on the well posedness of the euler equations in the triebel lizorkin spaces we prove local in time unique existence and a blowup criterion for solutions in the triebel lizorkin space for the euler equations of inviscid incompressible fluid flows in n n 2 as a corollary we obtain global persistence of the initial regularity characterized by the triebel lizorkin spaces for the solutions of two dimensional euler equations to prove the results we establish the logarithmic inequality of the beale kato majda type the moser type of inequality as well as the commutator estimate in the triebel lizorkin spaces the key methods of proof used are the littlewood paley decomposition and the paradifferential calculus by j m bony 2002 john wiley sons inc

self stretching of a perturbed vortex filament i the asymptotic equation for deviations from a straight line a new asymptotic equation is derived for the motion of thin vortex filaments in an incompressible fluid at high reynolds numbers this equation differs significantly from the familiar local self induction equation in that it includes self stretching of the filament in a nontrivial but to some extent analytically tractable fashion under the same change of variables as employed by hasimoto 1972 to convert the local self induction equation to the cubic nonlinear schrodinger equation the new asymptotic propagation law becomes a cubic nonlinear schrodinger equation perturbed by an explicit nonlocal linear operator explicit formulae are developed which relate the rate of local self stretch along the vortex filament to a particular quadratic functional of the solution of the perturbed schrodinger equation the asymptotic equation is derived systematically from suitable solutions of the navier stokes equations by the method of matched asymptotic expansions based on the limit of high reynolds numbers the key idea in the derivation is to consider a filament whose core deviates initially from a given smooth curve only by small amplitude but short wavelength displacements balanced so that the axial length scale of these perturbations is small compared to an integral length of the background curve but much larger than a typical core size o re 1 2 of the filament in a particular distinguished limit of wavelength preturbation amplitude and filament core size the nonlocal induction integral has a simplified asymptotic representation and yields a contribution in the schrodinger equation that directly competes with the cubic nonlinearity

integral and integrable algorithms for a nonlinear shallow water wave equation an asymptotic higher order model of wave dynamics in shallow water is examined in a combined analytical and numerical study with the aim of establishing robust and efficient numerical solution methods based on the hamiltonian structure of the nonlinear equation an algorithm corresponding to a completely integrable particle lattice is implemented first each particle in the particle method travels along a characteristic curve the resulting system of nonlinear ordinary differential equations can have solutions that blow up in finite time we isolate the conditions for global existence and prove l1 norm convergence of the method in the limit of zero spatial step size and infinite particles the numerical results show that this method captures the essence of the solution without using an overly large number of particles a fast summation algorithm is introduced to evaluate the integrals of the particle method so that the computational cost is reduced from o n2 to o n where n is the number of particles the method possesses some analogies with point vortex methods for 2d euler equations in particular near singular solutions exist and singularities are prevented from occurring in finite time by mechanisms akin to those in the evolution of vortex patches the second method is based on integro differential formulations of the equation two different algorithms are proposed based on different ways of extracting the time derivative of the dependent variable by an appropriately defined inverse operator the integro differential formulations reduce the order of spatial derivatives thereby relaxing the stability constraint and allowing large time steps in an explicit numerical scheme in addition to the cauchy problem on the infinite line we include results on the study of the nonlinear equation posed in the quarter space time plane we discuss the minimum number of boundary conditions required for solution uniqueness and illustrate this with numerical examples

symmetric operations in algebraic cobordism abstract in this article we describe certain new cohomological operations in algebraic cobordisms these operations give the natural obstructions for the cobordism class to be represented by the embedding also they permit to work with algebraic cobordisms and chow groups in a more subtle way than the landweber novikov operations related to 2 torsion effects we describe applications to the computation of the algebraic cobordisms of a pfister quadrics and to the problem of rationality of cycles

velocity vorticity helicity formulation and a solver for the navier stokes equations for the three dimensional incompressible navier stokes equations we present a formulation featuring velocity vorticity and helical density as independent variables we find the helical density can be observed as a lagrange multiplier corresponding to the divergence free constraint on the vorticity variable similar to the pressure in the case of the incompressibility condition for velocity as one possible practical application of this new formulation we consider a time splitting numerical scheme based on an alternating procedure between vorticity helical density and velocity bernoulli pressure systems of equations results of numerical experiments include a comparison with some well known schemes based on pressure velocity formulation and illustrate the competitiveness on the new scheme as well as the soundness of the new formulation

involutions of varieties and rost s degree formula to an algebraic variety equipped with an involution we associate a cycle class in the modulo two chow group of its fixed locus this association is functorial with respect to proper morphisms having a degree and preserving the involutions specialising to the exchange involution of the square of a complete variety we obtain rost s degree formula in arbitrary characteristic this formula was proved by rost and merkurjev in characteristic not two

incompressible viscous flows in borderline besov spaces we establish two new estimates for a transport diffusion equation as an application we treat the problem of global persistence of the besov regularity b p 1 frac 2 p 1 with p in 2 infty for the two dimensional navier stokes equations with uniform bounds on the viscosity we provide also an inviscid global result

the distribution of all french communes a composite parametric approach abstract the distribution of the size of all french settlements communes from 1962 to 2012 is examined by means of a three parameter composite lognormal pareto distribution this model is based on a lognormal density up to an unknown threshold value and a pareto density thereafter recent findings have shown that the untruncated settlement size data is in excellent agreement with the lognormal distribution in the lower and central parts of the empirical distribution but it follows a power law in the upper tail for that reason this probabilistic family that nests both models seems appropriate to describe urban agglomeration in france the outcomes of this paper reveal that for the early periods 1962 1975 the upper quartile of the commune size data adheres closely to a power law distribution whereas for later periods 2006 2012 most of the city size dynamics is explained by a lognormal model

local regularity for the modified sqg patch equation we study the patch dynamics on the whole plane and on the half plane for a family of active scalars called modified surface quasi geostrophic sqg equations these involve a parameter that appears in the power of the kernel in their biot savart laws and describes the degree of regularity of the equation the values 0 and correspond to the two dimensional euler and sqg equations respectively we establish here local in time regularity for these models for all 0 on the whole plane and for all small 0 on the half plane we use the latter result in 16 where we show existence of regular initial data on the half plane that lead to a finite time singularity 2016 wiley periodicals inc

axisymmetric solutions to the 3d euler equations abstract here we consider the 3 d incompressible euler equations with axisymmetric velocity without swirl first we will show that if u 0 c s l 2 s 1 and 0 x c x 1 2 x 2 2 then there exists a unique u c 0 c s that solves the equation this conclusion improves on the related results given by majda a l bertozzi a majda vorticity and incompressible flow in cambridge texts in applied mathematics vol 27 2002 a majda vorticity and the mathematical theory of incompressible fluid flow comm pure appl math 39 1986 s187 s220 and by raymond x saint raymond remarks on axisymmetric solutions of the incompressible euler system comm partial differential equations 19 1994 321 334 on the other hand if u 0 l 2 0 l and 0 x 1 2 x 2 2 l then there exists a unique quasilipschitzian solution u c 0 c 1 this improves on the corresponding results due to raymond x saint raymond remarks on axisymmetric solutions of the incompressible euler system comm partial differential equations 19 1994 321 334 and to chae and kim d chae n kim axisymmetric weak solutions of the 3 d euler equations for incompressible fluid flows nonlinear anal 29 12 1997 1393 1404

an eulerian lagrangian approach for incompressible fluids local theory we study a formulation of the incompressible euler equations in terms of the inverse lagrangian map in this formulation the equations become a first order advective nonlinear system of partial differential equations

on the use of the pareto arctan distribution for describing city size in australia and new zealand abstract the circular inverse of the tangent function is used to simply derive a generalization of the pareto distribution the pareto arctan pat distribution this model includes as limiting cases pareto and zipf distributions this new probabilistic family is used to describe city size data for australia and new zealand urban agglomerations of these two countries presents similar features a few large metropolitan areas with a steadily increasing population in the last years and many small cities the pat distribution improves the performance of other traditionally used models in urban agglomeration economics such as the classical pareto lognormal and the recently proposed pareto positive stable

on the blow up of solutions of the 3 d euler equations in a bounded domain it is shown that if 0 hat t is the maximal interval of existence of a smooth solutionu of the incompressible euler equations in a bounded simply connected domain subseteq r3 then int 0 hat t left omega cdot t right l infty omega dt infty where u is the vorticity crucial to this result is a special estimate proven in of the maximum velocity gradient in terms of the maximum vorticity and a logarithmic term involving a higher norm of the vorticity

the asymptotic behavior of the smooth solutions of the euler equations in r2 abstract in this paper we consider the smooth solutions u x t with suitable decay at infinity of the euler equations on r 2 0 we assume that the initial vorticity 0 u 0 has a compact support which leads by marchioro 1994 1996 to a consequence that the support of the vorticity t for any finite time is a bounded set then we will show that the decay growth rate in l p r 2 spaces for 2 p for 1 p 2 of the smooth solutions of the euler equations on r 2 cannot be faster than t 1 3 2 3 p as t we will see that the l p norm of the solution u x t is equivalent to the l q norm for any 1 q p

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 the euler equations in the critical triebel lizorkin spaces in this paper we study the initial value problem of the incompressible euler equations in n for initial data belonging to the critical triebel lizorkin spaces i e v0 fn 11 q q 1 we prove the blow up criterion of solutions in fn 11 q for n 2 3 for n 2 in particular we prove global well posedness of the euler equations in f31 q q 1 for the proof of these results we establish a sharp moser type inequality as well as a commutator type estimate in these spaces the key methods are the littlewood paley decomposition and the paradifferential calculus by j m bony

application of space time ce se method to shallow water magnetohydrodynamic equations in this article we apply a space time conservation element and solution element ce se method for the approximate solution of shallow water magnetohydrodynamic smhd equations in one and two space dimensions these equations model the dynamics of nearly incompressible conducting fluids for which the evolution is nearly two dimensional with magnetic equilibrium in the third direction in this article we are using a variant of the ce se method developed by zhang et al a space time conservation element and solution element method for solving the two dimensional unsteady euler equations using quadrilateral and hexahedral meshes j comput phys 175 2002 168 199 this method uses structured and unstructured quadrilateral and hexahedral meshes in two and three space dimensions respectively in this method a single conservation element at each grid point is employed for solving conservation laws no matter in one two and three space dimensions the present scheme use the conservation element to calculate flow variables only while the gradients of flow variables are calculated by central differencing reconstruction procedure we give both one and two dimensional test computations a qualitative comparison reveals an excellent agreement with previous published results of wave propagation method and evolution galerkin schemes the one and two dimensional computations reported in this paper demonstrate the remarkable versatility of the present ce se scheme

remarks on a nonlocal transport abstract we consider a one dimensional nonlocal transport equation and its natural multi dimensional analogues by using a new pointwise inequality for the hilbert transform we give a short proof of a nonlinear inequality first proved by cordoba cordoba and fontelos in 2005 we also prove several new weighted inequalities for the hilbert transform and various nonlinear versions some of these results generalize to a related family of nonlocal models

serfati solutions to the 2d euler equations on exterior domains abstract we prove existence and uniqueness of a weak solution to the incompressible 2d euler equations in the exterior of a bounded smooth obstacle when the initial data is a bounded divergence free velocity field having bounded scalar curl this work completes and extends the ideas outlined by p serfati for the same problem in the whole plane case with non decaying vorticity the biot savart integral does not converge and thus velocity cannot be reconstructed from vorticity in a straightforward way the key to circumventing this difficulty is the use of the serfati identity which is based on the biot savart integral but holds in more general settings

an initial boundary value problem for one dimensional shallow water magnetohydrodynamics in the solar tachocline abstract in this article we investigate the shallow water magnetohydrodynamic equations in space dimension one with dirichlet boundary conditions only for the velocity this model has been proposed to study the phenomena in the solar tachocline in this article the local well posedness in time of the model is proven by constructing the approximate solutions and showing the strong convergence of the approximate solutions

city size distributions for india and china this paper studies the size distributions of urban agglomerations for india and china we have estimated the scaling exponent for zipf s law with the indian census data for the years of 1981 2001 and the chinese census data for 1990 and 2000 along with the biased linear fit estimate the maximum likelihood estimate for the pareto and tsallis q exponential distribution has been computed for india the scaling exponent is in the range of 1 88 2 06 and for china it is in the interval 1 82 2 29 the goodness of fit tests of the estimated distributions are performed using the kolmogorov smirnov statistic

vortices and invariant surfaces generated by symmetries for the 3d navier stokes equations abstract we show that certain infinitesimal operators of the lie point symmetries of the incompressible 3d navier stokes equations give rise to vortex solutions with different characteristics this approach allows an algebraic classification of vortices and throws light on the alignment mechanism between the vorticity and the vortex stretching vector s where s is the strain matrix the symmetry algebra associated with the navier stokes equations turns out to be infinite dimensional new vortical structures generalizing in some cases well known configurations such as for example the burgers and lundgren solutions are obtained and discussed in relation to the value of the dynamic angle arctan s s a systematic treatment of the boundary conditions invariant under the symmetry group of the equations under study is also performed and the corresponding invariant surfaces are recognized

existence for the α patch model and the qg sharp front in sobolev spaces abstract we consider a family of contour dynamics equations depending on a parameter with 0 1 the vortex patch problem of the 2 d euler equation is obtained taking 0 and the case 1 corresponds to a sharp front of the qg equation we prove local in time existence for the family of equations in sobolev spaces

remarques sur l instabilite du probleme des poches de tourbillon resume nous etudions le probleme des poches de tourbillon pour des contours initiaux poches du cercle unite une approximation quadratique de l equation d evolution se revele instable en temps fini la norme lipschitz du contour devenant arbitrairement grande

on the regularity of the axisymmetric solutions of the navier stokes equations we obtain improved regularity criteria for the axisymmetric weak solutions of the three dimensional navier stokes equations with nonzero swirl in particular we prove that the integrability of single component of vorticity or velocity fields in terms of norms with zero scaling dimension give sufficient conditions for the regularity of weak solutions to obtain these criteria we derive new a priori estimates for the axisymmetric smooth solutions of the navier stokes equations

the pareto positive stable distribution a new descriptive model for city size data abstract the pareto positive stable pps distribution is introduced as a new model for describing city size data in a country the pps distribution provides a flexible model for fitting the entire range of a set of city size data where zero and unimodality are possible and the classical pareto and zipf distributions are included as a particular case expressions for the shape moments and other descriptive probabilistic measures are given estimation methods are discussed and a simple graphical method for studying the adequacy of the data to model is given finally we consider city size data for spain for several different years the new distribution is compared with three classical models pareto lognormal and tsallis distributions in all the data sets considered the pps distribution outperforms the fits of these three previous distributions

l p theory of the navier stokes flow in the exterior of a moving or rotating obstacle in this paper we describe two recent approaches for the lp theory of the navier stokes flow in the exterior of a moving or rotating obstacle

spectral dynamics of the velocity gradient field in restricted flows we study the velocity gradients of the fundamental eulerian equation t u u u f which shows up in different contexts dictated by the different modeling of f s to this end we utilize a basic description for the spectral dynamics of u expressed in terms of the possibly complex eigenvalues u which are governed by the ricatti like equation t u 2 l fr we focus our investigation on four prototype models associated with different forcing f ranging from simple linear damping and viscous dusty medium models to the main thrust of the paper the restricted models of euler navier stokes equations and euler poisson equations in particular we address the question of the time regularity for these models that is whether they admit a finite time breakdown a global smooth solution or an intermediate scenario of critical threshold phenomena where global regularity depends on initial configurations using the spectral dynamics as our essential tool in these investigations we obtain a simple form of a critical threshold for the linear damping model and we identify the 2d vanishing viscosity limit for the viscous irrotational dusty medium model moreover for the n dimensional restricted euler equations we obtain n 2 1 global invariants interesting for their own sake which enable us to precisely characterize the local topology at breakdown time extending previous studies in the n 3 dimensional case finally as a fourth model we introduce the n dimensional restricted euler poisson rep system identifying a set of n 2 global invariants which in turn yield i sufficient conditions for finite time breakdown and ii characterization of a large class of 2 dimensional

contour dynamics for the euler equations in two dimensions we present a contour dynamics algorithm for the euler equations of fluid dynamics in two dimensions this is applied to regions of piecewise constant vorticity withinfinite area vortexregions favrs essentially this reduces the dimensionality by one and we are treating the interaction of closed polygonal contours whose nodes are advected by the total fluid motion computed self consistently a leapfrog centered scheme is used for temporal advancement computer simulation results are given for two and four like signed interacting favrs in all cases wavelike surface deformations are observed if the distance between favrs is comparable to their extent diameter these surface deformations are large they play an essential role in the observed coalescence of favrs

well posedness theory for aggregation sheets in this paper we consider distribution solutions to the aggregation equation rho t mathrm div rho mathbf u 0 mathbf u nabla v rho in mathbb r d where the density concentrates on a co dimension one manifold we show that an evolution equation for the manifold itself completely determines the dynamics of such solutions we refer to such solutions aggregation sheets when the equation for the sheet is linearly well posed we show that the fully non linear evolution is also well posed locally in time for the class of bi lipschitz surfaces moreover we show that if the initial sheet is c1 then the solution itself remains c1 as long as it remains lipschitz lastly we provide conditions on the kernel g s frac mathrm d v mathrm d s that guarantee the solution remains a bi lipschitz surface globally in time and construct explicit solutions that either collapse or blow up in finite time when these conditions fail

mathematical analysis of a structure preserving approximation of the bidimensional vorticity equation we show the consistency and the convergence of a spectral approximation of the bidimensional vorticity equation proposed by v zeitlin in 13 and studied numerically by i szunyogh b kadar and d devenyi in 12 whose main feature is that it preserves the hamiltonian structure of the vorticity equation on demontre la consistance et la convergence d une approximation spectrale de l equation du tourbillon bidimensionnelle periodique proposee par v zeitlin dans 13 et etudiee numeriquement par i szunyogh b kadar et d devenyi dans 12 sa caracteristique principale est de preserver la structure hamiltonienne de l equation du tourbillon

on the dual systems to euler and navier stokes equations in r3 in this paper we consider two model systems of equations obtained by taking the hodge duals of the euler and the navier stokes systems respectively the vorticity formulation of the new system becomes a nonlinear non local scalar equation having similarities to the quasi geostrophic equations we prove the local existence of classical solutions and derive the beale kato majda type of blow up criterion for the dual euler system moreover we actually show that the finite time blow up occurs for generic non trivial initial data for this system for the dual navier stokes equations we construct an explicitly global in time smooth solution for general initial data using the hopf cole transform in the appendix we formulate the euler system and the dual euler system in terms of the clifford algebra

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

blow up of solutions for a 1d transport equation with nonlocal velocity and supercritical dissipation we study a 1d transport equation with nonlocal velocity and supercritical dissipation we show that for a certain class of initial data the solution blows up in finite time

expanding large global solutions of the equations of compressible fluid mechanics without any symmetry assumptions on the initial data we construct global in time unique solutions to the vacuum free boundary three dimensional isentropic compressible euler equations when the adiabatic exponent gamma lies in the interval 1 frac 5 3 our initial data lie sufficiently close to the expanding compactly supported affine motions recently constructed by sideris and they satisfy the physical vacuum boundary condition

nonexistence of self similar singularities for the 3d incompressible euler equations we prove that there exists no self similar finite time blowing up solution to the 3d incompressible euler equations if the vorticity decays sufficiently fast near infinity in mathbb r 3 by a similar method we also show nonexistence of self similar blowing up solutions to the divergence free transport equation in mathbb r n this result has direct applications to the density dependent euler equations the boussinesq system and the quasi geostrophic equations for which we also show nonexistence of self similar blowing up solutions

global regularity for vortex patches we present a proof of chemin s 4 result which states that the boundary of a vortex patch remains smooth for all time if it is initially smooth

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

integrality of the chern character in small codimension abstract we prove an integrality property of the chern character with values in chow groups as a consequence we obtain a construction of the p 1 first homological steenrod operations on chow groups modulo p and p primary torsion over an arbitrary field we provide applications to the study of correspondences between algebraic varieties

the beltrami spectrum for incompressible fluid flows recently v yakhot s orszag and their co workers have suggested that turbulent flows in various regions of space organize into a coherent hierarchy of weakly interacting superimposed approximate beltrami flows a mathematical framework is developed here to study organized beltrami hierarchies in a systematic fashion this framework is applied to several important classes of examples with universal beltrami hierarchies an analysis of the persistence of such beltrami hierarchies is also presented for general solutions of the navier stokes equations

on the null asymptotic stabilization of the two dimensional incompressible euler equations in a simply connected domain we study the asymptotic stabilization of the origin for the two dimensional 2 d euler equation of incompressible inviscid fluid in a bounded domain we assume that the controls act on an arbitrarily small nonempty open subset of the boundary we prove the null global asymptotic stabilizability by means of explicit feedback laws if the domain is connected and simply connected

multiply broken power law densities as survival functions an alternative to pareto and lognormal fits abstract survival functions defined by multiply broken power law densities are introduced to model firm size data sets of four historical censuses reported in montebruno et al physica a 523 2019 858 the survival functions complementary cumulative distributions are obtained by least squares regression the probability distributions are inferred from the analytic survival functions the mean firm growth the growth volatility and entropy evolution are calculated over a 30 year period covered by the censuses a representation of the survival functions as single power law density with varying exponent depending on firm size is derived index functions i e rescaled logarithmic derivatives of survival functions are used to demonstrate that neither pareto power laws nor lognormal densities can accurately reproduce the tails of the firm size distributions like the survival functions the empirical rank size relations of the four censuses also admit representation by broken power law densities a generating mechanism for empirically obtained broken power law densities based on the fokker planck equation is proposed as well

travelling wave like solutions of the navier stokes and the related equations abstract we present a new family of travelling wave like solutions to the navier stokes equations of incompressible fluid flows and other regularized equations of the euler equations obtain their trend to the solutions of the euler equations as the viscosity tends to zero and estimate the rate of convergence we also find a singularizing effect of the viscosity term in the navier stokes equations i e we have a local moving blow up of unbounded solutions with the blow up s speed depending on viscosity we demonstrate that if the initial function is the beltrami flow then the solution of the navier stokes equations conserves the beltrami flow property for all time

pareto tails and lognormal body of us cities size distribution abstract we consider a distribution which consists of lower tail pareto lognormal body and upper tail pareto to estimate the size distribution of all us cities this distribution fits the data more accurately than a distribution that comprises of only lognormal and the upper tail pareto

the size distributions of all indian cities abstract we apply five distributions lognormal double pareto lognormal lognormal upper tail pareto pareto tails lognormal and pareto tails lognormal with differentiability restrictions to estimate the size distribution of all indian cities since india contains numerous small cities it is important to explicitly model the lower tail behavior for studying the distribution of all indian cities our results rigorously confirm using both graphical and formal statistical tests that among these five distributions pareto tails lognormal is a better suited parametrization of the indian city size data verifying that the indian city size distribution exhibits a strong reverse pareto in the lower tail lognormal in the mid range body and pareto in the upper tail

isotropy of quadratic forms over the function field of a quadric in characteristic 2 abstract we extend to characteristic 2 a theorem by the first author which states that if and are anisotropic quadratic forms over a field f such that dim 2 n dim for some nonnegative integer n then stays anisotropic over the function field f of the case of singular forms is systematically included we give applications to the characterization of quadratic forms with maximal splitting we also prove a characteristic 2 version of a theorem by izhboldin on the isotropy of over f in the case dim 2 n 1 dim

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

size distribution of cities a kinetic explanation abstract we present a kinetic approach to the formation of urban agglomerations which is based on simple rules of immigration and emigration in most cases the boltzmann type kinetic description allows to obtain within an asymptotic procedure a fokker planck equation with variable coefficients of diffusion and drift which describes the evolution in time of some probability density of the city size it is shown that in dependence of the microscopic rules of migration the equilibrium density can follow both a power law for large values of the size variable which contains as particular case a zipf s law behavior and a lognormal law for middle and low values of the size variable in particular connections between the value of pareto index of the power law at equilibrium and the disposal of the population to emigration are outlined the theoretical findings are tested with recent data of the populations of italy and switzerland

global existence and uniqueness theorem to 2 d incompressible non resistive mhd system with non equilibrium background magnetic field abstract in this article we consider the global existence and uniqueness of the solution to the 2d incompressible non resistive mhd system with non equilibrium background magnetic field our result implies that a strong enough non equilibrium background magnetic field will guarantee the stability of the nonlinear mhd system beside the classical energy method the interpolation inequalities and the algebraic structure of the equations coming from the incompressibility of the fluid are crucial in our arguments

eigenvalues associated with the vortex patch in 2 d euler equations we consider the dirichlet eigenvalue problem associated with the vortex patch in 2 d euler equations we show that the eigenvalues grow at most doubly exponentially in time as an application we derive bounds on the growth of some geometric quantities like the diameter and the inscription radius of the patch we also discuss the growth of the perimeter of the patch in particular we give a double exponential bound of the growth of certain portion of the boundary of the patch

viscosity effect on the degenerate lake equations abstract this paper concerns the effect of viscosity on the degenerate lake equations anelastic limit when the bottom topography vanishes on the shore we establish the existence and uniqueness of a global weak solutions for various choices of viscosity term in weighted sobolev spaces where the weight is assumed to be a power type weight our solution is constructed by adapting carefully the known works on the incompressible navier stokes equations in unweighted context finally we study in one case the vanishing viscosity limit when the solution of the inviscid lake equations is regular enough generalizing the results by martino et al 2001 to the degenerate case

slow convergence to vortex patches in quasigeostrophic balance taking advantage of dispersive effects which supposes a flow extended in the whole space a sluggish convergence of the solutions of the boussinesq equations to a solution of the quasigeostrophic system when the rossby number tends to zero can be proved under weaker assumptions on the initial data than usual in particular no assumption of well preparedness is needed two examples are given both involving fields with striated potential vorticity a result of convergence to a vortex patch is deduced from one of them

concentrations in the one dimensional vlasov poisson equations ii screening and the necessity for measure valued solutions in the two component case abstract weak and measure valued solutions for the two component vlasov poisson equations in a single space dimension are proposed and studied here as a simpler analogue problem for the limiting behavior of approximations for the two dimensional euler equations with general vorticity of two signs from numerical experiments and mathematical theory it is known that much more complex behavior can occur in limiting processes for vortex sheets with general vorticity of two signs as compared with non negative vorticity here such behavior is confirmed rigorously for the simpler analogue problem through explicit examples involving singular charge concentration for the two component vlasov poisson equations the concepts of measure valued and weak solution are introduced explicit examples with charge concentration establish that the limit of weak solutions in a dynamic process is necessarily a measure valued solution in some cases rather than the anticipated weak solution i e no concentration cancellation occurs the limiting behavior of computational regularizations involving high resolution particle methods is presented here both for the instances with measure valued solutions and also for new examples with non unique weak solutions the authors demonstrate that different computational regularizations can exhibit completely different limiting behavior in situations with measure valued and or non unique weak solutions

on strong convergence to 3 d axisymmetric vortex sheets abstract we consider the 3 d axisymmetric incompressible euler equations without swirls with vortex sheets initial data it is proved that the approximate solutions generated by smoothing the initial data converge strongly in l 2 0 t l loc 2 r 3 provided that they have strong convergence in the region away from the symmetry axis this implies that if there would appear singularity or energy lost in the process of limit for the approximate solutions it then must happen in the region away from the symmetry axis there is no restriction on the signs of initial vorticity here in order to exclude the possible concentrations on the symmetry axis we use the special structure of the equations for axisymmetric flows and careful choice of test functions

local well posedness and regularity criterion for the density dependent incompressible euler equations in this paper local well posedness for the density dependent incompressible euler equations is established in besov spaces we also obtain a blow up criterion for the corresponding solution

analysis of the semi geostrophic shallow water equations abstract the semi geostrophic shallow water equations are an accurate approximation to the rotating shallow water equations on scales larger than the rossby radius of deformation the global existence of weak solutions of the lagrangian form of the equations is known however uniqueness is not known except for short times for smooth initial data similarly a rigorous proof of existence of solutions has not been achieved for variable rotation rate which is important for atmospheric applications the obstacle is the lack of regularity of solutions to the monge ampere equation which underlies the solution procedure in this note it is shown formally that the modified form of the monge ampere equation that appears in the solution procedure for the semi geostrophic shallow water equations is better controlled on large scales than that of the standard equation that is used in the incompressible case this suggests that more progress should be possible with the rigorous theory and in particular that it should be possible to prove uniqueness