local existence of solutions to the free boundary value problem for the primitive equations of the ocean lions temam and wang in probleme a frontiere libre pour les modeles couples de l ocean et de l atmosphere acad sci paris c r 318 12 1165 1171 1994 introduced a free surface model for the primitive equations of the ocean in this paper we establish the local well posedness of the model with analytic initial data

existence and regularity of invariant measures for the three dimensional stochastic primitive equations we establish the continuity of the markovian semigroup associated with strong solutions of the stochastic 3d primitive equations and prove the existence of an invariant measure the proof is based on new moment bounds for strong solutions the invariant measure is supported on strong solutions and is furthermore shown to have higher regularity properties

to what class of fractals does the alexander orbach conjecture apply discussion de la decouverte numerique recente d alexander et orbach selon laquelle pour un agregat infini naissant en percolation le rapport d f d w dimension fractale de l agregat dimension fractale d une marche aleatoire dans l agregat est approximativement independant de d pour d 1

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

uniqueness of the global weak solutions to 2d compressible primitive equations abstract in this paper we investigate an initial boundary value problem for 2d compressible primitive equations with some specific viscosities we will prove the uniqueness of the weak solutions obtained by b v gatapov and a v kazhikhov 12

nearest neighbour distance to a trap in a one dimensional smoluchowski model abstract we consider a one dimensional smoluchowski model for the reaction a t t with a single immobile trapping particle and noninteracting a particles which move by diffusion it is known that when there is an initial density of a particles the average value of the closest distance between the trap and an untrapped a is asymptotically proportional to t 1 4 we show that this dependence also holds when the initial distances between adjacent a particles are independent identically distributed random variables with a finite mean we also consider the case in which the trap is mobile and the a s are immobile here it is known that the asymptotic nearest neighbor distance is proportional to t 1 2 we show that for this to hold with a general initial distribution of a s one requires both the first and second moments of the initial distances to be finite

strong solutions to the 3d primitive equations with only horizontal dissipation near h 1 initial data abstract in this paper we consider the initial boundary value problem of the three dimensional primitive equations for oceanic and atmospheric dynamics with only horizontal viscosity and horizontal diffusivity we establish the local in time well posedness of strong solutions for any initial data v 0 t 0 h 1 by using the local in space type energy estimate we also establish the global well posedness of strong solutions for this system with any initial data v 0 t 0 h 1 l such that z v 0 l m for some m 2 by using the logarithmic type anisotropic sobolev inequality and a logarithmic type gronwall inequality this paper improves the previous results obtained in cao et al 2016 10 where the initial data v 0 t 0 was assumed to have h 2 regularity

rate processes on fractals theory simulations and experiments heterogeneous kinetics are shown to differ drastically from homogeneous kinetics for the elementary reaction a a products we show that the diffusion limited reaction rate is proportional tot h a 2 or to a x whereh 1 d s 2 x 1 2 d s h 2 h 1 andd s is the effective spectral dimension we note that ford d s 1 h 1 2 andx 3 for percolating clustersd s 4 3 h 1 3 andx 5 2 while for dust ds h 1 2 and x 3 scaling arguments supercomputer simulations and experiments give a consistent picture the interplay of energetic and geometric heterogeneity results in fractal like kinetics and is relevant to excitation fusion experiments in porous membranes films and polymeric glasses however in isotopic mixed crystals the geometric fractal nature percolation clusters dominates

random attractor for the 3d viscous primitive equations driven by fractional noises abstract we develop a new and general method to prove the existence of the random attractor strong attractor for the primitive equations pes of large scale ocean and atmosphere dynamics under non periodic boundary conditions and driven by infinite dimensional additive fractional wiener processes in contrast to our new method the common method compact sobolev embedding theorem is to obtain the time uniform a priori estimates in some sobolev space whose regularity is higher than the solution space but this method can not be applied to the 3d stochastic pes with the non periodic boundary conditions therefore the existence of universal attractor weak attractor was established in previous works see 15 16 the main idea of our method is that we first derive that p almost surely the solution operator of stochastic pes is compact then we construct a compact absorbing set by virtue of the compact property of the solution operator and the existence of a absorbing set we should point out that our method has some advantages over the common method of using compact sobolev embedding theorem i e using our method we only need to obtain time uniform a priori estimates in the solution space to prove the existence of random attractor for the corresponding stochastic system while the common method need to establish time uniform a priori estimates in a more regular functional space than the solution space take the stochastic pes for example as the unique strong solution to the stochastic pes belongs to c 0 t h 1 3 in view of our method we only need to obtain the time uniform a priori estimates in the solution space h 1 3 to prove the existence of random attractor for this stochastic system while the common method need to establish time uniform a priori estimates for the solution in the functional space h 2 3 however time uniform a priori estimates in h 2 3 for the solution to stochastic pes are too difficult to be established the present work provides a general way for proving the existence of random attractor for common classes of dissipative stochastic partial differential equations driven by wiener noises fractional noises and even jump noises in a forth coming paper using this new method we 46 prove the existence of random attractor for the stochastic nematic liquid crystals equations this is the first result about the long time behavior of stochastic nematic liquid crystals equations

on the h s theory of hydrostatic euler equations in this paper we study the two dimensional hydrostatic euler equations in a periodic channel we prove the local existence and uniqueness of hs solutions under the local rayleigh condition this extends brenier s nonlinearity 12 3 495 512 1999 existence result by removing an artificial condition and proving uniqueness in addition we prove weak strong uniqueness mathematical justification of the formal derivation and stability of the hydrostatic euler equations these results are based on weighted hs a priori estimates which come from a new type of nonlinear cancellation between velocity and vorticity

blowup of solutions of the unsteady prandtl s equation we prove that for certain class of compactly supported c initial data smooth solutions of the unsteady prandtl s equation blow up in nite time

hydrodynamics of stochastic cellular automata we investigate a stochastic version of cellular automata used for simulating hydrodynamical flows e g the hpp and fhp models the extra stochasticity consists of random exchanges between neighboring cells which conserve momentum we prove that in suitable limits these models satisfy the appropriate continuous boltzmann and hydrodynamic equations the same as those conjectured for the original models except that there is no negative viscosity contribution the results are obtained by proving a very strong form of propagation of chaos and by using hilbert chapman enskog type expansions explicit proofs are presented for the stochastic hpp model

generalized solutions and hydrostatic approximation of the euler equations abstract solutions to the euler equations on a 3d domain d 3 typically the unit cube or the periodic unit cube can be formally obtained by minimizing the action of an incompressible fluid moving inside d 3 between two given configurations when these two configurations are very close to each other classical solutions do exist as shown by ebin and marsden however shnirelman found a class of data essentially 2d in the sense that they trivially depend on the vertical coordinate for which there cannot be any classical minimizer for such data generalized solutions can be shown to exist as a substitute for classical solutions these generalized solutions have unusual features that look highly unphysical in particular different fluid parcels can cross at the same point and at the same time but the pressure field which does not depend on the vertical coordinate is well and uniquely defined in the present paper we show that these generalized solutions are actually quite conventional in the sense they obey up to a suitable change of variable a well known variant widely used for geophysical flows of the 3d euler equations for which the vertical acceleration is neglected according to the so called hydrostatic approximation

deterministic and stochastic models of enzymatic networks applications to pharmaceutical research the intracellular transducing device consists of complex networks of enzymatic reactions unfortunately the mathematical models commonly used to describe them are still unsatisfactory and unreliable even at the level of reproducing simple reaction schemes the improvement of mathematical models is necessary and can follow different approaches still poorly employed such as the modeling of spatial structures and phenomena time delays stochastic perturbations only to cite the most relevant ones in this paper we show some recent results related to the total quasi steady state approximation tqssa in a deterministic scenario moreover we show some possible applications of the tqssa in a stochastic scheme

global well posedness of the 3d primitive equations with partial vertical turbulence mixing heat diffusion the three dimensional incompressible viscous boussinesq equations under the assumption of hydrostatic balance govern the large scale dynamics of atmospheric and oceanic motion and are commonly called the primitive equations to overcome the turbulence mixing a partial vertical diffusion is usually added to the temperature advection or density stratification equation in this paper we prove the global regularity of strong solutions to this model in a three dimensional infinite horizontal channel subject to periodic boundary conditions in the horizontal directions and with no penetration and stress free boundary conditions on the solid top and bottom boundaries specifically we show that short time strong solutions to the above problem exist globally in time and that they depend continuously on the initial data

blowup in stagnation point form solutions of the inviscid 2d boussinesq equations abstract the 2d boussinesq equations model large scale atmospheric and oceanic flows whether its solutions develop a singularity in finite time remains a classical open problem in mathematical fluid dynamics in this work blowup from smooth nontrivial initial velocities in stagnation point form solutions of this system is established on an infinite strip x y 0 1 r we consider velocities of the form u f t x y f x t x with scalar temperature y t x assuming f x 0 x attains its global maximum only at points x i located on the boundary of 0 1 general criteria for finite time blowup of the vorticity y f x x t x i and the time integral of f x t x i are presented briefly for blowup to occur it is sufficient that 0 x 0 and f t x i 0 x i 0 while y f x x 0 x i 0 to illustrate how vorticity may suppress blowup we also construct a family of global exact solutions a local existence result and additional regularity criteria in terms of the time integral of f x t l 0 1 are also provided

axisymmetric flow of ideal fluid moving in a narrow domain a study of the axisymmetric hydrostatic euler equations abstract in this article we will introduce a new model to describe the leading order behavior of an ideal and axisymmetric fluid moving in a very narrow domain after providing a formal derivation of the model we will prove the well posedness and provide a rigorous mathematical justification for the formal derivation under a new sign condition finally a blowup result regarding this model will be discussed as well

transient a b 0 reaction on fractals stochastic and deterministic aspects in this paper we study the transient diffusion limited a b 0 a0 b0 annihilation via deterministic reaction diffusion equations and via simulation of the stochastic many particle problem we show that the two approaches are not equivalent and that the deterministic expressions capture only part of the picture a lower bound is derived for the density decay which indicates that the overall density follows the power law t x with mm d 4 1 hierarchical oscillations superimposed on the power law decay are observed for reactions on sierpinski gaskets

diffusion limited coalescence with finite reaction rates in one dimension we study the diffusion limited process a a to a in one dimension with finite reaction rates we develop an approximation scheme based on the method of inter particle distribution functions ipdf which was used formerly for the exact solution of the same process with an infinite reaction rate the approximation becomes exact in the very early time regime or the reaction controlled limit and in the long time diffusion controlled asymptotic limit for the intermediate time regime we obtain a simple interpolative behaviour between these two limits we also study the coalescence process with finite reaction rates with the back reaction a to a a and in the presence of particle input in each of these cases the system reaches a non trivial steady state with a finite concentration of particles theoretical predictions for the concentration time dependence and for the ipdf are compared with computer simulations

local in time existence and uniqueness of solutions to the prandtl equations by energy methods we prove local existence and uniqueness for the two dimensional prandtl system in weighted sobolev spaces under oleinik s monotonicity assumption in particular we do not use the crocco transform or any change of variables our proof is based on a new nonlinear energy estimate for the prandtl system this new energy estimate is based on a cancellation property that is valid under the monotonicity assumption to construct the solution we use a regularization of the system that preserves this nonlinear structure this new nonlinear structure may give some insight into the convergence properties from the navier stokes system to the euler system when the viscosity goes to 0 2015 wiley periodicals inc

global strong lp well posedness of the 3d primitive equations with heat and salinity diffusion abstract consider the full primitive equations i e the three dimensional primitive equations coupled to the equation for temperature and salinity and subject to outer forces it is shown that this set of equations is globally strongly well posed for arbitrary large initial data lying in certain interpolation spaces which are explicitly characterized as subspaces of h 2 p p 1 p satisfying certain boundary conditions in particular global well posedness of the full primitive equations is obtained for initial data having less differentiability properties than h 1 hereby generalizing a result by cao and titi 2007 5 to the case of non smooth data in addition it is shown that the solutions are exponentially decaying provided the outer forces possess this property

ill posedness of the hydrostatic euler and navier stokes equations we prove that the linearization of the hydrostatic euler equations at certain parallel shear flows is ill posed the result also extends to the hydrostatic navier stokes equations with a small viscosity

tagged particle diffusion in 3d lattice gas cellular automata abstract we report simulations of tagged particle diffusion in three dimensional lattice gas cellular automata lgca in particular we looked at the decay of the velocity autocorrelation function vacf using a new technique that is about a million times more efficient than the conventional techniques for longer times the simulations clearly show the algebraic t 3 2 tail of the vacf we compare the observed long time tail with the predictions of mode coupling theory in three dimensions the amplitude of this tail is found to agree within the small statistical error with these predictions

zero viscosity limit for analytic solutions of the navier stokes equation on a half space i existence for euler and prandtl equations this is the first of two papers on the zero viscosity limit for the incompressible navier stokes equations in a half space in this paper we prove short time existence theorems for the euler and prandtl equations with analytic initial data in either two or three spatial dimensions the main technical tool in this analysis is the abstract cauchy kowalewski theorem for the euler equations the projection method is used in the primitive variables to which the cauchy kowalewski theorem is directly applicable for the prandtl equations cauchy kowalewski is applicable once the diffusion operator in the vertical direction is inverted

ill posedness of the hydrostatic euler and singular vlasov equations in this paper we develop an abstract framework to establish ill posedness in the sense of hadamard for some nonlocal pdes displaying unbounded unstable spectra we apply this to prove the ill posedness for the hydrostatic euler equations as well as for the kinetic incompressible euler equations and the vlasov dirac benney system

rotation prevents finite time breakdown we consider a two dimensional 2d convection model augmented with the rotational coriolis forcing ut u xu 2ku w ith afi xed 2k being the inverse rossby number we ask whether the action of dispersive rotational forcing alone u prevents the generic finite time breakdown of the free nonlinear convection the answer provided in this work is a conditional yes namely we show that the rotating euler equations admit global smooth solutions for a subset of generic initial configurations with other configurations however finite time breakdown of solutions may and actually does occur thus global regularity depends on whether the initial configuration crosses an intrinsic o 1 critical threshold ct which is quantified in terms of the initial vorticity 0 u0 and the initial spectral gap associated with the 2 2 initial velocity gradient 0 2 0 1 0 j 0 j u0 specifically global regularity of the rotational euler equation is ensured if and only if 4k 0 2 4k 2 r 2 we also prove that the velocity field remains smooth if and only if it is periodic an equivalent lagrangian formulation reconfirms the ct and shows a global periodicity of velocity field as well as the associated particle orbits moreover we observe yet another remarkable periodic behavior exhibited by the gradient of the velocity field the spectral dynamics of the eulerian formulation siam j math anal 33 2001 930 reveals that the vorticity and the divergence of the flow evolve with their own path dependent period we conclude with a kinetic formulation of the rotating euler equation 2003 elsevier b v all rights reserved msc 35q35 35b30

long time correlations in a 2d lattice gas abstract we present the results of ca simulations of a moderately dense lattice gas using an extended version of the fhp model for colored automata by tracking a tagged particle we construct its velocity autocorrelation function and we show that the corresponding power spectrum exhibits a low frequency contribution characteristic of the long time power law behavior t 1 in 2d

fractal pharmacokinetics of the drug mibefradil in the liver we explore the ramifications of the fractal geometry of the key organ for drug elimination the liver on pharmacokinetic data analysis a formalism is developed for the use of a combination of well stirred euclidean and fractal compartments in the body perturbation analysis is carried out to obtain analytical solutions for the drug concentration time evolution these results are then fitted to experimental data collected from clinically instrumented dogs see a skerjanec et al j pharm sci 85 189 1995 using the drug mibefradil the thus obtained spectral fractal dimension has a range of values that is consistent with the value found in independently performed ultrasound experiments on the liver

monte carlo simulation of the reaction diffusion process of a complex chemical scheme on a fractal lattice abstract the reaction diffusion process is often assumed to form a markov chain at a mesoscopic description level and the chain is traditionally simulated by using the minimal process algorithm to overcome the difficulties of direct application of this method to large inhomogeneous systems we propose here a much improved version of this algorithm based on the concept of cascade classification which we introduced its efficiency was tested on a microcomputer by applying it to a complex chemical reaction scheme williamowski rossler reacting and diffusing on the sierpinski gasket concentration has been focused on exploring dynamic behavior when carrying out the simulation the case study indicated that the modified minimal process algorithm provides an efficient numerical technique to investigate complex phenomena of reaction and diffusion processes taking place in fractals as well as other systems of different physical nature

cellular automation model of reaction transport processes abstract the transport and chemical reactions of solutes are modelled as a cellular automation in which molecules of different species perform a random walk on a regular lattice and react according to a local probabilistic rule the model describes advection and diffusion in a simple way and as no restriction is placed on the number of particles at a lattice site it is also able to describe a wide variety of chemical reactions assuming molecular chaos and a smooth density function we obtain the standard reaction transport equations in the continuum limit simulations on one an two dimensional lattices show that the discrete model can be used to approximate the solutions of the continuum equations we discuss discrepancies which arise from correlations between molecules and how these disappear as the continuum limit is approached of particular interest are simulations displaying long time behaviour which depends on long wavelength statistical fluctuations not accounted for by the standard equations the model is applied to the reactions a b c and a b c with homogenous and inhomogenous initial conditions as well as to systems subject to autocatalytic reactions and displaying spontaneous formation of spatial concentration patterns

global well posedness of strong solutions to the 3d primitive equations with horizontal eddy diffusivity in this paper we consider the initial boundary value problem of the 3d primitive equations for oceanic and atmospheric dynamics with only horizontal diffusion in the temperature equation global well posedness of strong solutions are established with h2 initial data 2014 elsevier inc all rights reserved

local existence and uniqueness for the hydrostatic euler equations on a bounded domain abstract we address the question of well posedness in spaces of analytic functions for the cauchy problem for the hydrostatic incompressible euler equations inviscid primitive equations on domains with boundary by a suitable extension of the cauchy kowalewski theorem we construct a locally in time unique real analytic solution and give an explicit rate of decay of the radius of real analyticity

zero mach number limit of the compressible primitive equations well prepared initial data author s liu xin titi edriss s abstract this work concerns the zero mach number limit of the compressible primitive equations the primitive equations with the incompressibility condition are identified as the limiting equations the convergence with well prepared initial data i e initial data without acoustic oscillations is rigorously justified and the convergence rate is shown to be of order mathcal o varepsilon as varepsilon rightarrow 0 where varepsilon represents the mach number as a byproduct we construct a class of global solutions to the compressible primitive equations which are close to the incompressible flows

almost global existence for the prandtl boundary layer equations we consider the prandtl boundary layer equations on the half plane with initial datum that lies in a weighted h1 space with respect to the normal variable and is real analytic with respect to the tangential variable the boundary trace of the horizontal euler flow is taken to be a constant we prove that if the prandtl datum lies within varepsilon of a stable profile then the unique solution of the cauchy problem can be extended at least up to time t varepsilon geqq rm exp varepsilon 1 rm log varepsilon 1

application of a random network with a variable geometry of links to the kinetics of drug elimination in healthy and diseased livers this paper discusses an application of a random network with a variable number of links and traps to the elimination of drug molecules from the body by the liver the nodes and links represent the transport vessels and the traps represent liver cells with metabolic enzymes that eliminate drug molecules by varying the number and configuration of links and nodes different disease states of the liver related to vascular damage have been simulated and the effects on the rate of elimination of a drug have been investigated results of numerical simulations show the prevalence of exponential decay curves with rates that depend on the concentration of links in the case of fractal lattices at the percolation threshold we find that the decay of the concentration is described by exponential functions for high trap concentrations but transitions to stretched exponential behavior at low trap concentrations

pattern formation in diffusion limited reactions the conditions for macroscopic segregation ofa andb in a steady statea b 0 reaction are studied in infinite systems segregation occurs in one and two dimensions and is marginal ford 3 we note the dependence of these results on the precise experimental conditions assumed in the theory we also note the difference between these results and our earlier ones for finite systems where the critical dimension isd 2

stability issues in the quasineutral limit of the one dimensional vlasov poisson equation this work is concerned with the quasineutral limit of the one dimensional vlasov poisson equation for initial data close to stationary homogeneous profiles our objective is threefold first we provide a proof of the fact that the formal limit does not hold for homogeneous profiles that satisfy the penrose instability criterion second we prove on the other hand that the limit is true for homogeneous profiles that satisfy some monotonicity condition together with a symmetry condition we handle the case of well prepared as well as ill prepared data last we study a stationary boundary value problem for the formal limit the so called quasineutral vlasov equation we show the existence of numerous stationary states with a lot of freedom in the construction compared to that of bgk waves for vlasov poisson this illustrates the degeneracy of the limit equation

internal fluctuations period doubling and chemical chaos the effects of internal molecular fluctuations on period doubling bifurcations and chaotic band merging are studied in a well stirred reactive chemical system where spatial degrees of freedom play no role the calculations are carried out using a stochastic model based on reactive lattice gas cellular automata molecular fluctuations arising from reactive collisions are incorporated in the model and the mass action rate equations are recovered in the mean field limit large system sizes can be studied and permit the investigation of the internal noise scaling structure in the periodic and chaotic regimes for the willamowski r ossler system the noise scaling exponent is consistent with that found for one dimensional quadratic maps subject to external noise the model provides a means to study small non equilibrium reacting systems where fluctuations may not be neglected

a direct method to solve block banded block toeplitz systems with non banded toeplitz blocks a fast solution algorithm is proposed for solving block banded block toeplitz systems with non banded toeplitz blocks the algorithm constructs the circulant transformation of a given toeplitz system and then by means of the sherman morrison woodbury formula transforms its inverse to an inverse of the original matrix the block circulant matrix with toeplitz blocks is converted to a block diagonal matrix with toeplitz blocks and the resulting toeplitz systems are solved by means of a fast toeplitz solver the computational complexity in the case one uses fast toeplitz solvers is equal to x m n k o mn 3 o k 3n 3 flops there are m block rows and m block columns in the matrix n is the order of blocks 2k 1 is the bandwidth the validity of the approach is illustrated by numerical experiments

reactive lattice gas automata abstract a probabilistic lattice gas cellular automaton model of a chemically reacting system is constructed microdynamical equations for the evolution of the system are given the continuous and discrete boltzmann equations are developed and their reduction to a generalized reaction diffusion equation is discussed the microscopic reactive dynamics is consistent with any polynomial rate law up to the fourth order in the average particle density it is shown how several microscopic ca rules are consistent with a given rate law as most ca systems the present one has spurious properties whose effects are shown to be unimportant under appropriate conditions as an explicit example of the general formalism a ca dynamics is constructed for an autocatalytic reactive scheme known as the schlogl model simulations show that in spite of the simplicity of the underlying discrete dynamics the model exhibits the phase separation and wave propagation phenomena expected for this system because of the microscopic nature of the dynamics the role of internal fluctuations on the evolution process can be investigated

on the compatibility condition for linear systems and a factorization formula for wave functions the well known compatibility condition for linear systems wx gw and wt fw is considered and new results are obtained in this way a factorization formula for wave functions which is basic in the inverse spectral transform approach to initial boundary value problems is proved in greater generality than before some applications follow

zero viscosity limit for analytic solutions of the primitive equations the aim of this paper is to prove that the solutions of the primitive equations converge in the zero viscosity limit to the solutions of the hydrostatic euler equations we construct the solution of the primitive equations through a matched asymptotic expansion involving the solution of the hydrostatic euler equation and boundary layer correctors as the first order term and an error that we show to be o sqrt nu the main assumption is spatial analyticity of the initial datum

finite time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics in an earlier work we have shown the global for all initial data and all time well posedness of strong solutions to the three dimensional viscous primitive equations of large scale oceanic and atmospheric dynamics in this paper we show that for certain class of initial data the corresponding smooth solutions of the inviscid non viscous primitive equations if they exist they blow up in finite time specifically we consider the three dimensional inviscid primitive equations in a three dimensional infinite horizontal channel subject to periodic boundary conditions in the horizontal directions and with no normal flow boundary conditions on the solid top and bottom boundaries for certain class of initial data we reduce this system into the two dimensional system of primitive equations in an infinite horizontal strip with the same type of boundary conditions and then we show that for specific sub class of initial data the corresponding smooth solutions of the reduced inviscid two dimensional system develop singularities in finite time

anomalous coalescence phenomena under stationary regime condition through thompson s method abstract in this work anomalous coalescence phenomena of type a a a 0 are considered in the case of stationary regime namely the case of an external homogeneous source h of single particles a in the limit of zero external field rate the main purpose here is to pick up new critical exponents in the stationary regime for the phenomenon of anomalous coalescence of bubbles found in one dimension by josserand and rica jr phys rev lett 78 1997 1215 the critical exponents of statical concentration relaxation time and the exponent for concentration decay are evaluated in the limit h 0 zero field rate we show that such critical exponents depend on the parameter physica a 334 2004 335 which characterizes the anomalous diffusion condition in order to explain the anomalous coalescence of jr s work as already made before by the present author physica a 334 2004 335 the parameter must display an explicit dependence on the dimensionality d 4 d for d 2 we use such a result d in order to pick up new critical exponents for the special case of anomalous coalescence found by jr we find 5 4 5 and 1 4 in 1 d we also verify that the known scaling relations among such critical indexes phys rev a 32 2 1985 1129 are obeyed so that in the special case of 2 brownian diffusion condition given to mean field regime we naturally recover those results previously obtained by racz phys rev a 32 2 1985 1129 we go further to give a better explanation for such new results taking into account another exponent physica a 334 2004 335 to explain the behavior of diffusion constant which depends on concentration d d 0 n that is to say we obtain analytically new critical indexes which controll the diffusion behavior d for long time regime in the presence of an external homogeneous source

on the regularization mechanism for the periodic korteweg de vries equation in this paper we develop and use successive averaging methods for explaining the regularization mechanism in the the periodic korteweg de vries kdv equation in the homogeneous sobolev spaces s for s 0 specifically we prove the global existence uniqueness and lipschitz continuous dependence on the initial data of the solutions of the periodic kdv for the case where the initial data is in l2 we also show the lipschitz continuous dependence of these solutions with respect to the initial data as maps from s to s for s 1 0 2010 wiley periodicals inc

the van dommelen and shen singularity in the prandtl equations abstract in 1980 van dommelen and shen provided a numerical simulation that predicted the spontaneous generation of a singularity in the prandtl boundary layer equations from a smooth initial datum for a nontrivial euler background in this paper we prove this numerical conjecture by rigorously establishing the finite time blowup of the boundary layer thickness

a fast algorithm for solving banded toeplitz systems a fast algorithm for solving systems of linear equations with banded toeplitz matrices is presented this new approach is based on extending the given matrix with several rows on the top and several columns on the right and to assign zeros and some nonzero constants in each of these rows and columns in such a way that the augmented matrix has a lower triangular toeplitz structure stability of the algorithm is discussed and its performance is showed by numerical experiments

existence and non uniqueness of global weak solutions to inviscid primitive and boussinesq equations we consider the initial value problem for the inviscid primitive and boussinesq equations in three spatial dimensions we recast both systems as an abstract euler type system and apply the methods of convex integration of de lellis and szekelyhidi to show the existence of infinitely many global weak solutions of the studied equations for general initial data we also introduce an appropriate notion of dissipative solutions and show the existence of suitable initial data which generate infinitely many dissipative solutions

single random walker on disordered lattices random walks on square lattice percolating clusters were followed for up to 2 10 steps the mean number of distinct sites visited sn gives a spectral dimension of d s 1 30 5 0 03 consistent with superuniversality d s 4 3 but closer to the alternative ds 182 139 based on the low dimensionality correction simulations are also given for walkers on an energetically disordered lattice with a jump probability that depends on the local energy mismatch and the temperature an apparent fractal behavior is observed for a low enough reduced temperature above this temperature the walker exhibits a crossover from fractal to euclidean behavior walks on two and three dimensional lattices are similar except that those in three dimensions are more efficient

on the nonlinear instability of euler and prandtl equations in this paper we give examples of nonlinearly unstable solutions of euler equations in the whole space 2 the half space the periodic strip the strip 1 1 and the periodic torus 2 with an application to vortex sheets using the same methods we prove an instability result for prandtl type boundary layers that appear in and 2000 john wiley sons inc

on the ill posedness of the prandtl equation the concern of this paper is the cauchy problem for the prandtl equation this problem is known to be well posed for analytic data or for data with monotonicity properties we prove here that it is linearly ill posed in sobolev type spaces the key of the analysis is the construction at high tangential frequencies of unstable quasimodes for the linearization around solutions with non degenerate critical points interestingly the strong instability is due to vicosity which is coherent with well posedness results obtained for the inviscid version of the equation a numerical study of this instability is also provided

analyticity of solutions for a generalized euler equation abstract we consider the so called lake and great lake equations which are shallow water equations that describe the long time motion of an inviscid incompressible fluid contained in a shallow basin with a slowly spatially varying bottom a free upper surface and vertical side walls under the influence of gravity and in the limit of small characteristic velocities and very small surface amplitude if these equations are posed on a space periodic domain and the initial data are real analytic the solution remains real analytic for all times the proof is based on a characterization of gevrey classes in terms of decay of fourier coefficients in particular our result recovers known results for the euler equations in two and three spatial dimensions we believe the proof is new

global well posedness of the three dimensional viscous primitive equations of large scale ocean and atmosphere dynamics in this paper we prove the global existence and uniqueness regularity of strong solutions to the three dimensional viscous primitive equations which model large scale ocean and atmosphere dynamics 1 introduction large scale dynamics of oceans and atmosphere is governed by the primitive equations which are derived from the navier stokes equations with rotation coupled to thermodynamics and salinity diffusion transport equations which account for the buoyancy forces and stratification effects under the boussinesq approximation moreover and due to the shallowness of the oceans and the atmosphere i e the depth of the fluid layer is very small in comparison to the radius of the earth the vertical large scale motion in the oceans and the atmosphere is much smaller than the horizontal one which in turn leads to modeling the vertical motion by the hydrostatic balance as a result

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

williams watts dielectric relaxation a fractal time stochastic process dielectric relaxation in amorphous materials is treated in a defect diffusion model where relaxation occurs when a mobile defect such as a vacancy reaches a frozen in dipole the random motion of the defect is assumed to be governed by a fractal time stochastic process where the mean duration between defect movements is infinite when there are many more defects than dipoles the williams watts decaying fractional exponential relaxation law is derived the argument of the exponential is related to the number of distinct sites visited by the random walk of the defect for the same reaction dynamics but with more traps than walkers an algebraically decaying relaxation is found

well posedness in sobolev spaces of the full water wave problem in 3 d we consider the motion of the interface separating an inviscid incompressible irrotational fluid from a region of zero density in three dimensional space we assume that the fluid region is below the vacuum the fluid is under the influence of gravity and the surface tension is zero assume that the density of mass of the fluid is one the gravitational field is 0 0 1 the free interface is t at time t 0 and the fluid occupies the region t the motion of the fluid is described by vt v v 0 0 1 p on t t 0 euler s equation 1 1 div v 0 on t t 0 incompressible 1 2 curl v 0 on t t 0 irrotational 1 3 where v v1 v2 v3 is the fluid velocity p is the fluid pressure since we neglect the surface tension the pressure is zero on the interface so on the interface p 0 on t 1 4 1 v is tangent to the free surface t t 1 5 we want to find solutions of system 1 1 1 5 taking prescribed initial data such that for every fixed t 0 t approaches the xy plane at infinity and v x y z t 0 vt x y z t 0 as x y z since the fluid is assumed irrotational incompressible we can reduce the study of the entire motion to the motion of the free surface the above model is a 3 d water wave model it is generally known that when surface tension is neglected the motion of the interface between an inviscid fluid and vacuum under the influence of gravity can be subject to taylor instability 8 22 in a previous work 24 we studied the 2 dimensional water wave model we showed that for a 2 d water wave the sign condition relating to taylor instability always holds for nonself intersecting interface that is the motion of the interface is not subject to taylor instability we showed further that the 2 d full nonlinear water wave problem is uniquely solvable in sobolev spaces locally in time for any initially nonself intersecting interface earlier works on the well posedness in sobolev spaces of the 2 d water wave problem include nalimov 18 yosihara 25 and walter craig 6 where the main results concern the well posedness of the

global wellposedness of kdv in h 1 t r by the inverse method we show that the korteweg de vries equation kdv tv x t x3v x t 6v x t xv x t x t t r is globally in time wellposed in the sobolev space of distributions h t r for any 1

fractal behavior in trapping and reaction a random walk study we investigate the trapping of a random walker in fractal structures sierpinski gaskets with randomly distributed traps the survival probability is determined from the number of distinct sites visited in the trap free fractals we show that the short time behavior and the long time tails of the survival probability are governed by the spectral dimension d we interpolate between these two limits by introducing a scaling law an extension of the theory which includes a continuous time random walk on fractals is discussed as well as the case of direct trapping the latter case is shown to be governed by the fractal dimension d

global well posedness of the three dimensional primitive equations with only horizontal viscosity and diffusion 2015 wiley periodicals inc in this paper we consider the initial boundary value problem of the three dimen sional primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation global well posedness of the strong solution is established for any h2 initial data an n dimensional logarithmic sobolev embedding inequality which bounds the l norm in terms of the lq norms up to a logarithm of the lp norm for p n of the first order derivatives and a system version of the classic gronwall inequality are exploited to establish the required a priori h2 estimates for global regularity

on the hydrostatic approximation of the navier stokes equations in a thin strip abstract in this paper we first prove the global well posedness of a scaled anisotropic navier stokes system and the hydrostatic navier stokes system in a 2 d striped domain with small analytic data in the tangential variable then we justify the limit from the anisotropic navier stokes system to the hydrostatic navier stokes system with analytic data