bounds on coarsening rates for the lifschitz slyozov wagner equation this paper is concerned with the large time behavior of solutions to the lifschitz slyozov wagner lsw system of equations point wise in time upper and lower bounds on the rate of coarsening are obtained for solutions with fairly general initial data these bounds complement the time averaged upper bounds obtained by dai and pego and the point wise in time upper and lower bounds obtained by niethammer and velasquez for solutions with initial data close to a self similar solution

a new nonmonotone spectral residual method for nonsmooth nonlinear equations abstract in this paper a new spectral residual method is proposed to solve systems of large scale nonlinear equations where the steplength is obtained by minimizing the residue of an approximate secant equation especially the new steplength can be directly applied into solving strictly convex quadratic function combined with a new nonmonotone line search strategy a new derivative free algorithm called a nonmonotone spectral residual algorithm nsra is developed under mild assumptions global convergence is established for locally lipschitz continuous nonlinear systems compared with the state of the art algorithms available in the literatures the new algorithm is more efficient in solving large scale benchmark test problems

rayleigh benard convection improved bounds on the nusselt number we consider rayleigh benard convection as modelled by the boussinesq equations in the infinite prandtl number limit we are interested in the scaling of the average upward heat transport the nusselt number nu in terms of the non dimensionalized temperature forcing the rayleigh number ra experiments asymptotics and heuristics suggest that nu ra1 3 this work is mostly inspired by two earlier rigorous work on upper bounds of nu in terms of ra 1 the work of constantin and doering establishing nu ra1 3ln 2 3ra with help of a logarithmically failing maximal regularity estimate in l on the level of the stokes equation 2 the work of doering reznikoff and the first author establishing nu ra1 3ln 1 3ra with help of the background field method the paper contains two results 1 the background field method can be slightly modified to yield nu ra1 3ln 1 15ra 2 the estimates behind the background field method can be combined with the maximal regularity in l to yield nu ra1 3ln 1 3ln ra

upper bounds on coarsening rates we consider two standard models of surface energy driven coarsening a constant mobility cahn hilliard equation whose large time behavior corresponds to mullins sekerka dynamics and a degenerate mobility cahn hilliard equation whose large time behavior corresponds to motion by surface diffusion arguments based on scaling suggest that the typical length scale should behave as in the first case and in the second we prove a weak one sided version of this assertion showing roughly speaking that no solution can coarsen faster than the expected rate our result constrains the behavior in a time averaged sense rather than pointwise in time and it constrains not the physical length scale but rather the perimeter per unit volume the argument is simple and robust combining the basic dissipation relations with an interpolation inequality and an ode argument

on global stability for lifschitz slyozov wagner like equations this paper is concerned with the stability and asymptotic stability at large time of solutions to a system of equations which includes the lifschitz slyozov wagner lsw system in the case when the initial data has compact support the main result of the paper is a proof of weak global asymptotic stability for lsw like systems previously strong local asymptotic stability results were obtained by niethammer and velazquez for the lsw system with initial data of compact support comparison to a quadratic model plays an important part in the proof of the main theorem when the initial data is critical the quadratic model extends the linear model of carr and penrose and has a time invariant solution which decays exponentially at the edge of its support in the same way as the infinitely differentiable self similar solution of the lsw model

transient and self similar dynamics in thin film coarsening abstract we study coarsening in a simplified model of one dimensional thin films of viscous fluids on hydrophobic substrates lubrication theory shows that such films are unstable and dewet to form droplets that then aggregate over long timescales the masses and positions of the droplets can be described by a coarsening dynamical system cds consisting of odes and deletion rules we develop discrete and continuous mean field models that reproduce the well known n t o t 2 5 long time statistical power law for the number of drops a lifshitz slyozov wagner type lsw continuous model predicts the self similar distribution of drop masses matching with histograms produced by cds simulations and the discrete mean field model we also describe the distribution of drops following homogeneous versus heterogeneous dewetting and use these as initial conditions for the cds simulations that yield characteristic staircasing transients transients can also include recurring spike formation behavior in the mass distribution for idealized initial conditions we show that the transient dynamics can span the full coarsening process bypassing the power law regime entirely

second order convex splitting schemes for gradient flows with ehrlich schwoebel type energy application to thin film epitaxy we construct unconditionally stable unconditionally uniquely solvable and second order accurate in time schemes for gradient flows with energy of the form int omega f nabla phi bf x frac epsilon 2 2 delta phi bf x 2 d bf x the construction of the schemes involves the appropriate combination and extension of two classical ideas i appropriate convex concave decomposition of the energy functional and ii the secant method as an application we derive schemes for epitaxial growth models with slope selection f bf y frac14 bf y 2 1 2 or without slope selection f bf y frac12 ln 1 bf y 2 two types of unconditionally stable uniquely solvable second order schemes are presented the first type inherits the variational structure of the original continuous in time gradient flow while the second type does not preserve the variational structure we present numerical simulations for the case with slope selection which verify well known physical scaling laws for the long time coarsening process

phase segregation dynamics in particle systems with long range interactions ii interface motion we study properties of the solutions of a family of second order integrodifferential equations which describe the large scale dynamics of a class of microscopic phase segregation models with particle conserving dynamics we first establish existence and uniqueness as well as some properties of the instantonic solutions then we concentrate on formal asymptotic sharp interface limits we argue that the obtained interface evolution laws a stefan like problem and the mullins sekerka solidification model coincide with the ones which can be obtained in the analogous limits from the cahn hilliard equation the fourth order pde which is the standard macroscopic model for phase segregation with one conservation law

grain boundary motion arising from the gradient flow of the aviles giga functional abstract this paper considers the singular limit of the equation t 2 1 2 1 grain boundaries limiting discontinuities in form networks that coarsen over time a matched asymptotic analysis is used to derive a free boundary problem consisting of curve motion coupled along hyperbolic characteristics and junction conditions an intermediate boundary layer near extrema junctions is discovered along with the relevant nonlocal junction conditions the limiting dynamics can be viewed in the context of a gradient flow of the sharp interface energy on an attracting manifold dynamic scaling of the long time coarsening process can be explained by dimensional analysis of the reduced problem

interpolation inequalities in pattern formation abstract we prove some interpolation inequalities which arise in the analysis of pattern formation in physics they are the strong version of some already known estimates in weak form that are used to give a lower bound of the energy in many contexts coarsening and branching in micromagnetics and superconductors the main ingredient in the proof of our inequalities is a geometric construction which was first used by choksi conti kohn and one of the authors in 4 in the study of branching in superconductors

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

derivation of the lsw theory for ostwald ripening by homogenization methods the coarsening of a solid phase in an undercooled liquid is described by a stefan problem with surface tension the solid phase is characterized by a large number of balls with small volume fraction and small capacity we solve the quasi static equation as well as the parabolic problem and construct approximate solutions by means of comparison principles the framework of young measures is used to pass to the limit of infinitely many particles we obtain that the capacity is the crucial quantity to get the conservation law for the particle size distribution which is studied in the classical theory for ostwald ripening by lifshitz slyozov wagner lsw

upper bounds for coarsening for the deep quench obstacle problem the deep quench obstacle problem models phase separation at low temperatures during phase separation domains of high and low concentration are formed then coarsen or grow in average size of interest is the time dependence of the dominant length scales of the system relying on recent results by novick cohen and shishkov discrete contin dyn syst b 25 251 272 2009 we demonstrate upper bounds for coarsening for the deep quench obstacle problem with either constant or degenerate mobility for the case of constant mobility we obtain upper bounds of the form t1 3 at early times as well as at times t for which e t le frac 1 overline u 2 4 where e t denotes the free energy for the case of degenerate mobility we get upper bounds of the form t1 3 or t1 4 at early times depending on the value of e 0 as well as bounds of the form t1 4 whenever e t le frac 1 overline u 2 4

upper bounds on the coarsening rate of discrete ill posed nonlinear diffusion equations we prove a weak upper bound on the coarsening rate of the discrete in space version of an ill posed nonlinear diffusion equation the continuum version of the equation violates parabolicity and lacks a complete well posedness theory in particular numerical simulations indicate very sensitive dependence on initial data nevertheless models based on its discrete in space version which we study are widely used in a number of applications including population dynamics chemotactic movement of bacteria granular flow formation of shear bands and computer vision image denoising and segmentation our bounds have implications for all three applications 2008 wiley periodicals inc

geometric motion for a degenerate allen cahn cahn hilliard system the partial wetting case abstract using formal asymptotics we demonstrate that in a low temperature coarsening limit a degenerate allen cahn cahn hilliard system yields a geometric problem in which small particles whose shape evolves according to surface diffusion move along a surface where the chemical potential is quasi static which itself moves by motion by mean curvature the degenerate allen cahn cahn hilliard system was developed in j w cahn a novick cohen evolution equations for phase separation and ordering in binary alloys j stat phys 76 1994 877 909 to describe simultaneous ordering and phase separation and within this context the particles which contain a minor disordered phase are embedded along grain boundaries which partition the system into two ordered phase variants the limiting problem though can also be viewed as a diffuse interface approximation for various problems in materials science in which surface diffusion and motion by mean curvature are coupled see for example j kanel a novick cohen a vilenkin a numerical study of grain boundary motion in bicrystals acta mat 53 2004 227 235 a novick cohen order disorder and phase separation modeling grain sintering in e inan k z markov eds proceedings of the ninth international symposium cmds9 istanbul turkey world scientific publishing co 1999 pp 56 68 the present analysis extends a previous study a novick cohen triple junction motion for an allen cahn cahn hilliard system physica d 137 2000 1 24 which focused on the complete wetting limit and on motions in the plane here we treat the partial wetting case and our analysis accommodates motion in three dimensions

configuration design sensitivity analysis of built up structures part i theory a continuum based configuration design sensitivity analysis method is developed for built up structures that include truss beam plane elastic solid and plate design components the configuration design variation of a structural component can be characterized by changes in the domain shape and in the orientation of the component a variational approach is then used to incorporate both shape and orientation effects in the same energy equation a continuum shape design sensitivity analysis method that utilizes the material derivative idea of continuum mechanics is employed to account for effects of shape variation an approach that is similar to the continuum shape design sensitivity analysis method is developed in this paper to account for the effect of orientation variation variations of energy bilinear and load linear forms with respect to both shape and orientation design variables are derived for each structural component using the adjoint variable or direct differentiation method configuration design sensitivity results for built up structures are obtained in terms of the design velocity fields configuration design sensitivity analyses of both static and eigenvalue responses are considered

a multi phase mullins sekerka system matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem we propose a generalisation of the mullins sekerka problem to model phase separation in multi component systems the model includes equilibrium equations in bulk the gibbs thomson relation on the interfaces young s law at triple junctions together with a dynamic law of stefan type using formal asymptotic expansions we establish the relationship to a transition layer model known as the cahn hilliard system we introduce a notion of weak solutions for this sharp interface model based on integration by parts on manifolds together with measure theoretical tools through an implicit time discretisation we construct approximate solutions by stepwise minimisation under the assumption that there is no loss of area as the time step tends to zero we show the existence of a weak solution

upper bound on the coarsening rate for an epitaxial growth model we study a specific example of energy driven coarsening in two space dimensions the energy is u 2 1 u 2 2 the evolution is the fourth order pde representing steepest descent this equation has been proposed as a model of epitaxial growth for systems with slope selection numerical simulations and heuristic arguments indicate that the standard deviation of u grows like t1 3 and the energy per unit area decays like t 1 3 we prove a weak one sided version of the latter statement the time averaged energy per unit area decays no faster than t 1 3 our argument follows a strategy introduced by kohn and otto in the context of phase separation combining i a dissipation relation ii an isoperimetric inequality and iii an ode lemma the interpolation inequality is new and rather subtle our proof is by contradiction relying on recent compactness results for the aviles giga energy 2003 wiley periodicals inc

nonlinear approximation and the space bv r2 given a function 2 0 q 0 l 2 and a real number t 0 let u f t infg bv g 11 ll vgig where the infimum is taken over all functions g g bv of bounded variation on this and related extremal problems arise in several areas of mathematics such as interpolation of operators and statistical estimation as well as in digital image processing techniques for finding minimizers g for u f t based on variational calculus and nonlinear partial differential equations have been put forward by several authors dms ro ms cl the main disadvantage of these approaches is that they are numerically intensive on the other hand it is well known that more elementary methods based on wavelet shrinkage solve related extremal problems for example the above problem with bv replaced by the besov space b l i see e g cdll however since bv has no simple description in terms of wavelet coefficients it is not clear that minimizers for u f t can be realized in this way we shall show in this paper that simple methods based on haar thresholding provide near minimizers for u f t our analysis of this extremal problem brings forward many interesting relations between haar decompositions and the space bv 1 introduction nonlinear approximation has recently played an impor tant role in several problems of image processing including compression noise removal and feature extraction we have in mind techniques such as wavelet compression djl wavelet shrinkage or thresholding djkp1 wavelet packets cw and greedy algorithms mz dt there has also been an impressive contribution of techniques based on variational calculus and nonlinear partial dif ferential equations see e g dms ro ms cl especially to the problems of noise removal and image segmentation the common point between these two approaches is their ability to adapt to the composite nature of images edge tex tures and smooth regions should be treated adaptively a requirement which is certainly not fulfilled by the classical linear filtering techniques

error analysis of a mixed finite element method for the molecular beam epitaxy model this paper investigates the error analysis of a mixed finite element method with crank nicolson time stepping for simulating the molecular beam epitaxy mbe model the fourth order differential equation of the mbe model is replaced by a system of equations consisting of one nonlinear parabolic equation and an elliptic equation then a mixed finite element method requiring only continuous elements is proposed to approximate the resulting system it is proved that the semidiscrete and fully discrete versions of the numerical schemes satisfy the nonlinearity energy stability property which is important in the numerical implementation moreover detailed analysis is provided to obtain the convergence rate numerical experiments are carried out to validate the theoretical results

self similarity in two phase curvature flow in this work we present a family of exact self similar solutions for the distribution of enclosed areas under mean curvature flow on the union of disjoint closed curves in the plane to our knowledge this is the first example of closed form exact self similar solutions for the mean curvature flow we perform numerical simulations for large sets of initial data obtained as the zero level set of gaussian random fields and observe numerical convergence to one particular member of this family

domain branching in uniaxial ferromagnets a scaling law for the minimum energy we address the branching of magnetic domains in a uniaxial ferromagnet our thesis is that branching is required by energy minimization to show this we consider the nonlocal nonconvex variational problem of micromagnetics we identify the scaling law of the minimum energy by proving a rigorous lower bound which matches the already known upper bound we further show that any domain pattern achieving this scaling law must have average width of order l 2 3 where l is the length of the magnet in the easy direction finally we argue that branching is required by considering the constrained variational problem in which branching is prohibited and the domain structure is invariant in the easy direction its scaling law is different

a second order in time uniquely solvable unconditionally stable numerical scheme for cahn hilliard navier stokes equation we propose a novel second order in time numerical scheme for cahn hilliard navier stokes phase field model with matched density the scheme is based on second order convex splitting for the cahn hilliard equation and pressure projection for the navier stokes equation we show that the scheme is mass conservative satisfies a modified energy law and is therefore unconditionally stable moreover we prove that the scheme is unconditionally uniquely solvable at each time step by exploring the monotonicity associated with the scheme thanks to the simple coupling of the scheme we design an efficient picard iteration procedure to further decouple the computation of cahn hilliard equation and navier stokes equation we implement the scheme by the mixed finite element method ample numerical experiments are performed to validate the accuracy and efficiency of the numerical scheme

front migration in the nonlinear cahn hilliard equation the method of matched asymptotic expansions is used to describe solutions of the nonlinear cahn hilliard equation for phase separation in n 1 space dimensions the expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature on the dominant slowest timescale the interface velocity is determined by the mean curvature of the interface by a non local relation which is identical to that in a well known quasi static model of solidification which exhibits a shape instability discovered by mullins sekerka j appl phys 34 323 329 1963 on a faster timescale the cahn hilliard equation regularizes a classic two phase stefan problem similarity solutions of the two phase stefan problem should describe boundary layers existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

non self similar behavior in the lsw theory of ostwald ripening the classical lifshitz slyozov wagner theory of domain coarsening predicts asymptotically self similar behavior for the size distribution of a dilute system of particles that evolve by diffusional mass transfer with a common mean field here we consider the long time behavior of measure valued solutions for systems in which particle size is uniformly bounded i e for initial measures of compact support we prove that the long time behavior of the size distribution depends sensitively on the initial distribution of the largest particles in the system convergence to the classically predicted smooth similarity solution is impossible if the initial distribution function is comparable to any finite power of distance to the end of the support we give a necessary criterion for convergence to other self similar solutions and conditional stability theorems for some such solutions for a dense set of initial data convergence to any self similar solution is impossible

well posedness and spectral properties of heat and wave equations with non local conditions abstract we consider the one dimensional heat and wave equations but instead of boundary conditions we impose on the solution certain non local integral constraints an appropriate hilbert setting leads to an integration by parts formula in sobolev spaces of negative order and eventually allows us to use semigroup theory leading to analytic well posedness hence sharpening regularity results previously obtained by other authors in doing so we introduce a parametrization of such integral conditions that includes known cases but also shows the connection with more usual boundary conditions like periodic ones in the self adjoint case we even obtain eigenvalue asymptotics of so called weyl s type

computational studies of coarsening rates for the cahn hilliard equation with phase dependent diffusion mobility we study computationally coarsening rates of the cahn hilliard equation with a smooth double well potential and with phase dependent diffusion mobilities the latter is a feature of many materials systems and makes accurate numerical simulations challenging our numerical simulations confirm earlier theoretical predictions on the coarsening dynamics based on asymptotic analysis we demonstrate that the numerical solutions are consistent with the physical gibbs thomson effect even if the mobility is degenerate in one or both phases for the two sided degenerate mobility we report computational results showing that the coarsening rate is on the order of l c t 1 4 independent of the volume fraction of each phase for the one sided degenerate mobility that is non degenerate in the positive phase but degenerate in the negative phase we illustrate that the coarsening rate depends on the volume fraction of the positive phase for large positive volume fractions the coarsening rate is on the order of l c t 1 3 and for small positive volume fractions the coarsening rate becomes l c t 1 4

a multigrid finite element solver for the cahn hilliard equation a multigrid finite element solver for the cahn hilliard equation is presented that has mesh independent convergence rates for any time step size including in the important limit e 0 which is examined via numerical examples numerics are performed for a number of test problems which show that the features of the cahn hilliard equation minimising interface measure lyapunov energy functional etc are preserved we also explore the use of this solver in conjunction with adaptive time stepping and adaptive mesh strategies

higher order cahn hilliard equations with dynamic boundary conditions abstract our aim in this paper is to study the well posedness and the dissipativity of higher order cahn hilliard equations with dynamic boundary conditions more precisely we prove the existence and uniqueness of solutions and the existence of the global attractor