the generalized buckley leverett system solvability we propose a new approach to the mathematical modeling of the buckley leverett system which describes two phase flows in porous media considering the initial boundary value problem for a deduced model we prove the solvability of the problem the solvability result relies mostly on the kinetic method

a boundedness criterion for generalized calderon zygmund operators in 1965 a p calderon showed the l2 boundedness of the so called first calderon commutator this is one of the first examples of a non convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the hilbert transform these properties together with the l2 boundedness imply the lp boundedness for all p s in 1 oo many operators in analysis such as certain classes of pseudo differential operators and the cauchy integral operator on a curve are associated with kernels having these properties for these operators one of the major questions is if they are bounded on l2 we are going to give necessary and sufficient conditions for such an operator to be bounded on l2 they are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in bmo in the case of the aforementioned first commutator this can be checked by an integration by parts in the first section we present some basic notions and state the theorem which is proved in sections 2 and 3 in section 4 we show how to recover some classical results in sections 5 and 6 we construct a functional calculus for small perturbations of a and in section 7 we show a connection between the theory of calderon zygmund operators and kato s conjecture it is a pleasure to express our thanks to r r coifman and y meyer for suggesting many elegant simplifications in our proofs and most of the applications we also wish to thank stephen semmes for several pertinent remarks

global solutions of the super critical 2d quasi geostrophic equation in besov spaces abstract in this paper we study the super critical 2d dissipative quasi geostrophic equation we obtain some regularization effects allowing us to prove a global well posedness result for small initial data lying in critical besov spaces constructed over lebesgue spaces l p with p 1 local results for arbitrary initial data are also given

adaptive observer for multiple input multiple output mimo linear time varying systems for joint state parameter estimation in linear time varying ltv multiple input multiple output mimo systems an approach to the design of adaptive observers is proposed it is conceptually simple and computationally efficient its global exponential convergence is established for noise free systems in the presence of noises it is proved that the estimation errors are bounded and converge in the mean to zero if the noises are bounded and have zero means potential applications are fault detection and isolation and adaptive control

topological interference management through index coding this paper studies linear interference networks both wired and wireless with no channel state information at the transmitters except a coarse knowledge of the end to end one hop topology of the network that only allows a distinction between weak zero and significant nonzero channels and no further knowledge of the channel coefficients realizations the network capacity wired and degrees of freedom dof wireless are found to be bounded above by the capacity of an index coding problem for which the antidote graph is the complement of the given interference graph the problems are shown to be equivalent under linear solutions an interference alignment perspective is then used to translate the existing index coding solutions into the wired network capacity and wireless network dof solutions as well as to find new and unified solutions to different classes of all three problems

global well posedness for the critical 2d dissipative quasi geostrophic equation we give an elementary proof of the global well posedness for the critical 2d dissipative quasi geostrophic equation the argument is based on a non local maximum principle involving appropriate moduli of continuity

full diversity blind space time block codes in this paper the systematic design of a space time block code is considered for a wireless communication system with multiple transmitter receiver antennas and flat fading in which channel state information is completely unknown from the viewpoint of blind signal processing a necessary and sufficient condition is given for the unique identification of the multi input multi output flat fading channel and transmitted signal then some novel unique factorizations for a pair of coprime p ary and q ary phase shift keying psk constellations are established with this and currently available coherent space time block code designs a method is developed to systematically construct full diversity blind nonunitary space time block codes as well as unitary codes by just performing the qr decomposition of the nonunitary codes it is proved that under a noise free environment the code design proposed in this paper can guarantee that the transmitted signals and the channel coefficients are uniquely identified and that under a complex gaussian noise environment in which pth order and q th order statistics p and q are co prime of the received signals are available the channel coefficients can be still uniquely identified in addition a closed form solution to determine the channel coefficients is obtained

remarks on the method of modulus of continuity and the modified dissipative porous media equation abstract we employ besov space techniques and the method of modulus of continuity to obtain the global well posedness of the modified porous media equation

a new recurrent neurofuzzy network for identification of dynamic systems in this paper a new structure of a recurrent neurofuzzy network is proposed the network is based on two interconnected fuzzy inference systems fiss one recurrent and another static that intend to model the behavior of an unknown dynamic system from input output data in the proposed structure each rule involves a linear system in a controllable canonical form in order to reduce the online computational load and facilitate the online checking of the stability of the resulted network the training for the recurrent fis is made by a gradient based real time recurrent learning algorithm rtrla while the training for the static fis is based on a simple gradient method the initial parameter conditions prior to training are obtained by extracting information from a static fis trained with delayed input output signals to demonstrate the effectiveness of the proposed structure two nonlinear systems are identified

fuzzy identification using fuzzy neural networks with stable learning algorithms in general fuzzy neural networks cannot match nonlinear systems exactly unmodeled dynamic leads parameters drift and even instability problem according to system identification theory robust modification terms must be included in order to guarantee lyapunov stability this paper suggests new learning laws for mamdani and takagi sugeno kang type fuzzy neural networks based on input to state stability approach the new learning schemes employ a time varying learning rate that is determined from input output data and model structure stable learning algorithms for the premise and the consequence parts of fuzzy rules are proposed the calculation of the learning rate does not need any prior information such as estimation of the modeling error bounds this offer an advantage compared to other techniques using robust modification

representation theory for high rate multiple antenna code design multiple antennas can greatly increase the data rate and reliability of a wireless communication link in a fading environment but the practical success of using multiple antennas depends crucially on our ability to design high rate space time constellations with low encoding and decoding complexity it has been shown that full transmitter diversity where the constellation is a set of unitary matrices whose differences have nonzero determinant is a desirable property for good performance we use the powerful theory of fixed point free groups and their representations to design high rate constellations with full diversity furthermore we thereby classify all full diversity constellations that form a group for all rates and numbers of transmitter antennas the group structure makes the constellations especially suitable for differential modulation and low complexity decoding algorithms the classification also reveals that the number of different group structures with full diversity is very limited when the number of transmitter antennas is large and odd we therefore also consider extensions of the constellation designs to nongroups we conclude by showing that many of our designed constellations perform excellently on both simulated and real wireless channels

global existence singular solutions and ill posedness for the muskat problem the muskat or muskat leibenzon problem describes the evolution of the interface between two immiscible fluids in a porous medium or hele shaw cell under applied pressure gradients or fluid injection extraction in contras t to the hele shaw problem the one phase version of the muskat problem there are few nontrivial exact solutions or analytic results for the muskat problem for the stable forward muskat problem in which the higher viscosity fluid ex pands into the lower viscosity fluid we show global in time existence for initial data that is a small perturbation of a flat interface the initial data in this result ma y contain weak e g curvature singularities for the unstable backwa rd problem in which the higher viscosity fluid contracts we construct singular solution s that start off with smooth initial data but develop a point of infinite curvature at fi nite time c 2004 wiley periodicals inc

global existence of quasi stratified solutions for the confined ipm equation in this paper we consider a confined physical scenario to prove the global existence of smooth solutions with bounded density and finite energy for the inviscid incompressible porous media ipm equation the result is proved using the stability of stratified solutions combined with an additional structure of our initial perturbation which allows us to get rid of the boundary terms in the energy estimates

unitary space time modulation for multiple antenna communications in rayleigh flat fading motivated by information theoretic considerations we propose a signaling scheme unitary space time modulation for multiple antenna communication links this modulation is ideally suited for rayleigh fast fading environments since it does not require the receiver to know or learn the propagation coefficients unitary space time modulation uses constellations of t spl times m space time signals spl phi sub i l 1 l where t represents the coherence interval during which the fading is approximately constant and m t is the number of transmitter antennas the columns of each spl phi sub i are orthonormal when the receiver does not know the propagation coefficients which between pairs of transmitter and receiver antennas are modeled as statistically independent this modulation performs very well either when the signal to noise ratio snr is high or when t spl gt m we design some multiple antenna signal constellations and simulate their effectiveness as measured by bit error probability with maximum likelihood decoding we demonstrate that two antennas have a 6 db diversity gain over one antenna at 15 db snr

blind ml detection of orthogonal space time block codes identifiability and code construction the blind maximum likelihood ml detection of a general space time block code stbc is considered a challenging implementation problem recent work has revealed that for the orthogonal stbcs ostbcs their special code structures can be exploited to formulate highly effective blind ml based algorithms attracted by this realization merit this paper investigates the blind ml identifiability of ostbcs with an emphasis on the binary psk bpsk and quaternary psk qpsk constellations we find a class of ostbcs called the nonrotatable ostbcs that can be uniquely identified up to a sign uiuts almost surely under a few mild assumptions for example for an independently distributed rayleigh channel with any number of receiver antennas a nonrotatable ostbc can be uiuts with probability 1 while this identifiability looks appealing already we further examine a subclass of nonrotatable ostbcs called the nonintersecting subspace nis ostbcs we prove that nis ostbcs are uiuts for any nonzero channel however nis ostbcs are not available in the existing literature to fill this gap we devise a code construction procedure that can convert any bpsk or qpsk ostbc to an nis ostbc

black box identification of a class of nonlinear systems by a recurrent neurofuzzy network this brief presents a structure for black box identification based on continuous time recurrent neurofuzzy networks for a class of dynamic nonlinear systems the proposed network catches the dynamics of a system by generating its own states using only input and output measurements of the system the training algorithm is based on adaptive observer theory the stability of the network the convergence of the training algorithm and the ultimate bound on the identification error as well as the parameter error are established experimental results are included to illustrate the effectiveness of the proposed method

index coding an interference alignment perspective author s maleki h cadambe vr jafar sa abstract the index coding problem is studied from an interference alignment perspective providing new results as well as new insights into and generalizations of previously known results an equivalence is established between the capacity of multiple unicast index coding where each message is desired by exactly one receiver and groupcast index coding where a message can be desired by multiple receivers which settles the heretofore open question of insufficiency of linear codes for the multiple unicast index coding problem by equivalence with groupcast settings where this question has previously been answered necessary and sufficient conditions for the achievability of rate half per message in the index coding problem are shown to be a natural consequence of interference alignment constraints and generalizations to feasibility of rate 1 l 1 per message when each destination desires at least l messages are similarly obtained finally capacity optimal solutions are presented to a series of symmetric index coding problems inspired by the local connectivity and local interference characteristics of wireless networks the solutions are based on vector linear coding 1963 2012 ieee

stable adaptive observers for nonlinear time varying systems an adaptive observer identifier for single input single output observable nonlinear systems that can be transformed to a certain observable canonical form is described sufficient conditions for stability of this observer are provided these conditions are in terms of the structure of the system and canonical form the boundedness of the parameter variations and the sufficient richness of some signals the scope of the canonical form and the use of the observer identifier is motivated by the presentation of applications to time invariant bilinear systems nonlinear systems in phase variable form a biotechnological process and a robot manipulator in each case the specific stability conditions are presented

maximum rank distance codes as space time codes the critical design criterion for space time codes in asymptotically good channels is the minimum rank between codeword pairs rank codes are a two dimensional matrix code construction where by the rank is the metric of merit we look at the application of rank codes to space time code design in particular we provide construction methods of full rank codes over different complex signal constellations for arbitrary numbers of antennas and codeword periods we also derive a singleton type bound on the rate of a code for the rank metric and we show that rank codes satisfy this bound with equality

existence and partial regularity results for the heat flow for harmonic maps on considere le probleme d evolution t u m u n u 0 dans m r u 0 u 0 sur m ou m et n sont des varietes de riemann lisses compactes et u 0 l application harmonique de m dans n on demontre l existence d une solution globale faible

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

limiting case of the sobolev inequality in bmo with application to the euler equations we shall prove a logarithmic sobolev inequality by means of the bmo norm in the critical exponents as an application we shall establish a blow up criterion of solutions to the euler equations

bilinear estimates in bmo and the navier stokes equations we prove that the bmo norm of the velocity and the vorticity controls the blow up phenomena of smooth solutions to the navier stokes equations our result is applied to the criterion on uniqueness and regularity of weak solutions in the marginal class

global well posedness of incompressible flow in porous media with critical diffusion in besov spaces abstract in this paper we study the model of heat transfer in a porous medium with a critical diffusion we obtain global existence and uniqueness of solutions to the equations of heat transfer of incompressible fluid in besov spaces b p 1 3 p r 3 with 1 p by the method of modulus of continuity and fourier localization technique

nonintersecting subspaces based on finite alphabets two subspaces of a vector space are here called nonintersecting if they meet only in the zero vector motivated by the design of noncoherent multiple antenna communications systems we consider the following question how many pairwise nonintersecting m sub t dimensional subspaces of an m dimensional vector space v over a field f can be found if the generator matrices for the subspaces may contain only symbols from a given finite alphabet a spl sube f the most important case is when f is the field of complex numbers c then m sub t is the number of antennas if a f gf q it is shown that the number of nonintersecting subspaces is at most q sup m 1 q sup mt 1 and that this bound can be attained if and only if m is divisible by m sub t furthermore these subspaces remain nonintersecting when lifted to the complex field it follows that the finite field case is essentially completely solved in the case when f c only the case m sub t 2 is considered it is shown that if a is a psk configuration consisting of the 2 sup r complex roots of unity the number of nonintersecting planes is at least 2 sup r m 2 and at most 2 sup r m 1 1 the lower bound may in fact be the best that can be achieved

rate diversity tradeoff of space time codes with fixed alphabet and optimal constructions for psk modulation we show that for any q spl times m space time code s having a fixed finite signal constellation there is a tradeoff between the transmission rate r and the transmit diversity gain spl nu achieved by the code the tradeoff is characterized by r spl les q spl nu 1 where q is the number of transmit antennas when either binary phase shift keying bpsk or quaternary phase shift keying qpsk is used as the signal constellation a systematic construction is presented to achieve the maximum possible rate for every possible value of transmit diversity gain

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

partial regularity for stationary harmonic maps into spheres in an interesting recent paper 12 f hi lein has shown that any weakly harmonic mapping from a two dimensional surface into a sphere is smooth i present here a kind of generalization to higher dimensions asserting in effect that a stationary harmonic mapping from an open subset of nn n 3 into a sphere is smooth except possibly for a closed singular set of n 2 dimensional hausdorff measure zero my proof crucially depends upon several of h l in s observations as streamlined by p l lions to state the result precisely let us suppose that m n 2 u is a smooth open subset of nn and s m 1 denotes the unit sphere in nm a function u in the sobolev space hi u rm u u 1 urn belongs to h i u s ml provided t u 1 a e in u

on critical cases of sobolev s inequalities abstract we present a new form of the trudinger type inequality which shows an explicit dependence moreover we give an alternative proof of the brezis gallouet wainger inequality

global existence of weak solution to the heat and moisture transport system in fibrous porous media abstract this paper is concerned with theoretical analysis of a heat and moisture transfer model arising from textile industries which is described by a degenerate and strongly coupled parabolic system we prove the global in time existence of weak solution by constructing an approximate solution with some standard smoothing the proof is based on the physical nature of gas convection in which the heat energy flux in convection is determined by the mass vapor flux in convection

the critical sobolev inequalities in besov spaces and regularity criterion to some semi linear evolution equations we show the critical sobolev inequalities in the besov spaces with the logarithmic form such as brezis gallouet wainger and beale kato majda as an application of those inequalities the regularity problem under the critical condition to the navier stokes equations the euler equations in mathbb r n and the gradient flow to the harmonic map to the sphere are discussed namely the serrin ohyama type regularity criteria are improved in the terms of the besov spaces

a note on interface dynamics for convection in porous media we studied the fluid interface problem through porous media given by two incompressible 2 d fluids of different densities this problem is mathematically analogous to the dynamics interface for convection in porous media where the free boundary evolves between fluids with different temperatures we found a new formula for the evolution equation of the free boundary parameterized as a function in the periodic case in this formula there is not a principal value in the non local integral operator involved in the equation giving a simpler system using this formulation we perform numerical simulations in the stable case denser fluid below which shows a strong regularity effect in the periodic interface

remarks on a paper by j t beale t kato and a majda we prove that the maximum norm of the deformation tensor controls the possible breakdown of smooth solutions for the 3 dimensional euler equations more precisely the loss of regularity in a local smooth solution of the euler equations implies the growth without bound of the deformation tensor as the critical time approaches equivalently if the deformation tensor remains bounded the existence of a smooth solution is guaranteed

full diversity blind alamouti space time block codes for unique identification of flat fading channels in this paper we consider a systematic design of a full diversity blind alamouti space time block code for a wireless communication system having two transmitter antennas and a single receiver antenna in which the channel is completely unknown at both the transmitter and the receiver the key idea is to alternatively transmit alamouti codes with different phase shift keying psk constellations say with p psk and q psk to provide the theoretical analysis of its unique identification and full diversity conditions we first establish a general theorem for factorizing a pq psk constellation into a product of a p psk constellation and a q psk constellation then we prove that a necessary and sufficient condition for both the factorization and a set of solutions of the resulting alamouti diophantine equation from our code design to be unique is that p and q are coprime the same condition assures that the channel and the transmitted signal can be blindly and uniquely identified furthermore a closed form solution to determine the channel coefficients and the transmitted symbols are obtained also we prove that our code enables full diversity for the generalized likelihood ratio test glrt receiver if p and q are coprime when finite length of the received data is available the unitarity and linearity of our code design allow us to utilize the semi definite relaxation sdr decoder or the sphere decoder so that the joint estimation of the channel and symbols can be efficiently implemented

adaptive observers for single output nonlinear systems an adaptive version of the nonlinear observer obtained by a j krener et al 1983 is presented this version involves the cancellation of nonlinear terms by output injection as an intermediate step necessary and sufficient conditions are given for transforming a nonlinear system by state space change of coordinates into the special adaptive observer form that was used by y bastin et al 1988 to design adaptive observers

global well posedness of the critical burgers equation in critical besov spaces abstract we make use of the method of modulus of continuity a kiselev f nazarov r shterenberg blow up and regularity for fractal burgers equation dyn partial differ equ 5 2008 211 240 and fourier localization technique h abidi t hmidi on the global well posedness of the critical quasi geostrophic equation siam j math anal 40 1 2008 167 185 h abidi t hmidi on the global well posedness of the critical quasi geostrophic equation siam j math anal 40 1 2008 167 185 to prove the global well posedness of the critical burgers equation t u u x u u 0 in critical besov spaces b p 1 1 p r with p 1 where

analytical behavior of two dimensional incompressible flow in porous media in this paper we study the analytic structure of a two dimensional mass balance equation of an incompressible fluid in a porous medium given by darcy s law we obtain local existence and uniqueness by the particle trajectory method and we present different global existence criterions these analytical results with numerical simulations are used to indicate nonformation of singularities moreover blow up results are shown in a class of solutions with infinite energy

dynamic non singleton fuzzy logic systems for nonlinear modeling we investigate dynamic versions of fuzzy logic systems flss and specifically their non singleton generalizations nsflss and derive a dynamic learning algorithm to train the system parameters the history sensitive output of the dynamic systems gives them a significant advantage over static systems in modeling processes of unknown order this is illustrated through an example in nonlinear dynamic system identification since dynamic nsfls s can be considered to belong to the family of general nonlinear autoregressive moving average narma models they are capable of parsimoniously modeling narma processes we study the performance of both dynamic and static flss in the predictive modeling of a narma process

analyticity of lagrangian trajectories for well posed inviscid incompressible fluid models abstract we discuss general incompressible inviscid models including the euler equations the surface quasi geostrophic equation incompressible porous medium equation and boussinesq equations all these models have classical unique solutions at least for short time we show that they have real analytic lagrangian paths more precisely we show that as long as a solution of any of these equations is in a class of regularity that assures holder continuous gradients of velocity the corresponding lagrangian paths are real analytic functions of time the method of proof is conceptually straightforward and general and we address the combinatorial issues head on

well posedness of the cauchy problem for the fractional power dissipative equation in critical besov spaces abstract in this paper we study the cauchy problem for the semilinear fractional power dissipative equation u t u f u for the initial data u 0 in critical besov spaces b 2 r with n 2 2 d b where 0 f u p d u b 1 with p d being a homogeneous pseudo differential operator of order d 0 2 and b 0 being an integer making use of some estimates of the corresponding linear equation in the frame of mixed time space spaces the so called mono norm method which is different from the kato s double norm method fourier localization technique and littlewood paley theory we get the well posedness result in the case n 2

blow up criterion of strong solutions to the navier stokes equations in besov spaces with negative indices a holder type inequality in besov spaces is established and applied to show that every strong solution u t x on 0 t of the navier stokes equations can be continued beyond t t provided that the vorticity t x l22 0 t b r3 l21 0 t b 1 r3 for 0 1

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

differential unitary space time modulation we present a framework for differential modulation with multiple antennas across a continuously fading channel where neither the transmitter nor the receiver knows the fading coefficients the framework can be seen as a natural extension of standard differential phase shift keying commonly used in single antenna unknown channel systems we show how our differential framework links the unknown channel system with a known channel system and we develop performance design criteria as a special ease we introduce a class of diagonal signals where only one antenna is active at any time and demonstrate how these signals may be used to achieve full transmitter diversity and low probability of error

scalars convected by a two dimensional incompressible flow we provide a test for numerical simulations for several two dimensional incompressible flows that appear to develop sharp fronts we show that in order to have a front the velocity has to have uncontrolled velocity growth 2001 john wiley sons inc

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

self learning fuzzy controllers based on temporal backpropagation a generalized control strategy that enhances fuzzy controllers with self learning capability for achieving prescribed control objectives in a near optimal manner is presented this methodology termed temporal backpropagation is model sensitive in the sense that it can deal with plants that can be represented in a piecewise differentiable format such as difference equations neural networks gmdh structures and fuzzy models regardless of the numbers of inputs and outputs of the plants under consideration the proposed approach can either refine the fuzzy if then rules of human experts or automatically derive the fuzzy if then rules if human experts are not available the inverted pendulum system is employed as a testbed to demonstrate the effectiveness of the proposed control scheme and the robustness of the acquired fuzzy controller

communication on the grassmann manifold a geometric approach to the noncoherent multiple antenna channel we study the capacity of multiple antenna fading channels we focus on the scenario where the fading coefficients vary quickly thus an accurate estimation of the coefficients is generally not available to either the transmitter or the receiver we use a noncoherent block fading model proposed by marzetta and hochwald see ibid vol 45 p 139 57 1999 the model does not assume any channel side information at the receiver or at the transmitter but assumes that the coefficients remain constant for a coherence interval of length t symbol periods we compute the asymptotic capacity of this channel at high signal to noise ratio snr in terms of the coherence time t the number of transmit antennas m and the number of receive antennas n while the capacity gain of the coherent multiple antenna channel is min m n bits per second per hertz for every 3 db increase in snr the corresponding gain for the noncoherent channel turns out to be m 1 m t bits per second per hertz where m min m n t 2 the capacity expression has a geometric interpretation as sphere packing in the grassmann manifold

geometric statistics in turbulence the author presents results regarding certain average properties of incompressible fluids derived from the equations of motion the author estimates the average dissipation rate the average dimension of level sets the role played by the field of direction of vorticity in the three dimensional euler and navier stokes equations is discussed and a class of two dimensional equations that are useful models of incompressible dynamics is described the author presents results concerning scaling exponents in turbulence

gas kinetic derivation of navier stokes like traffic equations macroscopic traffic models have recently been severely criticized to base on lax analogies only and to have a number of deficiencies therefore this paper shows how to construct a logically consistent fluid dynamic traffic model from basic laws for the acceleration and interaction of vehicles these considerations lead to the gas kinetic traffic equation of paveri fontana its stationary and spatially homogeneous solution implies equilibrium relations for the fundamental diagram the variance density relation and other quantities which are partly difficult to determine empirically paveri fontana s traffic equation allows the derivation of macroscopic moment equations which build a system of non closed equations this system can be closed by the well proved method of chapman and enskog which leads to euler like traffic equations in zeroth order approximation and to navier stokes like traffic equations in first order approximation the latter are finally corrected for the finite space requirements of vehicles it is shown that the resulting model is able to withstand the above mentioned criticism

growth of solutions for qg and 2d euler equations the work of constantin majda tabak 1 developed an analogy between the quasi geostrophic and 3d euler equations constantin majda and tabak proposed a candidate for a singularity for the quasi geostrophic equation their numerics showed evidence of a blow up for a particular initial data where the level sets of the temperature contain a hyperbolic saddle the arms of the saddle tend to close in finite time producing a sharp front numerics studies done later by ohkitani yamada 8 and constantin nie schorghofer 2 with the same initial data suggested that instead of a singularity the derivatives of the temperature were increasing as double exponential in time the study of collapse on a curve was first studied in 1 for the quasi geostrophic equation where they considered a simplified ansatz for classical frontogenesis with trivial topology at the time of collapse the scalar is discontinuous across the curve x2 f x1 with different limiting values for the temperature on each side of the front they show that under this topology the directional field remains smooth up to the collapse which contradicts the following theorem proven in 1