rotational degeneracy of hyperbolic systems of conservation laws the purpose of the paper is to study wave patterns of hyperbolic systems in which rotational symmetry creates a specific kind of degeneracy in this situation hyperbolicity is necessarily non strict so that the elementary waves have interesting patterns the discussion is centered around a theorem on existence and uniqueness of solutions of the riemann problem we give a unified presentation of examples from continuum mechanics

the riemann problem near a hyperbolic singularity ii this paper is interested in classifying the solutions of riemann problems for the 2 times 2 conservation laws that have homogeneous quadratic flux functions such flux functions approximate an arbitrary 2 times 2 system in a neighborhood of an isolated point where strict hyperbolicity fails here the solution for the symmetric systems in region iii of the four region classification of schaeffer and shearer is given the solution is based on the qualitative shape of the integral curves described by schaeffer and shearer and a numerical calculation of the hugoniot loci and their shock types

stability of viscous shock waves associated with a system of nonstrictly hyperbolic conservations laws we introduce a simple model of two conservation laws which is strictly hyperbolic except for a degenerate parabolic line in the state space besides classical shock waves it also exhibits overcompressive marginal overcompressive and marginal undercompressive shock waves our purpose is to study the behavior of the corresponding viscous waves in particular the manner in which these waves are stable there are several basic differences between classical shock waves and other types of shock waves a perturbation of an overcompressive shock wave gives rise to a new wave monotone marginal overcompressive waves behave distinctly from the nonmonotone ones analytical techniques used in our study include characteristic energy and weighted energy methods and nonlinear superposition through time invariants although we carry out our analysis for a simple model the general phenomena would hold for overcompressive waves which occur in other physical models

an isometric study of the lindeberg feller central limit theorem via stein s method a b s t r a c t we use stein s method to prove a generalization of the lindeberg feller clt providing an upper and a lower bound for the superior limit of the kolmogorov distance between a normally distributed random variable and the rowwise sums of a rowwise independent triangular array of random variables which is asymptotically negligible in the sense of feller a natural example shows that the upper bound is of optimal order the lower bound improves a result by andrew barbour and peter hall

a global formalism for nonlinear waves in conservation laws we introduce a unifying framework for treating all of the fundamental waves occurring in general systems ofn conservation laws fundamental waves are represented as pairs of states statisfying the rankine hugoniot conditions after trivial solutions have been eliminated by means of a blow up procedure these pairs form an n 1 dimensional manifold the fundamental wave manifold there is a distinguishedn dimensional submanifold of containing a single one dimensonal foliation that represents the rarefaction curves for all families similarly there is a foliation of itself that represent shock curves we identify othern dimensional submanifolds of that are naturally interpreted as boundaries of regions of admissible shock waves these submanifolds also have one dimensional foliations which represent curves of composite waves this geometric framework promises to simplify greatly the study of the stability and bifurcation properties

knots and links in steady solutions of the euler equation given any possibly unbounded locally nite link we show that there exists a smooth dieomorphism transforming this link into a set of stream or vortex lines of a vector eld that solves the steady incompressible euler equation in r 3 furthermore the dieomorphism can be chosen arbitrarily close to the identity in any c r norm

existence of knotted vortex tubes in steady euler flows we prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible euler equation in mathbb r 3 more precisely given a finite collection of possibly linked and knotted disjoint thin tubes in mathbb r 3 we show that they can be transformed with a c m small diffeomorphism into a set of vortex tubes of a beltrami field that tends to zero at infinity the structure of the vortex lines in the tubes is extremely rich presenting a positive measure set of invariant tori and infinitely many periodic vortex lines the problem of the existence of steady knotted thin vortex tubes can be traced back to lord kelvin

the classification of 2 2 systems of non strictly hyperbolic conservation laws with application to oil recovery on considere un systeme 2 2 a une variable d espace u f u x 0 0 ou u u x t r 2 et f r 2 r 2 on montre que pour des equations hyperboliques des points ombilicaux sont typiquement isoles et que en un point ombilical isole df est typiquement diagonalisable on donne une classification des equations hyperboliques a points ombilicaux isoles

on a nonstrictly hyperbolic system of conservation laws a system of quasi linear hyperbolic conservation laws which is hyperbolic but not strictly hyperbolic is studied the system was derived as a model for the elastic string b keyfitz and kranzer arch rational mech anal 72 1980 210 241 and is assumed to be diagonizable interest is mainly in the large time behavior of the solution due to the nonlinearity of the system and the entropy condition solutions converge to very simple elementary waves nonstrict hyperbolicity of the system may cause stronger nonlinear interactions between waves pertaining to different families in particular such interactions may regularize linear waves in the solution the solutions are constructed using the random choice method

vanishing viscosity with short wave long wave interactions for systems of conservation laws motivated by benney s general theory we propose new models for short wave long wave interactions when the long waves are described by nonlinear systems of conservation laws we prove the strong convergence of the solutions of the vanishing viscosity and short wave long wave interactions systems by using compactness results from compensated compactness theory and new energy estimates obtained for the coupled systems we analyze several of the representative examples such as scalar conservation laws general symmetric systems nonlinear elasticity and nonlinear electromagnetism

on the riemann problem for a prototype of a mixed type conservation law we study the system of conservation laws given by with initial value the system is elliptic when u2 v2 2 and hyperbolic when u2 v2 2 following liu s construction it is found that the system always has a weak solution which however is not necessarily unique

beltrami fields with nonconstant proportionality factor we consider the question raised by enciso and peralta salas arch ration mech anal 220 1 243 260 2016 what nonconstant functions f can occur as the proportionality factor for a beltrami field mathbf u on an open subset u subset mathbb r 3 we also consider the related question for any such f how large is the space of associated beltrami fields by applying cartan s method of moving frames and the theory of exterior differential systems we are able to improve upon the results given in peralta salas 2016 in particular the answer to the second question depends crucially upon the geometry of the level surfaces of f we conclude by giving a complete classification of beltrami fields that possess either a translation symmetry or a rotation symmetry

a non uniform berry esseen bound via stein s method this paper is part of our efforts to develop stein s method beyond uniform bounds in normal approximation our main result is a proof for a non uniform berry esseen bound for independent and not necessarily identically distributed random variables without assuming the existence of third moments it is proved by combining truncation with stein s method and by taking the concentration inequality approach improved and adapted for non uniform bounds to illustrate the technique we give a proof for a uniform berry esseen bound without assuming the existence of third moments

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

stein s method and a quantitative lindeberg clt for the fourier transforms of random vectors abstract we use a multivariate version of stein s method to establish a quantitative lindeberg clt for the fourier transforms of random n vectors we achieve this conceptually mainly by constructing a natural approach structure on n random vectors overlying the topology of weak convergence and technically mainly by deducing a specific integral representation for the hessian matrix of a solution to the stein equation with test function e t x exp i k 1 n t k x k where t x r n

distances on probability measures and random variables in this paper we lift fundamental topological structures on probability measures and random variables in particular the weak topology convergence in law and finite dimensional convergence to an isometric level this allows for an isometric quantitative study of important concepts such as relative compactness tightness stochastic equicontinuity prohorov s theorem and smoothness in doing so we obtain numerical results which allow for the development of an intrinsic approximation theory and from which moreover all classical topological results follow as easy corollaries

dissipative continuous euler flows we show the existence of continuous periodic solutions of the 3d incompressible euler equations which dissipate the total kinetic energy

dissipative euler flows with onsager critical spatial regularity for any 0 we show the existence of continuous periodic weak solutions v of the euler equations that do not conserve the kinetic energy and belong to the space lt1 cx1 3 e namely x v x t is e holder continuous in space at a e time t and the integral t 1 3 edt is finite a well known open conjecture of l onsager claims that such solutions exist even in the class lt cx1 3 e 2016 wiley periodicals inc

beltrami fields with a nonconstant proportionality factor are rare we consider the existence of beltrami fields with a nonconstant proportionality factor f in an open subset u of mathbb r 3 by reformulating this problem as a constrained evolution equation on a surface we find an explicit differential equation that f must satisfy whenever there is a nontrivial beltrami field with this factor this ensures that there are no nontrivial regular solutions for an open and dense set of factors f in the c k topology k geqq 7 in particular there are no nontrivial beltrami fields whenever f has a regular level set diffeomorphic to the sphere this provides an explanation of the helical flow paradox of morgulis et al commun pure appl math 48 571 582 1995

on stationary solutions of two dimensional euler equation we study the geometry of streamlines and stability properties for steady state solutions of the euler equations for an ideal fluid

stein s method for diffusion approximations stein s method of obtaining distributional approximations is developed in the context of functional approximation by the wiener process and other gaussian processes an appropriate analogue of the one dimensional stein equation is derived and the necessary properties of its solutions are established the method is applied to the partial sums of stationary sequences and of dissociated arrays to a process version of the wald wolfowitz theorem and to the empirical distribution function

initial boundary value problem for conservation laws this paper concerns the initial boundary value problems for some systems of quasilinear hyperbolic conservation laws in the space of bounded measurable functions the main assumption is that the system under study admits a convex entropy extension it is proved that then any twicely differentiable entropy fluxes have traces on the boundary if the bounded solutions are generated by either godunov schemes or by suitable viscous approximations furthermore in the case that the weak interior solutions are generated by godunov schemes any lipschitz continuous entropy fluxes corresponding to convex entropies have traces on the boundary and the traces are bounded above by computable numerical boundary values this in particular gives a trace formula for the flux functions in terms of the numerical boundary data we also investigate the formulation of boundary conditions for systems of hyperbolic conservation laws it is shown that the set of expected boundary values derived from the viscous approximation contains the one derived in terms of the boundary riemann problems and the converse is not true in general the general theory is then applied to some specific examples first several new facts are obtained for convex scalar conservation laws for example we give example which show that godunov schemes produce numerical boundary layers it is shown that any continuous functions of density have traces on the boundary instead of only entropy fluxes we also obtain interior and boundary regularity of the weak solutions for bounded measurable initial and boundary data a generalized oleinik entropy condition is also obtained next we prove the existence of a weak solution to the initial boundary value problem for a family of quadratic system with a uniformly characteristic boundary condition

on the collapse of tubes carried by 3d incompressible flows we define the notion of a regular tube and prove that a regular tube convected by a 3d incompressible flow cannot collapse to zero thickness in finite time

multivariate normal approximation with stein s method of exchangeable pairs under a general linearity condition in this paper we establish a multivariate exchangeable pairs approach within the framework of stein s method to assess distributional distances to potentially singular multivariate normal distributions by extending the statistics into a higher dimensional space we also propose an embedding method which allows for a normal approximation even when the corresponding statistics of interest do not lend themselves easily to stein s exchangeable pairs approach to illustrate the method we provide the examples of runs on the line as well as double indexed permutation statistics

on the crossing rule abstract the paper gives conditions on a family of matrices which guarantee that some matrix in the family will have a multiple eigenvalue in particular the main theorem states exactly which dimensions admit k dimensional subspaces of matrices for which all nonzero elements have distinct eigenvalues this question arises naturally in the theory of first order hyperbolic systems of partial differential equations the main theorem in this context tells exactly for which integers n an n x n system in k space variables may be strictly hyperbolic author

divergence free fields energy and asymptotic crossing number in more than one context the equations of classical physics lead to a closed 2 form co on 3 dimensional space which is a conserved quantity in the sense that it pulls back under time evolution in the physics literature co is written as a divergence free vector field w appendix a describes two settings in which a divergence free w is conserved or frozen one is the magnetic field in ideal magneto hydrodynamics and the other is curl velocity in incompressible inviscid fluid flow if ht denotes the transport diffeomorphisms of a moving fluid in which the field is imbedded with ho id wt denotes the field in the fluid at time t and x0 is the space variable at time t 0 then

local representation and construction of beltrami fields ii solenoidal beltrami fields and ideal mhd equilibria abstract object of the present paper is the local theory of solution for steady ideal euler flows and ideal mhd equilibria the present analysis relies on the lie darboux theorem of differential geometry and the local theory of representation and construction of beltrami fields developed in sato and yamada 2019 a theorem for the construction of harmonic orthogonal coordinates is proved using such coordinates families of solenoidal beltrami fields with different topologies are obtained in analytic form existence of global solenoidal beltrami fields satisfying prescribed boundary conditions while preserving the local representation is considered it is shown that only singular solutions are admissible an explicit example is given in a spherical domain and a theorem on existence of singular solutions is proven local conditions for existence of solutions and local representation theorems are derived for generalized beltrami fields ideal mhd equilibria and general steady ideal euler flows the theory is applied to construct analytic examples

signature asymptotique d un champ de vecteurs en dimension 3 consider a volume preserving vector field defined in some compact domain of 3space and tangent to its boundary a long piece of orbit can be made into a knot by connecting its endpoints by some arc whose length is less than the diameter of the domain in this paper we study the behaviour of the signatures of these knots as the lengths of the pieces of orbits go to infinity we relate this asymptotic signature to the asymptotic hopf invariant that has been studied by arnold

maps of convex sets and invariant regions for finite difference systems of conservation laws general results about maps of convex sets in r n are proved we outline their extensions to an infinite dimensional context such extensions have applications in nonlinear analysis such as in the study of the invariance of convex sets under nonlinear maps here we explore applications only in the finite dimensional context more specifically we apply the general results to the problem of finding sufficient conditions for a region of the state space to be globally or locally invariant under finite difference schemes applied to systems of conservation laws in several space variables in particular we establish a final characterization of the invariant regions under the lax friedrichs scheme and also give sufficient conditions for the local invariance further we give sufficient conditions for the global and local invariance of regions under flux splitting finite difference schemes an example of the multi dimensional euler equations for non isentropic gas dynamics is discussed

on the rate of convergence in the multivariate clt berry esseen theorems are proved in the multidimensional central limit theorem without using fourier methods an effective and simple estimate of the error in the clt for sums and convex sets using stein s method and induction is derived furthermore the error in the clt for multivariate functions of independent random elements is estimated extending results of van zwet and friedrich to the multivariate case

riemann problems for nonstrictly hyperbolic 2 2 systems of conservation laws on resout le probleme de riemann pour des systemes 2 2 de lois de conservation hyperboliques ayant des fonctions flux quadratiques

hyperbolic conservation laws with umbilic degeneracy i in this paper a compactness framework for approximate solutions to nonlinear hyperbolic systems with umbilic degeneracy is established by combining techniques of compensated compactness with some classical methods and by a detailed analysis of a highly singular equation of euler poisson darboux type then this framework is successfully applied to prove the convergence of the viscosity method and to prove the existence of global entropy solutions for the cauchy problem with large initial data for a canonical class of the systems with quadratic flux form

euler equations on homogeneous spaces and virasoro orbits abstract we show that the following three systems related to various hydrodynamical approximations the korteweg de vries equation the camassa holm equation and the hunter saxton equation have the same symmetry group and similar bihamiltonian structures it turns out that their configuration space is the virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different right invariant metrics on this group or on appropriate homogeneous spaces in particular we describe how arnold s approach to the euler equations as geodesic flows of one sided invariant metrics extends from lie groups to homogeneous spaces we also show that the above three cases describe all generic bihamiltonian systems which are related to the virasoro group and can be integrated by the translation argument principle they correspond precisely to the three different types of generic virasoro orbits finally we discuss interrelation between the above metrics and kahler structures on virasoro orbits as well as open questions regarding integrable systems corresponding to a finer classification of the orbits

the beltrami spectrum for incompressible fluid flows recently v yakhot s orszag and their co workers have suggested that turbulent flows in various regions of space organize into a coherent hierarchy of weakly interacting superimposed approximate beltrami flows a mathematical framework is developed here to study organized beltrami hierarchies in a systematic fashion this framework is applied to several important classes of examples with universal beltrami hierarchies an analysis of the persistence of such beltrami hierarchies is also presented for general solutions of the navier stokes equations

on the existence of force free magnetic fields with small nonconstant α in exterior domains the existence of force free magnetic fields in the exterior domain of some compact simply connected surface s is proved via an iteration scheme the iteration starts with an arbitrary exterior vacuum field which contains flux tubes originating and ending on s at one cross section of such a flux tube with s an arbitrary function is prescribed for small values of in the holder norm 1 0 1 the iteration is shown to converge to a force free field with the prescribed values of in a flux tube which is close to the vacuum flux tube and 0 outside the force free field is close in the holder norm 1 to the starting vacuum field in particular it has the same field line topology the same boundary values on s and satisfies the same decay conditions in spatial infinity it is in general three dimensional and requires no continuous symmetries

nonstrictly hyperbolic nonlinear systems on etudie le caractere bien pose du probleme aux valeurs initiales localement en temps pour une classe de systemes hyperboliques non lineaires

two dimensional magnetohydrodynamic equilibria with prescribed topology we establish existence of weak solutions to the equilibrium equations of magnetohydrodynamics with prescribed topology this is carried out in two settings in the first we consider the variational problem of minimizing total energy in a torus under the assumption of axisymmetry and prescription of mass and flux profiles existence of weak solutions implies that the prescription of topology is a natural constraint both the compressible and incompressible cases are considered in the second setting we adapt examples of b c low and r wolfson 13 and j j aly and t amari 1 2 3 associated with parker s explanation of the extreme heating of the solar corona and other solar phenomena existence of solutions with fixed topology is a first crucial step in rigorously examining the relationship between topology and the existence of current sheets we use a decomposition introduced in 8 9 11 that captures much of the topology of level sets for certain classes of sobolev functions this decomposition is preserved under weak limits and so is useful for prescribing topological constraints the approach is especially suited to the use of domain deformations 2000 john wiley sons inc

the riemann problem near a hyperbolic singularity the classification of solutions of quadratic riemann problems 1 the purpose of this paper is to classify the solutions of riemann problems near a hyperbolic singularity in a nonlinear system of conservation laws hyperbolic singularities play the role in the theory of riemann problems that rest points play in the theory of ordinary differential equations indeed generically only a finite number of structures can appear in a neighborhood of such a singularity in this the first of three papers the program of classification is discussed in general and the simplest structure that occurs is characterized

an application of approach theory to the relative hausdorff measure of non compactness for the wasserstein metric abstract after shortly reviewing the fundamentals of approach theory as introduced by r lowen in 1989 we show that this theory is intimately related to the well known wasserstein metric on the space of probability measures with a finite first moment on a complete and separable metric space more precisely we introduce a canonical approach structure called the contractive approach structure and prove that it is metrized by the wasserstein metric the key ingredients of the proof of this result are dini s theorem ascoli s theorem and the fact that the class of real valued contractions on a metric space has some nice stability properties we then combine the obtained result with prokhorov s theorem to establish inequalities between the relative hausdorff measure of non compactness for the wasserstein metric and a canonical measure of non uniform integrability

compressible helical flows helical beltrami flow with nonuniform coefficient is considered for the case of compressible fluid and a class of exact solutions is proposed a paradox of helical flow is discussed and the compressibility is considered as a possible resolution of the paradox examples with different symmetries are given for the compressible helical flow and in particular the generalization of the abc arnold beltrami childress flow for the compressible case is proposed 1995 john wiley sons inc

local representation and construction of beltrami fields abstract a beltrami field is an eigenvector of the curl operator beltrami fields describe steady flows in fluid dynamics and force free magnetic fields in plasma turbulence by application of the lie darboux theorem of differential geometry we prove a local representation theorem for beltrami fields we find that locally a beltrami field has a standard form amenable to an arnold beltrami childress flow with two of the parameters set to zero furthermore a beltrami flow admits two local invariants a coordinate representing the physical plane of the flow and an angular momentum like quantity in the direction across the plane as a consequence of the theorem we derive a method to construct beltrami fields with given proportionality factor this method based on the solution of the eikonal equation guarantees the existence of beltrami fields for any orthogonal coordinate system such that at least two scale factors are equal we construct several solenoidal and non solenoidal beltrami fields with both homogeneous and inhomogeneous proportionality factors

beltrami fields with hyperbolic periodic orbits enclosed by knotted invariant tori abstract we prove that there exist beltrami fields in euclidean space with sharp decay at infinity which have a prescribed set of invariant tori possibly knotted or linked that enclose an arbitrarily large number of hyperbolic periodic orbits these hyperbolic orbits are cablings over the core curve of each torus moreover the domain bounded by each invariant torus is covered by an almost full measure set of invariant tori we show that an analogous result holds for high frequency beltrami fields on the flat torus t 3

algebraic conditions for hyperbolicity of systems of partial differential equations partial differential equations or systems of such equations are classified according to type such as elliptic parabolic hyperbolic etc hyperbolic systems with respect to the initial plane t 0 are those for which the initial value problem is well posed in the sense of hadamard see 8 general conditions that are both necessary and sufficient for hyperbolicity are complicated see 10 11 2 3 14 15 16 17 the situation however is rather simple for an m th order linear homogeneous system of equations with constant coefficients p left frac partial partial t frac partial partial x 1 frac partial partial x n right u 0 1 for a vector u u t x 1 x n u t x with n components here p is an n x n square matrix whose elements p ik are m th degree forms in their n 1 arguments we associate with 1 the matrix the symbol of 1 p lambda xi p lambda xi 1 cdot cdot cdot xi n pik lambda xi 1 cdot cdot cdot xi n 2 and the characteristic form q lambda xi q lambda xi 1 cdot cdot cdot xi n det p lambda xi 1 cdot cdot cdot xi n 3 of degree mn in its arguments

existence of chebyshev centers best n nets and best compact approximants in this paper we investigate the existence and continuity of chebyshev centers best n nets and best compact sets some of our positive results were obtained using the concept of quasi uniform convexity furthermore several examples of nonexistence are given e g a sublattice m of c o 1 and a bounded subset b c m is constructed which has no chebyshev center no best n net and not best compact set approximant

decay of entropy solutions of nonlinear conservation laws we are concerned with the asymptotic behavior of entropy solutions of nonlinear conservation laws the main objective of this paper is to present an analytical approach and to explore its applications to studying the large time behavior of periodic entropy solutions of hyperbolic conservation laws the asymptotic decay of periodic solutions of nonlinear hyperbolic conservation laws is an important nonlinear phenomenon it is observed that the genuine nonlinearity of equations forces the waves of each characteristic family to interact vigorously and to cancel each other the insightful analysis of glimm lax gl for scalar equations and 2 2 systems has indicated that the resultant mutual cancellation of interacting shock and rarefaction waves of the same family induces the decay of periodic solutions such a result was first shown by lax l1 in 1957 for one dimensional convex scalar conservation laws dafermos d1 applying his uniform processes proved the decay result for the case that the set of inflection points of the flux does not have an accumulation point the glimm lax theory gl indicates that for 2 2 strictly hyperbolic and genuinely nonlinear systems any periodic glimm solution decays like o 1 t this result was proved by using the approximate characteristic method in the glimm difference solutions provided that the oscillation of the corresponding initial data is small recently using the method of generalized characteristics dafermos d3 showed that any periodic solution with local bounded variation and small oscillation for 2 2 systems decays asymptotically with the same detailed structure pas found by lax l1 for the

analytic smoothing of geometric maps with applications to kam theory we show that finitely differentiable diffeomorphisms which are either symplectic volume preserving or contact can be approximated with analytic diffeomorphisms that are respectively symplectic volume preserving or contact we prove that the approximating functions are uniformly bounded on some complex domains and that the rate of convergence in cr norms of the approximation can be estimated in terms of the size of such complex domains and the order of differentiability of the approximated function as an application to this result we give a proof of the existence the local uniqueness and the bootstrap of regularity of kam tori for finitely differentiable symplectic maps the symplectic maps considered here are not assumed either to be written in action angle variables or to be perturbations of integrable systems our main assumption is the existence of a finitely differentiable parameterization of a maximal dimensional torus that satisfies a non degeneracy condition and that is approximately invariant the symplectic volume preserving and contact forms are assumed to be analytic

a variational principle for three dimensional water waves over beltrami flows abstract we consider steady three dimensional gravity capillary water waves with vorticity propagating on water of finite depth we prove a variational principle for doubly periodic waves with relative velocities given by beltrami vector fields under general assumptions on the wave profile

the euler equations as a differential inclusion we propose a new point of view on weak solutions of the euler equations describing the motion of an ideal incompressible fluid in r n with n 2 we give a reformulation of the euler equations as a differential inclusion and in this way we obtain transparent proofs of several celebrated results of v scheffer and a shnirelman concerning the non uniqueness of weak solutions and the existence of energy decreasing solutions our results are stronger because they work in any dimension and yield bounded velocity and pressure