some remark on the existence of infinitely many nonphysical solutions to the incompressible navier stokes equations abstract we prove that there exist infinitely many distributional solutions with infinite kinetic energy to both the incompressible navier stokes equations in r 2 and burgers equation in r with vanishing initial data

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

variational bounds on energy dissipation in incompressible flows ii channel flow a variational principle for lower bounds on the time averaged mass flux for newtonian fluids driven by a pressure gradient in a channel is derived from the incompressible navier stokes equations when supplied with appropriate test background flow fields the variational formulation produces explicit estimates for the friction coefficient these rigorous bounds are compared with the predictions of conventional turbulence theory

circuit satisfiability and constraint satisfaction around skolem arithmetic abstract we study interactions between skolem arithmetic and certain classes of circuit satisfiability and constraint satisfaction problems csps we revisit results of glaser et al 1 in the context of csps and settle the major open question from that paper finding a certain satisfiability problem on circuits involving complement intersection union and multiplication to be decidable this we prove using the decidability of skolem arithmetic then we solve a second question left open in 1 by proving a tight upper bound for the similar circuit satisfiability problem involving just intersection union and multiplication we continue by studying first order expansions of skolem arithmetic without constants n as csps we find already here a rich landscape of problems with non trivial instances that are in p as well as those that are np complete

bounds on dissipation for navier strokes flow with kologorov forcing abstract in this paper bounds on average viscous dissipation are derived for kolmogorov flow in a spatially periodic domain with steady and unsteady forcing at arbitrarily large grashof number g for a force of the form f 0 sin mz i or f 0 sin mz cos t i we derive various bounds on the total dissipation in the flow d u as well as on the dissipation d m obtained from the x velocity averaged over the x y plane the mean velocity of the flow we derive upper bounds on d u and d v d u d m as well as lower bounds on d m and d m d u adopting constraints of the kind introduced by howard and busse and assuming a steady force the background flow method introduced by doering and constantin is used to obtain an improved lower bound on d m d u of o g 1 and a lower bound on d u of o g 1 2 where g f0l3 2 is the grashof number some of these results are then generalized to time periodic forcing direct numerical simulation of the flow indicates that these bounds leave substantial gaps at large grashof number g the calculated d m g and d u g being o g 1 2 and o 1 respectively as g our theoretical bounds on d m d u are shown to be attained by steady laminar type flows for neighboring forcing functions which seems to indicate that these bounds cannot be improved by adding further dynamical constraints however our elementary upper bound on d v can probably be improved by placing more constraints on the flows these results serve to emphasize the difference between boundary driven turbulence and body force driven turbulence where the appropriate dissipation bound is believed saturated at least up to logarithms

robustness of one dimensional viscous fluid motion under multidimensional perturbations abstract we adapt the relative energy functional associated to the compressible navier stokes system to show stability of solutions emanating from 1 d initial data with respect to multidimensional n 2 3 perturbations besides the application of the relative energy inequality as a suitable distance between two solutions refined regularity estimates in l p based sobolev spaces are used

hard constraint satisfaction problems have hard gaps at location 1 an instance of the maximum constraint satisfaction problem max csp is a finite collection of constraints on a set of variables and the goal is to assign values to the variables that maximises the number of satisfied constraints max csp captures many well known problems such as maxk sat and max cut and is consequently np hard thus it is natural to study how restrictions on the allowed constraint types or constraint language affect the complexity and approximability of max csp the pcp theorem is equivalent to the existence of a constraint language for which max csp has a hard gap at location 1 i e it is np hard to distinguish between satisfiable instances and instances where at most some constant fraction of the constraints are satisfiable all constraint languages for which the csp problem i e the problem of deciding whether all constraints can be satisfied is currently known to be np hard have a certain algebraic property we prove that any constraint language with this algebraic property makes max csp have a hard gap at location 1 which in particular implies that such problems cannot have a ptas unless p np we then apply this result to max csp restricted to a single constraint type this class of problems contains for instance max cut and max dicut assuming p np we show that such problems do not admit ptas except in some trivial cases our results hold even if the number of occurrences of each variable is bounded by a constant finally we give some applications of our results

upper bounds on the heat transport in a porous layer abstract numerical and asymptotic analytical upper bounds on the convective heat transport in a fluid saturated porous layer of infinite darcy prandtl number are obtained by the howard busse variational method of the optimum theory of turbulence the nonlinear euler lagrange equations of the variational problem possess different numerical solutions as well as different approximate analytical solutions in the region of high rayleigh numbers we use symmetric fields for the numerical investigation and obtain slightly different numerical bounds on the heat transport in comparison to the numerical bounds obtained by gupta and joseph j fluid mech 57 1973 491 514 as well as different numerical values of the wave numbers connected to the optimum fields for the analytical investigation we use new approximate solution of the euler lagrange equations for the intermediate layers of the optimum fields this solution leads to different thickness of the boundary layers of the optimum fields as well as to a lower analytical upper bounds on the heat transport in comparison to the boundary layer thicknesses and analytical upper bounds obtained by gupta and joseph

the complexity of general valued csps an instance of the valued constraint satisfaction problem vcsp is given by a finite set of variables a finite domain of labels and a sum of functions each function depending on a subset of the variables each function can take finite values specifying costs of assignments of labels to its variables or the infinite value which indicates an infeasible assignment the goal is to find an assignment of labels to the variables that minimizes the sum we study assuming that p ne np how the complexity of this very general problem depends on the set of functions allowed in the instances the so called constraint language the case when all allowed functions take values in 0 infty corresponds to ordinary csps where one deals only with the feasibility issue and there is no optimization this case is the subject of the algebraic csp dichotomy conjecture predicting for which constraint languages csps are tractable i e solvable in polynomial time and for which they are np hard the case when all

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

on the backward behavior of some dissipative evolution equations abstract we prove that every solution of a kdv burgers sivashinsky type equation blows up in the energy space backward in time provided the solution does not belong to the global attractor this is a phenomenon contrast to the backward behavior of the periodic 2d navier stokes equations studied by constantin et al 1997 but analogous to the backward behavior of the kuramoto sivashinsky equation discovered by kukavica and malcok 2005 also we study the backward behavior of solutions to the damped driven nonlinear schrodinger equation the complex ginzburg landau equation and the hyperviscous navier stokes equations in addition we provide some physical interpretation of various backward behaviors of several perturbations of the kdv equation by studying explicit cnoidal wave solutions furthermore we discuss the connection between the backward behavior and the energy spectra of the solutions the study of backward behavior of dissipative evolution equations is motivated by the investigation of the bardos tartar conjecture on the navier stokes equations stated in bardos and tartar 1973

kolmogorov s theory of turbulence and inviscid limit of the navier stokes equations in mathbb r 3 we are concerned with the inviscid limit of the navier stokes equations to the euler equations in mathbb r 3 we first observe that a pathwise kolmogorov hypothesis implies the uniform boundedness of the th order fractional derivatives of the velocity for some 0 in the space variables in l2 which is independent of the viscosity 0 then it is shown that this key observation yields the l2 equicontinuity in the time variable and the uniform bound in lq for some q 2 of the velocity independent of 0 these results lead to the strong convergence of solutions of the navier stokes equations to a solution of the euler equations in mathbb r 3 we also consider passive scalars coupled to the incompressible navier stokes equations and in this case find the weak star convergence for the passive scalars with a limit in the form of a young measure pdf depending on space and time not only do we offer a framework for mathematical existence theories but also we offer a framework for the interpretation of numerical solutions through the identification of a function space in which convergence should take place with the bounds that are independent of 0 that is in the high reynolds number limit

classifying the complexity of constraints using finite algebras many natural combinatorial problems can be expressed as constraint satisfaction problems this class of problems is known to be np complete in general but certain restrictions on the form of the constraints can ensure tractability here we show that any set of relations used to specify the allowed forms of constraints can be associated with a finite universal algebra and we explore how the computational complexity of the corresponding constraint satisfaction problem is connected to the properties of this algebra hence we completely translate the problem of classifying the complexity of restricted constraint satisfaction problems into the language of universal algebra we introduce a notion of tractable algebra and investigate how the tractability of an algebra relates to the tractability of the smaller algebras which may be derived from it including its subalgebras and homomorphic images this allows us to reduce significantly the types of algebras which need to be classified using our results we also show that if the decision problem associated with a given collection of constraint types can be solved efficiently then so can the corresponding search problem we then classify all finite strictly simple surjective algebras with respect to tractability obtaining a dichotomy theorem which generalizes schaefer s dichotomy for the generalized satisfiability problem finally we suggest a possible general algebraic criterion for distinguishing the tractable and intractable cases of the constraint satisfaction problem

weak near unanimity functions and digraph homomorphism problems for a fixed digraph h the problem of deciding whether there exists a homomorphism from an input digraph g to h is known as the h colouring problem an algebraic approach to this problem was pioneered by jeavons et al in the context of the more general constraint satisfaction problem results by larose and zadori and by maroti and mckenzie allow one to interpret the algebraic approach in terms of so called weak near unanimity functions wnufs in this paper we focus on weak near unanimity functions and how they apply to the h colouring problem in particular our results range from non existence results of wnufs for certain digraphs h to the existence of wnufs for the well known polynomial cases of the h colouring problem treated by gutjahr woeginger and welzl along the way we develop wnuf analogs of the indicator and sub indicator constructions of hell and nesetril these results provide evidence to the conjecture that weak near unanimity functions are the right measure for the complexity of the h colouring problem

an algebraic theory of complexity for discrete optimization discrete optimization problems arise in many different areas and are studied under many different names in many such problems the quantity to be optimized can be expressed as a sum of functions of a restricted form here we present a unifying theory of complexity for problems of this kind we show that the complexity of a finite domain discrete optimization problem is determined by certain algebraic properties of the objective function which we call weighted polymorphisms we define a galois connection between sets of rational valued functions and sets of weighted polymorphisms and show how the closed sets of this galois connection can be characterized these results provide a new approach to studying the complexity of discrete optimization we use this approach to identify certain maximal tractable subproblems of the general problem and hence derive a complete classification of complexity for the boolean case

a dichotomy theorem for constraint satisfaction problems on a 3 element set the constraint satisfaction problem csp provides a common framework for many combinatorial problems the general csp is known to be np complete however certain restrictions on a possible form of constraints may affect the complexity and lead to tractable problem classes there is therefore a fundamental research direction aiming to separate those subclasses of the csp that are tractable and those which remain np complete schaefer gave an exhaustive solution of this problem for the csp on a 2 element domain in this article we generalise this result to a classification of the complexity of the csp on a 3 element domain the main result states that every subproblem of the csp is either tractable or np complete and the criterion separating them is that conjectured in bulatov et al 2005 and bulatov and jeavons 2001b we also characterize those subproblems for which standard constraint propagation techniques provide a decision procedure finally we exhibit a polynomial time algorithm which for a given set of allowed constraints outputs if this set gives rise to a tractable problem class to obtain the main result and the algorithm we extensively use the algebraic technique for the csp developed in jeavons 1998b bulatov et al 2005 and bulatov and jeavons 2001b

h coloring dichotomy revisited the h coloring problem can be expressed as a particular case of the constraint satisfaction problem csp whose computational complexity has been intensively studied under various approaches in the last several years we show that the dichotomy theorem proved by hell and neset il on the complexity of h coloring j combin theory ser b 48 1990 92 110 for the complexity of the h coloring problem for undirected graphs can be obtained using general methods for studying csp and that the criterion distinguishing the tractable cases of the h coloring problem agrees with that conjectured in a a bulatov p g jeavons a a krokhin constraint satisfaction problems and finite algebras in proc 27th internat colloq on automata languages and programming icalp 00 lecture notes in computer science vol 1853 springer berlin 2000 pp 272 282 for the complexity of the general csp

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

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

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

an efficient qr based method for the computation of lyapunov exponents abstract an efficient and numerically stable method to determine all the lyapunov characteristic exponents of a dynamical system is developed numerical experiments are presented highlighting some aspects of convergence accuracy and efficiency in the computation of the lyapunov characteristic exponents

optimization with affine homogeneous quadratic integral inequality constraints we introduce a new technique to optimize a linear cost function subject to an affine homogeneous quadratic integral inequality i e the requirement that a homogeneous quadratic integral functional affine in the optimization variables is nonnegative over a space of functions defined by homogeneous boundary conditions such problems arise in control and stability or input to state output analysis of systems governed by partial differential equations pdes particularly fluid dynamical systems we derive outer approximations for the feasible set of a homogeneous quadratic integral inequality in terms of linear matrix inequalities lmis and show that a convergent sequence of lower bounds for the optimal cost can be computed with a sequence of semidefinite programs sdps we also obtain inner approximations in terms of lmis and sum of squares constraints so upper bounds for the optimal cost and strictly feasible points for the integral inequality can be computed with sdps we present quinopt an open source add on to yalmip to aid the formulation and solution of our sdps and demonstrate our techniques on problems arising from the stability analysis of pdes

a new proof of the caffarelli kohn nirenberg theorem here we give a self contained new proof of the partial regularity theorems for solutions of incompressible navier stokes equations in three spatial dimensions these results were originally due to scheffer and caffarelli kohn and nirenberg our proof is much more direct and simpler 1998 john wiley sons inc

variational bounds on shear driven turbulence and turbulent boussinesq convection abstract based on earlier studies by hopf 1941 doering and constantin 1992 1994 1995 have recently formulated a new background technique for obtaining upper bounds on turbulent fluid flow quantities this method produces upper bounds on the limit supremum of long time averages making no statistical assumptions about the flow in contrast to the well known howard busse approach the full optimisation problems posed by this method for the momentum transport in turbulent couette flow and the heat transport with zero background flow in turbulent boussinesq convection are solved here for the first time at asymptotically large reynolds number and rayleigh number within busse s multiple boundary layer approximation to extract the best lowest possible upper bounds available intriguingly the original bounds isolated by busse 1969 1970 within the confines of statistical stationarity are recovered exactly using this new formalism the optimal background velocity profile for turbulent couette flow is found to be shearless in the interior thus differing from busse s 1 4 mean shear result in the convective case an interesting degeneracy in the formulation of the background variational problem leads to an indeterminacy in the optimal background temperature profile only for one special choice is the isothermal core feature of busse s mean profile recovered

a note on the computation of the largest p lces of discrete dynamical systems two efficient and numerically stable methods for the computation of the largest p lyapunov characteristic exponents of an n dimensional discrete dynamical systems are presented the efficiencies of the proposed methods are compared with the efficiencies of other methods through an operation count and illustrated with an example

the power of linear programming for general valued csps let d called the domain be a fixed finite set and let gamma called the valued constraint language be a fixed set of functions of the form f d m to mathbb q cup infty where different functions might have different arity m we study the valued constraint satisfaction problem parametrized by gamma denoted by vcsp gamma these are minimization problems given by n variables and the objective function given by a sum of functions from gamma each depending on a subset of the n variables for example if d 0 1 and gamma contains all ternary 0 infty valued functions vcsp gamma corresponds to 3 sat more generally if gamma contains only 0 infty valued functions vcsp gamma corresponds to csp gamma if d 0 1 and gamma contains all ternary 0 1 valued functions vcsp gamma corresponds to min 3 sat in which the goal is to minimize the number of unsatisfied clauses in a 3 cnf instance finite valued constraint languages contain functions that take on only rational values and not infinite values our main result is a precise algebraic characterization of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation blp for a valued constraint language gamma blp is a decision procedure for gamma if and only if gamma admits a symmetric fractional polymorphism of every arity for a finite valued constraint language gamma blp is a decision procedure if and only if gamma admits a symmetric fractional polymorphism of some arity or equivalently if gamma admits a symmetric fractional polymorphism of arity 2 using these results we obtain tractability of several novel classes of problems including problems over valued constraint languages that are 1 submodular on arbitrary lattices 2 k submodular on arbitrary finite domains 3 weakly and hence strongly tree submodular on arbitrary trees

gap theorems for robust satisfiability boolean csps and beyond abstract a computational problem exhibits a gap property when there is no tractable boundary between two disjoint sets of instances we establish a gap trichotomy theorem for a family of constraint problem variants completely classifying the complexity of possible np hard gaps in the case of boolean domains as a consequence we obtain a number of dichotomies for the complexity of specific variants of the constraint satisfaction problem all are either polynomial time tractable or np complete schaefer s original dichotomy for sat variants is a notable particular case universal algebraic methods have been central to recent efforts in classifying the complexity of constraint satisfaction problems a second contribution of the article is to develop aspects of the algebraic approach in the context of a number of variants of the constraint satisfaction problem in particular this allows us to lift our results on boolean domains to many templates on non boolean domains

convective heat transport in a rotating fluid layer of infinite prandtl number optimum fields and upper bounds on nusselt number by means of the howard busse method of the optimum theory of turbulence we investigate numerically upper bounds on convective heat transport for the case of infinite fluid layer with stress free vertical boundaries rotating about a vertical axis we discuss the case of infinite prandtl number 1 ensuremath ensuremath alpha solution of the obtained variational problem and optimum fields possessing internal intermediate and boundary layers we investigate regions of rayleigh and taylor numbers r and ta where no analytical bounds can be derived and compare the analytical and numerical bounds for these regions of r and ta where such comparison is possible the increasing rotation has a different influence on the rescaled optimum fields of velocity w 1 temperature ensuremath theta 1 and the vertical component of the vorticity f 1 the increasing ta for fixed r leads to vanishing of the boundary layers of w 1 and ensuremath theta 1 opposite to this the increasing ta leads first to a formation of boundary layers of the field f 1 but further increasing the rotation causes vanishing of these boundary layers we obtain optimum profiles of the horizontal averaged total temperature field which could be used as hints for construction of the background fields when applying doering constantin method to the problems of rotating convection the wave number ensuremath alpha 1 corresponding to the optimum fields follows the asymptotic relationship ensuremath alpha 1 r 5 1 4 for intermediate rayleigh numbers however when r becomes large with respect to ta after a transition region the power law for ensuremath alpha 1 becomes close to the power law for the case without rotation the nusselt number nu is close to the nonrotational bound 0 32r 1 3 for the case of large r and small ta nu decreases with increasing taylor number thus the upper bounds reflect the tendency of inhibiting thermal convection by increasing rotation for a fixed rayleigh number for the regions of rayleigh and taylor numbers where the numerical and asymptotic bounds on nu can be compared the numerical bounds are about 70 lower than the asymptotic bounds

compressible fluids driven by stochastic forcing the relative energy inequality and applications we show the relative energy inequality for the compressible navier stokes system driven by a stochastic forcing as a corollary we prove the weak strong uniqueness property pathwise and in law and convergence of weak solutions in the inviscid incompressible limit in particular we establish a yamada watanabe type result in the context of the compressible navier stokes system that is pathwise weak strong uniqueness implies weak strong uniqueness in law

duality and polynomial testing of tree homomorphisms let h be a fixed digraph we consider the h colouring problem i e the problem of deciding which digraphs g admit a homomorphism to h we are interested in a characterization in terms of the absence in g of certain tree like obstructions specifically we say that h has tree duality if for all digraphs g g is not homomorphic to h if and only if there is an oriented tree which is homomorphic to g but not to h we prove that if h has tree duality then the h colouring problem is polynomial we also generalize tree duality to bounded treewidth duality and prove a similar result we relate these duality concepts to the notion of the x property studied by gutjahr welzl and woeginger we then focus on the case when h itself is an oriented tree in fact we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two such trees are called triads we have shown in a companion paper that there exist oriented triads h for which the h colouring problem is np complete we contrast these with several families of oriented triads h which have tree duality or bounded treewidth duality and hence polynomial h colouring problems if p 6 np then no oriented triad h with an np complete h colouring problem can have bounded treewidth duality however no proof of this is known for any oriented triad h we prove that none of the oriented triads h with np complete hcolouring problems given in the companion paper has tree duality

the complexity of counting quantifiers on equality languages an equality language is a relational structure with infinite domain whose relations are first order definable in equality we classify the extensions of the quantified constraint satisfaction problem over equality languages in which the native existential and universal quantifiers are augmented by some subset of counting quantifiers in doing this we find ourselves in various worlds in which dichotomies or trichotomies subsist

self similar 2d euler solutions with mixed sign vorticity we construct a class of self similar 2d incompressible euler solutions that have initial vorticity of mixed sign the boundaries between regions of positive and negative vorticity form algebraic spirals similar to the kaden spiral and as opposed to prandtl s logarithmic vortex spirals also unlike the prandtl case spirals are not initially present

non existence of strong regular reflections in self similar potential flow abstract we consider shock reflection which has a well known local non uniqueness the reflected shock can be either of two choices called weak and strong we consider cases where existence of a global solution with weak reflected shock has been proven for compressible potential flow if there was a global strong shock solution as well then potential flow would be ill posed however we prove non existence of strong shock analogues in a natural class of candidates

a possible counterexample to well posedness of entropy solutions and to godunov scheme convergence a particular case of initial data for the two dimensional euler equations is studied numerically the results show that the godunov method does not always converge to the physical solution at least not on feasible grids moreover they suggest that entropy solutions in the weak entropy inequality sense are not well posed

computational explorations into the dynamics of rings of coupled oscillators this paper deals with the response of homogeneous and inhomogeneous rings of coupled oscillators where each individual oscillator when uncoupled from the others is chaotic it is shown that coupling can bring about a wide variety of global responses and that there is a significant range of coupling values when the response of the ring is periodic despite the fact that each oscillator is chaotic in fact numerous periodic solutions can be found depending on the initial conditions the response of a coupled set of homogeneous and non homogeneous rings is also investigated showing that the behavior of such coupled compartmental models can be quite counterintuitive and sensitive to the parameters that describe the extent and nature of coupling

congruence distributivity implies bounded width we show that a constraint language with compatible jonsson terms or equivalently associated with an algebra generating a congruence distributive variety defines a constraint satisfaction problem solvable by the local consistency checking algorithm

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

gibbs states and the set of solutions of random constraint satisfaction problems abstract an instance of a random constraint satisfaction problem defines a random subset the set of solutions of a large product space x n the set of assignments we consider two prototypical problem ensembles random k satisfiability and q coloring of random regular graphs and study the uniform measure with support on s as the number of constraints per variable increases this measure first decomposes into an exponential number of pure states clusters and subsequently condensates over the largest such states above the condensation point the mass carried by the n largest states follows a poisson dirichlet process for typical large instances the two transitions are sharp we determine their precise location further we provide a formal definition of each phase transition in terms of different notions of correlation between distinct variables in the problem the degree of correlation naturally affects the performances of many search sampling algorithms empirical evidence suggests that local monte carlo markov chain strategies are effective up to the clustering phase transition and belief propagation up to the condensation point finally refined message passing techniques such as survey propagation may also beat this threshold message passing algorithms phase transitions random graphs

the complexity of conservative valued csps we study the complexity of valued constraint satisfaction problems vcsps parametrized by a constraint language a fixed set of cost functions over a finite domain an instance of the problem is specified by a sum of cost functions from the language and the goal is to minimize the sum under the unique games conjecture the approximability of finite valued vcsps is well understood see raghavendra 2008 however there is no characterization of finite valued vcsps let alone general valued vcsps that can be solved exactly in polynomial time thus giving insights from a combinatorial optimization perspective we consider the case of languages containing all possible unary cost functions in the case of languages consisting of only 0 valued cost functions i e relations such languages have been called conservative and studied by bulatov 2003 2011 and recently by barto 2011 since we study valued languages we call a language conservative if it contains all finite valued unary cost functions the computational complexity of conservative valued languages has been studied by cohen et al 2006 for languages over boolean domains by deineko et al 2008 for 0 1 valued languages a k a max csp and by takhanov 2010a for 0 valued languages containing all finite valued unary cost functions a k a min cost hom we prove a schaefer like dichotomy theorem for conservative valued languages if all cost functions in the language satisfy a certain condition specified by a complementary combination of stp and mjn multimorphisms then any instance can be solved in polynomial time via a new algorithm developed in this article otherwise the language is np hard this is the first complete complexity classification of general valued constraint languages over non boolean domains it is a common phenomenon that complexity classifications of problems over non boolean domains are significantly harder than the boolean cases the polynomial time algorithm we present for the tractable cases is a generalization of the submodular minimization problem and a result of cohen et al 2008 our results generalize previous results by takhanov 2010a and a subset of results by cohen et al 2006 and deineko et al 2008 moreover our results do not rely on any computer assisted search as in deineko et al 2008 and provide a powerful tool for proving hardness of finite valued and general valued languages

weak solutions with decreasing energy of incompressible euler equations weak solution of the euler equations is an l2 vector field u x t satisfying certain integral relations which express incompressibility and the momentum balance our conjecture is that some weak solutions are limits of solutions of viscous and compressible fluid equations as both viscosity and compressibility tend to zero thus we believe that weak solutions describe turbulent flows with very high reynolds numbers

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

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

kolmogorov theory via finite time averages abstract several relations from the kolmogorov theory of fully developed three dimensional turbulence are rigorously established for finite time averages over leray hopf weak solutions of the navier stokes equations the navier stokes equations are considered with periodic boundary conditions and an external forcing term the main parameter is the grashof number associated with the forcing term the relations rigorously proved in this article include estimates for the energy dissipation rate the kolmogorov wavenumber the taylor wavenumber the reynolds number and the energy cascade process for some estimates the averaging time depends on the macroscale wavenumber and the kinematic viscosity alone while for others such as the kolmogorov energy dissipation law and the energy cascade the estimates depend also on the grashof number as compared with earlier works by some of the authors the more physical concept of finite time average is replacing the concept of infinite time average used before

closure properties of constraints many combinatorial search problems can be expressed as constraint satisfaction problems and this class of problems is known to be np complete in general in this paper we investigate the subclasses that arise from restricting the possible constraint types we first show that any set of constraints that does not give rise to an np complete class of problems must satisfy a certain type of algebraic closure condition we then investigate all the different possible forms of this algebraic closure property and establish which of these are sufficient to ensure tractability as examples we show that all known classes of tractable constraints over finite domains can be characterized by such an algebraic closure property finally we describe a simple computational procedure that can be used to determine the closure properties of a given set of constraints this procedure involves solving a particular constraint satisfaction problem which we call an indicator problem

attractor dimension estimates for two dimensional shear flows abstract we study the large time behavior of boundary and pressure gradient driven incompressible fluid flows in elongated two dimensional channels with emphasis on estimates for their degrees of freedom i e the dimension of the attractor for the solutions of the navier stokes equations for boundary driven shear flows and flux driven channel flows we present upper bounds for the degrees of freedom of the form c re 3 2 where c is a universal constant denotes the aspect ratio of the channel length width and re is the reynolds number based on the channel width and the imposed outer velocity scale for fixed pressure gradient driven channel flows we obtain an upper bound of form c re2 where c is another universal positive constant and the reynolds number is based on a velocity defined by the infimum over all possible trajectories of the time averaged mass flux per unit channel width we discuss these results in terms of physical arguments based on small length scales in turbulent flows

counterexamples to the sonic criterion we consider self similar pseudo steady shock reflection at an oblique wall there are three parameters wall corner angle mach number angle of incident shock ever since ernst mach discovered the irregular reflection named after him researchers have sought to predict precisely for which parameters the reflection is regular three conflicting proposals the detachment sonic and von neumann criteria have been studied extensively without a clear result we demonstrate that the sonic criterion is not correct we consider polytropic potential flow and prove that there is an open nonempty set of parameters that admit a global regular reflection with a reflected shock that is transonic we also provide a clear physical reason the flow type sub or supersonic is not decisive instead the reflected shock type weak or strong determines whether structural perturbations decay towards the reflection point

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

spontaneous stochasticity and anomalous dissipation for burgers equation we develop a lagrangian approach to conservation law anomalies in weak solutions of inviscid burgers equation motivated by previous work on the kraichnan model of turbulent scalar advection we show that the entropy solutions of burgers possess markov stochastic processes of generalized lagrangian trajectories backward in time for which the burgers velocity is a backward martingale this property is shown to guarantee dissipativity of conservation law anomalies for general convex functions of the velocity the backward stochastic burgers flows with these properties are not unique however we construct infinitely many such stochastic flows both by a geometric construction and by the zero noise limit of the constantin iyer stochastic representation of viscous burgers solutions the latter proof yields the spontaneous stochasticity of lagrangian trajectories backward in time for burgers at unit prandtl number it is conjectured that existence of a backward stochastic flow with the velocity as martingale is an admissibility condition which selects the unique entropy solution for burgers we also study linear transport of passive densities and scalars by inviscid burgers flows we show that shock solutions of burgers exhibit spontaneous stochasticity backward in time for all finite prandtl numbers implying conservation law anomalies for linear transport we discuss the relation of our results for burgers with incompressible navier stokes turbulence especially lagrangian admissibility conditions for euler solutions and the relation between turbulent cascade directions and time asymmetry of lagrangian stochasticity

variational bounds in dissipative systems abstract we present a general method for estimating the space time average dissipation in a large class of dissipative pdes the method leads to variational problems with spectral side conditions

improved variational principle for bounds on energy dissipation in turbulent shear flow we extend the doering constantin approach to upper bounds on energy dissipation in turbulent flows by introducing a balance parameter into the variational principle this parameter governs the relative weight of different contributions to the dissipation rate its optimization leads to improved bounds without entailing additional technical difficulties for plane couette flow the high re bounds obtainable with one dimensional background flows are methodically lowered by a factor of 2732

some specific mathematical constraints on 2d turbulence abstract we derive upper and lower bounds for ensemble averages of energy enstrophy and palinstrophy for the 2d periodic navier stokes equations this is carried out both in the general case and in the case where the energy power law for fully developed turbulence holds in the turbulent case the bounds are sharp up to a logarithm and provide a new lower bound on the landau lifschitz degrees of freedom we also prove two properties of the inertial term under the turbulence assumption one is that as the grashof number is increased the ensemble average of this term approaches the force the other is that an estimate of it via the ladyzhenskaya inequality is sharp on a considerable portion of the global attractor

the complexity of finite valued csps we study the computational complexity of exact minimization of rational valued discrete functions let be a set of rational valued functions on a fixed finite domain such a set is called a finite valued constraint language the valued constraint satisfaction problem vcsp is the problem of minimizing a function given as a sum of functions from we establish a dichotomy theorem with respect to exact solvability for all finite valued constraint languages defined on domains of arbitrary finite size we show that every constraint language either admits a binary symmetric fractional polymorphism in which case the basic linear programming relaxation solves any instance of vcsp exactly or satisfies a simple hardness condition that allows for a polynomial time reduction from max cut to vcsp

a proof of the csp dichotomy conjecture many natural combinatorial problems can be expressed as constraint satisfaction problems this class of problems is known to be np complete in general but certain restrictions on the form of the constraints can ensure tractability the standard way to parameterize interesting subclasses of the constraint satisfaction problem is via finite constraint languages the main problem is to classify those subclasses that are solvable in polynomial time and those that are np complete it was conjectured that if a constraint language has a weak near unanimity polymorphism then the corresponding constraint satisfaction problem is tractable otherwise it is np complete in the article we present an algorithm that solves constraint satisfaction problem in polynomial time for constraint languages having a weak near unanimity polymorphism which proves the remaining part of the conjecture 1

unification of variational principles for turbulent shear flows the background method of doering constantin and the mean fluctuation formulation of howard busse abstract the recent discovery of a new variational formulation for bounding turbulent flow quantities has rekindled interest in the whole field of producing rigorous inequality results directly from the navier stokes equations initiated by howard and busse in the 1960s the new background method introduced by doering and constantin 1992 1994 1995 1996 for bounding turbulent viscous dissipation appears to differ so fundamentally from that of howard and busse that any connection between the two methods has been fascinatingly unclear recently highly suggestive results obtained asymptotically kerswell 1997 have indicated that the variational problems produced for both cases of shear flow and convection may be equivalent here we confirm this for the shear flow case by uncovering the common underlying functional which both approaches seek to make stationary busse s upper bound problem is found to be the optimal dual or complementary variational problem to that which emerges from the doering constantin approach this ensures that busse s best bound exactly coincides with that available within the doering constantin formalism for all reynolds numbers

on self similar solutions to the incompressible euler equations abstract recent numerical simulations have shown the existence of multiple self similar solutions to the cauchy problem for the 2 dimensional incompressible euler equation with initial vorticity in l l o c p r 2 1 p toward a rigorous validation of these computations in this paper we analytically construct self similar solutions i on an outer domain of the form x r and ii in a neighborhood of the points where the solution exhibits a spiraling vortex singularity the outer solution is obtained as the fixed point of a contractive transformation based on the biot savart formula and integration along characteristics the inner solution is constructed using a system of adapted coordinates following the approach of v elling 2016 17

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

navier stokes equations and area of interfaces we present new a priori estimates for the vorticity of solutions of the three dimensional navier stokes equations these estimates imply that thel 1 norm of the vorticity is a priori bounded in time and that the time average of the 4 3 e power of thel 4 3 e spatial norm of the gradient of the vorticity is a priori bounded using these bounds we construct global leray weak solutions of the navier stokes equations which satisfy these inequalities in particular it follows that vortex sheet vortex line and even more general vortex structures with arbitrarily large vortex strengths are initial data which give rise to global weak solutions of this type of the navier stokes equations next we apply these inequalities in conjunction with geometric measure theoretical arguments to study the two dimensional hausdorff measure of level sets of the vorticity magnitude we obtain a priori bounds on an average such measure when expressed in terms of the reynolds number and the kolmogorov dissipation length these bounds are left langle mu right rangle leqq frac l 3 eta left 1 operatorname re 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 right 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 the right hand side of this inequality has a simple geometrical interpretation it represents the area of a union of non overlapping spheres of radii which fill a fraction of the spatial domain as the reynolds number increases this fraction decreases we study also the area of level sets of scalars and in particular isotherms in rayleigh benard convection we define a quantity r t x 0 describing an average value of the area of a portion of a level set contained in a small ball of radiusr about the pointx 0 we obtain the inequality left langle mu right rangle r t left x 0 right leqq c kappa 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 r 5 mathord left vphantom 5 2 right kern nulldelimiterspace 2 left langle v left x 0 right right rangle 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 wherek is the diffusivity constant and is the velocity the inequality is valid forr small but larger than a small scale k 1 in the case of turbulent velocities this scale is smaller than the smallest significant physical scale suggesting that 2 5 is a lower bound for the fractal dimension of an ensemble average interface in turbulent flow a fact which agrees with the experimental lower bound of 2 35 we define a similar quantity t representing an average value of the area of a portion of a level set contained in the region of spaced x d dist x d whered is the domain of aspect ratio of order one and diameterl where the convection takes place we obtain the inequality left langle mu right rangle delta t leqq c left l 3 delta 1 l 5 mathord left vphantom 5 2 right kern nulldelimiterspace 2 delta 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 ra 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 right where ra is the rayleigh number

the computational structure of monotone monadic snp and constraint satisfaction a study through datalog and group theory this paper starts with the project of finding a large subclass of np which exhibits a dichotomy the approach is to find this subclass via syntactic prescriptions while the paper does not achieve this goal it does isolate a class of problems specified by monotone monadic snp without inequality which may exhibit this dichotomy we justify the placing of all these restrictions by showing essentially using ladner s theorem that classes obtained by using only two of the above three restrictions do not show this dichotomy we then explore the structure of this class we show that all problems in this class reduce to the seemingly simpler class csp we divide csp into subclasses and try to unify the collection of all known polytime algorithms for csp problems and extract properties that make csp problems np hard this is where the second part of the title a study through datalog and group theory comes in we present conjectures about this class which would end in showing the dichotomy

the csp dichotomy holds for digraphs with no sources and no sinks a positive answer to a conjecture of bang jensen and hell bang jensen and hell conjectured in 1990 using the language of graph homomorphisms a constraint satisfaction problem csp dichotomy for digraphs with no sources or sinks the conjecture states that the csp for such a digraph is tractable if each component of its core is a cycle and is np complete otherwise in this paper we prove this conjecture and as a consequence a conjecture of bang jensen hell and macgillivray from 1995 classifying hereditarily hard digraphs further we show that the csp dichotomy for digraphs with no sources or sinks agrees with the algebraic characterization conjectured by bulatov jeavons and krokhin in 2005

constraint satisfaction problems over semilattice block mal tsev algebras abstract the dichotomy conjecture for the constraint satisfaction problem csp was recently settled independently by zhuk and the author the proofs of this conjecture are rather sophisticated and require deep understanding of the algebraic structure of csps this paper is a precursor of the author s proof of the dichotomy conjecture and represents its main ideas in a simpler and clearer form in a more restricted class of the csp there are two well known types of algorithms for solving csps local propagation and generating a basis of the solution space for several years the focus of the csp research has been on hybrid algorithms that somehow combine the two approaches in this paper we present a new method of such hybridization that allows us to solve certain csps that has been out of reach for a quite a while and eventually leads to resolving the dichotomy conjecture we apply this method to csps parametrized by a universal algebra an approach that has been very popular in the last decade or so specifically we consider a fairly restricted class of algebras we will call semilattice block mal tsev an algebra a is semilattice block mal tsev if it has a binary operation f a ternary operation m and a congruence such that the quotient a with operation f is a semilattice f is a projection on every block of and every block of is a mal tsev algebra with mal tsev operation m this means that the domain in such a csp is partitioned into blocks such that if the problem is considered on the quotient set a it can be solved by a simple constraint propagation algorithm on the other hand if the problem is restricted on individual blocks it can be solved by generating a basis of the solution space we show that the two methods can be combined in a highly nontrivial way and therefore the constraint satisfaction problem over a semilattice block mal tsev algebra is solvable in polynomial time

fluids elasticity geometry and the existence of wrinkled solutions we are concerned with underlying connections between fluids elasticity isometric embedding of riemannian manifolds and the existence of wrinkled solutions of the associated nonlinear partial differential equations in this paper we develop such connections for the case of two spatial dimensions and demonstrate that the continuum mechanical equations can be mapped into a corresponding geometric framework and the inherent direct application of the theory of isometric embeddings and the gauss codazzi equations through examples for the euler equations for fluids and the euler lagrange equations for elastic solids these results show that the geometric theory provides an avenue for addressing the admissibility criteria for nonlinear conservation laws in continuum mechanics

statistical estimates for the navier stokes equations and the kraichnan theory of 2 d fully developed turbulence a mathematical formulation of the kraichnan theory for 2 d fully developed turbulence is given in terms of ensemble averages of solutions to the navier stokes equations a simple condition is given for the enstrophy cascade to hold for wavenumbers just beyond the highest wavenumber of the force up to a fixed fraction of the dissipation wavenumber up to a logarithmic correction this is followed by partial rigorous support for kraichnan s eddy breakup mechanism a rigorous estimate for the total energy is found to be consistent with kraichnan s theory finally it is shown that under our conditions for fully developed turbulence the fractal dimension of the attractor obeys a sharper upper bound than in the general case

variational bounds on energy dissipation in incompressible flows iii convection building on a method of analysis for the navier stokes equations introduced by hopf math ann 117 764 1941 a variational principle for upper bounds on the largest possible time averaged convective heat flux is derived from the boussinesq equations of motion when supplied with appropriate test background fields satisfying a spectral constraint reminiscent of an energy stability condition the variational formulation produces rigorous upper bounds on the nusselt number nu as a function of the rayleigh number ra for the case of vertical heat convection between parallel plates in the absence of sidewalls a simplified but rigorous formulation of the optimization problem yields the large rayleigh number bound nu ensuremath le 0 167 mathrm ra 1 2 1 nonlinear euler lagrange equations for the optimal background fields are also derived which allow us to make contact with the upper bound theory of howard j fluid mech 17 405 1963 for statistically stationary flows the structure of solutions of the euler lagrange equations are elucidated from the geometry of the variational constraints which sheds light on busse s j fluid mech 37 457 1969 asymptotic analysis of general solutions to howard s euler lagrange equations the results of our analysis are discussed in the context of theory recent experiments and direct numerical simulations textcopyright 1996 the american physical society

complexity of counting csp with complex weights we give a complexity dichotomy theorem for the counting constraint satisfaction problem csp in short with algebraic complex weights to this end we give three conditions for its tractability let f be any finite set of algebraic complex valued functions defined on an arbitrary finite domain we show that csp f is solvable in polynomial time if all three conditions are satisfied and is p hard otherwise our dichotomy theorem generalizes a long series of important results on counting problems and reaches a natural culmination a the problem of counting graph homomorphisms is the special case when f has a single symmetric binary function dyer and greenhill 2000 bulatov and grohe 2005 goldberg et al 2010 cai et al 2013 b the problem of counting directed graph homomorphisms is the special case when f has a single but not necessarily symmetric binary function dyer et al 2007 cai and chen 2010 c the unweighted form of csp is when all functions in f take values in 0 1 bulatov 2008 dyer and richerby 2013

kraichnan turbulence via finite time averages relations central to kraichnan s theory of fully developed two dimensional turbulence are rigorously established for finite time averages in particular we prove that if the ratio of the averages of palinstrophy to enstrophy is large then a large inertial range displaying an enstrophy cascade exists moreover if this ratio is comparable up to a logarithm to the dissipation wave number a necessary condition for turbulence then the power law for the energy spectrum until now derived only heuristically is rigorously shown to provide up to a logarithm an upper bound for the energy spectrum finally we show that deep in the dissipation range the palinstrophy contributed by eddies smaller than a specified length decays exponentially in the corresponding wave number the averaging times needed for these relations are bounded in terms of the generalized grashof number independent of the solution for which the time averages are taken solutions are not assumed to be on the global attractor merely within the absorbing ball an easily verified condition

statistical estimates for channel flows driven by a pressure gradient we present rigorous estimates for some physical quantities related to turbulent and non turbulent channel flows driven by a uniform pressure gradient such results are based on the concept of stationary statistical solutions which is related to the notion of ensemble averages for flows in statistical equilibrium we provide a lower bound estimate for the mean skin friction coefficient and improve on a previous upper bound estimate for the same quantity both estimates are derived in terms of the reynolds number we also present lower and upper bound estimates for the mean rate of energy dissipation the mean longitudinal bulk velocity in the direction of the pressure gradient and the mean kinetic energy in terms of various physical parameters in particular we obtain an upper bound related to the energy dissipation law namely that the mean rate of energy dissipation is essentially bounded by a non dimensional universal constant times the cube of the mean longitudinal bulk velocity over a characteristic macro scale length finally we investigate the scale by scale energy injection due to the pressure gradient proving an upper bound estimate for the decrease of this energy injection as the scale length decreases

on the asymptotic behavior of average energy and enstrophy in 3d turbulent flows abstract rigorous upper and lower bounds are proved for the taylor and the kolmogorov wavenumbers for the three dimensional space periodic navier stokes equations under the assumption that kolmogorov s two thirds power law holds the bounds sharpen to t gr 1 4 and gr 3 8 respectively where gr is the grashof number this provides a rigorous proof that the power law implies 1 the energy cascade 2 kolmogorov dissipation law and 3 a connection between t and the portion of phase space where a key a priori estimate on the nonlinear term is sharp is shown to be significant by means of a lower bound on any probability measure associated with an infinite time average

a fundamental limit on the heat flux in the control of incompressible channel flow this paper proves that there are no zero net wall transpiration control strategies that can sustain net heat flux below the laminar level in an incompressible channel flow with constant temperature walls the result represents a fundamental limit on the performance of a controlled nonlinear system as measured by a linear cost function over a broad class of admissible initial conditions and control inputs not a zero sum tradeoff in the frequency domain or time domain both buoyancy effects via the boussinesq approximation and viscous heating effects are accounted for and phenomenological justification for the result is also given the boundedness of solutions of the two way coupled navier stokes energy equations when both buoyancy and viscous heating are accounted for is also discussed and a new proof of existence under an appropriate small data assumption is provided

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