mathematical and numerical analysis of a simplified time dependent viscoelastic flow a time dependent model corresponding to an oldroyd b viscoelastic fluid is considered the convective terms being disregarded global existence in time is proved in banach spaces provided the data are small enough using the implicit function theorem and a maximum regularity property for a three fields stokes problem a finite element discretization in space is then proposed existence of the numerical solution is proved for small data so as a priori error estimates using again an implicit function theorem

abstract quasilinear parabolic equations on considere des problemes paraboliques quasi lineaires abstraits p 0 u t a t u t u t f t u t t 0 u o u 0 ou u t est une fonction a valeurs dans un espace de banach x on donne des theoremes d existence locale et d unicite pour des solutions classiques et strictes de p 0

fuzzy nonlinear regression with fuzzified radial basis function network a fuzzified radial basis function network frbfn is a kind of fuzzy neural network that is obtained by direct fuzzification of the well known neural model rbfn a frbfn contains fuzzy weights and can handle fuzzy in fuzzy out data this paper shows that a frbfn can also be interpreted as a kind of fuzzy expert system hence it owns the advantages of simple structure and clear physical meaning some metrics for fuzzy numbers have been extended to the metrics for n dimensional fuzzy vectors which are applicable to computations in frbfns the corresponding metric spaces for n dimensional fuzzy vectors are proved to be complete further frbfns are proved to be able to act as universal function approximators for any continuous fuzzy function defined on a compact set this paper applies the proposed frbfn to nonparametric fuzzy nonlinear regression problems for multidimensional lr type fuzzy data fuzzy nonlinear regression with frbfns can be formulated as a nonlinear mathematical programming problem two training algorithms are proposed to quickly solve the two types of problems under different criteria and constraint conditions namely the two stage and bp back propagation training algorithms simulation studies are carried out to verify the feasibility and demonstrate the advantages of the proposed approaches

on the quasi geostrophic equations on compact closed surfaces in r3 abstract the quasi geostrophic equations on compact surfaces in r 3 without boundary are studied by means of the theory of quasi linear parabolic evolution equations based on maximal l p regularity it is shown that the problem is globally strongly well posed in l q the solutions regularize instantly and form a global semiflow in the proper state manifold and as time goes to infinity a solution converges exponentially to a constant in the topology of the state space

quenching for a one dimensional fully nonlinear parabolic equation in detonation theory we study and describe the quenching phenomenon for the fully nonlinear parabolic equation u t frac 1 2 u x 2 f c u u xx rm ln u quad x in 0 l t 0 which for f s rm ln e s 1 s represents the evolution of the perturbations of the zel dovich von neuman doering square wave occurring during a detonation in a duct in the general case the function f mathbb r rightarrow mathbb r is smooth and satisfies the parabolicity condition f s 0 in mathbb r c and l are positive constants and we impose neumann boundary conditions ux 0 t ux l t 0 for t 0 and take initial data u x 0 0 with inverse bell shaped form the phenomenon of quenching is characterized by the existence of a finite time t at which the solution u ceases to exist as a classical solution because rm min x u x t rightarrow 0 as t rightarrow t then the equation degenerates and forms a singularity at the level u 0 due to the presence of the logarithmic zero order term we

quasilinear evolutionary equations and continuous interpolation spaces in this paper we analyze the abstract parabolic evolutionary equations dt u x a u u f u h t u 0 x in continuous interpolation spaces allowing a singularity as t 0 here dt denotes the time derivative of order 0 2 we first give a treatment of fractional derivatives in the spaces lp 0 t x and then consider these derivatives in spaces of continuous functions having at most a prescribed singularity as t 0 the corresponding trace spaces are characterized and the dependence on is demonstrated via maximal regularity results on the linear equation dt u x au f u 0 x we arrive at results on existence uniqueness and continuation on the quasilinear equation finally an example is presented

motion of sets by curvature and derivative of capacity potential abstract we study a geometric flow where the motion of a set is driven by the mean curvature of its boundary and the normal derivative of its capacity potential we establish local well posedness and propose two possible weak formulations that exist after singularities

nonlinear analytic semiflows in this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems x t f x t and quasilinear initial value problems x t a x t x t f x t this theory extends the theory of daprato and grisvard of 1979 and shows how various properties like analyticity of solutions can be derived as a direct corollary of the existence theorem

on uniqueness of weak solutions for the thin film equation abstract in any number of space variables we study the cauchy problem related to the fourth order degenerate diffusion equation s h h h 0 this equation derived from a lubrication approximation also models the surface tension dominated flow of a thin viscous film in the hele shaw cell our focus is on uniqueness of weak solutions in the complete wetting regime when a zero contact angle between liquid and solid is imposed in this case we transform the problem by zooming into the free boundary and look at small lipschitz perturbations of a quadratic stationary solution in the perturbational setting the main difficulty is to construct scale invariant function spaces in such a way that they are compatible with the structure of the nonlinear equation here we rely on the theory of singular integrals in spaces of homogeneous type to obtain linear estimates in these functions spaces which provide optimal conditions on the initial data under which a unique solution exists in fact this solution can be used to define a class of functions in which the original initial value problem has a unique weak solution moreover we show that the interface between empty and wetted regions is an analytic hypersurface in time and space

well posedness and stability results for a quasilinear periodic muskat problem abstract we study the muskat problem describing the spatially periodic motion of two fluids with equal viscosities under the effect of gravity in a vertical unbounded two dimensional geometry we first prove that the classical formulation of the problem is equivalent to a nonlocal and nonlinear evolution equation expressed in terms of singular integrals and having only the interface between the fluids as unknown secondly we show that this evolution equation has a quasilinear structure which is at a formal level not obvious and we also disclose the parabolic character of the equation exploiting these aspects we establish the local well posedness of the problem for arbitrary initial data in h s s with s 3 2 2 determine a new criterion for the global existence of solutions and uncover a parabolic smoothing property besides we prove that the zero steady state solution is exponentially stable

the gradient flow of o hara s knot energies jun o hara invented a family of knot energies e j p j p in 0 infty o hara in topology hawaii honolulu hi 1990 world science publication river edge 1992 we study the negative gradient flow of the sum of one of the energies e alpha e alpha 1 alpha in 2 3 and a positive multiple of the length showing that the gradients of these knot energies can be written as the normal part of a quasilinear operator we derive short time existence results for these flows we then prove long time existence and convergence to critical points

on the flow of non axisymmetric perturbations of cylinders via surface diffusion we study the surface diusion ow acting on a class of general non axisymmetric perturbations of cylinders cr in ir 3 using tools from parabolic theory on uniformly regular manifolds and maximal regularity we establish existence and uniqueness of solutions to surface diusion ow starting from spatially unbounded surfaces dened over cr via scalar height functions which are uniformly bounded away from the central cylindrical axis addition ally we show that cr is normally stable with respect to 2 axially periodic perturbations if the radius r 1 and unstable if 0 r 1 stability is also shown to hold in settings with axial neumann boundary conditions

recognition of poly slender context free languages by trellis automata abstract a poly slender context free language is a context free language whose number of words of length n is polynomially bounded its structure has been thoroughly characterized by ilie rozenberg and salomaa thanks to this characterization we show that every poly slender context free language is recognizable by a trellis automaton if the languages of words with as many a as b and at most a constant number of alternations between a and b are recognizable by trellis automata then we construct a family of trellis automata that recognizes such languages

multiplicity results for interfaces of ginzburg landau allen cahn equations in periodic media the ginzburg landau allen cahn equation is a variational model for phase coexistence and for other physical problems it contains a term given by a kinetic part of elliptic type plus a double well potential we assume that the functional depends on the space variables in a periodic way we show that given a plane with rational normal there are minimal solutions satisfying the following properties these solutions are asymptotic to the pure phases and are separated by an interface the convergence to the pure phases is exponentially fast the interface lies at a finite distance m from the chosen plane where m is a universal constant furthermore these solutions satisfy some monotonicity properties with respect to integer translations namely integer translations are always comparable to the function we then show that all the interfaces of the global periodic minimizers satisfy similar monotonicity and plane like properties we also consider the case of possibly irrationally oriented planes we show that either there is a one parameter family of minimizers whose graphs provide a field of extremals or there are at least two solutions one which is a minimizer and another one which is not these solutions also have interfaces bounded by a universal constant they enjoy monotonicity properties with respect to integer translations and the nonminimal solutions are trapped inside a gap of the lamination induced by the minimizers

invariant manifolds and bifurcation for quasilinear reaction diffusion systems in this paper we consider parameter dependent quasilinear reaction diffusion equations on a bounded domain of r it is our purpose to study the dynamical behavior of solutions to these equations in particular we are interested in the behavior of the solution set if the parameter is changed to be more precise let us assume that sz c r is a bounded domain having a smooth boundary then we consider the parameter dependent equation

global solutions of abstract quasilinear parabolic equations abstract local and global existence and uniqueness for strict solutions of abstract quasilinear parabolic equations are studied applications to quasilinear parabolic partial differential equations are also given

gevrey regularity for a class of dissipative equations with applications to decay abstract in this paper following the techniques of foias and temam we establish gevrey class regularity of solutions to a class of dissipative equations with a general quadratic nonlinearity and a general dissipation including fractional laplacian the initial data is taken to be in besov type spaces defined via caloric extension we apply our result to the navier stokes equations the surface quasi geostrophic equations the kuramoto sivashinsky equation and the barotropic quasi geostrophic equation consideration of initial data in critical regularity spaces allow us to obtain generalizations of existing results on the higher order temporal decay of solutions to the navier stokes equations in the 3d case we extend the class of initial data where such decay holds while in 2d we provide a new class for such decay similar decay result and uniform analyticity band on the attractor is also proven for the sub critical 2d surface quasi geostrophic equation

classical solutions for diffusion induced grain boundary motion we prove existence and uniqueness of classical solutions for the motion of hypersurfaces driven by mean curvature and diffusion of a solute along the surface this free boundary problem involves solving a coupled system of nonlinear partial differential equations

optimal regularity for a class of singular abstract parabolic equations a general class of singular abstract cauchy problems is considered which naturally arises in applications to certain free boundary problems existence of an associated evolution operator characterizing its solutions is established and is subsequently used to derive optimal regularity results the latter are well known to be important basic tools needed to deal with corresponding nonlinear cauchy problems such as those associated to free boundary problems

multiphase thermomechanics with interfacial structure 2 evolution of an isothermal interface paper 1 1988 g 1 of this series began an investigation whose goal is a thermomechanics of two phase continua based on gibbs s notion of a sharp phase interface endowed with thermomechanical structure in that paper a balance law balance of capillary forces was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics the chief results are thermodynamic restrictions on constitutive equations exact and approximate free boundary conditions at the interface and a hierarchy of free boundary problems the simplest versions of these problems the mullins sekerka problems are essentially the classical stefan problem with the free boundary condition u 0 for the temperature replaced by the condition u h k where k is the curvature of the free boundary and h 0 is a material constant this dependence on curvature renders the problem difficult and apart from numerical studies involving linearization stability there are almost no supporting theoretical results

local existence and regularity for a class of degenerate parabolic equations on etudie des equations paraboliques scalaires a une variable d espace de la forme u t f u u x uu xx ou f f u p s est une fonction lisse de u p s r 3 avec f s strictement positive

time regularity and long time behavior of parabolic p laplace equations on infinite graphs abstract we consider the so called discrete p laplacian a nonlinear difference operator that acts on functions defined on the nodes of a possibly infinite graph we study the corresponding cauchy problem and identify the generator of the associated nonlinear semigroups we prove higher order time regularity of the solutions we investigate the long time behavior of the solutions and discuss in particular finite extinction time and conservation of mass namely on one hand for small p if an infinite graph satisfies some isoperimetric inequality then the solution to the parabolic p laplace equation vanishes in finite time on the other hand for large p these parabolic p laplace equations always enjoy conservation of mass

analyticity of solutions for quasilinear wave equations and other quasilinear systems we prove the persistence of analyticity for classical solution of the cauchy problem for quasilinear wave equations with analytic data our results show that the analyticity of solutions stated by the cauchy kowalewski and ovsiannikov nirenberg theorems lasts till a classical solution exists moreover they show that if the equation and the cauchy data are analytic only in a part of space variables then a classical solution also is analytic in these variables the approach applies to other quasilinear equations and implies the persistence of the space analyticity and the partial space analyticity of their classical solutions

quasilinear evolution equations and parabolic systems it is shown that general quasilinear parabolic systems possess unique maximal classical solutions for sufficiently smooth initial values provided the boundary conditions are time independent moreover it is shown that in the autonomous case these equations generate local semi flows on appropriate sobolev spaces our results apply in particular to the case of prescribed boundary values dirichlet boundary conditions

on the abstract cauchy problem of parabolic type in spaces of continuous functions on etudie l existence l unicite et les proprietes de regularite des solutions du probleme de cauchy u t au r f t o t t u o u 0 ou a d a ce e est un operateur lineaire clos dans l espace de banach e

geometric flows for quadrature identities a new geometric flow describing the motion of quadrature surfaces is introduced this characterization enables us to study quadrature surfaces through the investigation of the flow it is proved that the flow is uniquely solvable under the geometric condition that the initial surface has positive mean curvature as a consequence a bifurcation criterion for quadrature surfaces is obtained in fact we study a generalized flow which includes the geometric flow and the hele shaw flow as special cases

regularity of solutions to a free boundary problem modeling tumor growth by stokes equation in this paper we investigate regularity of solutions to a free boundary problem modeling tumor growth in fluid like tissues the model equations include a quasi stationary diffusion equation for the nutrient concentration and a stokes equation with a source representing the proliferation density of the tumor cells subject to a boundary condition with stress tensor effected by surface tension this problem is a fully nonlinear problem involving nonlocal terms based on the employment of the functional analytic method and the theory of maximal regularity we prove that the free boundary of this problem is real analytic in temporal and spatial variables for initial data of less regularity

existence of analytic solutions for the classical stefan problem we prove that under mild regularity assumptions on the initial data the two phase classical stefan problem admits a unique solution that is analytic in space and time

kahler non collapsing eigenvalues and the calabi flow abstract we first prove a new compactness theorem of kahler metrics which confirms a prediction in 17 then we establish several eigenvalue estimates along the calabi flow combining the compactness theorem and these eigenvalue estimates we generalize the method developed for the kahler ricci flow in 22 to obtain several new small energy theorems of the calabi flow

regularity of the free boundary for the one phase hele shaw problem abstract in this paper we prove that in a local neighborhood lipschitz continuous free boundary of a solution of the one phase hele shaw problem is indeed smooth if the solution is lipschitz continuous and non degenerate in the neighborhood

ostwald ripening in two dimensions the rigorous derivation of the equations from the mullins sekerka dynamics we consider a dilute mixture of a finite number of particles and we are interested in the coarsening of the spatial distribution in two space dimensions under mullins sekerka dynamics under the appropriate scaling hypotheses we associate radii and centers to each particle and derive equations for the whole evolution

parabolic equations for curves on surfaces part i curves with p integrable curvature this is the first of a two part paper in which we develop a theory of parabolic equations for curves on surfaces which can be applied to the so called curve shortening of flow by mean curvature problem as well as to a number of models for phase transitions in two dimensions we introduce a class of equations for which the initial value problem is solvable for initial data with p integrable curvature and we also give estimates for the rate at which the p norms of the curvature must blow up if the curve becomes singular in finite time a detailed discussion of the way in which solutions can become singular and a method for continuing the solution through a singularity will be the subject of the second part

evolution of plane curves driven by a nonlinear function of curvature and anisotropy in this paper we study evolution of plane curves satisfying a geometric equation v beta k nu where v is the normal velocity and k and nu are the curvature and tangential angle of a plane curve gamma we follow the direct approach and we analyze the so called intrinsic heat equation governing the motion of plane curves obeying such a geometric equation the intrinsic heat equation is modified to include an appropriate nontrivial tangential velocity functional alpha we show how the presence of a nontrivial tangential velocity can prevent numerical solutions from forming various instabilities from an analytical point of view we present some new results on short time existence of a regular family of evolving curves in the degenerate case when beta k nu gamma nu k m 0 m le 2 and the governing system of equations includes a nontrivial tangential velocity functional

analyticity of the maximal solution of an abstract nonlinear parabolic equation where u t x 0 7 j x r r satisfies other suitable conditions problem p is studied under the assumption that af au generates an analytic semigroup in e and has domain f which corresponds to a parabolicity condition on equation p problems of this kind have been studied by monotonicity methods see brezis 111 crandall ligget 3 and crandall pazy 4 to obtain global but weak solutions of p here we find a maximally defined strict solution by means of a linearization method and by virtue of maximal regularity results for the linear case this method has been used by da prato grisvard in 5 to obtain strict solutions of p with arguments which seem to be more simple and suitable to our purpose we get also the analyticity of u with respect to 1 w when f is analytic in section 1 some notation and assumptions are given problem p is studied in section 2 where we prove existence and uniqueness of the strict solution of p which is defined on a maximal time interval contained in 0 t and it is continuous with respect to t 0 in section 3 we give conditions to get global existence of the solution in section 4 we study analyticity with respect to t ug of the maximally defined solution of p and in section 5 a simple example is given in a subsequent paper we will show how this theory applies to the study of the analyttctty of the solutions of nonlinear parabol ic partial differential equations in spaces of holder continuous functions

l p maximal regularity for second order cauchy problems we introduce the concept of l p maximal regularity for second order cauchy problems we prove l p maximal regularity for an abstract model problem and we apply the abstract results to prove existence uniqueness and regularity of solutions for nonlinear wave equations

space time regularity of the koch tataru solutions to navier stokes equations abstract in koch tataru 2001 the authors have proved local and global well posedness of the navier stokes equations with small initial data u 0 b m o 1 and then the spatial analyticity of the koch tataru solution stated as t k 2 k u x t for any positive integer k has been presented by germain pavlovic staffilani 2007 where x t is the space defined by koch tataru 2001 see also definition 1 2 in this paper we shall present the space time regularity more precisely for any positive integer m k we have t m k 2 t m k u x t

on the evolution operator for a class of non autonomous abstract parabolic equations abstract we state existence and several properties of the evolution operator relevant to a family a t 0 t t of generators of analytic semigroups in a banach space the assumption that certain interpolation spaces do not depend on t allows us to weaken the usual regularity hypotheses on t a t requiring only continuity in suitable topologies