c1 α isometric embeddings of polar caps abstract we study isometric embeddings of c 2 riemannian manifolds in the euclidean space and we establish that the holder space c 1 1 2 is critical in a suitable sense in particular we prove that for 1 2 the levi civita connection of any isometric immersion is induced by the euclidean connection whereas for any 1 2 we construct c 1 isometric embeddings of portions of the standard 2 dimensional sphere for which such property fails

on nonperiodic euler flows with holder regularity in isett regularity in time along the coarse scale flow for the euler equations 2013 the first author proposed a strengthening of onsager s conjecture on the failure of energy conservation for incompressible euler flows with holder regularity not exceeding 1 3 this stronger form of the conjecture implies that anomalous dissipation will fail for a generic euler flow with regularity below the onsager critical space l t infty b 3 infty 1 3 due to low regularity of the energy profile this paper is the first and main paper in a series of two the results of which may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with holder exponent less than 1 5 the main result of the present paper shows that any given smooth euler flow can be perturbed in c 1 5 epsilon t x on any pre compact subset of mathbb r times mathbb r 3 to violate energy conservation furthermore the perturbed solution is no smoother than c 1 5 epsilon t x as a corollary of this theorem we show the existence of nonzero c 1 5 epsilon t x solutions to euler with compact space time support generalizing previous work of the first author isett holder continuous euler flows in three dimensions with compact support in time 2012 to the nonperiodic setting

totally asymmetric exclusion processes with particles of arbitrary size the steady state currents and densities of a one dimensional totally asymmetric exclusion process tasep with particles that occlude an integer number d of lattice sites are computed using various mean field approximations and monte carlo simulations taseps featuring particles of arbitrary size are relevant for modelling systems such as mrna translation vesicle locomotion along microtubules and protein sliding along dna we conjecture that the nonequilibrium steady state properties separate into low density high density and maximal current phases similar to those of the standard d 1 tasep a simple mean field approximation for steady state particle currents and densities is found to be inaccurate however we find local equilibrium particle distributions derived from a discrete tonks gas partition function yield apparently exact currents within the maximal current phase for the boundary limited phases the equilibrium tonks gas distribution cannot be used to predict currents phase boundaries or the order of the phase transitions however we employ a refined mean field approach to find apparently exact expressions for the steady state currents boundary densities and phase diagrams of the d 1 tasep extensive monte carlo simulations are performed to support our analytic mean field results

exact steady states of disordered hopping particle models with parallel and ordered sequential dynamics a one dimensional driven lattice gas with disorder in the particle hopping probabilities is considered it has previously been shown that in the version of the model with random sequential updating a phase transition occurs from a low density inhomogeneous phase to a high density congested phase here the steady states for both parallel fully synchronous updating and ordered sequential updating are solved exactly the phase transition is shown to persist in both cases with the critical densities being higher than that for random sequential dynamics the steady state velocities are related to the fugacity of a bose system suggesting a principle of minimization of velocity a generalization of the dynamics to the case where the hopping probabilities depend on the number of empty sites in front of the particles is also solved exactly in the case of parallel updating the models have natural interpretations as simplistic descriptions of traffic flow the relation to more sophisticated traffic flow models is discussed

exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra we study the partially asymmetric exclusion process with open boundaries we generalize the matrix approach previously used to solve the special case of total asymmetry and derive exact expressions for the partition sum and currents valid for all values of the asymmetry parameter q due to the relationship between the matrix algebra and the q deformed quantum harmonic oscillator algebra we find that q hermite polynomials along with their orthogonality properties and generating functions are of great utility we employ two distinct sets of q hermite polynomials one for q 1 it turns out that these correspond to two distinct regimes the previously studied case of forward bias q 1 where the boundaries support a current opposite in direction to the bulk bias for the forward bias case we confirm the previously proposed phase diagram whereas the case of reverse bias produces a new phase in which the current decreases exponentially with system size

multiple shocks in a driven diffusive system with two species of particles abstract a one dimensional driven diffusive system with two types of particles and nearest neighbors interactions has been considered on a finite lattice with open boundaries the particles can enter and leave the system from both ends of the lattice and there is also a probability for converting the particle type at the boundaries we will show that on a special manifold in the parameters space multiple shocks evolve in the system for both species of particles which perform continuous time random walks on the lattice

global ill posedness of the isentropic system of gas dynamics we consider the isentropic compressible euler system in 2 space dimensions with pressure law p 2 and we show the existence of classical riemann data i e pure jump discontinuities across a line for which there are infinitely many admissible bounded weak solutions bounded away from the void we also show that some of these riemann data are generated by a 1 dimensional compression wave our theorem leads therefore to lipschitz initial data for which there are infinitely many global bounded admissible weak solutions c 2015 wiley periodicals inc

matrix product representation of the stationary state of the open zero range process many one dimensional lattice particle models with open boundaries like the paradigmatic asymmetric simple exclusion process asep have their stationary states represented in the form of a matrix product with matrices that do not explicitly depend on the lattice site in contrast the stationary state of the open 1d zero range process zrp takes an inhomogeneous factorized form with site dependent probability weights we show that in spite of the absence of correlations the stationary state of the open zrp can also be represented in a matrix product form where the matrices are site independent non commuting and determined from algebraic relations resulting from the master equation we recover the known distribution of the open zrp in two different ways first using an explicit representation of the matrices and boundary vectors second from the sole knowledge of the algebraic relations satisfied by these matrices and vectors finally an interpretation of the relation between the matrix product form and the inhomogeneous factorized form is proposed within the framework of hidden markov chains

one dimensional partially asymmetric simple exclusion process with open boundaries orthogonal polynomials approach the stationary state of the partially asymmetric simple exclusion process with open boundaries is reconsidered the so called matrix product ansatz is employed this enables us to construct the stationary state in the form of the product of the matrices d and e noticing the fact that the matrix c d e for the model is related to certain q orthogonal polynomials the model is analysed for a wide range of parameters the current and the correlation length are evaluated in the thermodynamic limit it turns out that the phase diagram for the correlation length is richer than that for the totally asymmetric case

representations of the quadratic algebra and partially asymmetric diffusion with open boundaries we consider the one dimensional partially asymmetric exclusion model with open boundaries the model describes a system of hard core particles that hop stochastically in both directions with different rates at both boundaries particles are injected and extracted by means of the method of derrida et al the stationary probability measure can be expressed as a matrix product state involving two matrices forming a fock like representation of a general quadratic algebra we obtain the representations of this algebra which were unknown in the mathematical literature and use the two dimensional one to derive exact expressions for the density profile and correlation functions using the correspondence between the stochastic model and a quantum spin chain we obtain exact correlation functions for a spin heisenberg xxz chain with non diagonal boundary terms generalizations to other reaction diffusion models are discussed

bethe ansatz and q operator for the open asep in this paper we look at the asymmetric simple exclusion process with open boundaries with a current counting deformation we construct a two parameter family of transfer matrices which commute with the deformed markov matrix of the system we show that these transfer matrices can be factorized into two commuting matrices with one parameter each which can be identified with baxter s q operator and that for certain values of the product of those parameters they decompose into a sum of two commuting matrices one of which is the usual one parameter transfer matrix for a given dimension of the auxiliary space using this we find the t q equation for the open asep and through functional bethe ansatz techniques we obtain an exact expression for the dominant eigenvalue of the deformed markov matrix

general existence theorems for hamilton jacobi equations in the scalar and vectorial cases keywords dirichlet problem hamilton jacobi equation existence of solutions reference caa article 1997 002doi 10 1007 bf02392708view record in web of science record created on 2008 11 25 modified on 2017 05 12

onsager s conjecture on the energy conservation for solutions of euler s equation we give a simple proof of a result conjectured by onsager 1 on energy conservation for weak solutions of euler s equation

on admissibility criteria for weak solutions of the euler equations we consider solutions to the cauchy problem for the incompressible euler equations satisfying several additional requirements like the global and local energy inequalities using some techniques introduced in an earlier paper we show that for some bounded compactly supported initial data none of these admissibility criteria singles out a unique weak solution as a byproduct in more than one space dimension we show bounded initial data for which admissible solutions to the p system of isentropic gas dynamics in eulerian coordinates are not unique

exact large deviation functional of a stationary open driven diffusive system the asymmetric exclusion process we consider the asymmetric exclusion process asep in one dimension on sites i 1 n in contact at sites i 1 and i n with infinite particle reservoirs at densities a and b as a and b are varied the typical macroscopic steady state density profile x x a b obtained in the limit n l b a exhibits shocks and phase transitions here we derive an exact asymptotic expression for the probability of observing an arbitrary macroscopic profile rho x text p n rho x sim exp l mathcal f a b rho x rho a rho b so that mathcal f is the large deviation functional a quantity similar to the free energy of equilibrium systems we find as in the symmetric purely diffusive case q 1 treated in an earlier work that mathcal f is in general a non local functional of x unlike the symmetric case however the asymmetric case exhibits ranges of the parameters for which mathcal f rho x is not convex and others for which mathcal f rho x has discontinuities in its second derivatives at x x in the latter ranges the fluctuations of order 1 sqrt n in the density profile near x are then non gaussian and cannot be calculated from the large deviation function

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

high dimensionality and h principle in pde in this note we present an analyst s point of view on the nash kuiper theorem and in particular highlight the very close connection to turbulence a paradigm example of a high dimensional phenomenon our aim is to explain recent applications of nash s ideas in connection with the incompressible euler equations and onsager s famous conjecture on the energy dissipation in 3d turbulence

partially asymmetric exclusion process with open boundaries exclusive diffusion on a one dimensional lattice is studied in the model particles hop stochastically in both directions but with different rates at the ends of the lattice particles are injected and removed the exact stationary probability measure is represented in the form of a matrix product as a generalization of the solution given by derrida et al j phys a 26 1493 1993 for the fully asymmetric process the phase diagram of the current on the infinite lattice is obtained analytic expressions for the current in the different phases are derived the model is equivalent to an xxz heisenberg chain with a certain type of boundary terms the ground state of which corresponds to the stationary solution of the master equation

vorticity measures and the inviscid limit we consider a sequence of leray hopf weak solutions of the 2d navier stokes equations on a bounded domain in the vanishing viscosity limit we provide sufficient conditions on the associated vorticity measures away from the boundary which ensure that as the viscosity vanishes the sequence converges to a weak solution of the euler equations the main assumptions are local interior uniform bounds on the l 1 norm of vorticity and the local uniform convergence to zero of the total variation of vorticity measure on balls in the limit of vanishing ball radii

onsager s conjecture almost everywhere in time in recent works by isett holder continuous euler flows in three dimensions with compact support in time pp 1 173 2012 and later by buckmaster et al ann math 2015 iterative schemes were presented for constructing solutions belonging to the holder class c 1 5 e of the 3d incompressible euler equations which do not conserve the total kinetic energy the cited work is partially motivated by a conjecture of lars onsager in 1949 relating to the existence of c 1 3 e solutions to the euler equations which dissipate energy in this note we show how the later scheme can be adapted in order to prove the existence of non trivial holder continuous solutions which for almost every time belong to the critical onsager holder regularity c 1 3 e and have compact temporal support

the asymmetric exclusion process comparison of update procedures the asymmetric exclusion process asep has attracted a lot of interest not only because of its many applications e g in the context of the kinetics of biopolymerization and traffic flow theory but also because it is a paradigmatic model for nonequilibrium systems here we study the asep for different types of updates namely random sequential sequential sublattice parallel and parallel in order to compare the effects of the different update procedures on the properties of the stationary state we use large scale monte carlo simulations and analytical methods especially the so called matrix product ansatz mpa we present in detail the exact solution for the model with sublattice parallel and sequential updates using the mpa for the case of parallel update which is important for applications like traffic flow theory we determine the phase diagram the current and density profiles based on monte carlo simulations we furthermore suggest an mpa for that case and derive the corresponding matrix algebra

convex integration for a class of active scalar equations we show that a general class of active scalar equa tions including porous media and certain magnetostrophic tur bulence models admit non unique weak solutions in the class of bounded functions the proof is based upon the method of convex integration recently implemented for equations of fluid dynamics in 2 3

on the euler equations of incompressible fluids euler equations of incompressible fluids use and enrich many branches of mathematics from integrable systems to geometric analysis they present important open physical and mathematical problems examples include the stable statistical behavior of ill posed free surface problems such as the rayleightaylor and kelvin helmholtz instabilities the paper describes some of the open problems related to the incompressible euler equations with emphasis on the blowup problem the inviscid limit and anomalous dissipation some of the recent results on the quasigeostrophic model are also mentioned

nonequilibrium temperatures in steady state systems with conserved energy we study a class of nonequilibrium lattice models describing local redistributions of a globally conserved quantity which is interpreted as an energy a particular subclass can be solved exactly allowing us to define a statistical temperature t th along the same lines as in the equilibrium micro canonical ensemble we compute the response function and find that when the fluctuation dissipation relation is linear the slope t fd 1 of this relation differs from the inverse temperature t th 1 we argue that t th is physically more relevant than t fd since in the steady state regime it takes equal values in two subsystems of a large isolated system finally a numerical renormalization group procedure suggests that all models within the class behave similarly at a coarse grained level leading to a parameter that describes the deviation from equilibrium quantitative predictions concerning this parameter are obtained within a mean field framework

on the nonuniqueness of weak solution of the euler equation weak solution of the euler equations is defined as an l2 vector field satisfying the integral relations expressing the mass and momentum balance their general nature has been quite unclear in this work an example of a weak solution on a 2 dimensional torus is constructed that is identically zero outside a finite time interval this example is simpler and more transparent than the previous example of v scheffer j geom anal 3 4 1993 pp 343 401 1997 john wiley sons inc

the tree property abstract we construct a model in which there are no n aronszajn trees for any finiten 2 starting from a model with infinitely many supercompact cardinals we also construct a model in which there is no aronszajn tree for a strong limit cardinal of cofinality starting from a model with a supercompact cardinal and a weakly compact cardinal above it

on the energy dissipation rate of solutions to the compressible isentropic euler system in this paper we extend and complement the results in chiodaroli et al global ill posedness of the isentropic system of gas dynamics 2014 on the well posedness issue for weak solutions of the compressible isentropic euler system in 2 space dimensions with pressure law p 1 first we show that every riemann problem whose one dimensional self similar solution consists of two shocks admits also infinitely many two dimensional admissible bounded weak solutions not containing vacuum generated by the method of de lellis and szekelyhidi ann math 170 1417 1436 2009 arch ration mech anal 195 225 260 2010 moreover we prove that for some of these riemann problems and for 1 3 such solutions have a greater energy dissipation rate than the self similar solution emanating from the same riemann data we therefore show that the maximal dissipation criterion proposed by dafermos in j diff equ 14 202 212 1973 does not favour the classical self similar solutions

h principles for the incompressible euler equations in dissipative euler flows and onsager s conjecture arxiv 1205 3626 preprint 2012 de lellis and szekelyhidi construct holder continuous dissipative weak solutions to the incompressible euler equations in the torus mathbb t 3 the construction consists of adding fast oscillations to the trivial solution we extend this result by establishing optimal h principles in two and three space dimensions specifically we identify all subsolutions defined in a suitable sense which can be approximated in the h 1 norm by exact solutions furthermore we prove that the flows thus constructed on mathbb t 3 are genuinely three dimensional and are not trivially obtained from solutions on mathbb t 2

the energy measure for the euler and navier stokes equations the potential failure of energy equality for a solution u of the euler or navier stokes equations can be quantified using a so called energy measure the weak limit of the measures u t 2 rm d x as t approaches the first possible blowup time we show that membership of u in certain weak or strong l q l p classes gives a uniform lower bound on the lower local dimension of mathcal e more precisely it implies uniform boundedness of a certain upper s density of mathcal e we also define and give lower bounds on the concentration dimension associated to mathcal e which is the hausdorff dimension of the smallest set on which energy can concentrate both the lower local dimension and the concentration dimension of mathcal e measure the departure from energy equality as an application of our estimates we prove that any solution to the 3 dimensional navier stokes equations which is type i in time must satisfy the energy equality at the first blowup time

finite dimensional representations of the quadratic algebra applications to the exclusion process we study the one dimensional partially asymmetric simple exclusion process asep with open boundaries that describes a system of hard core particles hopping stochastically on a chain coupled to reservoirs at both ends derrida and coworkers showed in 1993 that the stationary probability distribution of this model can be represented as a trace on a quadratic algebra closely related to the deformed oscillator algebra we construct all finite dimensional irreducible representations of this algebra this enables us to compute the stationary bulk density as well as all correlation lengths for the asep on a set of special curves of the phase diagram

young measures generated by ideal incompressible fluid flows in their seminal paper diperna and majda commun math phys 108 4 667 689 1987 introduced the notion of a measure valued solution for the incompressible euler equations in order to capture complex phenomena present in limits of approximate solutions such as persistence of oscillation and development of concentrations furthermore they gave several explicit examples exhibiting such phenomena in this paper we show that any measure valued solution can be generated by a sequence of exact weak solutions in particular this gives rise to a very large arguably too large set of weak solutions of the incompressible euler equations

exact solution of the master equation for the asymmetric exclusion process using the bethe ansatz we obtain the exact solution of the master equation for the totally asymmetric exclusion process on an infinite one dimensional lattice we derive explicit expressions for the conditional probabilitiesp x1 xn t y1 yn 0 of findingn particles on lattices sitesx1 xn at timet with initial occupationy1 yn at timet 0

hydrodynamic mean field solutions of 1d exclusion processes with spatially varying hopping rates we analyse the open boundary partially asymmetric exclusion process with smoothly varying internal hopping rates in the infinite size mean field limit the mean field equations for particle densities are written in terms of ricatti equations with the steady state current j as a parameter these equations are solved both analytically and numerically upon imposing the boundary conditions set by the injection and extraction rates the currents j are found self consistently we find a number of cases where analytic solutions can be found exactly or approximated results for j from asymptotic analyses for slowly varying hopping rates agree extremely well with those from extensive monte carlo simulations suggesting that mean field currents asymptotically approach the exact currents in the hydrodynamic limit as the hopping rates vary slowly over the lattice if the forward hopping rate is greater than or less than the backward hopping rate throughout the entire chain the three standard steady state phases are preserved our analysis reveals the sensitivity of the current to the relative phase between the forward and backward hopping rate functions

cauchy problem for dissipative holder solutions to the incompressible euler equations we consider solutions to the cauchy problem for the incompressible euler equations on the 3 dimensional torus which are continuous or holder continuous for any exponent theta frac 1 16 using the techniques introduced in de lellis and szekelyhidi inventiones mathematicae 9 377 407 2013 dissipative euler flows and onsager s conjecture 2012 we prove the existence of infinitely many holder continuous initial vector fields starting from which there exist infinitely many holder continuous solutions with preassigned total kinetic energy

matrix product solution of the stationary state of two species open zero range processes using the matrix product ansatz we obtain solutions of the steady state distribution of the two species open one dimensional zero range process our solution is based on a conventionally employed constraint on the hop rates which eventually allows us to simplify the constituent matrices of the ansatz it is shown that the matrix at each site is given by the tensor product of two sets of matrices and the steady state distribution assumes an inhomogeneous factorized form our method can be generalized to the cases of more than two species of particles

the noncommutative space of stochastic diffusion systems in the matrix product ground states approach to n species diffusion processes the stationary probability distribution is expressed as a matrix product state with respect to a quadratic algebra determined by the dynamics of the process we show that the quadratic algebra defines a noncommutative space with a glq n quantum group action as its symmetry boundary processes account for the appearance of parameter dependent linear terms in the algebraic relations we argue that for systems with boundary conditions the diffusion algebras are also obtained either by a shift of basis in the n dimensional quantum plane or by an appropriate change of basis in a lower dimensional one which leads to a reduction of the glq n symmetry

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

lack of uniqueness for weak solutions of the incompressible porous media equation in this work we consider weak solutions of the incompressible two dimensional porous media ipm equation by using the approach of de lellis szekelyhidi we prove non uniqueness for solutions in l in space and time

the generalized continuum hypothesis can fail everywhere in 1874 cantor cl showed that every set has cardinality strictly smaller than the cardinality of its power set cantor asked c2 whether for infinite sets x there is a set y of cardinality strictly between cardinality x and cardinality 2x the special case of x z the integers was hilbert s first problem in his famous list hi the continuum hypothesis in this paper we show that it is consistent with zermelo frankel set theory with the full axiom of choice modulo large cardinals that for every set x there is a set y such that the cardinality of y lies strictly between the cardinality of x and the cardinality of the power set of x it was previously shown in gcdel g that the generalized continuum hypothesis g c h was consistent i e for every infinite cardinal x the cardinal successor to x was 2x in 1963 cohen co showed that it was consistent that the continuum hypothesis failed easton e showed that subject to relatively mild restrictions kdnig s theorem essentially arbitrary behavior of the power set operation could occur at regular cardinals singular cardinals presented a significantly more difficult matter the first work on them was done by prikry and silver p and sil who showed that the g c h can fail at singular strong limit cardinals magidor ml showed that it was consistent that it fail at the first singular strong limit and even that one could have the first failure of the g c h at a singular strong limit after magidor s work it was generally believed that arbitrary behavior was possible silver however showed si that if the g c h holds below a singular cardinal k of uncountable cofinality then it holds at k galvin and hajnal g h showed that under more general conditions the behavior of the power set below a singular cardinal k of uncountable cofinality strongly affects the power set at k these results are all local results the consistent global behavior of the power set operation was not settled we prove the following theorem

anomalous dissipation for 1 5 holder euler flows recently the second and fourth authors developed an iterative scheme for obtaining rough solutions of the 3d incompressible euler equations in holder spaces the motivation comes from onsager s conjecture the construction involves a superposition of weakly interacting perturbed beltrami flows on infinitely many scales an obstruction to better regularity arises from the errors in the linear transport of a fast periodic flow by a slow velocity field in a recent paper the third author has improved upon the methods introducing some novel ideas on how to deal with this obstruction thereby reaching a better holder exponent albeit weaker than the one conjectured by onsager in this paper we give a shorter proof of this final result adhering more to the original scheme of the second and fourth authors and introducing some new devices more precisely we show that for any positive e there exist periodic solutions of the 3d incompressible euler equations that dissipate the total kinetic energy and belong to the holder class c 1 5 epsilon

matrix product ground states for exclusion processes with parallel dynamics we show in the example of a one dimensional asymmetric exclusion process that stationary states of models with parallel dynamics may be written in a matrix product form the corresponding algebra is quadratic and involves three different matrices using this formalism we prove previous conjectures for the equal time correlation functions of the model

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