on a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients we develop a perturbation theory for stochastic differential equations sdes by which we mean both stochastic ordinary differential equations sodes and stochastic partial differential equations spdes in particular we estimate the l p distance between the solution process of an sde and an arbitrary ito process which we view as a perturbation of the solution process of the sde by the l q distances of the differences of the local characteristics for suitable p q 0 as one application of the developed perturbation theory we establish strong convergence rates for numerical approximations of a class of sodes with nonglobally monotone coefficients as another application of the developed perturbation theory we prove strong convergence rates for spatial spectral galerkin approximations of solutions of semilinear spdes with nonglobally monotone nonlinearities including cahn hilliard cook type equations and stochastic burgers equations further applications of the developed perturbation theory include regularity analyses of solutions of sdes with respect to their initial values as well as small noise analyses for ordinary and partial differential equations

optimal boundary control of a viscous cahn hilliard system with dynamic boundary condition and double obstacle potentials in this paper we investigate optimal boundary control problems for cahn hilliard variational inequalities with a dynamic boundary condition involving double obstacle potentials and the laplace beltrami operator the cost functional is of standard tracking type and box constraints for the controls are prescribed we prove existence of optimal controls and derive first order necessary conditions of optimality the general strategy which follows the lines of the recent approach by colli farshbaf shaker and sprekels appl math optim 71 2015 pp 1 24 to the simpler allen cahn case is the following we use the results that were recently established by colli gilardi and sprekels appl math optim online first doi 10 1007 s00245 015 9299 z 2015 for the case of differentiable logarithmic potentials and perform a so called deep quench limit using compactness and monotonicity arguments it is shown that this strategy leads to the desired first order necessary optimality conditions for t

ergodicity of the 2d navier stokes equations with degenerate stochastic forcing the stochastic 2d navier stokes equations on the torus driven by degenerate noise are studied we characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic in particular our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in l 2 0 t 2 unlike previous works this class is independent of the viscosity and the strength of the noise the two main tools of our analysis are the asymptotic strong feller property introduced in this work and an approximate integration by parts formula the first when combined with a weak type of irreducibility is shown to ensure that the dynamics is ergodic the second is used to show that the first holds un

absolute continuity and numerical approximation of stochastic cahn hilliard equation with unbounded noise diffusion abstract in this article we consider the absolute continuity and numerical approximation of the solution of the stochastic cahn hilliard equation with unbounded noise diffusion we first obtain the holder continuity and malliavin differentiability of the solution of the stochastic cahn hilliard equation by using the strong convergence of the spectral gakerkin approximation then we prove the existence and strict positivity of the density function of the law of the exact solution for the stochastic cahn hilliard equation with sublinear growth diffusion coefficient which fills a gap for the existed result when the diffusion coefficient satisfies a growth condition of order 1 3 1 to approximate the density function of the exact solution we propose a full discretization based on the spatial spectral galerkin approximation and the temporal drift implicit euler scheme furthermore a general framework for deriving the strong convergence rate of the full discretization is developed based on the variation approach and the factorization method consequently we obtain the sharp mean square convergence rates in both time and space via sobolev interpolation inequalities and semigroup theories to the best of our knowledge this is the first result on the convergence rate of full discretizations for the considered equation

singular stochastic equations on hilbert spaces harnack inequalities for their transition semigroups we consider stochastic equations in hilbert spaces with singular drift in the framework of g da prato m rockner singular dissipative stochastic equations in hilbert spaces probab theory related fields 124 2 2002 261 303 we prove a harnack inequality in the sense of f y wang logarithmic sobolev inequalities on noncompact riemannian manifolds probab theory related fields 109 1997 417 424 for its transition semigroup and exploit its consequences in particular we prove regularizing and ultraboundedness properties of the transition semigroup as well as that the corresponding kolmogorov operator has at most one infinitesimally invariant measure satisfying some mild integrability conditions finally we prove existence of such a measure for noncontinuous drifts

continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations abstract we prove convergence of the solutions x n of semilinear stochastic evolution equations on a banach space b driven by a cylindrical brownian motion in a hilbert space h d x n t a n x t f n t x n t d t g n t x n t d w h t x n 0 n assuming that the operators a n converge to a and the locally lipschitz functions f n and g n converge to the locally lipschitz functions f and g in an appropriate sense moreover we obtain estimates for the lifetime of the solution x of the limiting problem in terms of the lifetimes of the approximating solutions x n we apply the results to prove global existence for reaction diffusion equations with multiplicative noise and a polynomially bounded reaction term satisfying suitable dissipativity conditions the operator governing the linear part of the equation can be an arbitrary uniformly elliptic second order elliptic operator

on the instability of flow in a streamwise corner the linear stability of an incompressible laminar flow in the blending boundary layer between the boundary layer in a 90 streamwise corner and a blasius boundary layer well away from the corner is examined using a locally parallel flow approximation it is shown that the magnitude of the cross flow in the boundary layer is too small to be a significant factor in the observed early transition in the blending layer however the influence of the outer boundary conditions associated with oblique modes of disturbances which are anti symmetric about the bisector plane are shown to have a profound effect on the stability of the flow as a result the square root of the critical streamwise reynolds number r er associated with a spanwise location is significantly reduced as the corner is approached being r er 54 approximately for spanwise distance of z 6 x r 1 from the corner compared with r er 322 approximately for z 20 x r 1 where x measures downstream distance from the leading edges and r 2 is the streamwise reynolds number at r 600 the growth rate of the most amplified mode of disturbance at the former location is over six times greater than that at the latter the corresponding wave angle at the two locations is respectively 44 and 5 approximately

malliavin calculus for the stochastic cahn hilliard allen cahn equation with unbounded noise diffusion abstract the stochastic partial differential equation analyzed in this work is motivated by a simplified mesoscopic physical model for phase separation it describes pattern formation due to adsorption and desorption mechanisms involved in surface processes in the presence of a stochastic driving force this equation is a combination of cahn hilliard and allen cahn type operators with a multiplicative white space time noise of unbounded diffusion we apply malliavin calculus in order to investigate the existence of a density for the stochastic solution u in dimension one according to the regularity result in 5 u admits continuous paths a s using this property and inspired by a method proposed in 8 we construct a modified approximating sequence for u which properly treats the new second order allen cahn operator under a localization argument we prove that the malliavin derivative of u exists locally and that the law of u is absolutely continuous establishing thus that a density exists

hypercontractivity for space time white noise driven spdes with reflection abstract the hypercontractive property of the markov semigroup associated with the reflected stochastic partial differential equation driven by the additive space time white noise is mainly investigated the main tool for its proof is the general criterion presented recently by f y wang 29 in particular we obtain the hyperboundedness and the compactness of the markov semigroup the exponential convergences of the entropy the exponential convergences of the markov semigroup to its unique invariant measure in both l 2 and the total variation norm

approximation of quasi potentials and exit problems for multidimensional rde s with noise we deal with a class of reaction diffusion equations in space dimension d 1 perturbed by a gaussian noise w t which is white in time and colored in space we assume that the noise has a small correlation radius so that it converges to the white noise w t as 0 by using arguments of convergence we prove that under suitable assumptions the quasi potential v converges to the quasi potential v corresponding to space time white noise in spite of the fact that the equation perturbed by space time white noise has no solution we apply these results to the asymptotic estimate of the mean of the exit time of the solution of the stochastic problem from a basin of attraction of an asymptotically stable point for the unperturbed problem

distributed optimal control of the cahn hilliard system including the case of a double obstacle homogeneous free energy density in this paper we study the distributed optimal control for the cahn hilliard system a general class of free energy potentials is allowed which in particular includes the double obstacle potential the latter potential yields an optimal control problem of a parabolic variational inequality which is of fourth order in space we show the existence of optimal controls to approximating problems where the potential is replaced by a mollified version of its moreau yosida approximation corresponding first order optimality conditions for the mollified problems are given for this purpose a new result on the continuous frechet differentiability of superposition operators with values in sobolev spaces is established besides the convergence of optimal controls of the mollified problems to an optimal control of the original problem we also derive first order optimality conditions for the original problem by a limit process the newly derived stationarity system corresponds to a function space version of c stationarity

finite element approximation of the cahn hilliard cook equation we study the nonlinear stochastic cahn hilliard equation perturbed by additive colored noise we show almost sure existence and regularity of solutions we introduce spatial approximation by a standard finite element method and prove error estimates of optimal order on sets of probability arbitrarily close to 1 we also prove strong convergence without known rate

non modal stability and breakdown in corner and three dimensional boundary layers two classes of three dimensional boundary layer are studied namely the flow in the neighbourhood of a corner of general apex angle and the flow over a wall with spanwise varying porosity stre

large deviations for stochastic reaction diffusion systems with multiplicative noise and non lipshitz reaction term following classical work by freidlin trans amer math soc 1988 305 665 657 and subsequent works by sowers ann probab 1992 20 504 537 and peszat probab theory related fields 1994 98 113 136 we prove large deviation estimates for the small noise limit of systems of stochastic reaction diffusion equations with globally lipschitz but unbounded diffusion coefficients however assuming the reaction terms to be only locally lipschitz with polynomial growth this generalizes results of the above mentioned authors our results apply in particular to systems of stochastic ginzburg landau equations with multiplicative noise

the dynamics of nucleation for the cahn hilliard equation when a constant metastable solution of the cahn hilliard equation is subjected to a spatially localized large amplitude perturbation a transition process may be triggered leading to a globally stable stationary solution in one space dimension the existence and instability of a third stationary solution with the same mass is proved a spike like solution called a canonical nucleus within the class of solutions which are even with respect to the center of the spike it has a one dimensional unstable manifold in addition the process of nucleation by formal arguments using two space scales and two timescales is described the last stage in the process can be approximated by a nonlinear stefan free boundary problem

fast transport asymptotics for stochastic rdes with boundary noise we consider a class of stochastic reaction diffusion equations also having a stochastic perturbation on the boundary and we show that when the diffusion rate is much larger than the rate of reaction it is possible to replace the spde by a suitable one dimensional stochastic differential equation this replacement is possible under the assumption of spectral gap for the diffusion and is a result of averaging in the fast spatial transport we also study the fluctuations around the averaged motion

a note on nonlinear biharmonic equations with negative exponents abstract in this paper we study radially symmetric entire solutions of 2 u u q u 0 in r 3 we find the asymptotic behavior of these solutions for any q 1 and prove that for any q 3 there exists a solution with exactly linear growth

uniqueness theorem for the entire positive solutions of biharmonic equations in r n the entire positive solutions of a conformally invariant biharmonic equation in r n will be classified using the method of moving spheres as a byproduct one also shows that any entire non negative solution of the equation 2 u u p with 1 p n 4 n 4 with n 2 is zero

nonlinear aspects of the cahn hilliard equation abstract this paper treats phase separation within the context of the phenomenological cahn hilliard equation c t m c f c k 2 c where c is the concentration m c is the mobility and f c is homogeneou s free energy which is assumed here to be a fourth degree polynomial natural boundary conditions are introduced the full set of equilibrium solutions is specified a comparison theorem for stability criteria which was postulated by langer is proved here within the framework of the natural boundary conditions energy methods are used to define and estimate the limit of monotonic global stability it is pointed out that within the parameter region where the uniform homogeneous state is the only equilibrium solution there may still exist some internal excitable region in which the homogeneous solution possesses growing fluctuations furthermore a periodic instability is shown to exist in the metastable region in addition to the well known nucleation instability

a new approach to front propagation problems theory and applications in this paper we present a new definition for the global in time propagation motion of fronts hypersurfaces boundaries with a prescribed normal velocity past the first time they develop singularities we show that if this propagation satisfies a geometric maximum principle inclusion avoidance type property then the normal velocity must depend only on the position of the front and its normal direction and principal curvatures this new approach which is more geometric and as it turns out equivalent to the level set method is then used to develop a very general and simple method to rigorously validate the appearance of moving interfaces at the asymptotic limit of general evolving systems like interacting particles and reaction diffusion equations we finally present a number of new asymptotic results among them are the asymptotics of i reaction diffusion equations with rapidly oscillating coefficients ii fully nonlinear nonlocal integral differential equations and iii stochastic ising models with long range anisotropic interactions and general spin flip dynamics

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

logarithmic sobolev inequalities on noncompact riemannian manifolds this paper presents a dimension free harnack type inequality for heat semigroups on manifolds from which a dimension free lower bound is obtained for the logarithmic sobolev constant on compact manifolds and a new criterion is proved for the logarithmic sobolev inequalities abbrev lsi on noncompact manifolds as a result it is shown that lsi may hold even though the curvature of the operator is negative everywhere

strong and weak convergence rates of a spatial approximation for stochastic partial differential equation with one sided lipschitz coefficient strong and weak approximation errors of a spatial finite element method are analyzed for the stochastic partial differential equations spdes with one sided lipschitz coefficients including the s

strong convergence rate of finite difference approximations for stochastic cubic schrodinger equations abstract in this paper we derive a strong convergence rate of spatial finite difference approximations for both focusing and defocusing stochastic cubic schrodinger equations driven by a multiplicative q wiener process beyond the uniform boundedness of moments for high order derivatives of the exact solution the key requirement of our approach is the exponential integrability of both the exact and numerical solutions by constructing and analyzing a lyapunov functional and its discrete correspondence we derive the uniform boundedness of moments for high order derivatives of the exact solution and the first order derivative of the numerical solution which immediately yields the well posedness of both the continuous and discrete problems the latter exponential integrability is obtained through a variant of a criterion given by cox hutzenthaler and jentzen arxiv 1309 5595 as a by product of this exponential integrability we prove that the exact and numerical solutions depend continuously on the initial data and obtain a large deviation type result on the dependence of the noise with first order strong convergence rate

multiplicity of solutions for a fourth order equation with power type nonlinearity let b be the unit ball in mathbb r n n 3 and n be the exterior unit normal vector on the boundary we consider radial solutions to delta 2 u lambda 1 rm sign p u p quad rm in b quad u 0 quad frac partial u partial n 0 quad rm on partial b where 0 for positive p we assume 5 n 12 and p frac n 4 n 4 or n 13 and frac n 4 n 4 0 such that if s there exists a radial weakly singular solution for s there exist infinitely many regular radial solutions and the number of radial regular solutions goes to infinity as s

wang s harnack inequalities for space time white noises driven spdes with two reflecting walls and their applications abstract in this paper we establish wang s harnack inequalities for gaussian space time white noises driven the stochastic partial differential equation with double reflecting walls which is of the infinite dimensional skorokhod equation we first establish both the harnack inequality with power and the log harnack inequality for the special case of additive noises by the coupling approach then we investigate the log harnack inequality for the markov semigroup associated with the reflected spde driven by multiplicative noises using the penalization method and the comparison principle for spdes as their applications we study the strong feller property uniqueness of invariant measures the entropy cost inequality and some other important properties of the transition density

a new proof of i guerra s results concerning nonlinear biharmonic equations with negative exponents abstract here we give a self contained new proof of the asymptotic behavior of the radially symmetric entire solution of 2 u u p in r 3 these results were obtained by i guerra in 4 our proof is much more direct and simpler

on the convergence of a fourth order evolution equation to the allen cahn equation abstract we construct special sequences of solutions to a fourth order nonlinear parabolic equation of cahn hilliard allen cahn type converging to the second order allen cahn equation we consider the evolution equation without boundary as well as the stationary case on domains with dirichlet boundary conditions the proofs exploit the equivalence of the fourth order equation with a system of two second order elliptic equations with good signs

strong convergence rate in averaging principle for stochastic fitzhugh nagumo system with two time scales abstract this article deals with averaging principle for stochastic fitzhugh nagumo system with different time scales under suitable conditions the existence of an averaging equation eliminating the fast variable for this coupled system is proved and as a consequence the system can be reduced to a single stochastic ordinary equation with a modified coefficient moreover the rate of convergence for the slow component towards the solution of the averaging equation is of order 1 2

the scaling limit for a stochastic pde and the separation of phases we investigate the problem of singular perturbation for a reaction diffusion equation with additive noise or a stochastic partial differential equation of ginzburg landau type under the situation that the reaction term is determined by a potential with double wells of equal depth as the parameter e the temperature of the system tends to 0 the solution converges to one of the two stable phases and consequently the phase separation is formed in the limit we derive a stochastic differential equation which describes the random movement of the phase separation point the proof consists of two main steps we show that the solution stays near a manifoldm e of minimal energy configurations based on a lyapunov type argument then the limit equation is identified by introducing a nice coordinate system in a neighborhood ofm e

transient growth in developing plane and hagen poiseuille flow summary the problem of the stability of developing entry flow in both two dimensional channels and circular pipes is investigated the basic flow is generated by uniform flow entering a channel pipe which then provokes the growth of boundary layers on the walls until far downstream fully developed poiseuille flow is attained the length for this development to be reynolds number the channel pipe width diameter this enables the use of high reynolds number theory leading to boundary layer type equations as such there is no need to impose heuristic parallel flow approximations the resulting flow is shown to be susceptible to significant three dimensional transient initially algebraic growth in the streamwise direction and as such large amplifications to flow disturbances are shown to occur followed by ultimate decay far downstream it is suggested that this initial amplification of disturbances is a possible mechanism for flow transition with steady disturbances being the most dangerous introduction and formalism the stability of hagen poiseuille flow has intrigued scientists for more than a century since reynolds experimental investigations it is reasonably well accepted that fully developed hagen poiseuille flow is linearly stable and yet in practice most pipe flows are turbulent in a carefully controlled experimental study of this flow the following sequence of events is usually observed below a reynolds number of any disturbance introduced into the flow will decay and the flow can be said to be stable in practical terms above this value it is possible to maintain laminar flow and the highest recorded for laminar flow is which was achieved with extraordinary amounts of care in this paper we consider the stability of developing flow inside two dimensional channels and circular pipes at the inlet the flow is taken to be uniform and then boundary layers form on the channel pipe walls which downstream eventually merge to form fully developed flow the lengthscale for fully developed flow to become established is very long at large reynolds numbers width diameter of the channel pipe close to the inlet the boundary layers resemble flat plate blasius boundary layers which are susceptible to streamwise algebraic growth of three dimensional disturbances 1 2 in this paper it is shown how this class of disturbance is important in the flow development problem consider the two dimensional channel case the pipe case develops in an analogous manner we take coordinates origin at the leading edge of the lower wall which lies along with the flow directed along the positive direction is the crossflow direction here denotes the semi width of the channel and the reynolds number with the incoming uniform flow velocity and the kinematic viscosity of the fluid assumed to be incompressible throughout it is assumed enabling the use of the boundary layer equations to be entirely rational the flow velocity is then written in the form whilst the pressure develops in the form being the density of the fluid taking the leading order in terms in the navier stokes equations leads to

on the stochastic cahn hilliard equation with a singular double well potential abstract we prove well posedness and regularity for the stochastic pure cahn hilliard equation under homogeneous neumann boundary conditions with both additive and multiplicative wiener noise in contrast with great part of the literature the double well potential is treated as generally as possible its convex part being associated to a multivalued maximal monotone graph everywhere defined on the real line on which no growth nor smoothness assumptions are assumed the regularity result allows to give appropriate sense to the chemical potential and to write a natural variational formulation of the problem the proofs are based on suitable monotonicity and compactness arguments in a generalized variational framework

well posedness for a class of doubly nonlinear stochastic pdes of divergence type abstract we prove well posedness for doubly nonlinear parabolic stochastic partial differential equations of the form d x t div x t d t x t d t b t x t d w t where and are the two nonlinearities assumed to be multivalued maximal monotone operators everywhere defined on r d and r respectively and w is a cylindrical wiener process using variational techniques suitable uniform estimates both pathwise and in expectation and some compactness results well posedness is proved under the classical leray lions conditions on and with no restrictive smoothness or growth assumptions on the operator b is assumed to be hilbert schmidt and to satisfy some classical lipschitz conditions in the second variable

stability of the laminar boundary layer in a streamwise corner this work examines the stability of viscous incompressible flow along a streamwise corner often called the corner boundary layer problem the semi infinite boundary value problem satisfied by small amplitude disturbances in the blending boundary layer region is obtained the mean secondary flow induced by the corner exhibits a flow reversal in this region uniformly valid first approximations to solutions of the governing differential equations are derived uniformity at infinity is achieved by a suitable choice of the large parameter and use of an appropriate langer variable approximations to solutions of balanced type have a phase shift across the critical layer which is associated with instabilities in the case of two dimensional boundary layer profiles viscous incompressible laminar flow along a streamwise corner formed by the intersection of two solid surfaces is often referred to as the corner boundary layer problem in the theoretical model of this problem two flat plates which are infinite in the streamwise and spanwise directions intersect such that the line of intersection the corner line is normal to the plane containing their leading edges the basic flow is along the line of intersection most theoretical work assumes a rectangular corner this situation is shown in figure 1

stabilization by noise for a class of stochastic reaction diffusion equations we prove uniqueness ergodicity and strongly mixing property of the invariant measure for a class of stochastic reaction diffusion equations with multiplicative noise in which the diffusion term in front of the noise may vanish and the deterministic part of the equation is not necessary asymptotically stable to this purpose we show that the l1 norm of the difference of two solutions starting from any two different initial data converges a s to zero as time goes to infinity

a stefan problem with surface tension as the sharp interface limit of a nonlocal system of phase field type a model for the evolution of phase boundaries reminiscent of the phase field model is considered the equation related to conservation of thermal energy is diffusive and coupled to an equation for the order parameter which contains a nonlinear convolution operator related to the limit of an interacting particle model with kac potential under diffusive rescaling the solutions converge to solutions of the stefan problem with kinetic undercooling and surface tension

strong convergence of a fully discrete finite element approximation of the stochastic cahn hilliard equation we consider the stochastic cahn hilliard equation driven by additive gaussian noise in a convex domain with polygonal boundary in dimension d le 3 we discretize the equation using a standard fi

strong convergence rate of splitting schemes for stochastic nonlinear schrodinger equations abstract in this paper we show that solutions of stochastic nonlinear schrodinger nls equations can be approximated by solutions of coupled splitting systems based on these systems we propose a new kind of fully discrete splitting schemes which possess algebraic strong convergence rates for stochastic nls equations key ingredients of our approach are using the exponential integrability and stability of the corresponding splitting systems and numerical approximations in particular under very mild conditions we derive the optimal strong convergence rate o n 2 1 2 of the spectral splitting crank nicolson scheme where n and denote the dimension of the approximate space and the time step size respectively

a nonlocal stochastic cahn hilliard equation abstract we consider a stochastic extension of the nonlocal convective cahn hilliard equation containing an additive wiener process noise we first introduce a suitable analytical setting and make some mathematical and physical assumptions we then establish in a variational context the existence of a weak statistical solution for this problem finally we prove existence and uniqueness of a strong solution

white noise driven spdes with reflection we study reflected solutions of a nonlinear heat equation on the spatial interval 0 1 with dirichlet boundary conditions driven by space time white noise the nonlinearity appears both in the drift and in the diffusion coefficient roughly speaking at any point t x where the solutionu t x is strictly positive it obeys the equation and at a point t x whereu t x is zero we add a force in order to prevent it from becoming negative this can be viewed as an extension both of one dimensional sdes reflected at 0 and of deterministic variational inequalities existence of a minimal solution is proved the construction uses a penalization argument a new existence theorem for spdes whose coefficients depend on the past of the solution and a comparison theorem for solutions of white noise driven spdes

spinodal decomposition for the cahn hilliard equation in higher dimensions part i probability and wavelength estimate this paper is the first in a series of two papers addressing the phenomenon of spinodal decomposition for the cahn hilliard equation where is a bounded domain with sufficiently smooth boundary and f is cubic like for example f u u u3 we will present the main ideas of our approach and explain in what way our method differs from known results in one space dimension due to grant 26 furthermore we derive certain probability and wavelength estimates the probability estimate is needed to understand why in a neighborhood of a homogeneous equilibrium u0 of the cahn hilliard equation with mass in the spinodal region a strongly unstable manifold has dominating effects this is demonstrated for the linearized equation but will be essential for the nonlinear setting in the second paper 37 as well moreover we introduce the notion of a characteristic wavelength for the strongly unstable directions

the effects of freestream pressure gradient on a corner boundary layer the incompressible boundary layer in the corner formed by two intersecting perpendicular semi infinite planes is investigated the freestream flow aligned with the corner is of the form xn x repr

the role of multiple microscopic mechanisms in cluster interface evolution abstract in this paper we discuss mesoscopic models describing pattern formation mechanisms for a prototypical model of surface processes that involves multiple microscopic mechanisms we focus on a mean field partial differential equation which contains qualitatively microscopic information on particle particle interactions and multiple particle dynamics and we rigorously derive the macroscopic cluster evolution laws and transport structure we show that the motion by mean curvature is given by v where k is the mean curvature is the surface tension and is an effective mobility that depends on the presence of the multiple mechanisms and speeds up the cluster evolution this is in contrast with the allen cahn equation where v

a numerical analysis of the cahn hilliard equation with non permeable walls in this article we consider the numerical analysis of the cahn hilliard equation in a bounded domain with non permeable walls endowed with dynamic type boundary conditions the dynamic type boundary conditions that we consider here have been recently proposed in ruiz goldstein et al phys d 240 8 754 766 2011 in order to describe the interactions of a binary material with the wall the equation is semi discretized using a finite element method for the space variables and error estimates between the exact and the approximate solution are obtained we also prove the stability of a fully discrete scheme based on the backward euler scheme for the time discretization numerical simulations sustaining the theoretical results are presented

harnack inequality and applications for stochastic generalized porous media equations by using coupling and girsanov transformations the dimension free harnack inequality and the strong feller property are proved for transition semigroups of solutions to a class of stochastic generalized porous media equations as applications explicit upper bounds of the l p norm of the density as well as hypercontractivity ultracontractivity and compactness of the corresponding semigroup are derived

second phase spinodal decomposition for the cahn hilliard cook equation we consider the dynamics of a nonlinear partial differential equation perturbed by additive noise assuming that the underlying deterministic equation has an unstable equilibrium we show that the nonlinear stochastic partial differential equation exhibits essentially linear dynamics far from equilibrium more precisely we show that most trajectories starting at the unstable equilibrium are driven away in two stages after passing through a cylindrical region most trajectories diverge from the deterministic equilibrium through a cone shaped region which is centered around a finite dimensional subspace corresponding to strongly unstable eigenfunctions of the linearized equation and on which the influence of the nonlinearity is surprisingly small this abstract result is then applied to explain spinodal decomposition in the stochastic cahn hilliard cook equation on a domain g this equation depends on a small interaction parameter e 0 and one is generally interested in asymptotic results as e 0 specifically we show that linear behavior dominates the dynamics up to distances from the deterministic equilibrium which can reach e 2 dim g 2 with respect to the h 2 g norm

ergodicity of stochastic cahn hilliard equations with logarithmic potentials driven by degenerate or nondegenerate noises abstract we study the asymptotic properties of the stochastic cahn hilliard equation with the logarithmic free energy by establishing different dimension free harnack inequalities according to various kinds of noises the main characteristics of this equation are the singularities of the logarithmic free energy at 1 and 1 and the conservation of the mass of the solution in its spatial variable both the space time colored noise and the space time white noise are considered for the highly degenerate space time colored noise the asymptotic log harnack inequality is established under the so called essentially elliptic conditions and the harnack inequality with power is established for non degenerate space time white noise

group representations arising from lorentz conformal geometry abstract it is shown that there exist conformally covariant differential operators d 2 l k of all even orders 2 l on differential forms of all orders k in the double cover n of the n dimensional compactified minkowski space n these act as intertwining differential operators for natural representations of o 2 n the conformal group of n for even n the resulting decompositions of differential form representations of o 2 n the orthochronous conformal group produce infinite families of unitary representations the most interesting of which are carried by positive mass squared positive frequency quotients for 2l n 2k physically these generalize unitary representations of the conformal group associated with the modified wave operator d 2 0 n 2 2 2 and the maxwell operator on vector potentials d 2 n 2 2 d all the representation spaces produced unitary and nonunitary may be viewed as infinite systems of harmonic oscillators as a by product of the spectral resolution of the d 2 l k one gets some striking wave propagative properties for all of the equations d 2 l k 0 including huygens principle in the curved spacetime n the operators d 2 l k have not been seen before except in the special cases k 0 or n and k n 2 2 l 1 the maxwell operator thus much new information is obtained even in the physical case n 4

radial singular solutions for a fourth order equation with negative exponents abstract in this paper we consider the radial singular solution of 2 u 1 u p in r n with u 0 p 0 by using some elementary ordinary differential equation arguments we prove the above equation admits no radial singular solution for n 3 p 3 in addition the exact asymptotic behaviors of the radial singular solution as r 0 is established for p 1 n 4 which have a significant difference with the explicit singular solution as a product the singularity of any radial singular solution is of type ii if p 1 n 4

cahn hilliard equation with dynamic boundary conditions and mass constraint on the boundary abstract the well known cahn hilliard equation entails mass conservation if a suitable boundary condition is prescribed in the case when the equation is also coupled with a dynamic boundary condition including the laplace beltrami operator on the boundary the total mass on the inside of the domain and its trace on the boundary should be conserved the new issue of this paper is the setting of a mass constraint on the boundary the effect of this additional constraint is the appearance of a lagrange multiplier in fact two lagrange multipliers arise one for the bulk the other for the boundary the well posedness of the resulting cahn hilliard system with dynamic boundary condition and mass constraint on the boundary is obtained the theory of evolution equations governed by subdifferentials is exploited and a complete characterization of the solution is given

a variational approach to dissipative spdes with singular drift we prove global well posedness for a class of dissipative semilinear stochastic evolution equations with singular drift and multiplicative wiener noise in particular the nonlinear term in the drift is the superposition operator associated to a maximal monotone graph everywhere defined on the real line on which neither continuity nor growth assumptions are imposed the hypotheses on the diffusion coefficient are also very general in the sense that the noise does not need to take values in spaces of continuous or bounded functions in space and time our approach combines variational techniques with a priori estimates both pathwise and in expectation on solutions to regularized equations

existence of strong solutions for stochastic porous media equation under general monotonicity conditions this paper addresses existence and uniqueness of strong solutions to stochastic porous media equations dx x dt b x dw t in bounded domains of r d with dirichlet boundary conditions here is a maximal monotone graph in r r possibly multivalued with

analysis of a splitting scheme for damped stochastic nonlinear schrodinger equation with multiplicative noise in this paper we investigate the damped stochastic nonlinear schrodinger nls equation with multiplicative noise and its splitting based approximation when the damped effect is large enough we

generalized motion by mean curvature as a macroscopic limit of stochastic ising models with long range interactions and glauber dynamics we study themacroscopic limit of an appropriately rescaledstochastic ising model withlong range interactions evolving withglauber dynamics as well as the correspondingmean field equation which is nonlinear and nonlocal in the limit we obtain an interface evolving with normal velocity k wherek isthe mean curvature and thetransport coefficient is identified by aneffective green kubo type formula the above assertions are valid for all positive times the motion of the interface being interpreted in theviscosity sense after the onset of the geometric singularities

global solution to the allen cahn equation with singular potentials and dynamic boundary conditions abstract we prove well posedness results for the solution to an initial and boundary value problem for an allen cahn type equation describing the phenomenon of phase transitions for a material contained in a bounded and regular domain the dynamic boundary conditions for the order parameter have been recently proposed by some physicists to account for interactions with the walls in fischer 1997 13 kenzler 2001 14 we show our results using suitable regularizations of the nonlinearities of the problem and performing some a priori estimates which allow us to pass to the limit thanks to compactness and monotonicity arguments

a basic identity for kolmogorov operators in the space of continuous functions related to rdes with multiplicative noise we consider the kolmogorov operator associated with a reaction diffusion equation having polynomially growing reaction coefficient and perturbed by a noise of multiplicative type in the banach space e of continuous functions by analyzing the smoothing properties of the associated transition semigroup we prove a modification of the classical identite du carre des champs that applies to the present non hilbertian setting as an application of this identity we construct the sobolev space w e where is an invariant measure for the system and we prove the validity of the poincare inequality and of the spectral gap

singular stochastic allen cahn equations with dynamic boundary conditions abstract we prove a well posedness result for stochastic allen cahn type equations in a bounded domain coupled with generic boundary conditions the nonlinear flux at the boundary aims at describing the interactions with the hard walls and is motivated by some recent literature in physics the singular character of the drift part allows for a large class of maximal monotone operators generalizing the usual double well potentials one of the main novelties of the paper is the absence of any growth condition on the drift term of the evolution neither on the domain nor on the boundary a well posedness result for variational solutions of the system is presented using a priori estimates as well as monotonicity and compactness techniques a vanishing viscosity argument for the dynamic on the boundary is also presented

a particle model for spinodal decomposition we study a one dimensional lattice gas where particles jump stochastically obeying an exclusion rule and having a small drift toward regions of higher concentration we prove convergence in the continuum limit to a nonlinear parabolic equation whenever the initial density profile satisfies suitable conditions which depend on the strengtha of the drift there is a critical valuea c ofa fora a c the density values are unrestricted while fora a c they should all be to the right or to the left of a given interval a the diffusion coefficient of the limiting equation can be continued analytically to a and in the interior of a it has negative values which should correspond to particle aggregation phenomena we also show that the dynamics can be obtained as a limit of a kawasaki evolution associated to a kac potential the coefficienta plays the role of the inverse temperature the critical value ofa coincides with the critical inverse temperature in the van der waals limit and a with the spinodal region it is finally seen that in a scaling intermediate between the microscopic and the hydrodynamic the system evolves according to an integrodifferential equation the instanton solutions of this equation as studied by dal passo and de mottoni are then related to the phase transition region in the thermodynamic phase diagram analogies with the cahn hilliard equations are also discussed

stochastic reaction diffusion systems with multiplicative noise and non lipschitz reaction term we study existence and uniqueness of a mild solution in the space of continuous functions and existence of an invariant measure for a class of reaction diffusion systems on bounded domains of d perturbed by a multiplicative noise the reaction term is assumed to have polynomial growth and to be locally lipschitz continuous and monotone the noise is white in space and time if d 1 and coloured in space if d 1 in any case the covariance operator is never assumed to be hilbert schmidt the multiplication term in front of the noise is assumed to be lipschitz continuous and no restrictions are given either on its linear growth or on its degenaracy our results apply in particular to systems of stochastic ginzburg landau equations with multiplicative noise

on the cahn hilliard equation with dynamic boundary conditions and a dominating boundary potential the cahn hilliard and viscous cahn hilliard equations with singular and possibly nonsmooth potentials and dynamic boundary condition are considered and some well posedness and regularity results are proved

on quasi linear stochastic partial differential equations we prove existence and uniqueness of the solution of a parabolic spde in one space dimension driven by space time white noise in the case of a measurable drift and a constant diffusion coefficient as well as a comparison theorem

on the regularization effect of space time white noise on quasi linear parabolic partial differential equations we show that an existence and uniqueness and a comparison theorem hold if we add a space time white noise to a quasi linear parabolic equation in one space dimension even if the nonlinearity is only measurable and not even locally bounded

conservative stochastic cahn hilliard equation with reflection we consider a stochastic partial differential equation with reflection at 0 and with the constraint of conservation of the space average the equation is driven by the derivative in space of a space time white noise and contains a double laplacian in the drift due to the lack of the maximum principle for the double laplacian the standard techniques based on the penalization method do not yield existence of a solution we propose a method based on infinite dimensional integration by parts formulae obtaining existence and uniqueness of a strong solution for all continuous nonnegative initial conditions and detailed information on the associated invariant measure and dirichlet form

stability of strong solutions for a model of incompressible two phase flow under thermal fluctuations abstract we consider a model of a two phase flow based on the phase field approach where the fluid bulk velocity obeys the standard navier stokes system while the concentration difference of the two fluids plays a role of order parameter governed by the allen cahn equations possible thermal fluctuations are incorporated through a random forcing term in the allen cahn equation we show that suitable dissipative martingale solutions satisfy a stochastic version of the relative energy inequality this fact is used for showing the weak strong uniqueness principle both pathwise and in law

nonlinear diffusion equations as asymptotic limits of cahn hilliard systems an asymptotic limit of a class of cahn hilliard systems is investigated to obtain a general nonlinear diffusion equation the target diffusion equation may reproduce a number of well known model equations stefan problem porous media equation hele shaw profile nonlinear diffusion of singular logarithmic type nonlinear diffusion of penrose fife type fast diffusion equation and so on namely by setting the suitable potential of the cahn hilliard systems all these problems can be obtained as limits of the cahn hilliard related problems convergence results and error estimates are proved

existence and regularity of solution for a stochastic cahn hilliard allen cahn equation with unbounded noise diffusion the cahn hilliard allen cahn equation with noise is a simplified mean field model of stochastic microscopic dynamics associated with adsorption and desorption spin flip mechanisms in the context of surface processes for such an equation we consider a multiplicative space time white noise with diffusion coefficient of sub linear growth using technics from semigroup theory we prove existence and path regularity of stochastic solution depending on that of the initial condition our results are also valid for the stochastic cahn hilliard equation with unbounded noise diffusion for which previous results were established only in the framework of a bounded diffusion coefficient we prove that the path regularity of stochastic solution depends on that of the initial condition and are identical to those proved for the stochastic cahn hilliard equation and a bounded noise diffusion coefficient if the initial condition vanishes they are strictly less than 2 d 2 in space and 1 2 d 8 in time as expected from the theory of parabolic operators in the sense of petrovski the bi laplacian operator seems to be dominant in the combined model

equation and dynamic boundary condition of cahn hilliard type with singular potentials abstract the well posedness of a system of partial differential equations and dynamic boundary conditions both of cahn hilliard type is discussed the existence of a weak solution and its continuous dependence on the data are proved using a suitable setting for the conservation of a total mass in the bulk plus the boundary a very general class of double well like potentials is allowed moreover some further regularity is obtained to guarantee the strong solution

smoothing properties of transition semigroups relative to sdes with values in banach spaces in the present paper we consider the transition semigroup pt related to some stochastic reaction diausion equations with the non linear term f having polynomial growth and satisfying some dis sipativity conditions we are proving that it has a regularizing eaect in the banach space of continuous functions coou where o r d is a bounded open set in l 2 oou the only result proved is the strong feller property for da 1 here we are able to prove that if f 2 c 1 oru and d 3 then ptu2 c 1 b ocoouu for any u2 bbocoouu and t 0 an important application is to the study of the ergodic properties of the system these results are also of interest for some problem in sto chastic control

hypercontractivity and applications for stochastic hamiltonian systems abstract the hypercontractivity is proved for the markov semigroup associated with a class of stochastic hamiltonian systems on hilbert spaces consequently the markov semigroup converges exponentially to the invariant probability measure in entropy and is compact for large time these strengthen the hypocoercivity results derived in the literature since the log sobolev inequality is invalid we introduce a new argument to prove the hypercontractivity using coupling and dimension free harnack inequality the main results are illustrated by concrete examples of the kinetic fokker planck equation and highly degenerate diffusion processes

spinodal decomposition for the cahn hilliard cook equation this paper gives theoretical results on spinodal decomposition for the stochastic cahn hilliard cook equation which is a cahn hilliard equation perturbed by additive stochastic noise we prove that most realizations of the solution which start at a homogeneous state in the spinodal interval exhibit phase separation leading to the formation of complex patterns of a characteristic size