lifespan theorem for constrained surface diffusion flows we consider closed immersed hypersurfaces in mathbb r 3 and mathbb r 4 evolving by a class of constrained surface diffusion flows our result similar to earlier results for the willmore flow gives both a positive lower bound on the time for which a smooth solution exists and a small upper bound on a power of the total curvature during this time by phrasing the theorem in terms of the concentration of curvature in the initial surface our result holds for very general initial data and has applications to further development in asymptotic analysis for these flows

one dimensional behavior of singular n dimensional solutions of semilinear heat equations we consider u x t a solution of u t u u p 1 u that blows up at time t where u n 0 t p 1 n 2 p 0 in particular if in addition p 3 then the solution is very close to a superposition of one dimensional solutions as functions of the distance to the blow up set finally we prove that the mere hypothesis that the blow up set is continuous implies that it is c 1 1 2 for any 0

a confidence limit for the empirical mode decomposition and hilbert spectral analysis the confidence limit is a standard measure of the accuracy of the result in any statistical analysis most of the confidence limits are derived as follows the data are first divided into subsections and then under the ergodic assumption the temporal mean is substituted for the ensemble mean next the confidence limit is defined as a range of standard deviations from this mean however such a confidence limit is valid only for linear and stationary processes furthermore in order for the ergodic assumption to be valid the subsections have to be statistically independent for non stationary and nonlinear processes such an analysis is no longer valid the confidence limit of the method here termed emd hsa for empirical mode decomposition hilbert spectral analysis is introduced by using various adjustable stopping criteria in the sifting processes of the emd step to generate a sample set of intrinsic mode functions imfs the emd technique acts as a pre processor for hsa on the original data producing a set of components imfs from the original data that equal the original data when added back together each imf represents a scale in the data from smallest to largest the ensemble mean and standard deviation of the imf sample sets obtained with different stopping criteria are calculated and these form a simple random sample set the confidence limit for emd hsa is then defined as a range of standard deviations from the ensemble mean without evoking the ergodic assumption subdivision of the data stream into short sections is unnecessary hence the results and the confidence limit retain the full frequency resolution of the full dataset this new confidence limit can be applied to the analysis of nonlinear and non stationary processes by these new techniques data from length of day measurements and a particularly violent recent earthquake are used to demonstrate how the confidence limit is obtained and applied by providing a confidence limit for this new approach a stable range of stopping criteria for the decomposition or sifting phase emd has been established making the results of the final processing with hsa and the entire emd hsa method more definitive

space time finite element methods for surface diffusion with applications to the theory of the stability of cylinders a family of space time finite element approximation schemes is presented for the nonlinear partial differential equations governing diffusion in the surface of a body of revolution the schemes share with the partial differential equations properties of conservation of volume and decrease of area numerical experiments are described showing that the result of the linear theory of small amplitude longitudinal perturbations of a cylinder to the effect that a long cylinder is stable against all perturbations with spatial fourier spectra containing only wavelengths less than the circumference of the cylinder does not hold in the full nonlinear theory examples are given of cases in which longitudinal perturbations with high wave number spectra grow in amplitude after an initial rapid decay followed by a long incubation period and result in break up of the body into a necklace of beads the results of finite element calculations are compared with the predictions of a perturbation analysis

analytic solutions for a stefan problem with gibbs thomson correction we provide existence of a unique smooth solution for a class of oneand two phase stefan problems with gibbs thomson correction in arbitrary space dimensions in addition it is shown that the moving interface depends analytically on the temporal and spatial variables of crucial importance for the analysis is the property of maximal lpregularity for the linearized problem which is fully developed in this paper as well

on convergence of solutions to equilibria for quasilinear parabolic problems abstract we show convergence of solutions to equilibria for quasilinear parabolic evolution equations in situations where the set of equilibria is non discrete but forms a finite dimensional c 1 manifold which is normally hyperbolic our results do not depend on the presence of an appropriate lyapunov functional as in the ojasiewicz simon approach but are of local nature

optimal l p l q estimates for parabolic boundary value problems with inhomogeneous data in this paper we investigate vector valued parabolic initial boundary value problems mathcal a t x d mathcal b j t x d subject to general boundary conditions in domains g in mathbb r n with compact c 2m boundary the top order coefficients of mathcal a are assumed to be continuous we characterize optimal l p l q regularity for the solution of such problems in terms of the data we also prove that the normal ellipticity condition on mathcal a and the lopatinskii shapiro condition on mathcal a mathcal b 1 dots mathcal b m are necessary for these l p l q estimates as a byproduct of the techniques being introduced we obtain new trace and extension results for sobolev spaces of mixed order and a characterization of triebel lizorkin spaces by boundary data

qualitative behaviour of solutions for the two phase navier stokes equations with surface tension the two phase free boundary value problem for the isothermal navier stokes system is studied for general bounded geometries in absence of phase transitions external forces and boundary contacts it is shown that the problem is well posed in an l p setting and that it generates a local semiflow on the induced state manifold if the phases are connected the set of equilibria of the system forms a n 1 dimensional manifold each equilibrium is stable and it is shown that global solutions which do not develop singularities converge to an equilibrium as time goes to infinity the latter is proved by means of the energy functional combined with the generalized principle of linearized stability

critical spaces for quasilinear parabolic evolution equations and applications abstract we present a comprehensive theory of critical spaces for the broad class of quasilinear parabolic evolution equations the approach is based on maximal l p regularity in time weighted function spaces it is shown that our notion of critical spaces coincides with the concept of scaling invariant spaces in case that the underlying partial differential equation enjoys a scaling invariance applications to the vorticity equations for the navier stokes problem convection diffusion equations the nernst planck poisson equations in electro chemistry chemotaxis equations the mhd equations and some other well known parabolic equations are given

refined asymptotics for the blowup of ut δu up this work is concerned with positive blowing up solutions of the semilinear heat equation ut u up in rn our main contribution is a sort of center manifold analysis for the equation in similarity variables leading to refined asymptotics for u in a backward space time parabola near any blowup point we also explore a connection between the asymptotics of u and the local geometry of the blowup set

a general approach to stability in free boundary problems abstract every solution of a linear elliptic equation on a bounded domain may be considered as an equilibrium of a free boundary problem the free boundary problem consists of the corresponding parabolic equation on a variable unknown domain with free boundary conditions prescribing both dirichlet and neumann data we establish a rigorous stability analysis of such equilibria including the construction of stable and unstable manifolds for this purpose we transform the free boundary problem to a fully nonlinear and nonlocal parabolic problem on a fixed domain with fully nonlinear lateral boundary conditions and we develop the general theory for such problems as an illustration we give two examples the second being the focussing flame problem in combustion theory

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

energy levels of steady states for thin film type equations abstract we study the phase space of the evolution equationht f h hxxx x g h hx x by means of a dissipated energy here h x t 0 and at h 0 the coefficient functions f 0 and g can either degenerate to 0 or blow up to or tend to a nonzero constant we first show that all positive periodic steady states are saddles in the energy landscape with respect to zero mean perturbations if g f 0 or if the perturbations are allowed to have period longer than that of the steady state for power law coefficients f y yn and g y b ym for some b 0 we analytically determine the relative energy levels of distinct steady states for example with m n 1 2 and for suitable choices of the period and mean value we find three fundamentally different steady states the first is a constant steady state that is stable and is a local minimum of the energy the second is a positive periodic steady state that is linearly unstable and has higher energy than the constant steady state it is a saddle point for the energy the third is a periodic collection of droplet compactly supported steady states having lower energy than either the positive steady state or the constant one since the energy must decrease along every orbit these results significantly constrain the dynamics of the evolution equation

variational composition of a monotone operator and a linear mapping with applications to elliptic pdes with singular coefficients abstract this paper proposes a regularized notion of a composition of a monotone operator with a linear mapping this new concept called variational composition can be shown to be maximal monotone in many cases where the usual composition is not the two notions coincide however whenever the latter is maximal monotone the utility of the variational composition is demonstrated by applications to subdifferential calculus theory of measurable multifunctions and elliptic pdes with singular coefficients

a high order boundary integral method for surface diffusions on elastically stressed axisymmetric rods many applications in materials involve surface diffusion of elastically stressed solids study of singularity formation and long time behavior of such solid surfaces requires accurate simulations in both space and time here we present a high order boundary integral method for an elastically stressed solid with axi symmetry due to surface diffusions in this method the boundary integrals for isotropic elasticity in axi symmetric geometry are approximated through modified alternating quadratures along with an extrapolation technique leading to an arbitrarily high order quadrature in addition a high order temporal integration factor method based on explicit representation of the mean curvature is used to reduce the stability constraint on time step to apply this method to a periodic in axial direction and axi symmetric elastically stressed cylinder we also present a fast and accurate summation method for the periodic green s functions of isotropic elasticity using the high order boundary integral method we demonstrate that in absence of elasticity the cylinder surface pinches in finite time at the axis of the symmetry and the universal cone angle of the pinching is found to be consistent with the previous studies based on a self similar assumption in the presence of elastic stress we show that a finite time geometrical singularity occurs well before the cylindrical solid collapses onto the axis of symmetry and the angle of the corner singularity on the cylinder surface is also estimated

a center manifold analysis for the mullins sekerka model abstract the mullins sekerka model is a nonlocal evolution model for hypersurfaces which arises as a singular limit for the cahn hilliard equation we show that classical solutions exist globally and tend to spheres exponentially fast provided that they are close to a sphere initially our analysis is based on center manifold theory and on maximal regularity

dynamics of the ericksen leslie equations with general leslie stress ii the compressible isotropic case in this article the non isothermal compressible ericksen leslie system for nematic liquid crystals subject to general leslie stress is considered it is shown that this system is locally well posed within the l q setting and that for initial data close to equilibria points which are identical with the ones for the incompressible situation the solution exists globally moreover any global solution which does not develop singularities converges to an equilibrium in the topology of the natural state manifold note that no structural assumptions on the leslie coefficients are imposed and in particular parodi s relation is not being assumed the results can be viewed as an extension of the studies in hieber and pruss math ann 369 977 996 2017 for the incompressible case to the compressible situation

extended forward backward algorithm abstract we propose an extended forward backward algorithm for approximating a zero of a maximal monotone operator which can be split as the extended sum of two maximal monotone operators we establish the weak convergence in average of the sequence generated by the algorithm under assumptions similar to those used in classical forward backward algorithms this provides as a special case an algorithm for solving convex constrained minimization problems without qualification condition

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

the geometric triharmonic heat flow of immersed surfaces near spheres abstract we consider closed immersed surfaces in r 3 evolving by the geometric triharmonic heat flow using local energy estimates we prove interior estimates and a positive absolute lower bound on the lifespan of solutions depending solely on the local concentration of curvature of the initial immersion in l 2 we further use an e regularity type result to prove a gap lemma for stationary solutions using a monotonicity argument we then prove that a blowup of the flow approaching a singular time is asymptotic to a non umbilic embedded stationary surface this allows us to conclude that any solution with initial l 2 norm of the tracefree curvature tensor smaller than an absolute positive constant converges exponentially fast to a round sphere with radius equal to 3 v 0 4 3 where v 0 denotes the signed enclosed volume of the initial data

the empirical mode decomposition and the hilbert spectrum for nonlinear and non stationary time series analysis a new method for analysing nonlinear and non stationary data has been developed the key part of the method is the empirical mode decomposition method with which any complicated data set can be dec

variational discretization of axisymmetric curvature flows we present natural axisymmetric variants of schemes for curvature flows introduced earlier by the present authors and analyze them in detail although numerical methods for geometric flows have been used frequently in axisymmetric settings numerical analysis results so far are rare in this paper we present stability equidistribution existence and uniqueness results for the introduced approximations numerical computations show that these schemes are very efficient in computing numerical solutions of geometric flows as only a spatially one dimensional problem has to be solved the good mesh properties of the schemes also allow them to compute in very complex axisymmetric geometries

quasi static motion of a capillary drop ii the three dimensional case abstract a theory is presented to analyze the nonlinear stability of a drop of incompressible viscous fluid with negligible inertia the theory is developed here on the three dimensional version of the relevant free boundary model for stokes equations within this context we show that if the free boundary initiates close to a sphere r 1 e 0 e small then there exists a global in time solution with free boundary r 1 t e 1 n 1 n t e n which approaches a sphere exponentially fast as t moreover we prove that if 0 is analytic resp c in then the velocity u x t e the pressure p x t e and the free boundary are all jointly analytic resp c in x e in an earlier paper we considered the analogous problem for a two dimensional drop although the three dimensional problem proceeds along similar lines the analysis is more complicated due to the fact that we work here with spherical harmonics and vector spherical harmonics

long wave instabilities and saturation in thin film equations hocherman and rosenau conjectured that long wave unstable cahn hilliard type interface models develop nite time singularities when the nonlinearity in the destabilizing term grows faster at large amplitudes than the nonlinearity in the stabilizing term phys d 67 113 125 1993 we consider this conjecture for a class of equations often used to model thin lms in a lubrication context by showing that if the solutions are uniformly bounded above or below e g are nonnegative then the destabilizing term can be stronger than previously conjectured yet the solution still remains globally bounded for example they conjecture that for the long wave unstable equation ht h n hxxx x h m hx x m n leads to blow up using a conservation of volume constraint for the case m n 0 we conjecture a dierent critical exponent for possible singularities of nonnegative solutions we prove that nonlinearities with exponents below the conjectured critical exponent yield globally bounded solutions specically for the above equation solutions are bounded if m n 2 the bound is proved using energy methods and is then used to prove the existence of non negative weak solutions in the sense of distributions we present preliminary numerical evidence suggesting that m n 2 can allow blow up

stability of spherical caps under the volume preserving mean curvature flow with line tension abstract we show stability of spherical caps scs lying on a flat surface where the motion is governed by the volume preserving mean curvature flow mcf moreover we introduce a dynamic boundary condition that models a line tension effect on the boundary the proof is based on the generalized principle of linearized stability

stability of a liquid drop trapped between two parallel planes ii general contact angles the physical problem considered is that of a drop of liquid forming a bridge between two parallel planar plates where the contact curves between the drop and the planes are not prescribed a priori it is shown that for sufficiently large volumes a stable drop must exist and that no bifurcation can occurwithin the family of curves without inflection points it is also shown that if the drop is a convex set satisfying the correct differential equations then it is stable some numerical results are presented

solving monotone inclusions with linear multi step methods in this paper a new class of proximal like algorithms for solving monotone inclusions of the form t x 0 is derived it is obtained by applying linear multi step methods lmm of numerical integration in order to solve the differential inclusion x t t x t which can be viewed as a generalization of the steepest decent method for a convex function it is proved that under suitable conditions on the parameters of the lmm the generated sequence converges weakly to a point in the solution set t 1 0 the lmm is very similar to the classical proximal point algorithm in that both are based on approximately evaluating the resol vants of t consequently lmm can be used to derive multi step versions of many of the optimization methods based on the classical proximal point algorithm the convergence analysis allows errors in the computation of the iterates and two different error criteria are analyzed namely the classical scheme with summable errors and a recently proposed more constructive criterion

enlargements and sums of monotone operators in this paper we study two important notions related to monotone operators one is the concept of enlargement of a given monotone operator which has turned out to be a useful tool in the analysis of approximate solutions to problems involving monotone operators the second one is the notion of sum of monotone operators for the latter we introduce and study a kind of extended sum of two monotone operators which in several important cases turns out to be a maximal monotone operator

maximal lp regularity of parabolic problems with boundary dynamics of relaxation type in this paper we investigate vector valued parabolic initial boundary value problems of relaxation type typical examples for such boundary conditions are dynamic boundary conditions or linearized free boundary value problems like in the stefan problem we present a complete lp theory for such problems which is based on maximal regularity of certain model problems

symmetric singularity formation in lubrication type equations for interface motion fourth order degenerate diffusion equations arise in a lubrication approximation of a thin film or neck driven by surface tension numerical studies of the lubrication equation le h t h n h xxx x 0 with various boundary conditions indicate that singularity formation in which h x t t to 0 occurs for small enough n with anomalous or second type scaling inconsistent with usual dimensional analysis this paper considers locally symmetric or even singularities in the le and in the modified lubrication equation mle h t h n h xxxx 0 both equations have the property that entropy bounds forbid finite time singularities when n is sufficiently large power series expansions for local symmetric similarity solutions are proposed for equation le with n 1 and mle for all n in mathbb r in the latter case special boundary conditions that force singularity formation are required to produce singularities when n is large matching conditions at higher order terms in the

on the maximality of the sum of two maximal monotone operators abstract a wide variety of problems involving nonlinear partial differential equations subject to boundary conditions may be shown to have solutions by establishing that the associated differential operators satisfy a certain technical condition this condition called maximal monotonicity allows the use of a well developed abstract theory which includes results about existence regularity etc of solutions of operator equations it is quite useful therefore to have easily verified conditions which imply that an operator is maximal monotone frequently an operator may be regarded as the sum of simpler components this paper gives a new sufficient condition which guarantees that the sum of two maximal monotone operators is again maximal monotone author

lie group action and stability analysis of stationary solutions for a free boundary problem modelling tumor growth abstract in this paper we study asymptotic behavior of solutions for a free boundary problem modelling tumor growth we first establish a general result for differential equations in banach spaces possessing a lie group action which maps a solution into new solutions we prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either d a or d a type by using this general result and making delicate analysis of the spectrum of the linearization of the stationary free boundary problem we prove that if the surface tension coefficient is larger than a threshold value then the unique stationary solution is asymptotically stable modulo translations provided the constant c is sufficiently small whereas if then this stationary solution is unstable

invariant manifolds for parabolic partial differential equations on unbounded domains in this paper finite dimensional invariant manifolds for nonlinear parabolic partial differential equations of the form

non linear higher order boundary value problems describing thin viscous flows near edges abstract two boundary value problems for non linear higher order ordinary differential equations are analyzed which have been recently proposed in the modeling of steady and quasi steady thin viscous flows over a bounded solid substrate the first problem concerns steady states and consists of a third order ode for the height of the liquid the ode contains an unknown parameter the flux and the boundary conditions relate near the edges of the substrate the height and its second derivative to the flux itself for this problem non existence and non uniqueness results are proved depending on the behavior as the flux approaches zero of the height function the function which relates the height to the flux near the edge out of which the liquid flows the second problem concerns quasi steady states and consists of a fourth order ode for the suitably scaled height of the liquid non linear boundary conditions relate the height to the flux near the edges of the substrate for this problem the existence of a solution is proved for a suitable class of height functions

axisymmetric surface diffusion dynamics and stability of self similar pinchoff the dynamics of surface diffusion describes the motion of a surface with its normal velocity given by the surface laplacian of its mean curvature this flow conserves the volume enclosed inside the surface while minimizing its surface area we review the axisymmetric equilibria the cylinder sphere and the delaunay unduloid the sphere is stable while the cylinder is long wave unstable a subcritical bifurcation from the cylinder produces a continuous family of unduloid solutions we present computations that suggest that the stable manifold of the unduloid forms a separatrix between states that relax to the cylinder in infinite time and those that tend toward finite time pinchoff we examine the structure of the pinchoff showing it has self similar structure using asymptotic numerical and analytical methods in addition to a previously known similarity solution we find a countable set of similarity solutions each with a different asymptotic cone angle we develop a stability theory in similarity variables that selects the original similarity solution as the only linearly stable one and consequently the only observable solution we also consider similarity solutions describing the dynamics after the topological transition

linking anisotropic sharp and diffuse surface motion laws via gradient flows we compare four surface motion laws for sharp surfaces with their diffuse interface counterparts by means of gradient flows on corresponding energy functionals the energy functionals can be defined to give the same dependence on normal direction for the energy of sharp plane surfaces as for their diffuse counterparts the anisotropy of the kinetics can be incorporated into the inner product without affecting the energy functional

dynamics of the ericksen leslie equations with general leslie stress i the incompressible isotropic case the ericksen leslie model for nematic liquid crystals in a bounded domain with general leslie and isotropic ericksen stress tensor is studied in the case of a non isothermal and incompressible fluid this system is shown to be locally strongly well posed in the l p setting and a dynamical theory is developed the equilibria are identified and shown to be normally stable in particular a local solution extends to a unique global strong solution provided the initial data are close to an equilibrium or the solution is eventually bounded in the topology of the natural state manifold in this case the solution converges exponentially to an equilibrium in the topology of the state manifold the above results are proven without any structural assumptions on the leslie coefficients and in particular without assuming parodi s relation

self similar blowup solutions to an aggregation equation in r n we present numerical simulations of radially symmetric finite time blowup for the aggregation equation ut u k u where the kernel k x x the dynamics of the blowup exhibits self similar behavior in which zero mass concentrates at the core at the blowup time computations are performed in r n for n between 2 and 10 using a method based on characteristics in all cases studied the self similarity exhibits second kind anomalous scaling

a numerical study of electro migration voiding by evolving level set functions on a fixed cartesian grid a numerical method for studying migration of voids driven by surface diffusion and electric current in a metal conducting line is developed the mathematical model involves moving boundaries governed by a fourth order nonlinear partial differential equation which contains a nonlocal term corresponding to the electrical field and a nonlinear term corresponding to the curvature numerical challenges include efficient computation of the electrical field with sufficient accuracy to afford fourth order differentiation along the void boundary and to capture singularities arising in topological changes we use the modified immersed interface method with a fixed cartesian grid to solve for the electrical field and the fast local level set method to update the position of moving voids numerical examples are performed to demonstrate the physical mechanisms by which voids interact under electromigration

singularities in a modified kuramoto sivashinsky equation describing interface motion for phase transition abstract phase transitions can be modeled by the motion of an interface between two locally stable phases a modified kuramoto sivashinsky equation h t 2 h 4 h 1 h 2 2 h 2 h xx h yy h xy 2 describes near planar interfaces which are marginally long wave unstable we study the question of finite time singularity formation in this equation in one and two space dimensions on a periodic domain such singularity formation does not occur in the kuramoto sivashinsky equation 0 for all 1 0 we provide sufficient conditions on the initial data and size of the domain to guarantee a finite time blow up in which a second derivative of h becomes unbounded using a bifurcation theory analysis we show a parallel between the stability of steady periodic solutions and the question of finite time blow up in one dimension finally we consider the local structure of the blow up in the one dimensional case via similarity solutions and numerical simulations that employ a dynamically adaptive self similar grid the simulations resolve the singularity to over 25 decades in h xx l and indicate that the singularities are all locally described by a unique self similar profile in h xx we discuss the relevance of these observations to the full intrinsic equations of motion and the associated physics

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

linear and quasilinear parabolic equations in sobolev space abstract we consider linear parabolic equations of second order in a sobolev space setting we obtain existence and uniqueness results for such equations on a closed two dimensional manifold with minimal assumptions about the regularity of the coefficients of the elliptic operator in particular we derive a priori estimates relating the sobolev regularity of the coefficients of the elliptic operator to that of the solution the results obtained are used in conjunction with an iteration argument to yield existence results for quasilinear parabolic equations

evolution of geometric statistics a model for the description of the geometry in multiphase mechanics is derived and studied equations for the rate of change of mean and gauss curvature are derived these coupled equations show that the evolution of these curvatures depend on the velocity of the interface normal to itself examples are given to show the utility of these equations to apply these concepts to multiphase mechanical situations an ensemble average is applied to the curvature equations equations for the evolution of the volume fraction and the interfacial area density are also derived these equations also depend on the curvatures and the average interfacial velocity examples are given to show that these equations give a description of the geometry of multiphase mechanics that is superior to the present description which describes the force between materials as depending on two geometric parameters the volume fraction and the average radius of the dispersed material these examples also suggest constitutive equations to d

semi global existence and convergence of solutions of the robinson trautman 2 dimensional calabi equation it is shown that for smooth initial data solutions of the robinson trautman equation also known as the two dimensional calabi equation exist for all positive times and asymptotically converge to a constant curvature metric

subdifferential calculus using ϵ subdifferentials in applications of convex analysis it is important to be able to calculate the subdifferentials of various combinations of proper and lower semicontinuous convex functions such as the sum of two such functions or their inf convolution epi sum as well as the pre composition of a convex function with an affine map or the marginal function obtained from a convex function and a linear map the classical formulas for such calculations all require additional hypotheses same of which may be difficult to check or are not always satisfied in a given variational problem in this paper we present formulas for the subdifferentials of such combinations without assuming any additional hypotheses by utilizing subdifferentials this wider applicability comes at the price of somewhat more complicated formulas

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

on the necessity of the moreau rockafellar robinson qualification condition in banach spaces as well known the moreau rockafellar robinson internal point qualification condition is sufficient to ensure that the infimal convolution of the conjugates of two extended real valued convex lower semi continuous functions defined on a locally convex space is exact and that the subdifferential of the sum of these functions is the sum of their subdifferentials this note is devoted to proving that this condition is in a certain sense also necessary provided the underlying space is a banach space our result is based upon the existence of a non supporting weak closed hyperplane to any weak closed and convex unbounded linearly bounded subset of the topological dual of a banach space

stability of cylindrical bodies in the theory of surface diffusion abstract perturbation arguments and finite element calculations are employed to study the nonlinear partial differential equations governing morphological changes induced by curvature driven diffusion of mass in the surface of an axially symmetric body isotropy of surface properties is assumed second and higher order perturbation analyses indicate that the familiar result of the linear theory of small amplitude longitudinal perturbations of a cylinder to the effect that a long cylinder is stable against all perturbations with spatial fourier spectra containing only wavelengths less than the circumference of the cylinder does not hold in the full nonlinear theory the perturbation analyses yield criteria for determining when longitudinal perturbations with high wave number spectra grow in amplitude after an initial decay followed by an incubation time and result in break up of the body into a necklace of beads the principal conclusions of the formal perturbation analyses are found to be in good accord with numerical solutions obtained by finite element methods

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

reaction diffusion systems of maxwell stefan type with reversible mass action kinetics abstract the mass based maxwell stefan approach to one phase multicomponent reactive mixtures is mathematically analyzed it is shown that the resulting quasilinear strongly coupled reaction diffusion system is locally well posed in an l p setting and generates a local semiflow on its natural state space solutions regularize instantly and become strictly positive if their initial components are all nonnegative and nontrivial for a class of reversible mass action kinetics the positive equilibria are identified these are precisely the constant chemical equilibria of the system which may form a manifold here the total free energy of the system is employed which serves as a lyapunov function for the system by the generalized principle of linearized stability positive equilibria are proved to be normally stable

fully discrete finite element approximation for anisotropic surface diffusion of graphs we analyze a fully discrete numerical scheme for approximating the evolution of graphs for surfaces evolving by anisotropic surface diffusion the scheme is based on the idea of second order operator splitting for the nonlinear geometric fourth order equation this yields two coupled spatially second order problems which are approximated by linear finite elements the time discretization is semi implicit we prove error bounds for the resulting scheme and present numerical test calculations that confirm our analysis and illustrate surface diffusion

on a curvature flow model for embryonic epidermal wound healing abstract the paper is a mathematical investigation of a curvature flow model for embryonic epidermal wound healing proposed by ravasio et al 2015 under the flow we show that a closed initially convex or close to convex curve shrinks to a round point in finite time we also study the singularity showing that the singularity profile after continuous rescaling is that of a circle one of the key new results we require is a maximal time estimate which is also of independent interest

what shape is your conjugate a survey of computational convex analysis and its applications computational convex analysis algorithms have been rediscovered several times in the past by researchers from different fields to further communications between practitioners we review the field of computational convex analysis which focuses on the numerical computation of fundamental transforms arising from convex analysis current models use symbolic numeric and hybrid symbolic numeric algorithms our objective is to disseminate widely the most efficient numerical algorithms useful for applications in image processing computing the distance transform the generalized distance transform and mathematical morphology operators partial differential equations solving hamilton jacobi equations and using differential equations numerical schemes to compute the convex envelope max plus algebra computing the equivalent of the fast fourier transform multifractal analysis etc the fields of applications include among others computer vision robot navigation thermodynamics electrical networks medical imaging and network communication

well posedness and qualitative behaviour of the mullins sekerka problem with ninety degree angle boundary contact we show local well posedness for a mullins sekerka system with ninety degree angle boundary contact we will describe the motion of the moving interface by a height function over a fixed reference surface using the theory of maximal regularity together with a linearization of the equations and a localization argument we will prove well posedness of the full nonlinear problem via the contraction mapping principle here one difficulty lies in choosing the right space for the neumann trace of the height function and showing maximal l p l q regularity for the linear problem in the second part we show that solutions starting close to certain equilibria exist globally in time are stable and converge to an equilibrium solution at an exponential rate

lifespan theorem for simple constrained surface diffusion flows abstract we consider closed immersed hypersurfaces in r 3 and r 4 evolving by a special class of constrained surface diffusion flows this class of constrained flows includes the classical surface diffusion flow in this paper we present a lifespan theorem for these flows which gives a positive lower bound on the time for which a smooth solution exists and a small upper bound on the total curvature during this time the hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature this turns out to be a deep property of the initial manifold as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in l 2 for m 2 immersed in r 3 and additionally on the concentration in l 3 for m 3 immersed in r 4 this is stronger than a previous result on a different class of constrained surface diffusion flows as here we obtain an improved lower bound on maximal time a better estimate during this period and eliminate any assumption on the area of the evolving hypersurface

stability and clustering of self similar solutions of aggregation equations in this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation t k in rd d 2 where k r r with 2 it was previously observed y huang and a l bertozzi self similar blowup solutions to an aggregation equation in rn j siam appl math 70 2582 2603 2010 10 1137 090774495 that radially symmetric solutions are attracted to a self similar collapsing shell profile in infinite time for 2 in this paper we compute the stability of the similarity solution and show that the collapsing shell solution is stable for 2 4 we show that the shell solution is always unstable and destabilizes into clusters that form a simplex which we observe to be the long time attractor we then classify the stability of these simplex solutions and prove that two dimensional in stability implies n dimensional in stability