global regularity for a class of 2d generalized tropical climate models abstract this paper establishes the global existence and regularity of solutions to a two dimensional 2d tropical climate model tcm with fractional dissipation the inviscid counterpart of this model was derived by frierson majda and pauluis 8 as a model for tropical geophysical flows this model reflects the interaction and coupling among the barotropic mode u the first baroclinic mode v of the velocity and the temperature the systems with fractional dissipation studied here may arise in the modeling of geophysical circumstances mathematically these systems allow simultaneous examination of a family of systems with various levels of regularization the aim here is the global regularity with the least dissipation we prove two main results first the global regularity of the system with v and for 1 and 3 2 and second the global regularity of the system with v for 3 2 the proofs of these results are not trivial and the requirements on the fractional indices appear to be optimal the key tools employed here include the maximal regularity for general fractional heat operators the littlewood paley decomposition and besov space techniques lower bounds involving fractional laplacian and simultaneous estimates of several coupled quantities

the variational iteration method for solving two forms of blasius equation on a half infinite domain the variational iteration method is applied for a reliable treatment of two forms of the third order nonlinear blasius equation which comes from boundary layer equations the study shows that the series solution is obtained without restrictions on the nonlinearity behavior the obtained series solution is combined with the diagonal pade approximants to handle the boundary condition at infinity for only one of these forms

symmetries local names and dynamic de allocation of names the semantics of name passing calculi is often defined employing coalgebraic models over presheaf categories this elegant theory lacks finiteness properties hence it is not apt to implementation coalgebras over named sets called history dependent automata are better suited for the purpose due to locality of names a theory of behavioural functors for named sets is still lacking the semantics of each language has been given in an ad hoc way and algorithms were implemented only for the p calculus existence of the final coalgebra for the p calculus was never proved we introduce a language of accessible functors to specify history dependent automata in a modular way leading to a clean formulation and a generalisation of previous results and to the proof of existence of a final coalgebra in a wide range of cases

the l2 boundedness of pseudodifferential operators on donne une nouvelle demonstration du theoreme de calderon vaillancourt on obtient la l 2 continuite de a x d si le symbole a x satisfait a certaines conditions

local well posedness for the maxwell schrodinger equation time local well posedness for the maxwell schrodinger equation in the coulomb gauge is studied in sobolev spaces by the contraction mapping principle the lorentz gauge and the temporal gauge cases are also treated by the gauge transform

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

on the amplitude equations for weakly nonlinear surface waves nonlocal generalizations of burgers equation were derived in earlier work by hunter contemp math vol 100 pp 185 202 ams 1989 and more recently by benzoni gavage and rosini comput math appl 57 3 4 1463 1484 2009 as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves the local in time well posedness of such equations in sobolev spaces was proved by benzoni gavage differ integr equ 22 3 4 303 320 2009 under an appropriate stability condition originally pointed out by hunter the same stability condition has also been shown to be necessary for well posedness in sobolev spaces in a previous work of the authors in collaboration with tzvetkov benzoni gavage et al in adv math 227 6 2220 2240 2011 in this article we show how the verification of hunter s stability condition follows from natural stability assumptions on the original hyperbolic boundary value problem thus avoiding lengthy computations in each particular situation we also show that the resulting amplitude equation has a hamiltonian structure when the original boundary value problem has a variational origin our analysis encompasses previous equations derived for nonlinear rayleigh waves in elasticity

variational iteration method for autonomous ordinary differential systems in this paper a new iteration technique is proposed to solve autonomous ordinary differential systems in this method general lagrange multipliers are introduced to construct correction functionals for the systems the multipliers in the functionals can be identified by the variational theory the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary initial conditions some examples are given the results reveal that the method is very effective and convenient

analytic structure of two 1d transport equations with nonlocal fluxes abstract we replace the flux term in burger s equation by two simple alternates that contain contributions depending globally on the solution in one case the term is in the form of a hyperbolic equation where the characteristic speed is nonlocal and in the other the term is in conservation form in both cases the nonanalytic is due to the presence of the hilbert transform the equations have a loose analogy to the motion of vortex sheets in particular they both form singularities in finite time in the absence of viscous effects our motivation then is to study the influence of viscosity in one case viscosity does not prevent singularity formation in the other we can prove solutions exist for all time and determine the likely weak solution as viscosity vanishes an interesting aspect of our work is that singularity formation can be viewed as the motion of singularities in the complex physical plane that reach the real axis in finite time in one case the singularity is a pole and causes the solution to blow up when it reaches the real axis in the other numerical solutions and an asymptotic analysis suggest that the weak solution contains a square root singularity that reaches the real axis in finite time and then propagates along it we hope our results will spur further interest in the role of singularities in the complex spatial plane in solutions to transport equations

on the fractional fisher information with applications to a hyperbolic parabolic system of chemotaxis we introduce new lower bounds for the fractional fisher information equipped with these bounds we study a hyperbolic parabolic model of chemotaxis and prove the global existence of solutions in certain dissipation regimes

splitting methods for the nonlocal fowler equation we consider a nonlocal scalar conservation law proposed by andrew c fowler to describe the dynamics of dunes and we develop a numerical procedure based on splitting methods to approximate its solutions we begin by proving the convergence of the well known lie formula which is an approximation of the exact solution of order one in time we next use the split step fourier method to approximate the continuous problem using the fast fourier transform and the finite difference method our numerical experiments confirm the theoretical results

ill posedness of nonlocal burgers equations nonlocal generalizations of burgers equation were derived in earlier work by hunter contemp math 1989 and more recently by benzoni gavage and rosini comput math appl 2009 as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves the local in time well posedness of such equations in sobolev spaces was proved by benzoni gavage diff int eq 2009 under an appropriate stability condition originally pointed out by hunter in this article it is shown that the latter condition is not only sufficient for well posedness in sobolev spaces but also necessary the main point of the analysis is to show that when the stability condition is violated nonlocal burgers equations reduce to second order elliptic pdes the resulting ill posedness result encompasses various cases previously studied in the literature

on bounded pseudodifferential operators in wiener spaces we aim at extending the definition of the weyl calculus to an infinite dimensional setting by replacing the phase space r2n by b2 where i h b is an abstract wiener space a first approach is to generalize the integral definition using the wigner function the symbol is then a function defined on b2 and belonging to a l1 space for a gaussian measure the weyl operator is defined as a quadratic form on a dense subspace of l2 b for example the symbol can be the stochastic extension on b2 in the sense of l gross of a function f which is continuous and bounded on h2 in the second approach this function f defined on h2 satisfies differentiability conditions analogous to the finite dimensional ones one needs to introduce hybrid operators acting as weyl operators on the variables of finite dimensional subset of h and as anti wick operators on the rest of the variables the final weyl operator is then defined as a limit and it is continuous on a l2 space under rather weak conditions it is an extension of the operator defined by the first approach we give examples of monomial symbols linking this construction to the classical pseudodifferential operators theory and other examples related to other fields or previous works on this subject

higher order resonances and instability of high frequency wkb solutions abstract this paper focuses on the destabilizing role of resonances in high frequency wkb solutions specifically we study higher order resonances associated with higher order harmonics generated by nonlinearities we give examples of systems and solutions for which such resonances generate instantaneous instabilities even though the equations linearized around the leading wkb terms are initially stable meaning in particular that the key destabilizing terms are not present in the data

modified variational iteration method nonlinear homogeneous initial value problem the modified variational iteration method is applied for analytical treatment of nonlinear homogeneous initial value problem the modified variational iteration method accelerates the convergence of the power series solution and reduces the size of work a comparison between modified variational iteration method mvim and variational iteration method vim was made the comparison enhances the use of the modified variational iteration method if we wish to obtain an approximate power series solution that converges faster to the closed form solution the method is very simple and easy

the mach stem equation and amplification in strongly nonlinear geometric optics we study highly oscillating solutions to a class of weakly well posed hyperbolic initial boundary value problems weak well posedness is associated with an amplification phenomenon of oscillating waves on the boundary in the previous works cgw14 cw14 we have rigorously justified a weakly nonlinear regime for semilinear problems in that case the forcing term on the boundary has amplitude o e 2 and oscillates at a frequency o 1 e the corresponding exact solution which has been shown to exist on a time interval that is independent of e 0 1 has amplitude o e in this paper we deal with the exact same scaling namely o e 2 forcing term on the boundary and o e solution for quasilinear problems in analogy with cgm03 this corresponds to a strongly nonlinear regime and our main result proves solvability for the corresponding wkb cascade of equations which yields existence of approximate solutions on a time interval that is independent of e 0 1 existence of exact solutions close to approximate ones is a stability issue which as shown in cgm03 highly depends on the hyperbolic system and on the boundary conditions we do not address that question here this work encompasses previous formal expansions in the case of weakly stable shock waves mr83 and two dimensional compressible vortex sheets am87 in particular we prove well posedness for the leading amplitude equation the mach stem equation of mr83 and generalize its derivation to a large class of hyperbolic boundary value problems and to periodic forcing terms the latter case is solved under a crucial nonresonant assumption and a small divisor condition

stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem we determine sharp lp estimates on the linearized solution operator about a multidimensional planar periodic wave of a system of conservation laws with viscosity yielding linearized l1 lp lp stability for all p geqq 2 and dimensions d geqq 1 and nonlinear l1 hs lp hs stability and l2 asymptotic behavior for p geqq 2 and d geqq 3 the behavior can in general be rather complicated involving both convective that is wave like and diffusive effects

multilinear pseudodifferential operators beyond calderon zygmund theory abstract we consider two types of multilinear pseudodifferential operators first we prove the boundedness of multilinear pseudodifferential operators with symbols which are only measurable in the spatial variables in lebesgue spaces these results generalise earlier work of the present authors concerning linear pseudo pseudodifferential operators secondly we investigate the boundedness of bilinear pseudodifferential operators with symbols in the hormander s m classes these results are new in the case 1 that is outwith the scope of multilinear calderon zygmund theory

toward a modified variational iteration method the variational iteration method vim attracted much attention in the past few years as a promising method for solving nonlinear differential equations it is shown in this paper that the application of vim to a special kind of nonlinear differential equations leads to calculation of unneeded terms and more time consumed in repeated calculations for series solutions a modified vim is introduced to eliminate the shortcomings and its effectiveness is illustrated by some examples

sharp estimates for pseudo differential operators with symbols of limited smoothness and commutators we consider here pseudo differential operators whose symbol x is not infinitely smooth with respect to x decomposing such symbols into four sometimes five components and using tools of paradifferential calculus we derive sharp estimates on the action of such pseudo differential operators on sobolev spaces and give explicit expressions for their operator norm in terms of the symbol x we also study commutator estimates involving such operators and generalize or improve the so called kato ponce and calderon coifman meyer estimates in various ways

nonselfadjoint schrodinger operators with singular first order coefficients schrodinger operators of the form t i grad b x 2 a x grad q x in r m are considered where a b ate real vector valued functions and q is a scalar complex valued function it is shown that t is essentially quasi m accretive in l 2 r m if 1 x2223 1 a l 4 l div a l and re q 0 the proof is elementary

periodic solutions for a 1d model with nonlocal velocity via mass transport abstract this paper concerns periodic solutions for a 1d model with nonlocal velocity given by the periodic hilbert transform there is a rich literature showing via numerics and rigorous analysis that this model presents singular behavior of solutions for instance they can blow up by forming mass concentration we develop a global well posedness theory for periodic measure initial data that allows in particular to analyze how the model evolves from those singularities our results are based on periodic mass transport theory and the abstract gradient flow theory in metric spaces developed by ambrosio et al 2005 a viscous version of the model is also analyzed and inviscid limit properties are obtained

dissipative property for higher order nonlinear schrodinger equation abstract we consider the cauchy problem for the higher order nonlinear schrodinger equation 0 1 i t u 1 2 x 2 u 1 4 x 4 u i 3 1 u 3 2 u 2 u t 1 x r u 1 x u 0 x x r where the coefficients 0 1 2 the aim of the present paper is to prove the global existence of solutions to also we find the large time decay estimates for the solutions

a new modification of the variational iteration method this paper presents a new variational iteration algorithm which provides great freedom in choosing linear operators for various nonlinear equations so that the lagrange multiplier can be effectively identified

l2 estimates for weyl quantization abstract sharp l 2 estimates for weyl quantized pseudodifferential operators are established in the framework of hormander s classes of symbols s m in particular the case of forbidden symbols is discussed

multiscale methods for shape constraints in deconvolution confidence statements for qualitative features we derive multiscale statistics for deconvolution in order to detect qualitative features of the unknown density an important example covered within this framework is to test for local monotonicity on all scales simultaneously we investigate the moderately ill posed setting where the fourier transform of the error density in the deconvolution model is of polynomial decay for multiscale testing we consider a calibration motivated by the modulus of continuity of brownian motion we investigate the performance of our results from both the theoretical and simulation based point of view a major consequence of our work is that the detection of qualitative features of a density in a deconvolution problem is a doable task although the minimax rates for pointwise estimation are very slow

global existence singularities and ill posedness for a nonlocal flux in this paper we study a one dimensional model equation with a nonlocal flux given by the hilbert transform that is related with the complex inviscid burgers equation this equation arises in different contexts to characterize nonlocal and nonlinear behaviors we show global existence local existence blow up in finite time and ill posedness depending on the sign of the initial data for classical solutions

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

on the maximal bochner riesz conjecture in the plane for p 2 we give a new estimate on the maximal bochner riesz operator in the plane for p 2 as a corollary we obtain an almost everywhere convergence result for certain bochner riesz means this work was inspired by discussions with michael christ and chris sogge

further properties of a continuum of model equations with globally defined flux to develop an understanding of singularity formation in vortex sheets we consider model equations that exhibit shared characteristics with the vortex sheet equation but are slightly easier to analyze a model equation is obtained by replacing the flux term in burgers equation by alternatives that contain contributions depending globally on the solution we consider the continuum of partial differential equationsut h u u x 1 u ux uxx 0 1 0 whereh u is the hilbert transform ofu we show that when 1 2 for 0 the solution of the equation exists for all time and is unique we also show with a combination of analytical and numerical means that the solution when 1 2 and 0 is analytic using a pseudo spectral method in space and the adams moulton fourth order predictor corrector in time we compute the numerical solution of the equation with 1 2 for various viscosities the results confirm that for 0 the solution is well behaved and analytic the numerical results also confirm that for 0 and 1 2 the solution becomes singular in finite time and finite viscosity prevents singularity formation we also present for a certain class of initial conditions solutions of the equation with 0 1 3 and 1 that become infinite for 0 in finite time

a mass transportation approach to a one dimensional fluid mechanics model with nonlocal velocity abstract we consider a one dimensional transport model with nonlocal velocity given by the hilbert transform and develop a global well posedness theory of probability measure solutions both the viscous and non viscous cases are analyzed both in original and in self similar variables we express the corresponding equations as gradient flows with respect to a free energy functional including a singular logarithmic interaction potential existence uniqueness self similar asymptotic behavior and inviscid limit of solutions are obtained in the space p 2 r of probability measures with finite second moments without any smallness condition our results are based on the abstract gradient flow theory developed by ambrosio et al 2005 2 an important byproduct of our results is that there is a unique up to invariance and translations global in time self similar solution with initial data in p 2 r which was already obtained by deslippe et al 2004 17 and biler et al 2010 6 by different methods moreover this self similar solution attracts all the dynamics in self similar variables the crucial monotonicity property of the transport between measures in one dimension allows to show that the singular logarithmic potential energy is displacement convex we also extend the results to gradient flow equations with negative power law locally integrable interaction potentials

solving nonlinear partial differential equations using the modified variational iteration pade technique in this paper the modified variational iteration method mvim is reintroduced with the enhancement of pade approximants to lengthen the interval of convergence of vim or mvim when used alone in solving nonlinear problems kdv mkdv burger s and lax s equations are used as examples to illustrate the effectiveness and convenience of the proposed technique

existence of a local smooth solution in probability to the stochastic euler equations in r3 abstract we establish the existence of a local smooth solution of the stochastic euler equations in r 3 probabilistic estimate of the random time interval for the existence of a local solution is expressed in terms of expected values of the initial data and the random noise there are numerous works on the stochastic euler equations in a two dimensional domain even for the deterministic euler equations in a three dimensional domain the only results are concerned with the local existence of smooth solutions our goal is to extend such local existence result to the stochastic equations

on the well posedness of various one dimensional model equations for fluid motion abstract we consider 1d equations with nonlocal velocity of the form w t u w x u x w w where the nonlocal velocity u is given by 1 u 1 x x w 0 or 2 u h w h is the hilbert transform in this paper we address several local well posedness results with blow up criteria for smooth initial data we then establish the global well posedness by using the blow up criteria

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

fractional nonlinear schrodinger equation of order α 0 1 abstract we consider the cauchy problem for the fractional nonlinear schrodinger equation i t u 1 x u u 2 u t 0 x r u 0 x u 0 x x r where r the order of the fractional derivative 0 1 we obtain the large time asymptoitic behavior of solutions which has a logarithmic phase modifications for large time comparing with the linear problem

harmonic approximation of difference operators for a general class of difference operators he te ve on l2 ez d where ve is a multi well potential and e is a small parameter we analyze the asymptotic behavior as e 0 of the low lying eigenvalues and eigenfunctions we show that the first n eigenvalues of he converge to the first n eigenvalues of the direct sum of harmonic oscillators on rd located at the several wells our proof is microlocal

modified variational iteration method for boussinesq equation this paper applies the modified variational iteration method to solve a class of nonlinear partial differential equations boussinesq equation is used as a case study to illustrate the simplicity and effectiveness of the method comparison between variational iteration method and adomian decomposition method is made

global solutions for a supercritical drift diffusion equation abstract we study the global existence of solutions to a one dimensional drift diffusion equation with logistic term generalizing the classical parabolic elliptic keller segel aggregation equation arising in mathematical biology in particular we prove that there exists a global weak solution if the order of the fractional diffusion 1 c 1 2 where c 1 0 is an explicit constant depending on the physical parameters present in the problem chemosensitivity and strength of logistic damping furthermore in the range 1 c 2 2 with 0 c 2 c 1 the solution is globally smooth let us emphasize that when 1 the diffusion is in the supercritical regime

well posedness of the muskat problem with h2 initial data abstract we study the dynamics of the interface between two incompressible fluids in a two dimensional porous medium whose flow is modeled by the muskat equations for the two phase muskat problem we establish global well posedness and decay to equilibrium for small h 2 perturbations of the rest state for the one phase muskat problem we prove local well posedness for h 2 initial data of arbitrary size finally we show that solutions to the muskat equations instantaneously become infinitely smooth

preface the variational iteration method reliable efficient and promising in the last few decades considerable work has been invested in developing new methods for analytical and numerical solutions of differential and integral equations linear or nonlinear there has been a great deal of research work done to address the issues of nonlinearity and singularity phenomena that arise in many scientific and engineering problems moreover there has been a great need for effective algorithms to avoid the onerous work required by traditional techniques as a result of this need many analytical and numerical tools such as grid points techniques perturbation techniques spline solutions the adomian method and others have been developed however each of these methods suffers from one or more limitations the grid points techniques define the solution at grid points only the spline solution requires restrictions on boundary points the perturbation method suffers from a high computational workload especially when the degree of nonlinearity increases in addition the adomian method suffers from the complication of the algorithms needed to calculate adomian polynomials that are necessary for nonlinear problems this special issue of computers and mathematics with applications conveys a strong reliable efficient and promising development ji huan he developed the variational iteration method vim in 1999 in this method the equations are initially approximated with possible unknowns a correction functional is established by a general lagrange multiplier which can be identified optimally via the variational theory the method gives rapidly convergent successive approximations of the exact solution the vim has no restrictions or unrealistic assumptions such as linearization or small parameters that are used for nonlinear operators the vim handles linear and nonlinear problems in a like manner the convergence concept has been proved a substantial amount of research work has been directed to the study of the variational iteration method by many well known researchers several studies have been conducted to compare the vim method with existing techniques and it was shown by all that this method gives exact solutions faster than other methods the method has been applied in a wide variety of scientific and engineering applications these works confirmed the fact that this methodology is reliable efficient and promising as well as having a wider applicability included herein is a collection of original refereed research papers by well established researchers in the field of applied mathematics we hope that this issue will prove to be a timely and valuable reference for researchers in this area special thanks go to the referees for their valuable work we here thank dr ervin y rodin for providing us with the opportunity to produce this special issue on this exciting method

bialgebras for structural operational semantics bialgebras and distributive laws are an abstract categorical framework to study various flavors of structural operational semantics this paper aims to introduce the reader to the basics of bialgebras for operational semantics and to sketch the state of the art in this research area

a new approach for a nonlocal nonlinear conservation law we describe an approach to nonlocal nonlinear advection in one dimension that extends the usual pointwise concepts to account for nonlocal contributions to the flux the spatially nonlocal operators we consider do not involve derivatives instead the spatial operator involves an integral that in a distributional sense reduces to a conventional nonlinear advective operator in particular we examine a nonlocal inviscid burgers equation which gives a basic form with which to characterize properties associated with well posedness and to examine numerical results for specific cases we describe the connection to a nonlocal viscous regularization which mimics the viscous burgers equation in an appropriate limit we present numerical results that compare the behavior of the nonlocal burgers formulation to the standard local case the developments presented in this paper form the preliminary building blocks upon which to build a theory of nonlocal advection phenomena consistent within the peridynamic theo

global existence for some transport equations with nonlocal velocity abstract in this paper we study transport equations with nonlocal velocity fields with rough initial data we address the global existence of weak solutions of a one dimensional model of the surface quasi geostrophic equation and the incompressible porous media equation and one dimensional and n dimensional models of the dissipative quasi geostrophic equations and the dissipative incompressible porous media equation

boundedness of pseudo differential operators of class _ 0 0 we study the lp boundedness of pseudo differential operators with the support of their symbols being contained in e x rn where e is a compact subset of rn and their symbols have derivatives with respect to x only up to order k in the holder continuous sense where k n 2 the case 1 n p the case 2 n 2 k nlp the case 1 nlp k nt2 the case 2 p oo

on the generalized buckley leverett equation in this paper we study the generalized buckley leverett equation with nonlocal regularizing terms one of these regularizing terms is diffusive while the other one is conservative we prove that if the regularizing terms have order higher than one combined there exists a global strong solution for arbitrarily large initial data in the case where the regularizing terms have combined order one we prove the global existence of solution under some size restriction for the initial data moreover in the case where the conservative regularizing term vanishes regardless of the order of the diffusion and under a certain hypothesis on the initial data we also prove the global existence of a strong solution and we obtain some new entropy balances finally we provide numerics suggesting that if the order of the diffusion is 0 1 a finite time blow up of the solution is possible

on a 1d nonlocal transport equation with nonlocal velocity and subcritical or supercritical diffusion abstract we study a 1d transport equation with nonlocal velocity with subcritical or supercritical dissipation for all data in the weighted sobolev space h k w l where k max 0 3 2 and w is a given family of muckenhoupt weights we prove a global existence result in the subcritical case 1 2 we also prove a local existence theorem for large data in h 2 w l in the supercritical case 0 1 the proofs are based on the use of the weighted littlewood paley theory interpolation along with some new commutator estimates

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 weak solution and blow up criterion of the general ericksen leslie system for nematic liquid crystal flows abstract in this paper we investigate the three dimensional general ericksen leslie e l system with ginzburg landau type approximation modeling nematic liquid crystal flows first by overcoming the difficulties from lack of maximum principle for the director equation and high order nonlinearities for the stress tensor we prove existence of global in time weak solutions under physically meaningful boundary conditions and suitable assumptions on the leslie coefficients which ensures that the total energy of the e l system is dissipated moreover for the e l system with periodic boundary conditions we prove the local well posedness of classical solutions under the so called parodi s relation and establish a blow up criterion in terms of the temporal integral of both the maximum norm of the curl of the velocity field and the maximum norm of the gradient of the liquid crystal director field

comparing operational models of name passing process calculi we study three operational models of name passing process calculi coalgebras on pre sheaves indexed labelled transition systems and history dependent automata the coalgebraic model is considered both for presheaves over the category of finite sets and injections and for its subcategory of atomic sheaves known as the schanuel topos each coalgebra induces an indexed labelled transition system such transition systems are characterised relating the coalgebraic approach to an existing model of name passing further we consider internal labelled transition systems within the sheaf topos and axiomatise a class that is in precise correspondence with the coalgebraic and the indexed labelled transition system models by establishing and exploiting the equivalence of the schanuel topos with a category of named sets these internal labelled transition systems are also related to the theory of history dependent automata

a pointwise estimate for fractionary derivatives with applications to partial differential equations this article emphasizes the role played by a remarkable pointwise inequality satisfied by fractionary derivatives in order to obtain maximum principles and lp decay of solutions of several interesting partial differential equations in particular there are applications to quasigeostrophic flows in two space variables with critical viscosity that model the eckman pumping see baroud ch n plapp b b she z s swinney h l 2002 phys rev lett 88 114501 and constantin p 2002 phys rev lett 89 184501

weakly nonlinear surface waves and subsonic phase boundaries the aim of this work is twofold in a first abstract part it is shown how to derive an asymptotic equation for the amplitude of weakly nonlinear surface waves associated with neutrally stable undercompressive shocks the amplitude equation obtained is a non local generalization of burgers equation for which an explicit stability condition is exhibited this is an extension of earlier results by j hunter the second part is devoted to ideal subsonic phase boundaries which were shown by the first author to be associated with linear surface waves the amplitude equation for corresponding weakly non linear surface waves is calculated explicitly and the stability condition is investigated analytically and numerically

finite time singularities in a 1d model of the quasi geostrophic equation in this paper we study 1d equations with nonlocal flux these models have resemblance of the 2d quasi geostrophic equation we show the existence of singularities in finite time and construct explicit solutions to the equations where the singularities formed are shocks for the critical viscosity case we show formation of singularities and global existence of solutions for small initial data

well posedness of the water waves equations 1 1 presentation of the problem the water waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect incompressible irrotational fluid under the influence of gravity in this paper we restrict our attention to the case when the surface is a graph parameterized by a function x where t denotes the time variable and x xi xd rd the horizontal spatial variables the method developed here works equally well for any integer d 1 but the only physically relevant cases are of course d 1 and d 2 the layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time independent function b x we denote by q t the fluid domain at time t the incompressibility of the fluids is expressed by 1 1 div v 0 in qu t 0

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

asymptotics frequency modulation and low regularity ill posedness for canonical defocusing equations in a recent paper kenig ponce and vega study the low regularity behavior of the focusing nonlinear schrodinger nls focusing modified korteweg de vries mkdv and complex korteweg de vries kdv equations using soliton and breather solutions they demonstrate the lack of local well posedness for these equations below their respective endpoint regularities in this paper we study the defocusing analogues of these equations namely defocusing nls defocusing mkdv and real kdv all in one spatial dimension for which suitable soliton and breather solutions are unavailable we construct for each of these equations classes of modified scattering solutions which exist globally in time and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts these solutions are used to demonstrate lack of local well posedness in certain sobolev spaces in the sense that the dependence of solutions upon initial data fails to be uniformly continuous in particular we show that the mkdv flow is not uniformly continuous in the l 2 topology despite the existence of global weak solutions at this regularity finally we investigate the kdv equation at the endpoint regularity h and construct solutions for both the real and complex kdv equations the construction provides a nontrivial time interval t t and a locally lipschitz continuous map taking the initial data in h to a distributional solution u c 0 t t h which is uniquely defined for all smooth data the proof uses a generalized miura transform to transfer the existing endpoint regularity theory for mkdv to kdv

a congruence rule format for name passing process calculi abstract we introduce a gsos like rule format for name passing process calculi specifications in this format correspond to theories in nominal logic the intended models of such specifications arise by initiality from a general categorical model theory for operational semantics given in this rule format a natural behavioural equivalence a form of open bisimilarity is a congruence

continuity properties of the solution map for the generalized reduced ostrovsky equation abstract it is shown that the data to solution map for the generalized reduced ostrovsky gro equation is not uniformly continuous on bounded sets in sobolev spaces on the circle with exponent s 3 2 considering that for this range of exponents the gro equation is well posed with continuous dependence on initial data this result makes the continuity of the solution map an optimal property however if a weaker h r topology is used then it is shown that the solution map becomes holder continuous in h s

exact solutions of some nonlinear partial differential equations using the variational iteration method linked with laplace transforms and the pade technique in this paper the variational iteration method vim is reintroduced with laplace transforms and the pade technique treatment to obtain closed form solutions of nonlinear equations some examples including the coupled burger s equation compacton k n n equation zakharov kuznetsov zk n n equation and kdv and mkdv equations are given to show the effectiveness of the coupled vim laplace pade and vim pade techniques

an aggregation equation with a nonlocal flux abstract in this paper we study an aggregation equation with a general nonlocal flux we study the local well posedness and some conditions ensuring global existence we are also interested in the differences arising when the nonlinearity in the flux changes thus we perform some numerics corresponding to different convexities for the nonlinearity in the equation

sharp local well posedness results for the nonlinear wave equation this article is concerned with local well posedness of the cauchy problem for second order quasilinear hyperbolic equations with rough initial data the new results obtained here are sharp in low dimension

the tanh method for traveling wave solutions of nonlinear equations in this work we employ the tanh method for traveling wave solutions of nonlinear equations the study is extended to equations that do not have tanh polynomial solutions the efficiency of the method is demonstrated by applying it for a variety of selected equations