strong convergence to global solutions for multidimensional flows of compressible viscous fluids with polytropic equations of state and discontinuous initial data we extend to general polytropic pressures p k 1 the existence theory of 8 for isothermal 1 flows of navier stokes fluids in two and three space dimensions with fairly general initial data specifically we require that the initial density be close to a constant in l2 and l and that the initial velocity be small in l2 and bounded in l2n in two dimensions the l2 norms must be weighted slightly solutions are obtained as limits of approximate solutions corresponding to mollified initial data the key point is that the approximate densities are shown to converge strongly so that nonlinear pressures can be accommodated even in the absence of any uniform regularity information for the approximate densities

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

adaptive methods for the computation of pagerank we observe that the convergence patterns of pages in the pagerank algorithm have a nonuniform distribution specifically many pages converge to their true pagerank quickly while relatively few pages take a much longer time to converge furthermore we observe that these slow converging pages are generally those pages with high pagerank we use this observation to devise a simple algorithm to speed up the computation of pagerank in which the pagerank of pages that have converged are not recomputed at each iteration after convergence this algorithm which we call adaptive pagerank speeds up the computation of pagerank by nearly 30

on navier stokes korteweg and euler korteweg systems application to quantum fluids models in this paper the main objective is to generalize to the navier stokes korteweg with density dependent viscosities satisfying the bd relation and euler korteweg systems a recent relative entropy proposed by d bresch p noble and j p vila 2016 introduced for the compressible navier stokes equations with a linear density dependent shear viscosity and a zero bulk viscosity as a concrete application this helps to justify mathematically the convergence between global weak solutions of the quantum navier stokes system recently obtained by i lacroix violet and a vasseur 2016 and dissipative solutions of the quantum euler system when the viscosity coefficient tends to zero this selects a dissipative solution as the limit of a viscous system our results are based on the fact that euler korteweg systems and corresponding navier stokes korteweg systems can be reformu lated through an augmented system as the compressible navier stokes system with density dependent viscosities satisfying the bd algebraic relation this was also observed recently by d bresch f couderc p noble and j p vila 2016 for the euler korteweg system as a by product of our analysis we show that this augmented formulation helps to define relative entropy estimates for the euler korteweg systems in a simplest way compared to recent works see d donatelli e feireisl p marcati 2015 and j giesselmann c lattanzio a e tzavaras 2017 and with less hypothesis required on the capillary coefficient no concavity assumption needed in our result

simulation of a cellular automat with an oriented bootstrap rule abstract the evolution of a cellular automat with a culling or bootstrap condition on the minimum number of neighbours for survival has been simulated the introduction of an orientational constraint converts the lattice from isotropic to semidirected simulations of systems of up to 230 million sites are found to be compatible with a threshold value of p 0 966 0 01

on global existence of weak solutions to a viscous capillary model of plasma abstract in this paper we study a viscous capillary model of plasma as a so called navier stokes poisson korteweg model our purpose is to prove the existence of global weak solutions for large data in a two dimensional torus we utilize the effective velocity and some interesting identities to remove the restrictions on the coefficients it is worth pointing out that we prove the critical case that the value of viscosity coefficient is equivalent to the capillary coefficient in this case the b d entropy and method used in antonelli and spirito 2017 cannot be applied directly moreover there is no friction term and cold pressure term in the model in some senses we improve the previous results

discontinuous solutions of the navier stokes equations for multidimensional flows of heat conducting fluids we prove the global existence of weak solutions of the navier stokes equations for compressible heat conducting fluids in two and three space dimensions when the initial density is close to a constant in l 2 l the initial temperature is close to a constant in l 2 and the initial velocity is small in h s l 4 where s 0 when n 2 and when n 3 the l p norms must be weighted slightly when n 2 in particular the initial data may be discontinuous across a hypersurface of n a great deal of qualitative information about the solution is obtained for example we show that the velocity vorticity and temperature are relatively smooth in positive time as is the effective viscous flux f which is the divergence of the velocity minus a certain multiple of the pressure we find that f plays a central role in the entire analysis particularly in closing the required energy estimates and in understanding rates of regularization near the initial layer moreover f is precisely the quantity through which the hyperbolicity of the corresponding equations for inviscid fluids shows itself an effect which is crucial for obtaining time independent pointwise bounds for the density

existence of global weak solutions for a 2d viscous shallow water equations and convergence to the quasi geostrophic model we consider a two dimensional viscous shallow water model with friction term existence of global weak solutions is obtained and convergence to the strong solution of the viscous quasi geostrophic equation with free surface term is proven in the well prepared case the ill prepared data case is also discussed

global existence of weak solutions to 3d compressible primitive equations with degenerate viscosity in this paper we investigate the compressible primitive equations cpes with density dependent viscosity for large initial data the cpe model can be derived from the 3d compressible and anisotropic navier stokes equations by hydrostatic approximation motivated by the work of vasseur and yu siam j math anal 48 1489 1511 2016 invent math 206 935 974 2016 in which the global existence of weak solutions to the compressible navier stokes equations with degenerate viscosity was obtained we construct approximate solutions and prove the global existence of weak solutions to the cpe in this paper in our proof we first present the vertical velocity as a function of density and horizontal velocity which plays a role in using the faedo galerkin method to obtain the global existence of the approximate solutions then we obtain the key estimates of lower bound of the density the bresch desjardins entropy on the approximate solutions finally we apply compactness arguments to obtain global existence of weak solutions by vanishing the parameters in our approximate system step by step

comparison of bootstrap percolation models new simulations involving up to nearly 109 sites for square lattice bootstrap percolation give a percolation threshold compatible with unity in this model sites in a square lattice are initially occupied randomly and then are removed if they have less than three occupied neighbours the results are compared with those of similar models such as that studied by frobose 1989 and the earlier bootstrap and diffusion percolation models in two three and four dimensions of adler and aharony 1986 8 in the latter models the effective threshold approaches its asymptotic limit one or zero as lambda ln l d 1 for samples of size ld in agreement with theoretical predictions the coefficient lambda decreases monotonically with the parameters which facilitate the removal addition of sites

rational extrapolation for the pagerank vector an important problem in web search is to determine the importance of each page from the mathematical point of view this problem consists in finding the nonnegative left eigenvector of a matrix corresponding to its dominant eigenvalue 1 since this matrix is neither stochastic nor irreducible the power method has convergence problems so the matrix is replaced by a convex combination depending on a parameter c with a rank one matrix its left principal eigenvector now depends on c and it is the pagerank vector we are looking for however when c is close to 1 the problem is ill conditioned and the power method converges slowly so the idea developed in this paper consists in computing the pagerank vector for several values of c and then to extrapolate them by a conveniently chosen rational function at a point near 1 the choice of this extrapolating function is based on the mathematical expression of the pagerank vector as a function of c numerical experiments end the paper

k core percolation in four dimensions the k core percolation on the bethe lattice has been proposed as a simple model of the jamming transition because of its hybrid first order second order nature we investigate numerically k core percolation on the four dimensional regular lattice for k 4 the presence of a discontinuous transition is clearly established but its nature is strictly first order in particular the k core density displays no singular behavior before the jump and its correlation length remains finite for k 3 the transition is continuous

metastability effects in bootstrap percolation bootstrap percolation models or equivalently certain types of cellular automata exhibit interesting finite volume effects these are studied at a rigorous level the authors find that for an initial configuration obtained by placing particles independently with probability p or 2 the density of the bootstrapped final configurations in the sequence of cubes l 2 l 2 d typically undergoes an abrupt transition as l is increased from being close to 0 to the value 1 with l fixed at a large value the mean final density as a function of p changes from 0 to 1 around a value which varies only slowly with l the pertinent parameter being lambda p1 d 1 ln l the driving mechanism is the capture of a critical droplet this behaviour is analogous to the decay of a metastable state near a first order phase transition for which the analysis offers some suggestive ideas

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

critical points of two dimensional bootstrap percolation like cellular automata cellular automata in two dimensions that generalize the bootstrap percolation dynamics are considered focusing on the thresholdp c of the initial density for convergence to total occupancy to occur these models are classified according top c being 0 1 or strictly between these extreme values explicit upper and lower bounds are provided in the third case

first passage times for threshold growth dynamics on bf z sp 2 in the threshold growth model on an integer lattice the occupied set grows according to a simple local rule a site becomes occupied iff it sees at least a threshold number of already occupied sites in its prescribed neighborhood in this paper we analyze the behavior of two dimensional threshold growth dynamics started from a sparse bernoulli density of occupied sites we explain how nucleation of rare centers invariant shapes and interaction between growing droplets influence the first passage time in the supercritical case we also briefly address scaling laws for the critical case

irreversible conversion of graphs given a graph g a function f v g z and an initial 0 1 vertex labelling c 1 v g 0 1 we study an iterative 0 1 vertex labelling process on g where in each round every vertex v never changes its label from 1 to 0 and changes its label from 0 to 1 if at least f v neighbours have label 1 such processes model opinion disease spreading or fault propagation and have been studied under names such as irreversible threshold majority processes in a large variety of contexts our contributions concern computational aspects related to the minimum cardinality irr f g of sets of vertices with initial label 1 such that during the process on g all vertices eventually change their label to 1 such sets are known as irreversible conversion sets dynamic irreversible monopolies or catastrophic fault patterns answering a question posed by dreyer and roberts p a dreyer jr f s roberts irreversible k threshold processes graph theoretical threshold models of the spread of disease and of opinion discrete appl math 157 2009 1615 1627 we prove a hardness result for irr f g where f v 2 for every email protected v g furthermore we describe a general reduction principle for irr f g which leads to efficient algorithms for graphs with simply structured blocks such as trees and chordal graphs

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

a sharp threshold for a modified bootstrap percolation with recovery bootstrap percolation is a type of cellular automaton on graphs introduced as a simple model of the dynamics of ferromagnetism vertices in a graph can be in one of two states healthy or infected and from an initial configuration of states healthy vertices become infected by local rules while the usual bootstrap processes are monotone in the sets of infected vertices in this paper a modification is examined in which infected vertices can return to a healthy state vertices are initially infected independently at random and the central question is whether all vertices eventually become infected the model examined here is such a process on a square grid for which healthy vertices with at least two infected neighbours become infected and infected vertices with no infected neighbours become healthy sharp thresholds are given for the critical probability of initial infections for all vertices eventually to become infected

existence of global weak solutions for 3d degenerate compressible navier stokes equations in this paper we prove the existence of global weak solutions for 3d compressible navier stokes equations with degenerate viscosity the method is based on the bresch and desjardins commun math phys 238 211 223 2003 entropy conservation the main contribution of this paper is to derive the mellet and vasseur commun partial differ equ 32 431 452 2007 type inequality for weak solutions even if it is not verified by the first level of approximation this provides existence of global solutions in time for the compressible barotropic navier stokes equations the result holds for any gamma 1 in two dimensional space and for 1 gamma 3 in three dimensional space in both case with large initial data possibly vanishing on the vacuum this solves an open problem proposed by lions mathematical topics in fluid mechanics vol 2 compressible models 1998

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

the stokes operator in weighted l q spaces ii weighted resolvent estimates and maximal l p regularity in this paper we establish a general weighted l q theory of the stokes operator mathcal a q omega in the whole space the half space and a bounded domain for general muckenhoupt weights omega in a q we show weighted l q estimates for the stokes resolvent system in bounded domains for general muckenhoupt weights these weighted resolvent estimates imply not only that the stokes operator mathcal a q omega generates a bounded analytic semigroup but even yield the maximal l p regularity of mathcal a q omega in the respective weighted l q spaces for arbitrary muckenhoupt weights omega in a q this conclusion is archived by combining a recent characterisation of maximal l p regularity by mathcal r bounded families due to weis operator valued fourier multiplier theorems and maximal l p regularity preprint 1999 with the fact that for l q spaces mathcal r boundedness is implied by weighted estimates

bootstrap percolation in a polluted environment let a low densityp of sites on the lattice z2 be occupied remove a proportionq of them and call the remaining sites empty then update this configuration in discrete time by iteration of the following synchronous rule an empty site becomes occupied by contact with at least two occupied nearest neighbors while occupied and removed sites nerver change their states ifq p2 is large most sites remain unoccupied forever while ifq p2 is small this dynamics eventually makes most sites occupied this demonstrates how sensitive the usual bootstrap percolation rule theq 0 case is to the pollution of space

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

sharp thresholds in bootstrap percolation in the standard bootstrap percolation on the d dimensional grid gnd in the initial position each of the nd sites is occupied with probability p and empty with probability 1 p independently of the state of every other site once a site is occupied it remains occupied for ever while an empty site becomes occupied if at least two of its neighbours are occupied if at the end of the process every site is occupied we say that the initial configuration percolates by making use of a theorem of friedgut and kalai proc amer math soc 124 1996 2993 we shall show that the threshold function of the percolation is sharp we shall prove similar results for three other models of bootstrap percolation as well

global existence of weak solutions to the compressible quantum navier stokes equations with degenerate viscosity we study the compressible quantum navier stokes qns equations with degenerate viscosity in the three dimensional periodic domains on the one hand we consider qns with additional damping terms motivated by the recent works of li and xin e print arxiv 1504 06826 and p antonelli and s spirito arch ration mech anal 225 1161 1199 2017 we construct a suitable approximate system which has smooth solutions satisfying the energy inequality and the bd entropy estimate using this system we obtain the global existence of weak solutions to the compressible qns equations with damping terms for large initial data moreover we obtain some new a priori estimates which can avoid using the assumption that the gradient of the velocity is a well defined function which was indeed used directly in the work of vasseur and yu siam j math anal 48 1489 1511 2016 invent math 206 935 974 2016 on the other hand in the absence of damping terms we also prove the global existence of weak solutions to the compressible qns equations without the lower bound assumption on the dispersive coefficient which improves the previous result of p antonelli and s spirito arch ration mech anal 225 1161 1199 2017

monte carlo methods in pagerank computation when one iteration is sufficient pagerank is one of the principle criteria according to which google ranks web pages pagerank can be interpreted as a frequency of visiting a web page by a random surfer and thus it reflects the popularity of a web page google computes the pagerank using the power iteration method which requires about one week of intensive computations in the present work we propose and analyze monte carlo type methods for the pagerank computation there are several advantages of the probabilistic monte carlo methods over the deterministic power iteration method monte carlo methods already provide good estimation of the pagerank for relatively important pages after one iteration monte carlo methods have natural parallel implementation and finally monte carlo methods allow one to perform continuous update of the pagerank as the structure of the web changes

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

well posedness in critical spaces for the system of compressible navier stokes in larger spaces this paper is dedicated to the study of viscous compressible barotropic fluids in dimension n geq2 we address the question of well posedness for it large data having critical besov regularity our result improves the analysis of r danchin in cite dw of chen et al in cite c2 and of the author in cite h1 h2 inasmuch as we may take initial density in b nn p 1 with 1 leq p infty our result relies on a new a priori estimate for the velocity where we introduce a new unknown called textit effective velocity to weaken one the coupling between the density and the velocity in particular for the first time we obtain uniqueness without imposing hypothesis on the gradient of the density

simulation of three dimensional bootstrap percolation on a simple cubic lattie which is initially occupied randomly with concentration p a site is emptied if it does not have at least four of its six neighbors occupied the threshold concentration below which all sites are emptied after sufficiently many iterations is found to vary logarithmically with l for l l l lattices and l between 32 and 704 it is extrapolated to about 0 936 for infinite lattices two other cases are roughly compatible with the theoretical prediction of unity as a threshold concentration

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

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

global existence for 1d compressible isentropic navier stokes equations with large initial data we prove the global existence of weak solutions of the cauchy problem for the navier stokes equations of compressible isentropic flow of a polytropic gas in one space dimension the initial velocity and density are assumed to be in l2 and l2 n bv respectively modulo additive constants in particular no smallness assumptions are made about the intial data in addition we prove a result concerning the asymptotic decay of discontinuities in the solution when the adiabatic constant exceeds 3 2

global solutions of the navier stokes equations for multidimensional compressible flow with discontinuous initial data abstract we prove the global existence of weak solutions of the navier stokes equations for compressible isothermal flow in two and three space dimensions when the initial density is close to a constant in l 2 and l and the initial velocity is small in l 2 and bounded in l 2 n in two dimensions the l 2 norms must be weighted slightly a great deal of qualitative information about the solution is obtained for example we show that the velocity and vorticity are relatively smooth in positive time as is the effective viscous flux f which is the divergence of the velocity minus a certain multiple of the pressure we find that f plays a crucial role in the entire analysis particularly in closing the required energy estimates understanding rates of regularization near the initial layer and most important obtaining time independent pointwise bounds for the density

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

global weak solutions to 3d compressible navier stokes poisson equations with density dependent viscosity abstract in this paper we study the global existence of weak solutions to the compressible navier stokes poisson n s p equations with density dependent viscosity and non monotone pressure in a three dimensional torus our approach is based on the faedo galerkin method and the compactness arguments motivated by vasseur yu 29 and 30 we construct the approximate solutions and the key estimates ling in the elementary energy estimates b d entropy and mellet vasseur type inequality for the weak solutions here we need the conditions that the adiabatic constant satisfies 4 3 3 for 1 or 1 3 for 1 where is a sign constant of poisson equation which determines the physical meaning of the n s p system

on perturbations of principal eigenvectors of substochastic matrices in this report we consider substochastic matrices with some zero rows and study the sensitivity of their eigenvectors to the modifications of zero entries the analysis is prompted by and directly applied to the google method of ranking the web sites which substitutes the pages without outlinks dangling nodes in the original web link matrix by non zero rows dangling vectors we present an analysis of the influence of artificial links attributed to the dangling nodes on the principal eigenvectors of the web matrix we clarify when the choice of the dangling vector does not change the original eigenvectors and give an evaluation for perturbations of the principal eigenvectors when they are subject to modification

classification of square lattice cellular automata abstract all 65 536 automata on the square lattice with nearest neighbor interactions ignoring the central spin were simulated we determined which percentage of them ended up in a fixed point with all spins parallel in other fixed points in up down oscillations with all spins parallel and in other oscillations of period two nine tenths of the rules did not fall into these classes we also checked for chaos i e how does the damage spread if one line of spins is reversed for 62 percent of the possible rules it spread over the lattice 31 percent had the damage remaining localized but nonzero and in 7 percent of all rules the damage healed out after some time these simulations were done with a speed of up to 120 updates per microsecond and cray xmp processor

proof of straley s argument for bootstrap percolation we prove thatpc 1 for bootstrap percolation with large void instabilities in particular ifm 3 on the square lattice

finite size scaling behavior of a biased majority rule cellular automaton abstract we consider the biased majority rule cellular automaton in which the sites take the majority of the states 0 or 1 of their neighbors to update their states with a bias towards 1 in case of ties we observe that in any dimension for any initial density of 1 s the probability of final complete occupancy by 1 s converges to 1 as the lattice grows in two dimensions the value p l of the initial density for which the probability of eventual complete occupancy of a lattice l l is 0 1 is shown to satisfy for fixed the bounds a log l b log l for large l where a and b are positive and finite constants

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

global stability of weak solutions for a multilayer saint venant model with interactions between the layers in this paper we investigate the existence of global weak solutions for the multilayer model introduced by audusse et al which is related to incompressible free surface flows we prove the global stability of weak solutions over the torus we observe that this model admits the so called bd entropy and a gain of integrability on the velocity in the spirit of the work of mellet and vasseur it allows us to obtain enough compactness estimates in order to show the stability of global weak solutions

pressureless euler with nonlocal interactions as a singular limit of degenerate navier stokes system abstract we show that weak solutions of degenerate navier stokes equations converge to the strong solutions of the pressureless euler system with linear drag term newtonian repulsion and quadratic confinement the proof is based on the relative entropy method using the artificial velocity formulation for the one dimensional navier stokes system

highly rotating viscous compressible fluids in presence of capillarity effects we study here a singular limit problem for a navier stokes korteweg system with coriolis force in the domain mathbb r 2 times 0 1 and for general ill prepared initial data taking the mach and the rossby numbers proportional to a small parameter varepsilon rightarrow 0 we perform the incompressible and high rotation limits simultaneously moreover we consider both the constant and vanishing capillarity regimes in this last case the limit problem is identified as a 2 d incompressible navier stokes equation in the variables orthogonal to the rotation axis if the capillarity is constant instead the limit equation slightly changes keeping however a similar structure due to the presence of an additional surface tension term in the vanishing capillarity regime various rates at which the capillarity coefficient goes to 0 are considered in general this produces an anisotropic scaling in the system the proof of the results is based on suitable applications of the rage theorem combined with microlocal symmetrization arguments

sharp metastability threshold for two dimensional bootstrap percolation in the bootstrap percolation model sites in an l by l square are initially independently declared active with probability p at each time step an inactive site becomes active if at least two of its four neighbours are active we study the behaviour as p 0 and l simultaneously of the probability i l p that the entire square is eventually active we prove that i l p 1 if and i l p 0 if where 2 18 we prove the same behaviour with the same threshold for the probability j l p that a site is active by time l in the process on the infinite lattice the same results hold for the so called modified bootstrap percolation model but with threshold 2 6 the existence of the thresholds settles a conjecture of aizenman and lebowitz 3 while the determination of their values corrects numerical predictions of adler stauffer and aharony 2

global well posedness for the lake equations abstract we prove global well posedness for the lake equations these equations arise to leading order in a low aspect ratio low froude number i e low wave speed and very small wave amplitude expansion of the three dimensional incompressible euler equations in a basin with a free upper surface and a spatially varying bottom topography well posedness means that there exists a solution that it is unique and that it depends continuously on the data i e the initial condition and the bottom topography our approach follows the works of yudovitch and bardos in constructing the solutions as the inviscid limit of solutions to a system with artificial viscosity which is the analog of the navier stokes with respect to the euler equations one of the main assumptions in the present work is a nondegenerate bottom topography

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

diffusion percolation i infinite time limit and bootstrap percolation several models for the dynamic growth of percolation clusters or diffusion percolation dp are introduced and analysed in these models a random walker an ant walks on percolation clusters which are occupied with initial site concentration pi the ant is allowed to step off such clusters and add new sites to them if certain conditions are met some of these models are shown to have a one to one correspondence with models of bootstrap percolation bp in which sites which do not have a required number of neighbours are successively culled two new percolation thresholds have been calculated for two diffusion percolation models on the square lattice

finite size scaling in three dimensional bootstrap percolation we consider the problem of bootstrap percolation on a three dimensional lattice and we study its finite size scaling behavior bootstrap percolation is an example of cellular automata defined on the d dimensional lattice 1 2 l d in which each site can be empty or occupied by a single particle in the starting configuration each site is occupied with probability p occupied sites remain occupied forever while empty sites are occupied by a particle if at least among their 2d nearest neighbor sites are occupied when d is fixed the most interesting case is the one d this is a sort of threshold in the sense that the critical probability p c for the dynamics on the infinite lattice z d switches from zero to one when this limit is crossed finite size effects in the three dimensional case are already known in the cases 2 in this paper we discuss the case 3 and we show that the finite size scaling function for this problem is of the form f l const in in l we prove a conjecture proposed by a c d van enter

asymptotic limit for rotational quantum compressible navier stokes equations with multiple scales abstract in this paper we consider the degenerate quantum compressible navier stokes equations giving rise to a variety of mathematical problems in many areas we study the asymptotic limit for the rotational compressible navier stokes equations with quantum term and the ill prepared initial data

global strong solutions to the two dimensional full compressible navier stokes equations with vacuum abstract in this paper we consider the global existence of strong solutions to the two dimensional viscous compressible and heat conducting navier stokes equations on the square domain 0 1 2 based on the blow up criterion and uniform estimates we prove that the strong solution exists globally in time if the initial mass is small for the constant coefficients of viscosity and heat conductivity

weak solutions of the time fractional navier stokes equations and optimal control abstract in this paper we deal with the navier stokes equations with the time fractional derivative of order 0 1 which can be used to simulate anomalous diffusion in fractal media we firstly give the concept of the weak solutions and establish the existence criterion of weak solutions by means of galerkin approximations in the case that the dimension n 4 moreover a complete proof of the uniqueness is given when n 2 at last we give a sufficient condition of optimal control pairs

weighted sobolev spaces and embedding theorems in the present paper we study embedding operators for weighted sobolev spaces whose weights satisfy the well known muckenhoupt a p condition sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities the proposed method is based on the concept of generalized quasiconformal homeomorphisms homeomorphisms with bounded mean distortion the choice of the homeomorphism type depends on the choice of the corresponding weighted sobolev space such classes of homeomorphisms induce bounded composition operators for weighted sobolev spaces with the help of these homeomorphism classes the embedding problem for non smooth domains is reduced to the corresponding classical embedding problem for smooth domains examples of domains with anisotropic holder singularities demonstrate the sharpness of our machinery comparatively with known results

finite size effects for some bootstrap percolation models the consequences of schonmann s new proof that the critical threshold is unity for certain bootstrap percolation models are explored it is shown that this proof provides an upper bound for the finite size scaling in these systems comparison with data for one case demonstrates that this scaling appears to give the correct asymptotics we show that the threshold for a finite system of sizel scales aso ln lnl for the isotropic model in three dimensions where sites that fail to have at least four neighbors are culled

global existence in critical spaces for compressible navier stokes equations we investigate global strong solutions for isentropic compressible fluids with initial data close to a stable equilibrium we obtain the existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations more precisely the initial velocity has the same critical regularity index as for the incompressible homogeneous navier stokes equations and one more derivative is needed for the density we point out a smoothing effect on the velocity and a l 1 decay on the difference between the density and the constant reference state the proof lies on uniform estimates for a mixed hyperbolic parabolic linear system with a convection term

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

statistical physics of immune networks abstract several discrete models of immune networks have been developed in the last few years to describe the dynamics of the various aspects of immune response the populations of the various types of cells involved in immune response are represented in these theries by automata and the population dynamics of these cells are formulated in terms of discrete maps in discrete time in contrast to the classical approaches where differential equations describe the time evolution of the populations which are taken to be real variables we summarize here the main results of these investigations and present a critical analysis of the successes and limitations of this discrete approach by comparing with the results of corresponding continuum models recently interclonal interactions have been taken into account by implementing jerne s idea of a functional network using a shape space approach we report some of the progress made recently in this area of interdisciplinary research and also outline a simple theory of t cell anergy

sharp metastability threshold for an anisotropic bootstrap percolation model bootstrap percolation models have been extensively studied during the two past decades in this article we study the following anisotropic boot strap percolation model the neighborhood of a point m n is the set m 2 n m 1 n m n 1 m 1 n m 2 n m n 1 at time 0 sites are occupied with probability p at each time step sites that are occupied remain occupied while sites that are not occupied become occupied if and only if three of more sites in their neighborhood are occupied we prove that it exhibits a sharp metastability threshold this is the first mathematical proof of a sharp threshold for an anisotropic bootstrap percolation model

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

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

on the flow of chemically reacting gaseous mixture abstract we consider the cauchy problem for the system of equations governing flow of isothermal reactive mixture of compressible gases our main contribution is to prove sequential stability of weak solutions when the state equation essentially depends on the species concentration and the viscosity coefficients vanish on vacuum moreover under additional assumption on the cold component of the pressure in the regions of small density we prove the existence of weak solutions for arbitrary large initial data

a dilute bootstrap percolation lattice model for unresponsiveness in t cell immunology the biased majority rule of cellular automata takes a spin up if and only if at least two of its four nearest neighbors on the square lattice are up we generalize this type of bootstrap percolation by introducing quenched site dilution as well as a random birth and decay process our monte carlo simulations then give first order transitions qualitatively similar to our results from meanfield reaction equations describing the induction of t cell unresponsiveness in the immune system