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

remarks on the helicity of the 3 d incompressible euler equations in this note we obtain a sufficient condition on the regularity of the weak solutions to guarantee the conservation of helicity for the 3 d incompressible euler equations as a corollary we obtain a lower bound of the vorticities for a weak solution of the euler equations

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

regularity of inviscid shell models of turbulence in this paper we continue the analytical study of the sabra shell model of energy turbulent cascade we prove the global existence of weak solutions of the inviscid sabra shell model and show that these solutions are unique for some short interval of time in addition we prove that the solutions conserve energy provided that the components of the solution satisfy ensuremath mid u n ensuremath mid ensuremath le c k n ensuremath 1 3 sqrt n phantom rule 0 2em 0ex mathrm log n 1 ensuremath 1 for some positive absolute constant c which is the analog of the onsager s conjecture for the euler s equations moreover we give a beal kato majda type criterion for the blow up of solutions of the inviscid sabra shell model and show the global regularity of the solutions in the two dimensional parameters regime

optimal besov differentiability for entropy solutions of the eikonal equation in this paper we study the eikonal equation in a bounded planar domain we prove the equivalence among optimal besov regularity the finiteness of every entropy production and the validity of a kinetic formulation

analysis of fastened structural connections an approach for modeling the transfer of forces in fastener groups is described which takes nonlinearities into account the method is used for analyzing two problems which involve design of repairs and damaged structures the model predictions are found to be in a good agreement with available experimental results the study demonstrates the importance of fastener stiffness and initial clearance on load distribution r eliable structural analysis of mechanically fastened connections is of major importance in modern design of aircraft structures where requirements for optimal design must be met in current practice extensive full scale testing is used mainly because the reliability of mathematical models has not been established to the satisfaction of engineers who have to make or approve design decisions improved reliability of mathematical models will make it possible to reduce the scope of experimental programs and the time required for developing information on which design decisions can be based with confidence also extensive testing programs can improve the safety of design at a substantial cost but cannot be used for optimizing design with respect to weight and durability only reliable mathematical models can do that in this paper the problem of simulating the elastostatic response of fastened structural connections with guarantee of reliability is discussed some aspects of this problem were presented in ref 1 where the interaction between the fasteners and the plates was simulated by distributed springs friction was treated by the addition of external tractions in an iterative process each fastener was represented by a nonlinear relation between the transferred force and the relative displacements the accuracy of the method was verified through tight quality control of the numerical approximation errors and comparisons with experimental results this paper is concerned with the accuracy and reliability of the mathematical model two kinds of error have to be considered2 1 the differences between the exact solution of the mathematical problem formulated to represent a physical system and the actual response or behavior of the physical system are called errors of idealization 2 the differences between the exact solution of the mathematical problem formulated to represent a physical system or process and its numerical approximation are called errors of discretization in some cases the two errors the errors of idealization and discretization may partially cancel one another therefore it is important to verify by means other than the experiment itself that the numerical solution is close to the exact solution of the model only then is it possible to investigate whether the errors of idealization are large or small by making comparisons with experimental observations the load distribution in bolted or riveted joints had been investigated both experimentally and theoretically e g refs 3 25 in these studies a number of assumptions had been

inviscid dyadic model of turbulence the fixed point and onsager s conjecture properties of an infinite system of nonlinearly coupled ordinary differential equations are discussed this system models some properties present in the equations of motion for an inviscid fluid such as the skew symmetry and the three dimensional scaling of the quadratic nonlinearity it is proved that the system with forcing has a unique equilibrium and that every solution blows up in finite time in h5 6 norm onsager s nuovo cimento 6 279 287 1949 conjecture is confirmed for the model system

energy and cross helicity conservation for the three dimensional ideal mhd equations in a bounded domain abstract in this paper we prove the energy and cross helicity conservation of weak solutions to the three dimensional ideal mhd equations in a bounded domain under the interior besov regularity conditions which are exactly same as the three dimensional periodic domain case in 3 and the boundedness and the besov type continuity conditions for both the velocity and magnetic fields near the boundary which seem crucial for the bounded domain case due to the boundary effect note that our besov type continuity conditions near the boundary are consistent with the interior besov regularity conditions

simple systems with anomalous dissipation and energy cascade we analyze a class of dynamical systems of the type dot a n t c n 1 a n 1 t c n a n 1 t f n t n epsilon mathbb n a 0 0 where fn t is a forcing term with f n t not 0 only for n le n star infty and the coupling coefficients cn satisfy a condition ensuring the formal conservation of energy frac12 sum n a n t 2 despite being formally conservative we show that these dynamical systems support dissipative solutions suitably defined and as a result may admit unique statistical steady states when the forcing term fn t is nonzero this claim is demonstrated via the complete characterization of the solutions of the system above for specific choices of the coupling coefficients cn the mechanism of anomalous dissipations is shown to arise via a cascade of the energy towards the modes with higher n this is responsible for solutions with interesting energy spectra namely mathbb e a n 2 scales as n alpha as n here the exponents depend on the coupling coefficients cn and mathbb e denotes expectation with respect to the equilibrium measure this is reminiscent of the conjectured properties of the solutions of the navier stokes equations in the inviscid limit and their accepted relationship with fully developed turbulence hence these simple models illustrate some of the heuristic ideas that have been advanced to characterize turbulence similar in that respect to the random passive scalar or random burgers equation but even simpler and fully solvable

locality of turbulent cascades in this paper we establish sufficient conditions for locality of turbulent cascades by an exact analysis of the fluid equations no assumptions of homogeneity or isotropy are invoked and the results apply to individual realizations with no resort to any statistical averaging the only requirement is suitable regularity of the turbulent solutions of the fluid equations in the high reynolds number limit corresponding to holder continuity but non differentiability in space we use a smooth filtering approach to resolve the turbulent fields both in space and in scale we discuss several physical cascades to exemplify and clarify the analysis including joint energy and helicity cascades in three space dimensions and the dual inverse energy cascade and direct enstrophy cascade in two dimensions

regularity and energy conservation for the compressible euler equations we give sufficient conditions on the regularity of solutions to the inhomogeneous incompressible euler and the compressible isentropic euler systems in order for the energy to be conserved our strategy relies on commutator estimates similar to those employed by constantin et al for the homogeneous incompressible euler equations

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

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

asymptotic normality of linear multiuser receiver outputs this paper proves large system asymptotic normality of the output of a family of linear multiuser receivers that can be arbitrarily well approximated by polynomial receivers this family of receivers encompasses the single user matched filter the decorrelator the minimum mean square error mmse receiver the parallel interference cancelers and many other linear receivers of interest both with and without the assumption of perfect power control we show that the output decision statistic for each user converges to a gaussian random variable in distribution as the number of users and the spreading factor both tend to infinity with their ratio fixed analysis reveals that the distribution conditioned on almost all spreading sequences converges to the same distribution which is also the unconditional distribution this normality principle allows the system performance e g the multiuser efficiency to be completely determined by the output signal to interference ratio sir for large linear systems

inviscid limit for damped and driven incompressible navier stokes equations in mathbb r 2 we consider the zero viscosity limit of long time averages of solutions of damped and driven navier stokes equations in mathbb r 2 we prove that the rate of dissipation of enstrophy vanishes stationary statistical solutions of the damped and driven navier stokes equations converge to renormalized stationary statistical solutions of the damped and driven euler equations these solutions obey the enstrophy balance

on the connection between kolmogorov microscales and friction in pipe flows of viscoplastic fluids abstract the present work extends kolmogorov s micro scales to a large family of viscoplastic fluids the new micro scales combined with gioia and chakaborty s 2006 friction phenomenology theory lead to a unified framework for the description of the friction coefficient in turbulent flows a resulting blasius type friction equation is tested against some available experimental data and shows good agreement over a significant range of hedstrom and reynolds numbers the work also comments on the role of the new expression as a possible benchmark test for the convergence of dns simulations the formula also provides limits for the maximum drag reduction of viscoplastic flows

rigidity of a non elliptic differential inclusion related to the aviles giga conjecture in this paper we prove sharp regularity for a differential inclusion into a set k subset mathbb r 2 times 2 that arises in connection with the aviles giga functional the set k is not elliptic and in that sense our main result goes beyond sverak s regularity theorem on elliptic differential inclusions it can also be reformulated as a sharp regularity result for a critical nonlinear beltrami equation in terms of the aviles giga energy our main result implies that zero energy states coincide modulo a canonical transformation with solutions of the differential inclusion into k this opens new perspectives towards understanding energy concentration properties for aviles giga quantitative estimates for the stability of zero energy states can now be approached from the point of view of stability estimates for differential inclusions all these reformulations of our results are strong improvements upon a recent work by the last two authors lorent and peng where the link between the differential inclusion into k and the aviles giga functional was first observed and used our proof relies moreover on new observations concerning the algebraic structure of entropies

onsager s conjecture in bounded domains for the conservation of entropy and other companion laws we show that weak solutions of general conservation laws in bounded domains conserve their generalized entropy and other respective companion laws if they possess a certain fractional differentiability of order one third in the interior of the domain and if the normal component of the corresponding fluxes tend to zero as one approaches the boundary this extends various recent results of the authors

on maximum likelihood detection and the search for the closest lattice point maximum likelihood ml decoding algorithms for gaussian multiple input multiple output mimo linear channels are considered linearity over the field of real numbers facilitates the design of ml decoders using number theoretic tools for searching the closest lattice point these decoders are collectively referred to as sphere decoders in the literature in this paper a fresh look at this class of decoding algorithms is taken in particular two novel algorithms are developed the first algorithm is inspired by the pohst enumeration strategy and is shown to offer a significant reduction in complexity compared to the viterbo boutros sphere decoder the connection between the proposed algorithm and the stack sequential decoding algorithm is then established this connection is utilized to construct the second algorithm which can also be viewed as an application of the schnorr euchner strategy to ml decoding aided with a detailed study of preprocessing algorithms a variant of the second algorithm is developed and shown to offer significant reductions in the computational complexity compared to all previously proposed sphere decoders with a near ml detection performance this claim is supported by intuitive arguments and simulation results in many relevant scenarios

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

no eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices let b n 1 n t n 1 2 x n x n t n 1 2 where x n is n times n with i i d complex standardized entries having finite fourth moment and t n 1 2 is a hermitian square root of the nonnegative definite hermitian matrix t n it is known that as n to infty if n n converges to a positive number and the empirical distribution of the eigenvalues of t n converges to a proper probability distribution then the empirical distribution of the eigenvalues of b n converges a s to a nonrandom limit in this paper we prove that under certain conditions on the eigenvalues of t n for any closed interval outside the support of the limit with probability 1 there will be no eigenvalues in this interval for all n sufficiently large

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

renormalization of active scalar equations abstract we consider transport equations with an incompressible transporting vector field whereas smooth solutions of such equations conserve every l p norm simply by the chain rule the question arises how regular a weak solution needs to be to guarantee this conservation property whereas the classical diperna lions theory gives sufficient conditions in terms of the regularity of the coefficients with no regularity requirement on the transported scalar we give here sufficient conditions in terms of the combined regularities of the coefficients and the scalar this is motivated by the case of active scalar equations where the transporting vector field has the same regularity as the transported scalar we use commutator estimates similar to those of constantin e titi in the context of onsager s conjecture but we require novel arguments to handle the case of l p norms when p 2

on the energy of inviscid singular flows abstract it is known that the energy of a weak solution to the euler equation is conserved if it is slightly more regular than the besov space b 3 1 3 when the singular set of the solution is or belongs to a smooth manifold we derive various l p space regularity criteria dimensionally equivalent to the critical one in particular if the singular set is a hypersurface the energy of u is conserved provided the one sided non tangential limits to the surface exist and the non tangential maximal function is l 3 integrable while the maximal function of the pressure is l 3 2 integrable the results directly apply to prove energy conservation of the classical vortex sheets in both 2d and 3d at least in those cases where the energy is finite

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

local energy flux and the refined similarity hypothesis in this paper we demonstrate the locality of energy transport for incompressible euler equations both in space and in scale the key to the proof is the proper definition of a local subscale flux t r which is supposed to be a measure of energy transfer to length scales l at the space point r kraichnan suggested that for such a quantity the refined similarity hypothesis will hold which kolmogorov originally assumed to hold instead for volume averaged dissipation we derive a local energy balance relation for the large scale motions which yields a natural definition of such a subscale flux for this definition a precise form of the refined similarity hypothesis is rigorously proved as a big o bound the established estimate is t r o l3h 1 in terms of the local holder exponenth at the point r which is also the estimate assumed in the parisi frisch multifractal model our method not only establishes locality of energy transfer but it also clarifies the physical reason that convection effects which naively violate locality do not contribute to the subscale flux in fact we show that as a consequence of incompressibility such effects enter into the local energy balance only as the divergence of a spatial current therefore they describe motion of energy in space and cancel in the integration over volume we also discuss theorems of onsager eyink and constantinet al on energy conservation for euler dynamics particularly to explain their relation with the parisi frisch model the constantinet al proof may be interpreted as giving a bound on the total flux t d d r t r of the form t r o lz3h 1 wherez3 is the third order scaling exponent or besov index in agreement with the multifractal model finally we discuss how the local estimates are related to the results of caffarelli kohn nirenberg on partial regularity for solutions of navier stokes equations they provide some heuristic support to a scenario proposed recently by pumir and siggia for singularities in the solutions of navier stokes with small enough viscosity

exact results on stationary turbulence in 2d consequences of vorticity conservation we establish a series of exact results for a model of stationary turbulence in two dimensions the incompressible navier stokes equation in a periodic domain with stochastic force white noise in time essentially all of our conclusions follow from the simple consideration of the simultaneous conservation of energy and enstrophy by the inertial dynamics our main results are as follows 1 we show the blow up of mean energy as l02e for 0 when there is no ir dissipation at the large length scale l0 2 with an additional ir dissipation we establish the validity of the traditional cascade directions and magnitudes of flux of energy and enstrophy for 0 assuming finite mean energy in the limit 3 we rigorously establish the balance equations for the energy and vorticity invariants in the 2d steady state and the forward cascade of the higher order vorticity invariants assuming finite mean values 4 we derive exact inequalities for scaling exponents in the 2d enstrophy range as follows if l p l p then 2 23 and p 0 for p 3 if the minimum holder exponent of the vorticity hmin 0 then we establish a 2d analogue of the refined similarity hypothesis which improves these bounds the most novel and interesting conclusion of this work is the connection established between intermittency in 2d and negative holder singularities of the vorticity we show that the latter are necessary for deviations from the 1967 kraichnan scaling to occur

on the energy equality for the 3d navier stokes equations abstract in this paper we study the problem of energy conservation for the solutions of the initial boundary value problem associated to the 3d navier stokes equations with dirichlet boundary conditions first we consider leray hopf weak solutions and we prove some new criteria involving the gradient of the velocity next we compare them with the existing literature in scaling invariant spaces and with the onsager conjecture then we consider the problem of energy conservation for very weak solutions proving energy equality for distributional solutions belonging to the so called shinbrot class a possible explanation of the role of this classical class of solutions which is not scaling invariant is also given

analysis of bonded and fastened repairs by the p version of the finite element method abstract the development of efficient and reliable methods for the design and analysis of fastened structural connections and bonded joints is among the most important and most challenging problems in aerospace applications because these connections are common sites of failure initiation and there is a complicated nonlinear interaction between the fasteners and the connected plates realistic representations of fastened connections must account for the diameter of fastener holes the stiffness of the fasteners the effects of interference fitting or looseness of fit contact between the fasteners and the connected plates and yielding of the plates similarly the treatment of bonded joints is complicated by the fact that bonded patches are typically composed of several thin layers of orthotropic material and the adhesive material may yield under load requiring a nonlinear analysis consequently the complexity of the models is such that it is difficult and time consuming to treat them using conventional finite element or boundary element procedures fortunately these problems lend themselves to standardization in fea based handbook libraries structural connections can be described in a parametric form therefore topologically similar families of connections need to be meshed only once the fea based handbook library provides for the archival and recall of entries for analysis users need only enter the parameters the finite element mesh being associative is adjusted automatically therefore users need not be concerned with meshing or other details of the analysis process the finite element solution and the computation of all data of interest are performed automatically and the results are displayed in tabular and graphical forms automatic quality control tests are performed and reported for purposes of documentation there are provisions for the computation of margins of safety the advanced fea based handbook library produces results of sufficient reliability to permit certification of repairs and joints based on computed information this will be demonstrated by examples

the vanishing viscosity limit for a dyadic model abstract a dyadic shell model for the navier stokes equations is studied in the context of turbulence the model is an infinite nonlinearly coupled system of odes it is proved that the unique fixed point is a global attractor which converges to the global attractor of the inviscid system as viscosity goes to zero this implies that the average dissipation rate for the viscous system converges to the anomalous dissipation rate for the inviscid system which is positive as viscosity goes to zero this phenomenon is called the dissipation anomaly predicted by kolmogorov s theory for the actual navier stokes equations

onsager s conjecture with physical boundaries and an application to the vanishing viscosity limit author s bardos claude titi edriss s wiedemann emil abstract we consider the incompressible euler equations in a bounded domain in three space dimensions recently the first two authors proved onsager s conjecture for bounded domains i e that the energy of a solution to these equations is conserved provided the solution is h quot older continuous with exponent greater than 1 3 uniformly up to the boundary in this contribution we relax this assumption requiring only interior h quot older regularity and continuity of the normal component of the energy flux near the boundary the significance of this improvement is given by the fact that our new condition is consistent with the possible formation of a prandtl type boundary layer in the vanishing viscosity limit

a posteriori regularity of the three dimensional navier stokes equations from numerical computations in this paper we consider the role that numerical computations in particular galerkin approximations can play in problems modeled by the three dimensional 3d navier stokes equations for which no rigorous proof of the existence of unique solutions is currently available we prove a robustness theorem for strong solutions from which we derive an a posteriori check that can be applied to a numerical solution to guarantee the existence of a strong solution of the corresponding exact problem we then consider galerkin approximations and show that if a strong solution exists the galerkin approximations will converge to it thus if one is prepared to assume that the navier stokes equations are regular one can justify this particular numerical method rigorously combining these two results we show that if a strong solution of the exact problem exists then this can be verified numerically using an algorithm that can be guaranteed to terminate in a finite time we thus introduce the possibility of rigorous com

the energy balance relation for weak solutions of the density dependent navier stokes equations abstract we consider the incompressible inhomogeneous navier stokes equations with constant viscosity coefficient and density which is bounded and bounded away from zero we show that the energy balance relation for this system holds for weak solutions if the velocity density and pressure belong to a range of besov spaces of smoothness 1 3 a density dependent version of the classical karman howarth monin relation is derived

on the relationship between mmse sic and bi gdfe receivers for large multiple input multiple output channels a minimum mean square error mmse based iterative soft interference cancellation mmse sic receiver has been proposed to mitigate the interferences of the multiple input multiple output mimo channels with reduced complexity as compared to maximum likelihood ml detection on the other hand the block iterative generalized decision feedback equalizer bi gdfe attains close to the performance of the mmse sic receivers with further reduced complexity the bi gdfe however requires an accurate estimate of the input decision correlation idc which is a statistical reliability metric of earlier made decisions to date the bi gdfe receiver is applicable only to phase shift keying psk modulations due to the absence of a method to estimate the idc for higher order quadrature amplitude modulations qams in this paper we establish the relationship between the mmse sic and bi gdfe receivers and propose an algorithm to determine the idc for bi gdfe from the unconditional mmse sic u mmse sic we further analyze and compare the asymptotic performances of the two receivers for large random mimo channels and prove that for the limiting case the output signal to interference plus noise ratios sinrs at each iteration for both receivers converge in probability to their respective deterministic limits our simulation results have shown that the bit error rate ber performance of the bi gdfe receiver with the proposed idc selection method achieves close to that of the u mmse sic receiver with similar convergence behavior and reaches the single user matched filter bound mfb with several iterations for high enough signal to noise ratio snr

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

asymptotic performance of mmse receivers for large systems using random matrix theory random matrix theory is used to derive the limit and asymptotic distribution of signal to interference plus noise ratio sir for a class of suboptimal minimum mean square error mmse receivers applied to large random systems with unequal power users we prove that the limiting sir converges to a deterministic value when k and n go to infinity with lim k n y being a positive constant where k is the number of users and n is the number of degrees of freedom we also prove that the sir of each particular user is asymptotically gaussian for large n and derive the closed form expressions of the variance for the sir variable under real spreading and complex spreading channel environments it is revealed that for a given k n pair under certain mild conditions the variance of the sir for complex spreading channels is half of that for the corresponding real spreading channels since the suboptimal mmse receiver becomes optimal for the case when the users are equally powered our results show that the conjecture made by tse and zeitouni for the complex spreading case is not affirmative we also derive the asymptotic distribution for sir in decibels which provides better description when n is small numerical results and computer simulations are provided to evaluate the accuracy of various limiting and asymptotic results obtained in this paper

besov spaces and the multifractal hypothesis parisi and frisch proposed some time ago an explanation for multiscaling of turbulent velocity structure functions in terms of a multifractal hypothesis i e they conjecture that the velocity field has local holder exponents in a range hmin hmax with exponents h occurring on a sets h with a fractal dimensiond h heuristic reasoning led them to an expression for the scaling exponentz p ofpth order as the legendre transform of the codimensiond d h we show here that a part of the multifractal hypothesis is correct under even weaker assumptions namely if the velocity field hasl p mean holder indexs i e if it lies in the besov spaceb p s then local holder regularity is satisfied ifs d p then the hypothesis is true in a generalized sense of holder space with negative exponents and we discuss the proper definition of local holder classes of negative index finally if a certain box counting dimension exists then the legendre transform of its codimension gives the scaling exponentz p and more generally the maximal besov index of order p ass p z p p our method of proof is derived from a recent paper of s jaffard using compactly supported orthonormal wavelet bases and gives an extension of his results we discuss implications of the theorems for ensemble average scaling and fluid turbulence

renormalization and energy conservation for axisymmetric fluid flows we study vanishing viscosity solutions to the axisymmetric euler equations without swirl with relative vorticity in l p with p 1 we show that these solutions satisfy the corresponding vorticity equations in the sense of renormalized solutions moreover we show that the kinetic energy is preserved provided that p 3 2 and the vorticity is nonnegative and has finite second moments

structure function and fractal dissipation for an intermittent inviscid dyadic model we study a generalization of the original tree indexed dyadic model by katz and pavlovic for the turbulent energy cascade of the three dimensional euler equation we allow the coefficients to vary with some restrictions thus giving the model a realistic spatial intermittency by introducing a forcing term on the first component the fixed point of the dynamics is well defined and some explicit computations allow us to prove the rich multifractal structure of the solution in particular the exponent of the structure function is concave in accordance with other theoretical and experimental models moreover anomalous energy dissipation happens in a fractal set of dimension strictly less than 3

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

non uniqueness of weak solutions to 2d hypoviscous navier stokes equations abstract through an adaption of the convex integration scheme in the two dimensional case a result of the h principle type is presented for the two dimensional hypoviscous incompressible navier stokes equations it is shown that the c t 0 l x 2 weak solutions can possess compact temporal supports and thus are not unique in general

output mai distributions of linear mmse multiuser receivers in ds cdma systems multiple access interference mai in a code division multiple access cdma system plays an important role in performance analysis and characterization of fundamental system limits we study the behavior of the output mai of the minimum mean square error mmse receiver employed in the uplink of a direct sequence ds cdma system we focus on imperfect power controlled systems with random spreading and establish that in a synchronous system 1 the output mai of the mmse receiver is asymptotically gaussian and 2 for almost every realization of the signatures and received powers the conditional distribution of the output mai converges weakly to the same gaussian distribution as in the unconditional case we also extend our study to asynchronous systems and establish the gaussian nature of the output interference these results indicate that in a large system the output interference is approximately gaussian and the performance of the mmse receiver is robust to the randomness of the signatures and received powers the gaussianity justifies the use of single user gaussian codes for cdma systems with linear mmse receivers and implies that from the viewpoints of detection and channel capacity signal to interference ratio sir is the key parameter that governs the performance of the mmse receiver in a cdma system

on the conserved quantities for the weak solutions of the euler equations and the quasi geostrophic equations in this paper we obtain sufficient conditions on the regularity of the weak solutions to guarantee conservation of the energy and the helicity for the incompressible euler equations the regularity of the weak solutions are measured in terms of the triebel lizorkin type of norms dot mathcal f s p q and the besov norms dot mathcal b s p q in particular in the besov space case our results refine the previous ones due to constantin e titi energy and the author of this paper helicity where the regularity is measured by a special class of the besov space norm dot mathcal b s p infty dot mathcal n s p which is the nikolskii space we also obtain a sufficient regularity condition for the conservation of the lp norm of the temperature function in the weak solutions of the quasi geostrophic equation

calculation of complex singular solutions to the 3d incompressible euler equations abstract this paper presents numerical computations of complex singular solutions to the 3d incompressible euler equations the euler solutions found here consist of a complex valued velocity field u that contains all positive wavenumbers u satisfies the usual euler equations but with complex initial data the real valued velocity u u u where u u is an approximate singular solution to the euler equations under moore s approximation the method for computing singular solutions is an extension of that in caflisch 1993 25 for axisymmetric flow with swirl but with several improvements that prevent the extreme magnification of round off error which affected previous computations this enables the first clean analysis of the singular surface in three dimensional complex space we find singularities in the velocity field of the form u 1 for near 3 2 and u log where 0 denotes the singularity surface the logarithmic singular surface is related to the double exponential growth of vorticity observed in recent numerical simulations

singular weak and absent solutions of the euler equations we will describe necessary and sufficient conditions for blowup and discuss weak solutions for the incompressible euler equations we will also describe a result concerning anomalous dissipation of energy

dissipative anomalies in singular euler flows abstract we discuss onsager s conjecture that non vanishing energy dissipation in high reynolds number turbulence is associated to singular distributional solutions of the incompressible euler equations we carefully explain the physical and mathematical meaning of the conjecture and also review relevant theoretical experimental and numerical work emphasizing some of the dramatic successes of onsager s point of view finally we present several new ideas and results on lagrangian dynamics of circulations and vortex lines that we believe will be important for future progress

holder continuous solutions of boussinesq equation with compact support abstract we show the existence of holder continuous solution of boussinesq equations in whole space which has compact support both in space and time

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

energy conservation for inhomogeneous incompressible and compressible euler equations abstract energy conservations are studied for inhomogeneous incompressible and compressible euler equations with general pressure law in a torus or a bounded domain we provide sufficient conditions for a weak solution to conserve the energy by exploiting a suitable test function the spatial regularity for the density is only required to be of order 2 3 in the incompressible case and of order 1 3 in the compressible case when the density is constant we recover the existing results for classical incompressible euler equation

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

iterative multiuser joint decoding optimal power allocation and low complexity implementation we consider a canonical model for coded code division multiple access cdma with random spreading where the receiver makes use of iterative belief propagation bp joint decoding we provide simple density evolution analysis in the large system limit large number of users of the performance of the bp decoder and of some suboptimal approximations based on interference cancellation ic based on this analysis we optimize the received user signal to noise ratio snr distribution in order to maximize the system spectral efficiency for given user channel codes channel load users per chip and target user bit error rate ber the optimization of the received snr distribution is obtained by solving a simple linear program and can be easily incorporated into practical power control algorithms remarkably under the optimized snr assignment the suboptimal minimum mean square error mmse ic based decoder performs almost as well as the more complex bp decoder moreover for a large class of commonly used convolutional codes we observe that the optimized snr distribution consists of a finite number of discrete snr levels based on this observation we provide a low complexity approximation of the mmse ic decoder that suffers from very small performance degradation while attaining considerable savings in complexity as by products of this work we obtain a closed form expression of the multiuser efficiency me of power mismatched mmse filters in the large system limit and we extend the analysis of the symbol by symbol maximum a posteriori probability map multiuser detector in the large system limit to the case of nonconstant user powers and nonuniform symbol prior probabilities

stability and spectral convergence of fourier method for nonlinear problems on the shortcomings of the 2 3 2 3 de aliasing method the high order accuracy of fourier method makes it the method of choice in many large scale simulations we discuss here the stability of fourier method for nonlinear evolution problems focusing on the two prototypical cases of the inviscid burgers equation and the multi dimensional incompressible euler equations the fourier method for such problems with quadratic nonlinearities comes in two main flavors one is the spectral fourier method the other is the 2 3 2 3 pseudo spectral fourier method where one removes the highest 1 3 1 3 portion of the spectrum this is often the method of choice to maintain the balance of quadratic energy and avoid aliasing errors two main themes are discussed in this paper first we prove that as long as the underlying exact solution has a minimal c 1 alpha c 1 spatial regularity then both the spectral and the 2 3 2 3 pseudo spectral fourier methods are stable consequently we prove their spectral convergence for smooth solutions of the inviscid burgers equation and the incompressible euler equations on the other hand we prove that after a critical time at which the underlying solution lacks sufficient smoothness then both the spectral and the 2 3 2 3 pseudo spectral fourier methods exhibit nonlinear instabilities which are realized through spurious oscillations in particular after shock formation in inviscid burgers equation the total variation of bounded pseudo spectral fourier solutions must increase with the number of increasing modes and we stipulate the analogous situation occurs with the 3d incompressible euler equations the limiting fourier solution is shown to enforce l 2 l 2 energy conservation and the contrast with energy dissipating onsager solutions is reflected through spurious oscillations

single scale wavelet representation of turbulence dynamics formulation and navier stokes regularity abstract the triad interactions arising from the spectral resolution of the nonlinear terms in the navier stokes equations have so far not been substantially modified in the wavelet representation in this paper the multiscale interactions are captured by exact expressions evaluated at a single scale of the mexican hat wavelet coefficients the larger scale terms as a volume integral of nearby wavelet coefficients and the smaller scale contributions as iterated laplacians of the coefficient at the point of interest as a result the navier stokes equations are expressed exactly at a single scale this facilitates the evaluation of the dominant holder exponent near singularities from the scaling properties of wavelet coefficients it is shown that euler dynamics would generate stronger singularities for any h 1 but that viscous dynamics would not unless h 1 a discontinuous case we discuss how this conclusion could be affected by boundary conditions

a sufficient condition for the kolmogorov 4 5 law for stationary martingale solutions to the 3d navier stokes equations we prove that statistically stationary martingale solutions of the 3d navier stokes equations on mathbb t 3 subjected to white in time colored in space forcing satisfy the kolmogorov 4 5 law in an averaged sense and over a suitable inertial range using only the assumption that the kinetic energy is o nu 1 as nu rightarrow 0 where is the inverse reynolds number this plays the role of a weak anomalous dissipation no energy balance or additional regularity is assumed aside from that satisfied by all martingale solutions from the energy inequality if the force is statistically homogeneous then any homogeneous martingale solution satisfies the spherically averaged 4 5 law pointwise in space an additional hypothesis of approximate isotropy in the inertial range gives the traditional version of the kolmogorov law we demonstrate a necessary condition by proving that energy balance and an additional quantitative regularity estimate as nu rightarrow 0 imply that the 4 5 law or any similar scaling law cannot hold

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

energy conservation for the 3d tropical climate model in bounded domains abstract in this paper we consider the energy conservation to the 3d tropical climate model in bounded domain by exploring the special structure of the nonlinear components and using the ideas introduced in 1 we can give some sufficient conditions for the weak solutions to ensure energy conservation

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

complete integrability of shock clustering and burgers turbulence we consider scalar conservation laws with convex flux and random initial data the hopf lax formula induces a deterministic evolution of the law of the initial data in a recent article we derived a kinetic theory and lax equations to describe the evolution of the law under the assumption that the initial datum is a spectrally negative markov process here we show that i the lax equations are hamiltonian and describe a principle of least action on the markov group ii the lax equations are completely integrable and linearized via a loop group factorization of operators iii the associated zero curvature equations can be solved via inverse scattering our results are rigorous for n dimensional approximations of the lax equations and yield formulas for the limit n the main observation is that the lax equations and zero curvature equations are a markovian analog of known integrable systems geodesic flow on lie groups and the n wave model respectively this allows us to introduce a variety of methods from the theory of integrable systems

energy conservation in two dimensional incompressible ideal fluids this note addresses the issue of energy conservation for the 2d euler system with an lp control on vorticity we provide a direct argument based on a mollification in physical space to show that the energy of a weak solution is conserved if omega nabla times u in l frac 3 2 an example of a 2d field in the class omega in l frac 3 2 epsilon for any epsilon 0 and u in b 1 3 3 infty onsager critical space see shvydkoy in discr contin dyn syst ser s 3 3 473 496 2010 is constructed with non vanishing energy flux this demonstrates sharpness of the kinematic argument which does not differentiate between 2d and 3d and requires onsager s regularity control on the solution next we show that for physically realizable solutions there is a mechanism preventing the anomalous dissipation in 2d that does not require such a control namely we prove that any solution to the euler equations produced via a vanishing viscosity limit from the navier stokes equations with omega in l p for p 1 conserves energy

statistical solutions and onsager s conjecture abstract we prove a version of onsager s conjecture on the conservation of energy for the incompressible euler equations in the context of statistical solutions as introduced recently by fjordholm et al 2017 as a byproduct we also obtain an alternative proof for the conservative direction of onsager s conjecture for weak solutions under a weaker besov type regularity assumption than previously known

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

sub grid effects of the voigt viscoelastic regularization of a singular dyadic model of turbulence in this work we investigate the spectral signature of navier stokes voigt nsv viscoelastic fluid flows by employing numerical simulations of a singular dyadic shell model our results clearly show that as the relaxation time is increased above a threshold the inertial range is reduced conserving part of the large scale statistics these results differ drastically from the two power law scenarios observed in a previous work where the nsv model was studied via sabra shell model simulations instead we also show that the additional elastic term regularizes the singular dyadic model which is the main reason behind this reduction of degrees of freedom the results of this work aim at proposing the nsv regularization as a sub grid model

intermittency and anomalous scaling of passive scalars in any space dimension we establish exact inequalities for the structure function scaling exponents of a passively advected scalar in both the inertial convective and viscous convective ranges these inequalities involve the scaling exponents of the velocity structure functions and in a refined form an intermittency exponent of the convective range scalar flux they are valid for three dimensional navier stokes turbulence and satisfied within errors by present experimental data the inequalities also hold for any synthetic turbulent velocity statistics with a finite correlation in time we show that for time correlation exponents of the velocity smaller than the local turnover exponent the scalar spectral exponent is strictly less than that in kraichnan s phys rev lett 72 1016 1994 phys fluids 11 945 1968 j fluid mech 64 737 1974 62 305 1974 soluble rapid change model with velocity ensuremath delta correlated in time our results include as a special case an exponent inequality derived previously by constantin and procaccia nonlinearity 7 1045 1994 but with a more direct proof the inequalities in their simplest form follow from a kolmogorov type relation for the turbulent passive scalar valid in each space dimension d our improved inequalities are based upon a rigorous version of the refined similarity hypothesis for passive scalars these are compared with the relations implied by fusion rules hypothesized for scalar gradients textcopyright 1996 the american physical society

on the weak solutions to the maxwell landau lifshitz equations and to the hall magneto hydrodynamic equations in this paper we deal with weak solutions to the maxwell landau lifshitz equations and to the hall magneto hydrodynamic equations first we prove that these solutions satisfy some weak strong uniqueness property then we investigate the validity of energy identities in particular we give a sufficient condition on the regularity of weak solutions to rule out anomalous dissipation in the case of the hall magneto hydrodynamic equations we also give a sufficient condition to guarantee the magneto helicity identity our conditions correspond to the same heuristic scaling as the one introduced by onsager in hydrodynamic theory finally we examine the sign locally of the anomalous dissipations of weak solutions obtained by some natural approximation processes

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

markov selections for the 3d stochastic navier stokes equations we investigate the markov property and the continuity with respect to the initial conditions strong feller property for the solutions to the navier stokes equations forced by an additive noise first we prove by means of an abstract selection principle that there are markov solutions to the navier stokes equations due to the lack of continuity of solutions in the space of finite energy the markov property holds almost everywhere in time then depending on the regularity of the noise we prove that any markov solution has the strong feller property for regular initial conditions we give also a few consequences of these facts together with a new sufficient condition for well posedness

design of block transceivers with decision feedback detection this paper presents a method for jointly designing the transmitter receiver pair in a block by block communication system that employs intrablock decision feedback detection we provide closed form expressions for transmitter receiver pairs that simultaneously minimize the arithmetic mean squared error mse at the decision point assuming perfect feedback the geometric mse and the bit error rate of a uniformly bit loaded system at moderate to high signal to noise ratios separate expressions apply for the zero forcing and minimum mse mmse decision feedback structures in the mmse case the proposed design also maximizes the gaussian mutual information and suggests that one can approach the capacity of the block transmission system using independent instances of the same gaussian code for each element of the block our simulation studies indicate that the proposed transceivers perform significantly better than standard transceivers and that they retain their performance advantages in the presence of error propagation

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