutions to the 3 d semi geostrophic equations with rigid boundaries was proved by benamou and brenier 3 using monge transport theory this paper extends the results to a free surface boundary condition which is more physically appropriate this extension is at present for the 2 d shallow water case only in addition we establish stronger time regularity than was possible in 3

on the singular local limit for conservation laws with nonlocal fluxes we give an answer to a question posed in amorim et al esaim math model numer anal 49 1 19 37 2015 which can loosely speaking be formulated as follows consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel in the singular limit where the convolution kernel is replaced by a dirac delta one formally recovers a conservation law can we rigorously justify this formal limit we exhibit counterexamples showing that despite numerical evidence suggesting a positive answer one does not in general have convergence of the solutions we also show that the answer is positive if we consider viscous perturbations of the nonlocal equations in this case in the singular local limit the solutions converge to the solution of the viscous conservation law

renormalized solutions to the vlasov equation with coefficients of bounded variation we prove that weak bounded solutions to the vlasov equation with bv coefficients have the renormalization property and we show that when the renormalization property holds for a general transport equation it also holds for only lipschitz nonlinearities

holder regularity of the 2d dual semigeostrophic equations via analysis of linearized monge ampere equations we obtain the holder regularity of time derivative of solutions to the dual semigeostrophic equations in two dimensions when the initial potential density is bounded away from zero and infinity our main tool is an interior holder estimate in two dimensions for an inhomogeneous linearized monge ampere equation with right hand side being the divergence of a bounded vector field as a further application of our holder estimate we prove the holder regularity of the polar factorization for time dependent maps in two dimensions with densities bounded away from zero and infinity our applications improve previous work by g loeper who considered the cases of densities sufficiently close to a positive constant

sobolev regular flows of non lipschitz vector fields abstract we show that vector fields with exponentially integrable derivatives admit a well defined flow of homeomorphisms x t w 1 p t l o c for some p t 1 at least for small times when the field is certain riesz potential of a bounded function the result becomes global in time due to techniques from geometric function theory the local result also applies to the flows arising from yudovich solutions to the planar euler system with bounded vorticity

the semigeostrophic equations discretized in reference and dual variables we study the evolution of a system of n particles x i v i i 1 n in mathbb r 2d that system is a conservative system with a hamiltonian of the form h mu w 2 2 mu nu n where w2 is the wasserstein distance and is a discrete measure concentrated on the set x i v i i 1 n typically 0 is a discrete measure approximating an initial l density and can be chosen randomly when d 1 our results prove convergence of the discrete system to a variant of the semigeostrophic equations we obtain that the limiting densities are absolutely continuous with respect to the lebesgue measure when nu n n 1 infty converges to a measure concentrated on a special d dimensional set we obtain the vlasov monge ampere vma system when d 1 the vma system coincides with the standard vlasov poisson system

hamiltonian odes in the wasserstein space of probability measures in this paper we consider a hamiltonian h on p 2 r 2d the set of probability measures with finite quadratic moments on the phase space r 2d r d r d which is a metric space when endowed with the wasserstein distance w 2 we study the initial value problem d t dt j d v t t 0 where j d is the canonical symplectic matrix 0 is prescribed and v t is a tangent vector to p 2 r 2d at t belonging to h t the subdifferential of h at t two methods for constructing solutions of the evolutive system are provided the first one concerns only the case where 0 is absolutely continuous it ensures that fit remains absolutely continuous and v t h t is the element of minimal norm in h t the second method handles any initial measure 0 if we further assume that h is convex proper and lower semicontinuous on p 2 r 2d we prove that the hamiltonian is preserved along any solution of our evolutive system h t h 0

uniqueness of continuous solutions for bv vector fields we consider a vector field whose coefficients are functions of bounded variation with a bounded divergence we prove the uniqueness of continuous solutions for the cauchy problem

structure of entropy solutions for multi dimensional scalar conservation laws an entropy solution u of a multi dimensional scalar conservation law is not necessarily in bv even if the conservation law is genuinely nonlinear we show that u nevertheless has the structure of a bv function in the sense that the shock location is codimension one rectifiable this result highlights the regularizing effect of genuine nonlinearity in a qualitative way it is based on the locally finite rate of entropy dissipation the proof relies on the geometric classification of blow ups in the framework of the kinetic formulation

free boundaries in optimal transport and monge ampere obstacle problems given compactly supported 0 f g 2 l 1 r n the problem of trans porting a fraction m min kfkl1 kgkl1 of the mass of f onto g as cheaply as possible is considered where cost per unit mass transported is given by a cost function c typically quadratic c x y x y 2 2 this question is shown to be equivalent to a double obstacle problem for the monge ampere equation for which sucient conditions are given to guarantee uniqueness of the solution such as f vanishing on sptg in the quadratic case the part of f to be transported increases monotonically with m and if sptf and sptg are separated by a hyperplane h then this part will separated from the balance of f by a semiconcave lipschitz graph over the hyperplane if f f and g g are bounded away from zero and infinity on separated strictly convex domains r n for the quadratic cost this graph is shown to be a c 1 loc hypersurface in whose normal coincides with the direction transported the optimal map between f and g is shown to be holder continuous up to this free bound ary and to those parts of the fixed boundary which map to locally convex parts of the path connected target region

relaxed notions of curvature and a lumped strain method for elastic plates the paper proposes an approximation method lumped strain method lsm for elastic plates based on generalized notions of the gaussian and mean curvatures and on a relaxation of the energy functional to the space of continuous piecewise linear functions the method adapts ideas from the theory of gamma convergence we restrict our attention to the case of the simply supported plate and state the proof of convergence of the method we give an application to the rhombic plate and compare the results with those obtained by standard finite element approximations

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

structure of solutions of multidimensional conservation laws with discontinuous flux and applications to uniqueness we investigate the structure of solutions of conservation laws with discontinuous flux under quite general assumption on the flux we show that any entropy solution admits traces on the discontinuity set of the coefficients and we use this to prove the validity of a generalized kato inequality for any pair of solutions applications to uniqueness of solutions are then given

existence of a weak solution for the semigeostrophic equation with integrable initial data in this article we present a proof of existence of a weak solution for the semigeostrophic system of equations formulated as an active scalar transport equation with monge ampere coupling with initial data in this is an extension of a 1998 result due to benamou and brenier who proved existence with initial data in

γ convergence of external approximations in boundary value problems involving the bi laplacian in addition to the various uses it was introduced for the theory of convergence offers a rather natural setting for discussing and developing nonorthodox approximation methods for variational problems for certain boundary value problems involving the bi laplacian sequences of discrete functionals are here defined and are shown to converge to the corresponding functionals of the continuous problems the minimizers of the discrete functionals provide converging approximations to the solution of the limit problem in question thus we obtain approximation schemes that are nonconforming but direct and that can be treated by current algorithms for symmetric and positive definite functionals the class of problems considered in this paper includes the stokes problem in fluid dynamics the loading problem of 2 d isotropic elastostatics and some boundary value problems of the kirchhoff love theory of plates also discussed is an extension of the discretization method that seems suitable for treating more general boundary value problems of elastic plates but whose convergence is conditional to a conjecture that remains to be proved a relevant application to the so called babuska paradox is presented

turing s method for the selberg zeta function in one of his final research papers alan turing introduced a method to certify the completeness of a purported list of zeros of the riemann zeta function in this paper we consider turing s method in the analogous setting of selberg zeta functions and we demonstrate that it can be carried out rigorously in the prototypical case of the modular surface

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

a note on stochastic semilinear equations and their associated fokker planck equations abstract the main purpose of this paper is to prove existence and uniqueness of probabilistically weak and strong solutions to stochastic differential equations sde on hilbert spaces under a new approximation condition on the drift recently proposed in 6 to solve fokker planck equations fpe extended in this paper to a considerably larger class of drifts as a consequence we prove existence of martingale solutions to the sde whose time marginals then solve the corresponding fpe applications include stochastic semilinear partial differential equations with white noise and a non linear drift part which is the sum of a burgers type part and a reaction diffusion part the main novelty is that the latter is no longer assumed to be of at most linear but of at most polynomial growth this case so far had not been covered by the existing literature we also give a direct and more analytic proof for existence of solutions to the corresponding fpe extending the technique from 6 to our more general framework which in turn requires to work on a suitable gelfand triple rather than just the hilbert state space

on the distribution of eigenvalues of maass forms on certain moonshine groups in this paper we study both analytically and numerically questions involving the distribution of eigen values of maass forms on the moonshine groups 0 n where n 1 is a square free integer after we prove that 0 n has one cusp we compute the constant term of the associated non holomorphic eisenstein series we then derive an average weyl s law for the distribution of eigenvalues of maass forms from which we prove the classical weyl s law as a special case the groups corresponding to n 5 and n 6 have the same signature however our analysis shows that asymptotically there are infinitely more cusp forms for 0 5 than for 0 6 we view this result as being consistent with the phillips sarnak philosophy since we have shown unconditionally the existence of two groups which have different weyl s laws in addition we employ hejhal s algorithm together with recently developed refinements from 32 and numerically determine the first 3557 of 0 5 and the first 12474 eigenvalues of 0 6 with this information we empirically verify some conjectured distributional properties of the eigenvalues

reduced hausdorff dimension and concentration cancellation for two dimensional incompressible flow where v t v1 v2 is the fluid velocity p is the scalar pressure dv dt av at v v v and vo is an initial incompressible velocity field i e div vo 0 in this paper we study the detailed limiting behavior of approximate solution sequences for 2 d euler with vortex sheet initial data a sequence of smooth velocity fields v8 x t is an approximate solution sequence for 2 d euler provided that v is incompressible i e div v 0 and satisfies the following properties 1 the velocity fields v6 have uniformly bounded local kinetic energy i e

an unconstrained mixed method for the biharmonic problem in this work we present a finite element method for the biharmonic problem based on the primal mixed formulation of ciarlet and raviart a mixed finite element method for the biharmonic equation in symposium on mathematical aspects of finite elements in partial differential equations c de boor ed academic press new york 1974 pp 125 143 we introduce a dual mesh and a suitable approximation of the constraint that enables us to eliminate the auxiliary variable with no computational effort thus the discrete problem turns to be governed by a system of linear equations with symmetric and positive definite coefficients and can be solved by classical algorithms the construction of the stiffness matrix is obtained by using courant triangles and can be done with great efficiency

moreau yosida approximation and convergence of hamiltonian systems on wasserstein space abstract in this paper we study the stability property of hamiltonian systems on the wasserstein space let h be a given hamiltonian satisfying certain properties we regularize h using the moreau yosida approximation and denote it by h we show that solutions of the hamiltonian system for h converge to a solution of the hamiltonian system for h as converges to zero we provide sufficient conditions on h to carry out this process

weak existence for the semigeostrophic equations formulated as a coupled monge ampere transport problem hoskins s semigeostrophic equations are reformulated as a coupled monge ampere transport problem b j hoskins quart j royal met soc 97 1971 pp 139 153 existence of global weak solutions is obtained for this formulation

the stability principle and global weak solutions of the free surface semi geostrophic equations in geostrophic coordinates the semi geostrophic equations are used widely in the modelling of large scale atmospheric flows in this note we prove the global existence of weak solutions of the incompressible semi geostrophic equations in geostrophic coordinates in a three dimensional domain with a free upper boundary the proof based on an energy minimization argument originally inspired by the stability principle as studied by cullen purser and others uses optimal transport techniques as well as the analysis of hamiltonian odes in spaces of probability measures as studied by ambrosio and gangbo we also give a general formulation of the stability principle in a rigorous mathematical framework

a new approach to variational problems with multiple scales we introduce a new concept the young measure on micropatterns to study singularly perturbed variational problems that lead to multiple small scales depending on a small parameter e this allows one to extract in the limit e 0 the relevant information at the macroscopic scale as well as the coarsest microscopic scale say e and to eliminate all finer scales to achieve this we consider rescaled functions rx t x s e t viewed as maps of the macroscopic variable s with values in a suitable function space the limiting problem can then be formulated as a variational problem on the young measures generated by rex as an illustration we study a one dimensional model that describes the competition between formation of microstructure and highest gradient regularization we show that the unique minimizer of the limit problem is a young measure supported on sawtooth functions with a given period 2001 john wiley sons inc

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

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

noise prevents singularities in linear transport equations a stochastic linear transport equation with multiplicative noise is considered and the question of no blow up is investigated the drift is assumed only integrable to a certain power opposite to the deterministic case where smooth initial conditions may develop discontinuities we prove that a certain sobolev degree of regularity is maintained which implies holder continuity of solutions the proof is based on a careful analysis of the associated stochastic flow of characteristics

weak solutions of 2 d euler equations with initial vorticity in l log l abstract in this note we prove global in time existence of weak solutions of 2 0 euler equations for incompressible fluid flows in the whole plane of r 2 when the initial vorticity is in the zygmund class l log l the solution is constructed by a vanishing viscosity limit of the sequence of solutions to the navier stokes equations with the same initial data this is a limiting case of the corresponding results of diperna and majda in which global existence of weak solutions was obtained for initial vorticity in l 1 l p 1 p

on the semi geostrophic system in physical space with general initial data in order to accommodate general initial data an appropriately relaxed notion of renormalized lagrangian solutions for the semi geostrophic system in physical space is introduced this is shown to be consistent with previous notions generalizing them a weak stability result is obtained first followed by a general existence result whose proof employs said stability and approximating solutions with regular initial data the renormalization property ensures the return from physical to dual space as consequences we get conservation of hamiltonian energy and some weak time regularity of solutions

geodesics in the space of measure preserving maps and plans we study brenier s variational models for incompressible euler equations these models give rise to a relaxation of the arnold distance in the space of measure preserving maps and more generally measure preserving plans we analyze the properties of the relaxed distance we show a close link between the lagrangian and the eulerian model and we derive necessary and sufficient optimality conditions for minimizers these conditions take into account a modified lagrangian induced by the pressure field moreover adapting some ideas of shnirelman we show that even for non deterministic final conditions generalized flows can be approximated in energy by flows associated to measure preserving maps

a geometrical approach to monotone functions in r n this paper is concerned with the fine properties of monotone func tions on r n we study the continuity and differentiability properties of these functions the approximability properties the structure of the distributional derivatives and of the weak jacobians moreover we exhibit an example of a monotone function u which is the gradient of a c 1 convex function and

spectral functions special functions and the selberg zeta function the functional determinant of an eigenvalue sequence as defined by zeta regularization can be simply evaluated by quadratures we apply this procedure to the selberg trace formula for a compact riemann surface to find a factorization of the selberg zeta function into two functional determinants respectively related to the laplacian on the compact surface itself and on the sphere we also apply our formalism to various explicit eigenvalue sequences reproducing in a simpler way classical results about the gamma function and the barnesg function concerning the latter our method explains its connection to the selberg zeta function and evaluates the related glaisher kinkelin constanta

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

selection problems for a discount degenerate viscous hamilton jacobi equation abstract we prove that the solution of the discounted approximation of a degenerate viscous hamilton jacobi equation with convex hamiltonians converges to that of the associated ergodic problem we characterize the limit in terms of stochastic mather measures by using the nonlinear adjoint method and deriving a commutation lemma this convergence result was first proven by davini fathi iturriaga and zavidovique for first order hamilton jacobi equations

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

a lumped stress method for plane elastic problems and the discrete continuum approximation this paper proposes a rational method to approximate a plane elastic body through a latticed structure composed of truss elements the method is based on the introduction of a relaxed stress energy that allows an extension of the original problem to a larger space of admissible stress fields including stresses concentrated along lines use is made of polyhedral approximations of the airy stress function the truss analogy is employed to obtain a displacement formulation the paper includes several numerical applications of the method to sample problems a numerical convergence study and comparisons with exact solutions and standard finite element approximations

detecting squarefree numbers we present an algorithm based on the explicit formula for l functions and conditional on grh for proving that a given integer is squarefree with little or no knowledge of its factorization we analyze the algorithm both theoretically and practically and use it to prove that several rsa challenge numbers are not squarefull f a non zero element of k x then it is well known that f is squarefree if and only if gcd f f 0 1 and the latter condition may be checked quickly in deterministic polynomial time by the euclidean algorithm it is a long standing question in algorithmic number theory whether there is a correspondingly simple procedure to test if a given integer is squarefree in particular can one determine whether n 2 z is squarefree more rapidly than by factoring it in this paper we describe an algorithm conditional on the generalized riemann hy pothesis grh for proving an integer squarefree with little or no knowledge of its factor ization and analyze the complexity of the algorithm both theoretically and practically in particular we present some heuristic evidence based on random matrix theory and other probabilistic calculations that our algorithm runs in deterministic subexponential time o exp logn 2 3 o 1 although this is poorer than the performance expected of the cur rent best known factoring algorithms our method is able to give partial results that one does not obtain from a failed attempt at factoring in particular we show the following see

multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice iii from ab initio models to wkb for schrodinger poisson this work is concerned with the semiclassical approximation of the schrodinger poisson equation modeling ballistic transport in a 1d periodic potential by means of wkb techniques it is derived by considering the mean field limit of a n body quantum problem then k multivalued solutions are adapted to the treatment of this weakly nonlinear system obtained after homogenization without taking into account for pauli s exclusion principle numerical experiments display the behaviour of self consistent wave packets and screening effects

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

duality theorems for marginal problems given topological spaces x1 xn with product space x probability measures i on xi together with a real function h on x define a marginal problem as well as a dual problem using an extended version of choquet s theorem on capacities an analogue of the classical duality theorem of linear programming is established imposing only weak conditions on the topology of the spaces xi and the measurability resp boundedness of the function h applications concern among others measures with given support stochastic order and general marginal problems

analysis of the semi geostrophic shallow water equations abstract the semi geostrophic shallow water equations are an accurate approximation to the rotating shallow water equations on scales larger than the rossby radius of deformation the global existence of weak solutions of the lagrangian form of the equations is known however uniqueness is not known except for short times for smooth initial data similarly a rigorous proof of existence of solutions has not been achieved for variable rotation rate which is important for atmospheric applications the obstacle is the lack of regularity of solutions to the monge ampere equation which underlies the solution procedure in this note it is shown formally that the modified form of the monge ampere equation that appears in the solution procedure for the semi geostrophic shallow water equations is better controlled on large scales than that of the standard equation that is used in the incompressible case this suggests that more progress should be possible with the rigorous theory and in particular that it should be possible to prove uniqueness

existence of global weak solutions to finitely extensible nonlinear bead spring chain models for dilute polymers with variable density and viscosity abstract we prove the existence of global in time weak solutions to a general class of coupled bead spring chain models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids with noninteracting polymer chains with finitely extensible nonlinear elastic fene spring potentials the class of models under consideration involves the unsteady incompressible navier stokes equations with variable density and density dependent dynamic viscosity in a bounded domain in r d d 2 or 3 for the density the velocity and the pressure of the fluid with an elastic extra stress tensor appearing on the right hand side in the momentum equation the extra stress tensor stems from the random movement of the polymer chains and is defined by the kramers expression through the associated probability density function that satisfies a fokker planck type parabolic equation a crucial feature of which is the presence of a centre of mass diffusion term and a nonlinear density dependent drag coefficient we require no structural assumptions on the drag term in the fokker planck equation in particular the drag term need not be corotational with initial density 0 min max for the continuity equation where min 0 a square integrable and divergence free initial velocity datum u 0 for the navier stokes equation and a nonnegative initial probability density function 0 for the fokker planck equation which has finite relative entropy with respect to the maxwellian m associated with the spring potential in the model we prove via a limiting procedure on certain regularization parameters the existence of a global in time weak solution t t u t t to the coupled navier stokes fokker planck system satisfying the initial condition 0 u 0 0 0 u 0 0 such that t t min max t u t belongs to the classical leray space and t t has bounded relative entropy with respect to m and t t m has integrable fisher information w r t the gibbs measure d m q d q d x over any time interval 0 t t 0

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159