extension of the berlekamp massey algorithm to n dimensions abstract we present an algorithm for finding a minimal set of linear recurring relations which are valid for a given n dimensional array over any field where the minimality is defined with respect to the partial order over the n dimensional lattice the algorithm is an extension of our two dimensional version of the berlekamp massey algorithm to more than two dimensions the n dimensional theory is based on more general concepts which can be reduced into those of the two dimensional theory in the previous paper in a typical case the resulting set of polynomials characterizing the minimal linear recurring relations proves to be a groebner basis of the ideal defined by the array and consequently the structure of an n dimensional linear feedback shift register with the minimum number of storage devices which can generate the array is determined by it

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

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

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

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

on the minimal partial realization of 2d discrete linear shift invariant systems the partial realization problem for 2d discrete linear shift invariant systems is discussed the authors attempt to establish a reasonable definition of minimality and propose a method for finding the degree and the system function of a minimal realization of a 2d linear system characterized by a given 2d impulse response array the method used is based on the 2d berlekamp massey algorithm which has a close connection with the 2d hankel matrix derived from the 2d array while it compares in efficiency with other methods for identification and approximation of 2d linear systems it gives a novel approach to the problem in the sense that one does not need any knowledge or assumptions concerning the system degree

topological sensitivity analysis in heterogeneous anisotropic elasticity problem theoretical and computational aspects the topological sensitivity analysis for the heterogeneous and anisotropic elasticity problem in two dimensions is performed in this work the main result of the paper is an analytical closed form of the topological derivative for the total potential energy of the problem this derivative displays the sensitivity of the cost functional the energy in this case when a small singular perturbation is introduced in an arbitrary point of the domain in this case we consider a small disc with a completely different elastic material full mathematical justification for the derived formula and derivations of precise estimates for the remainders of the topological asymptotic expansion are provided finally the influence of the heterogeneity and anisotropy is shown through some numerical examples of structural topology optimization

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

some convergence results in perimeter controlled topology optimization abstract the computation of optimal topologies of elastic continuum structures using a constraint on the perimeter is investigated predicting macroscopic black white topologies without the use of homogenization techniques this approach is presently one of the most attractive approaches in topology optimization mathematical justifications are given for both the relaxation of the discrete value constraint on the design variable and for the finite element discretizations it turns out that the way in which the perimeter has been calculated to date the numerical results will not approximate the intended original problem but one with a taxi cab perimeter which measures lengths of structural edges after projection onto the coordinate axes

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

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

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

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 algorithm for topology optimization using a level set method the level set method has been recently introduced in the field of shape optimization enabling a smooth representation of the boundaries on a fixed mesh and therefore leading to fast numerical algorithms however most of these algorithms use a hamilton jacobi equation to connect the evolution of the level set function with the deformation of the contours and consequently they can hardly create new holes in the domain at least in 2d in this work we propose an evolution equation for the level set function based on a generalization of the concept of topological gradient this results in a new algorithm allowing for all kinds of topology changes

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

filters in topology optimization in this article a modified filtered version of the minimum compliance topology optimization problem is studied the direct dependence of the material properties on its pointwise density is replaced by a regularization of the density field by the mean of a convolution operator in this setting it is possible to establish the existence of solutions moreover convergence of an approximation by means of finite elements can be obtained this is illustrated through some numerical experiments the filtering technique is also shown to cope with two important numerical problems in topology optimization checkerboards and mesh dependent designs copyright 2001 john wiley sons ltd

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

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

filters in topology optimization based on helmholtz type differential equations the aim of this paper is to apply a helmholtz type partial differential equation as an alternative to standard density filtering in topology optimization problems previously this approach has been successfully applied as a sensitivity filter the usual filtering techniques in topology optimization require information about the neighbor cells which is difficult to obtain for fine meshes or complex domains and geometries the complexity of the problem increases further in parallel computing when the design domain is decomposed into multiple non overlapping partitions obtaining information from the neighbor subdomains is an expensive operation the proposed filter technique requires only mesh information necessary for the finite element discretization of the problem the main idea is to define the filtered variable implicitly as a solution of a helmholtz type differential equation with homogeneous neumann boundary conditions the properties of the filter are demonstrated for various 2d and 3d topology optimization problems in linear elasticity solved on serial and parallel computers copyright 2010 john wiley sons ltd

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

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

rational approximation of 2 d linear discrete systems in this paper we present an efficient procedure to obtain a rational model for a 2 d linear shift invariant discrete system using first and second order data from it this procedure is a modification of the nonlinear least squares approximation and it generalizes the pade approximants and the spectral estimation modeling procedures the parameters of the approximating filter are obtained by solving a system of linear equations by means of an efficient recursive algorithm which is developed using the relation of the approximation problem with the theory of orthogonal polynomials on the unit bidisk we discuss some of the algebraic properties of the solution and apply them to define cases for which the bibo stability of the approximating filters is ensured the proposed procedure finds applications in the design and stabilization of 2 d recursive digital filters and in the autoregressive moving average arma modeling of stationary random fields

a homogenization method for shape and topology optimization abstract shape and topology optimization of a linearly elastic structure is discussed using a modification of the homogenization method introduced by bendsoe and kikuchi together with various examples which may justify validity and strength of the present approach for plane structures

a consistent relaxation of optimal design problems for coupling shape and topological derivatives in this article we introduce and analyze a general procedure for approximating a black and white shape and topology optimization problem with a density optimization problem allowing for the presence of grayscale regions our construction relies on a regularizing operator for smearing the characteristic functions involved in the exact optimization problem and on an interpolation scheme which endows the intermediate density regions with fictitious material properties under mild hypotheses on the smoothing operator and on the interpolation scheme we prove that the features of the approximate density optimization problem material properties objective function etc converge to their exact counterparts as the smoothing parameter vanishes in particular the gradient of the approximate objective functional with respect to the density function converges to either the shape or the topological derivative of the exact objective these results shed new light on the connections between these two different notions of sensitivities for functions of the domain and they give rise to different numerical algorithms which are illustrated by several experiments

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

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

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

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

damage and fracture evolution in brittle materials by shape optimization methods this paper is devoted to a numerical implementation of the francfort marigo model of damage evolution in brittle materials this quasi static model is based at each time step on the minimization of a total energy which is the sum of an elastic energy and a griffith type dissipated energy such a minimization is carried over all geometric mixtures of the two healthy and damaged elastic phases respecting an irreversibility constraint numerically we consider a situation where two well separated phases coexist and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy in the context of interface variations hadamard method and using a steepest descent algorithm we compute local minimizers of this quasi static damage model initially the damaged zone is nucleated by using the so called topological derivative we show that when the damaged phase is very weak our numerical method is able to predict crack propagation including kinking and branching several numerical examples in 2d and 3d are discussed

the shape and topological optimizations connection abstract in 1973 jean cea introduced a topological shape optimization algorithm based on a fixed point method the recent topological gradient concept provides a mathematical justification of cea s powerful method the aim of the topological gradient is to compute the sensitivity of a cost function when we drill a little hole in the domain some optimality conditions are derived when classical shape gradient and topological gradient are used simultaneously and numerical results are shown both in mechanics and electromagnetics

topology optimization of non linear elastic structures and compliant mechanisms abstract a linear elastic response is assumed in most structural topology optimization problems while this assumption is valid for a wide variety of problems it is not valid for structures undergoing large displacements the elastic structural analysis used here accommodates geometric and material non linearities the material density field is filtered to enforce a length scale on the field variation and is penalized to remove less effective intermediate densities the filtering scheme is embedded in the structural analysis to resolve the non existent solution to the solid void topology problem in this way we know precisely what optimization problem is being solved the structural topology optimization formulation is also used to design compliant mechanisms

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

lattice parameter autoregressive modeling of two dimensional fields part i the quarter plane case quarter plane autoregressive lattice parameter modeling of two dimensional fields is developed as a natural extension of one dimensional lattice parameter theory starting with a given data field four prediction error fields are generated their linear combination is used to define and calculate 2 d lattice parameter reflection factors for successive lattice parameter model stages in addition to developing the basic theory the presentation includes the relationships between the quarter plane two dimensional transfer function and the lattice parameters the lattice parameter synthesis model for generating a two dimensional field from a random field input and lattice parameter stability conditions the theory has been confirmed by computer simulations the extension of this lattice parameter theory for the asymmetric half plane case is to be presented in a subsequent paper

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

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

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