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

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

global regularity for a birkhoff rott α approximation of the dynamics of vortex sheets of the 2d euler equations abstract we present an regularization of the birkhoff rott equation induced by the two dimensional euler equations for the vortex sheet dynamics we show that an initially smooth self avoiding vortex sheet remains smooth for all times under the regularized dynamics provided the initial density of vorticity is an integrable function over the curve with respect to the arc length measure

an eulerian lagrangian approach to the navier stokes equations we present a formulation of the incompressible viscous navier stokes equation based on a generalization of the inviscid weber formula in terms of a diffusive back to labels map and a virtual velocity we derive a generalization of the inviscid cauchy formula and obtain certain bounds for the objects introduced

existence of global solutions of elliptic systems we study existence of entire solutions of elliptic systems whose prototype is given by div u u m 2 u

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

type i membranes phase resetting curves and synchrony type i membrane oscillators such as the connor model connor et al 1977 and the morris lecar model morris and lecar 1981 admit very low frequency oscillations near the critical applied current hansel et al 1995 have numerically shown that synchrony is difficult to achieve with these models and that the phase resetting curve is strictly positive we use singular perturbation methods and averaging to show that this is a general property of type i membrane models we show in a limited sense that so called type ii resetting occurs with models that obtain rhythmicity via a hopf bifurcation we also show the differences between synapses that act rapidly and those that act slowly and derive a canonical form for the phase interactions

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

hardy spaces and the two dimensional euler equations with nonnegative vorticity we prove that certain quadratic expressions involving the gradient of a weakly superharmonic function in r belong to a local hardy space as an application we provide a new proof of j m delort s convergence theorem for solutions of the two dimensional euler equations with vorticities of one sign department of mathematics university of california berkeley california 94720 sfb 256 institut fur angewandte mathematik beringstrasse 4 d5300 bonn 1 ger many this content downloaded from 207 46 13 101 on sat 08 oct 2016 05 44 35 utc all use subject to http about jstor org terms

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

quantifying strength of chaos in the population firing rate of neurons in this study i considered quantifying the strength of chaos in the population firing rate of a pulse coupled neural network in particular i considered the dynamics where the population firing rate is chaotic and the firing of each neuron is stochastic i calculated a time histogram of firings to show the variation in the population firing rate over time to smooth this histogram i used bayesian adaptive regression splines and a gaussian filter the nonlinear prediction method based on reconstruction was applied to a sequence of interpeak intervals in the smoothed time histogram of firings i propose the use of the sum of nonlinearity as a quantifier of the strength of chaos when applying this method to the firings of a pulse coupled neural network the sum of nonlinearity was seen to satisfy three properties for quantifying the strength of chaos first it can be calculated from spiking data alone second it takes large values when applied to firings that are confirmed theoretically or numerically to be chaotic third it reflects the strength of chaos of the original dynamics

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

rewiring induced chaos in pulse coupled neural networks the dependence of the dynamics of pulse coupled neural networks on random rewiring of excitatory and inhibitory connections is examined when both excitatory and inhibitory connections are rewired periodic synchronization emerges with a hopf like bifurcation and a subsequent period doubling bifurcation chaotic synchronization is also observed when only excitatory connections are rewired periodic synchronization emerges with a saddle node like bifurcation and chaotic synchronization is also observed this result suggests that randomness in the system does not necessarily contaminate the system and sometimes it even introduces rich dynamics to the system such as chaos

oscillations and concentrations in weak solutions of the incompressible fluid equations the authors introduce a new concept of measure valued solution for the 3 d incompressible euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations one application of the concepts developed here is the following important result a sequence of leray hopf weak solutions of the navier stokes equations converges in the high reynolds number limit to a measure valued solution of 3 d euler defined for all positive times the authors present several explicit examples of solution sequences for 3 d incompressible euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations an extensions of the concept of young measure is developed to incorporate these complex phenomena in the measure valued solutions constructed here

on a leray α model of turbulence in this paper we introduce and study a new model for three dimensional turbulence the leray model this model is inspired by the lagrangian averaged navier stokes model of turbulence also known navier stokes model or the viscous camassa holm equations as in the case of the lagrangian averaged navier stokes model the leray model compares successfully with empirical data from turbulent channel and pipe flows for a wide range of reynolds numbers we establish here an upper bound for the dimension of the global attractor the number of degrees of freedom of the leray model of the order of l l d 12 7 where l is the size of the domain and l d is the dissipation length scale this upper bound is much smaller than what one would expect for three dimensional models i e l l d 3 this remarkable result suggests that the leray model has a great potential to become a good sub grid scale large eddy simulation model of turbulence we support this observation by studying analytically and computationally the energy spectrum and show that in addition to the usual k 5 3 kolmogorov power law the inertial range has a steeper power law spectrum for wavenumbers larger than 1 finally we propose a prandtl like boundary layer model induced by the leray model and show a very good agreement of this model with empirical data for turbulent boundary layers

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

a criterion for the equivalence of the birkhoff rott and euler descriptions of vortex sheet evolution in this article we consider the evolution of vortex sheets in the plane both as a weak solution of the two dimensional incompressible euler equations and as a weak solution of the birkhoff rott equations we begin by discussing the classical birkhoff rott equations with respect to arbitrary parametrizations of the sheet we introduce a notion of weak solution to the birkhoff rott system and we prove consistency of this notion with the classical formulation of the equations our main purpose in this paper is to present a sharp criterion for the equivalence of the weak euler and weak birkhoff rott descriptions of vortex sheet dynamics

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

the camassa holm equations and turbulence abstract in this paper we will survey our results on the camassa holm equations and their relation to turbulence as discussed in s chen c foias d d holm e olson e s titi s wynne the camassa holm equations as a closure model for turbulent channel and pipe flow phys rev lett 81 1998 5338 s chen c foias d d holm e olson e s titi s wynne a connection between the camassa holm equations and turbulent flows in channels and pipes phys fluids in press in particular we will provide a more detailed mathematical treatment of those equations for pipe flows which yield accurate predictions of turbulent flow profiles for very large reynolds numbers there are many facts connecting the camassa holm equations to turbulent fluid flows the dimension of the attractor agrees with the heuristic argument based on the kolmogorov statistical theory of turbulence the statistical properties of the energy spectrum agree in numerical simulation with the kolmogorov power law furthermore comparison of mean flow profiles for turbulent flow in channels and pipes given by experimental and numerical data show acceptable agreement with the profile of the corresponding solution of the camassa holm equations

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

localised relative energy and finite speed of propagation for compressible flows abstract for the incompressible and the isentropic compressible euler equations in arbitrary space dimension we establish the principle of localised relative energy thus generalising the well known relative energy method to this end we adapt classical arguments of c dafermos to the euler equations we give several applications to the behaviour of weak solutions like local weak strong uniqueness local preservation of smoothness and finite speed of propagation for the isentropic system

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

global regularity and convergence of a birkhoff rott α approximation of the dynamics of vortex sheets of the two dimensional euler equations we present an regularization of the birkhoff rott equation br equation induced by the two dimensional euler equations for the vortex sheet dynamics we show the convergence of the solutions of euler equations to a weak solution of the euler equations for initial vorticity being a finite radon measure of fixed sign which includes the vortex sheets case we also show that provided the initial density of vorticity is an integrable function over the curve with respect to the arc length measure i an initially lipschitz chord arc vortex sheet curve evolving under the br equation remains lipschitz for all times ii an initially holder c1 0 1 chord arc curve remains in c1 for all times and finally iii an initially holder cn n 1 0 1 closed chord arc curve remains so for all times in all these cases the weak euler and the br descriptions of the vortex sheet motion are equivalent 2009 wiley periodicals inc

existence of vortex sheets with reflection symmetry in two space dimensions the main purpose of this work is to establish the existence of a weak solution to the incompressible 2d euler equations with initial vorticity consisting of a radon measure with distinguished sign in h 1 compactly supported in the closed right half plane superimposed on its odd reflection in the left half plane we make use of a new a priori estimate to control the interaction between positive and negative vorticity at the symmetry axis we prove that a weak limit of a sequence of approximations obtained by either regularizing the initial data or by using the vanishing viscosity method is a weak solution of the incompressible 2d euler equations we also establish the equivalence at the level of weak solutions between mirror symmetric flows in the full plane and flows in the half plane finally we extend our existence result to odd l 1 perturbations without distinguished sign of our original initial vorticity

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

bayesian curve fitting with free knot splines we describe a bayesian method for fitting curves to data drawn from an exponential family that uses splines for which the number and locations of knots are free parameters the method uses reversible jump markov chain monte carlo to change the knot configurations and a locality heuristic to speed up mixing for nonnormal models we approximate the integrated likelihood ratios needed to compute acceptance probabilities by using the bayesian information criterion bic under priors that make this approximation accurate our technique is based on a marginalised chain on the knot number and locations but we provide methods for inference about the regression coefficients and functions of them in both normal and nonnormal models simulation results suggest that the method performs well and we illustrate the method in two neuroscience applications

finite time analyticity for the two and three dimensional kelvin helmholtz instability the well posed property for the finite time vortex sheet problem with analytic initial data was first conjectured by birkhoff in two dimensions and is shown here to hold both in two and three dimensions incompressible inviscid and irrotational flow with a velocity jump across an interface is assumed in two dimensions global existence of a weak solution to the euler equation with such initial conditions is established in three dimensions a lagrangian representation of the vortex sheet analogous to the birkhoff equation in two dimensions is presented

the anisotropic lagrangian averaged euler and navier stokes equations the purpose of this paper is twofold first we give a derivation of the lagrangian averaged euler lae and navier stokes lans equations this theory involves a spatial scale and the equations are designed to accurately capture the dynamics of the euler and navier stokes equations at length scales larger than while averaging the motion at scales smaller than the derivation involves an averaging procedure that combines ideas from both the material lagrangian and spatial eulerian viewpoints this framework allows the use of a variant of g i taylor s frozen turbulence hypothesis as the foundation for the model equations more precisely the derivation is based on the strong physical assumption that fluctutations are frozen into the mean flow in this article we use this hypothesis to derive the averaged lagrangian for the theory and all the terms up to and including order 2 are accounted for

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

variational principles for lagrangian averaged fluid dynamics the lagrangian average la of the ideal fluid equations preserves their fundamental transport structure this transport structure is responsible for the kelvin circulation theorem of the la flow and hence for its potential vorticity convection and helicity conservation we show that lagrangian averaging also preserves the euler poincar e variational framework that implies the exact ideal fluid equations in the eulerian representation this is expressed in the lagrangian averaged euler poincar e laep theorem proved here we illustrate the laep theorem by applying it to incompressible ideal fluids to derive the lagrangian averaged euler equations and thereby recover the generalized lagrangian mean motion equation finally we discuss recent progress in applications of these equations as the basis for new la closure models of fluid turbulence

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

uniformly local l p estimate for 2 d vorticity equation and its application to euler equations with initial vorticity in bmo we shall prove the global existence theorem for the 2 dimensional euler equations in open image in new window with the initial vorticity in bmo containing functions which do not decay at infinity and have logarithmic singularities

a method for selecting the bin size of a time histogram the time histogram method is the most basic tool for capturing a time dependent rate of neuronal spikes generally in the neurophysiological literature the bin size that critically determines the goodness of the fit of the time histogram to the underlying spike rate has been subjectively selected by individual researchers here we propose a method for objectively selecting the bin size from the spike count statistics alone so that the resulting bar or line graph time histogram best represents the unknown underlying spike rate for a small number of spike sequences generated from a modestly fluctuating rate the optimal bin size may diverge indicating that any time histogram is likely to capture a spurious rate given a paucity of data the method presented here can nevertheless suggest how many experimental trials should be added in order to obtain a meaningful time dependent histogram with the required accuracy

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

on the convergence rate of the euler α an inviscid second grade complex fluid model to the euler equations we study the convergence rate of the solutions of the incompressible euler an inviscid second grade complex fluid equations to the corresponding solutions of the euler equations as the regularization parameter approaches zero first we show the convergence in hs s n 2 1 in the whole space and that the smooth euler solutions exist at least as long as the corresponding solution of the euler equations next we estimate the convergence rate for two dimensional vortex patch with smooth boundaries

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

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

global vortex sheet solutions of euler equations in the plane abstract for any curve close enough to a straight line there is a global solution of euler equations having for all positive time a vortex sheet and admitting the given curve as the initial sheet

an inviscid regularization for the surface quasi geostrophic equation inspired by recent developments in berdina like models for turbulence we propose an inviscid regularization for the surface quasi geostrophic sqg equations we are particularly interested in the celebrated question of blowup in finite time of the solution gradient of the sqg equations the new regularization yields a necessary and sufficient condition satisfied by the regularized solution when a regularization parameter tends to 0 for the solution of the original sqg equations to develop a singularity in finite time as opposed to the commonly used viscous regularization the inviscid equations derived here conserve a modified energy therefore the new regularization provides an attractive numerical procedure for finite time blowup testing in particular we prove that if the initial condition is smooth then the regularized solution remains as smooth as the initial data for all times 2007 wiley periodicals inc

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

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

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

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

surrogate time series before we apply nonlinear techniques e g those inspired by chaos theory to dynamical phenomena occurring in nature it is necessary to first ask if the use of such advanced techniques is justified by the data while many processes in nature seem very unlikely a priori to be linear the possible nonlinear nature might not be evident in specific aspects of their dynamics the method of surrogate data has become a very popular tool to address such a question however while it was meant to provide a statistically rigorous foolproof framework some limitations and caveats have shown up in its practical use in this paper recent efforts to understand the caveats avoid the pitfalls and to overcome some of the limitations are reviewed and augmented by new material in particular we will discuss specific as well as more general approaches to constrained randomisation providing a full range of examples new algorithms will be introduced for unevenly sampled and multivariate data and for surrogate spike trains the main limitation which lies in the interpretability of the test results will be illustrated through instructive case studies we will also discuss some implementational aspects of the realisation of these methods in the tisean software package

convergence of the 2d euler α to euler equations in the dirichlet case indifference to boundary layers abstract in this article we consider the euler system as a regularization of the incompressible euler equations in a smooth two dimensional bounded domain for the limiting euler system we consider the usual non penetration boundary condition while for the euler regularization we use velocity vanishing at the boundary we also assume that the initial velocities for the euler system approximate in a suitable sense as the regularization parameter 0 the initial velocity for the limiting euler system for small values of this situation leads to a boundary layer which is the main concern of this work our main result is that under appropriate regularity assumptions and despite the presence of this boundary layer the solutions of the euler system converge as 0 to the corresponding solution of the euler equations in l 2 in space uniformly in time we also present an example involving parallel flows in order to illustrate the indifference to the boundary layer of the 0 limit which underlies our work

entire radial solutions of elliptic systems and inequalities of the mean curvature type abstract in this paper we study first nonexistence of radial entire solutions of elliptic systems of the mean curvature type with a singular or degenerate diffusion depending on the solution u in particular we extend a previous result given in r filippucci nonexistence of radial entire solutions of elliptic systems j differential equations 188 2003 353 389 moreover in the scalar case we obtain nonexistence of all entire solutions radial or not of differential inequalities involving again operators of the mean curvature type and a diffusion term we prove that in the scalar case nonexistence of entire solutions is due to the explosion of the derivative of every nonglobal radial solution in the right extremum of the maximal interval of existence while in that point the solution is bounded this behavior is qualitatively different with respect to what happens for the m laplacian operator studied in r filippucci nonexistence of radial entire solutions of elliptic systems j differential equations 188 2003 353 389 where nonexistence of entire solutions is due even in the vectorial case to the explosion in norm of the solution at a finite point our nonexistence theorems for inequalities extend previous results given by naito and usami in y naito h usami entire solutions of the inequality div a d u d u f u math z 225 1997 167 175 and ghergu and radulescu in m ghergu v radulescu existence and nonexistence of entire solutions to the logistic differential equation abstr appl anal 17 2003 995 1003

blow up zero α limit and the liouville type theorem for the euler poincare equations in this paper we study the euler poincare equations in mathbb r n we prove local existence of weak solutions in w 2 p mathbb r n p n and local existence of unique classical solutions in h k mathbb r n k n 2 3 as well as a blow up criterion for the zero dispersion equation 0 we prove a finite time blow up of the classical solution we also prove that as the dispersion parameter vanishes the weak solution converges to a solution of the zero dispersion equation with sharp rate as 0 provided that the limiting solution belongs to c 0 t h k mathbb r n with k n 2 3 for the stationary weak solutions of the euler poincare equations we prove a liouville type theorem namely for 0 any weak solution mathbf u in h 1 mathbb r n is u 0 for 0 any weak solution mathbf u in l 2 mathbb r n is u 0

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