on the muskat problem with viscosity jump global in time results abstract the muskat problem models the filtration of two incompressible immiscible fluids of different characteristics in porous media in this paper we consider both the 2d and 3d setting of two fluids of different constant densities and different constant viscosities in this situation the related contour equations are non local not only in the evolution system but also in the implicit relation between the amplitude of the vorticity and the free interface among other extra difficulties no maximum principles are available for the amplitude and the slopes of the interface in l we prove global in time existence results for medium size initial stable data in critical spaces we also enhance previous methods by showing smoothing instant analyticity improving the medium size constant in 3d together with sharp decay rates of analytic norms the found technique is twofold giving ill posedness in unstable situations for very low regular solutions

dynamics near unstable interfacial fluids we study three examples of unstable interfacial fluid motions vortex sheets with surface tension hele shaw flows with surface tension and vortex patches in all three cases the nonlinear dynamics of a large class of smooth perturbations is proven to be characterized by the corresponding fastest linear growing mode s up to the time scale of log frac 1 delta where delta is the magnitude of the initial perturbation in all three cases the analysis is based on an unified analytical framework that includes precise bounds on the growth of the linearized operator given by an explicit solution formula as well as a special sharp nonlinear energy growth estimate our main contribution is establishing this nonlinear energy growth estimate for each interface problem in certain high energy norms

motion of a set by the curvature of its boundary abstract the study of a crystal shrinking or growing in a melt gives rise to equations relating the normal velocity of the motion to the curvature of the crystal boundary often these equations are anisotropic indicating the preferred directions of the crystal structure in the isotropic case this equation is called the curve shortening or the mean curvature flow equation and has been studied by differential geometric tools in general there are no classical solutions to these equations in this paper we develop a weak theory for the generalized mean curvature equation using the newly developed theory of viscosity solutions our approach is closely related to that of osher and sethian chen giga and goto and evans and spruck who view the boundary of the crystal as the level set of a solution to a nonlinear parabolic equation although we use their results in an essential way we give an intrinsic definition our main results are the existence of a solution large time asymptotics of this solution and its connection to the level set solution of osher and sethian chen giga and goto and evans and spruck in general there is no uniqueness even for classical solutions but we prove a uniqueness result under restrictive assumptions we also construct a class of explicit solutions which are dilations of wulff crystals

remarks on the application of the keldysh theorem to the completeness of root subspace of non self adjoint operators and comments on spectral operators generated by timoshenko beam model in this note we show by means of an example that the assumption of compactness in the keldysh theorem in proving the completeness of root subspaces of non self adjoint operators cannot be ignored this indicates a gap of the above mentioned paper and many other related papers in the proof of completeness for the root subspaces of beam related problems

orthogonality conditions and asymptotic stability in the stefan problem with surface tension we prove nonlinear asymptotic stability of steady spheres in the two phase stefan problem with surface tension our method relies on the introduction of appropriate orthogonality conditions in conjunction with a high order energy method

breakdown of smoothness for the muskat problem in this paper we show that there exists analytic initial data in the stable regime for the muskat problem such that the solution turns to the unstable regime and later breaks down that is no longer belongs to c4

riesz basis property of evolution equations in hilbert spaces and application to a coupled string equation suppose that lambda n is the set of zeros of a sine type generating function of the exponential system e i lambda n t in l2 0 t and is separated levin and golovin s classical theorem claims that e i lambda n t forms a riesz basis for l2 0 t in this article we relate this result with riesz basis generation of eigenvectors of the system operator of the linear time invariant evolution equation in hilbert spaces through its spectrum a practically favorable necessary and sufficient condition for the separability of zeros of function of sine type is derived the result is applied to get riesz basis generation of a coupled string equation with joint dissipative feedback control

contour dynamics of incompressible 3 d fluids in a porous medium with different densities we consider the problem of the evolution of the interface given by two incompressible fluids through a porous medium which is known as the muskat problem and in two dimensions it is mathematically analogous to the two phase hele shaw cell we focus on a fluid interface given by a jump of densities being the equation of the evolution obtained using darcy s law we prove local well posedness when the smaller density is above stable case and in the unstable case we show ill posedness

the zero surface tension limit of three dimensional interfacial darcy flow abstract we study the zero surface tension limit of three dimensional interfacial darcy flow we start with a proof of well posedness of three dimensional interfacial darcy flow for any positive value of the surface tension coefficient the primary tool for this well posedness proof is an energy estimate the time of existence for these solutions will in general go to zero with the surface tension parameter however in the case that a stability condition is satisfied by the initial data we prove an additional energy estimate establishing that the time of existence can be made uniform in the surface tension parameter then an additional estimate allows the limit to be taken as surface tension vanishes demonstrating that three dimensional interfacial darcy flow without surface tension is the limit of three dimensional interfacial darcy flow with surface tension as surface tension vanishes this provides a new proof of existence of solutions for the problem without surface tension

bending torsion vibration model with two end energy dissipation this research is devoted to the asymptotic and spectral analysis of a coupled euler bernoulli and timoshenko beam model the model is governed by a system of two coupled differential equations and a two parameter family of boundary conditions modelling the action of self straining actuators the aforementioned equations of motion together with a four parameter family of boundary conditions form a coupled linear hyperbolic system which is equivalent to a single operator evolution equation in the energy space that equation defines a semigroup of bounded operators the dynamics generator of the semigroup is our main object of interest for each set of the boundary parameters the dynamics generator has a compact inverse if all boundary parameters are not purely imaginary numbers then the dynamics generator is a non self adjoint operator in the energy space we calculate the spectral asymptotics of the dynamics generator we find that the spectrum lies in a strip parallel to the horizontal axis and is asymptotically close to the horizontal axis thus the system is stable but is not uniformly stable we have also found that the set of the eigenfunctions of the dynamics generator forms an unconditional basis the riesz basis in the state space of the system the riesz basis property allows us to solve explicitly boundary and or distributed controllability problems via the spectral decomposition method

the hele shaw problem with surface tension in a half plane abstract in this paper we consider the hele shaw problem in a 2 dimensional fluid domain t which is constrained to a half plane the boundary of t consist of two components 0 t which lies on the boundary of the half plane and t which lies inside the half plane on t we impose the classical boundary conditions with surface tension and on 0 t we prescribe the normal derivative of the fluid pressure at the point where 0 t and t meet there is an abrupt change in the boundary condition giving rise to a singularity in the fluid pressure we prove that the problem has a unique solution with smooth free boundary t for some small time interval

the polyharmonic heat flow of closed plane curves abstract in this paper we consider the polyharmonic heat flow of a closed curve in the plane our main result is that closed initial data with initially small normalised oscillation of curvature and isoperimetric defect flows exponentially fast in the c topology to a simple circle our results yield a characterisation of the total amount of time during which the flow is not strictly convex quantifying in a sense the failure of the maximum principle

on an evolution equation in a cell motility model abstract this paper deals with the evolution equation of a curve obtained as the sharp interface limit of a non linear system of two reaction diffusion pdes this system was introduced as a phase field model of crawling motion of eukaryotic cells on a substrate the key issue is the evolution of the cell membrane interface curve which involves shape change and net motion this issue can be addressed both qualitatively and quantitatively by studying the evolution equation of the sharp interface limit for this system however this equation is non linear and non local and existence of solutions presents a significant analytical challenge we establish existence of solutions for a wide class of initial data in the so called subcritical regime existence is proved in a two step procedure first for smooth h 2 initial data we use a regularization technique second we consider non smooth initial data that are more relevant from the application point of view here uniform estimates on the time when solutions exist rely on a maximum principle type argument we also explore the long time behavior of the model using both analytical and numerical tools we prove the nonexistence of traveling wave solutions with nonzero velocity numerical experiments show that presence of non linearity and asymmetry of the initial curve results in a net motion which distinguishes it from classical volume preserving curvature motion this is done by developing an algorithm for efficient numerical resolution of the non local term in the evolution equation

some mathematical aspects of the micro phase separation in diblock copolymers abstract a free energy functional of nonlocal type is considered that was originally introduced to describe the micro phase separation of diblock copolymer a mathematical framework is given to the issues of the scaling law for stationary states and the governing equation of morphology the scaling law is equivalent to finding a nice rescaling in order to have a well defined limiting interfacial problem which is free from the interfacial thickness and the total chain length of copolymer and the associated stationary problem becomes a morphology equation that governs the configuration of final patterns although the steady patterns become finer and finer in our scaling limit due to the mesoscopic nature a possible rigorous approach is presented to stability and morphological selection problems for them

turning waves and breakdown for incompressible flows we consider the evolution of an interface generated between two immiscible incompressible and irrotational fluids specifically we study the muskat and water wave problems we show that starting with a family of initial data given by f0 the interface reaches a regime in finite time in which is no longer a graph therefore there exists a time t where the solution of the free boundary problem parameterized as f t blows up f l t in particular for the muskat problem this result allows us to reach an unstable regime for which the rayleigh taylor condition changes sign and the solution breaks down

free boundary problems with surface tension conditions abstract in this paper we consider free boundary problems with a surface tension condition at the free boundary the simplest of these problems is the classical hele shaw problem which describes viscous flow between two parallel plates with small spacing we shall consider models which describe tumor growth and are expressed by systems of pdes we shall also consider the hele shaw problem constrained in a half space

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

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

interface evolution the hele shaw and muskat problems we study the dynamics of the interface between two incompressible 2 d ows where the evolution equation is obtained from darcy s law the free boundary is given by the discontinuity among the densities and viscosities of the uids this physical scenario is known as the two dimensional muskat problem or the two phase hele shaw ow we prove local existence in sobolev spaces when initially the dierence of the gradients of the pressure in the normal direction has the proper sign an assumption which is also known as the rayleigh taylor condition

a paradifferential approach for well posedness of the muskat problem we study the muskat problem for one fluid or two fluids with or without viscosity jump with or without rigid boundaries and in arbitrary space dimension d of the interface the muskat problem is scaling invariant in the sobolev space h s c mathbb r d where s c 1 frac d 2 employing a paradifferential approach we prove local well posedness for large data in any subcritical sobolev spaces h s mathbb r d s s c moreover the rigid boundaries are only required to be lipschitz and can have arbitrarily large variation the rayleigh taylor stability condition is assumed for the case of two fluids with viscosity jump but is proved to be automatically satisfied for the case of one fluid the starting point of this work is a reformulation solely in terms of the drichlet neumann operator the key elements of proofs are new paralinearization and contraction results for the drichlet neumann operator in rough domains

a maximum principle for the muskat problem for fluids with different densities we study the fluid interface problem given by two incompressible fluids with different densities evolving by darcy s law this scenario is known as the muskat problem for fluids with the same viscosities being in two dimensions mathematically analogous to the two phase hele shaw cell we prove in the stable case the denser fluid is below a maximum principle for the l norm of the free boundary

the hele shaw problem with surface tension in a half plane a model problem abstract consider the hele shaw problem with surface tension in the half plane y 1 0 when at time t 0 the domain t lies partly on the line y 1 0 and partly in y 1 0 in order to establish existence of a solution to this free boundary problem we need to study the linear model problem when the t is a fixed angular domain in this paper we consider this model problem and establish existence of a solution satisfying sharp weighted holder estimates these estimates will be used in subsequent work to solve the full hele shaw problem

on the muskat problem global in time results in 2d and 3d this paper considers the three dimensional muskat problem in the stable regime we obtain a conservation law which provides an l 2 maximum principle for the fluid interface we also show global in time existence for strong and weak solutions with initial data controlled by explicit constants furthermore we refine the available estimates to obtain global existence and uniqueness for strong solutions with larger initial data than we previously had in 2d finally we provide global in time results in spaces with critical regularity giving solutions with bounded slope and time integrable bounded curvature

riesz basis approach to the stabilization of a flexible beam with a tip mass using an abstract condition of riesz basis generation of discrete operators in the hilbert spaces we show in this paper that a sequence of generalized eigenfunctions of an euler bernoulli beam equation with a tip mass under boundary linear feedback control forms a riesz basis for the state hilbert space in the meanwhile an asymptotic expression of eigenvalues and the exponential stability are readily obtained the main results of siam j control optim 36 1998 pp 1962 1986 are concluded as a special case and the additional conditions imposed there are removed

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

global existence singular solutions and ill posedness for the muskat problem the muskat or muskat leibenzon problem describes the evolution of the interface between two immiscible fluids in a porous medium or hele shaw cell under applied pressure gradients or fluid injection extraction in contras t to the hele shaw problem the one phase version of the muskat problem there are few nontrivial exact solutions or analytic results for the muskat problem for the stable forward muskat problem in which the higher viscosity fluid ex pands into the lower viscosity fluid we show global in time existence for initial data that is a small perturbation of a flat interface the initial data in this result ma y contain weak e g curvature singularities for the unstable backwa rd problem in which the higher viscosity fluid contracts we construct singular solution s that start off with smooth initial data but develop a point of infinite curvature at fi nite time c 2004 wiley periodicals inc

ostwald ripening for dilute systems under quasistationary dynamics we consider a dilute mixture in 3d of a finite number of particles initially close to spherical but of varying size evolving according to quasistationary dynamics under the scaling hypotheses that 1 size distance 2 deviation from sphericity size are both initially small we show robustness of the almost spherical shape by associating centers and radii to each particle and then derive rigorously a set of o d e s for the radii which we relate to the lifschitz slyozov wagner theory of coarsening and also establish an estimate for the speed of the centers

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

ground state solutions to kirchhoff type transmission problems with critical perturbation abstract a class of kirchhoff type transmission problems with critical perturbation is studied unlike other studies on elliptical equations with critical growth a ground state solution is obtained using a perturbation method instead of verifying that the mountain pass level is lower than the critical energy which is used to verify the ps condition

darcy s flow with prescribed contact angle well posedness and lubrication approximation we consider the spreading of a thin two dimensional droplet on a solid substrate we use a model for viscous fluids where the evolution is governed by darcy s law at the contact point where air and liquid meet the solid substrate a constant non zero contact angle partial wetting is assumed we show local and global well posedness of this free boundary problem in the presence of the moving contact point our estimates are uniform in the contact angle assumed by the liquid at the contact point in the so called lubrication approximation long wave limit we show that the solutions converge to the solution of a one dimensional degenerate parabolic fourth order equation which belongs to a family of thin film equations the main technical difficulty is to describe the evolution of the non smooth domain and to identify suitable spaces that capture the transition to the asymptotic model uniformly in the small parameter varepsilon

well posedness and uniform bounds for a nonlocal third order evolution operator on an infinite wedge we investigate regularity and well posedness for a fluid evolution model in the presence of a three phase contact point we consider a fluid evolution governed by darcy s law after linearization we obtain a nonlocal third order operator which contains the dirichlet neumann operator on the wedge with opening angle epsilon 0 we show well posedness and regularity for this linear evolution equation in the limit of vanishing opening angle we show the convergence of solutions to a fourth order degenerate parabolic operator related to the thin film equation in the course of the analysis we introduce and characterize a new type of sum of weighted sobolev spaces which are suitable to capture the singular limit as epsilon to 0 in particular the nature of the problem requires the use of techniques that are adapted to the problem in the singular domain as well as the degenerate limit problem

a simplified version of the fast algorithms of brent and salamin we produce more elementary algorithms than those of brent and salamin for respectively evaluating e x and although the gauss arithmetic geometric process still plays a central role the elliptic function theory is now unnecessary

approximate solutions of the cahn hilliard equation via corrections to the mullins sekerka motion we develop an alternative method to matched asymptotic expansions for the construction of approximate solutions of the cahn hilliard equation suitable for the study of its sharp interface limit the method is based on the hilbert expansion used in kinetic theory besides its relative simplicity it leads to calculable higher order corrections to the interface motion

non splat singularity for the one phase muskat problem for the water waves equations the existence of splat singularities has been shown in 3 i e the interface self intersects along an arc in finite time the aim of this paper is to show the absence of splat singularities for the incompressible fluid dynamics in porous media

well posedness of the muskat problem with h2 initial data abstract we study the dynamics of the interface between two incompressible fluids in a two dimensional porous medium whose flow is modeled by the muskat equations for the two phase muskat problem we establish global well posedness and decay to equilibrium for small h 2 perturbations of the rest state for the one phase muskat problem we prove local well posedness for h 2 initial data of arbitrary size finally we show that solutions to the muskat equations instantaneously become infinitely smooth

an integral hidden in gradshetyn and ryzik abstract we provide a closed form expression for the integral n 0 4 a m 0 dz z 4 2az 2 1 m 1 where m n and a 1 n 0 4 a m 2 m 3 2 a 1 m 1 2 p m m 1 2 m 1 2 a 2 3m 3 2 a 1 m 1 2 x k 0 m 2k 2m 2k m k m k m a 1 k here p m m 1 2 m 1 2 a is the jacobi polynomial pm a with parameters m 1 2 and m 1 2

rayleigh taylor breakdown for the muskat problem with applications to water waves the muskat problem models the evolution of the interface between two different fluids in porous media the rayleigh taylor condition is natural to reach linear stability of the muskat problem we show that the rayleigh taylor condition may hold initially but break down in finite time as a consequence of the method used we prove the existence of water waves turning

the muskat problem with surface tension and a nonregular initial interface abstract in this paper we consider the two dimensional muskat problem with surface tension on a free boundary the initial shape of the unknown interface has a corner point we prove that the problem has a unique solution in the weighted holder classes locally in time and specify the sufficient conditions for the existence of the waiting time phenomenon

landen transformations and the integration of rational functions we present a rational version of the classical landen transformation for elliptic integrals this is employed to obtain explicit closed form expressions for a large class of integrals of even rational functions and to develop an algorithm for numerical integration of these functions

evolution of microstructure in unstable porous media flow a relaxational approach we study the flow of two immiscible fluids of different density and mobility in a porous medium if the heavier phase lies above the lighter one the interface is observed to be unstable the two phases start to mix on a mesoscopic scale and the mixing zone grows in time an example of evolution of microstructure a simple set of assumptions on the physics of this two phase flow in a porous medium leads to a mathematically ill posed problem when used to establish a continuum free boundary problem we propose and motivate a relaxation of this nonconvex constraint of a phase distribution with a sharp interface on a macroscopic scale we prove that this approach leads to a mathematically well posed problem that predicts shape and evolution of the mixing profile as a function of the density difference and mobility quotient 1999 john wiley sons inc

on the c0 semigroup generation and exponential stability resulting from a shear force feedback on a rotating beam in this paper we show that a linear unbounded operator associated with an euler bernoulli beam equation under shear boundary feedback generates a c0 semig roup in the underlyingstate hilbert space this provides an answer to a longtime unsolved problem due to the lack of dissipativity for the operator the main steps are a careful estimation of the green s function and the verification of the riesz basis property for the generalized eigenfunctions as a consequence we show that this semigroup is differentiable and exponentially stable which is in sharp contrast to the properties possessed by most feedback controlled beams based on a passive design principle 2005 elsevier b v all rights reserved

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

on the two dimensional muskat problem with monotone large initial data we consider the evolution of two incompressible immiscible fluids with different densities in porous media known as the muskat problem 21 which in two dimensions is analogous to the hele shaw cell 24 we establish for a class of large and monotone initial data the global existence of weak solutions the proof is based on a local well posedness result for the initial data with certain specific asymptotics at spatial infinity and a new maximum principle for the first derivative of the graph function 2016 wiley periodicals inc

absence of splash singularities for surface quasi geostrophic sharp fronts and the muskat problem in this paper for both the sharp front surface quasi geostrophic equation and the muskat problem we rule out the splash singularity blow up scenario in other words we prove that the contours evolving from either of these systems cannot intersect at a single point while the free boundary remains smooth splash singularities have been shown to hold for the free boundary incompressible euler equation in the form of the water waves contour evolution problem our result confirms the numerical simulations in earlier work in which it was shown that the curvature blows up because the contours collapse at a point here we prove that maintaining control of the curvature will remove the possibility of pointwise interphase collapse another conclusion that we provide is a better understanding of earlier work in which squirt singularities are ruled out in this case a positive volume of fluid between the contours cannot be ejected in finite time

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

viscous fingering an optimal bound on the growth rate of the mixing zone we consider the flow of two immiscible fluids of different mobility in a porous medium if the more mobile fluid displaces the other a macroscopically sharp interface is unstable by growing a network of fingers on a mesoscopic scale the two phases mix on a macroscopic scale we are interested in the evolution of this mixing zone we show that the effect of a large but finite mobility ratio lambda is strong enough to limit the growth rate of the mixing zone this is done by rigorously deriving an a priori estimate for the saffman taylor model in this geometry of an infinite channel the estimate essentially states that the mobility ratio lambda itself in the nondimensionalized setting with unit velocity imposed at infinity is the optimal bound on the velocity by which the penetrating phase progresses in direction of the channel since the introduction of diffusion limited aggregation various stochastic algorithms simulating this two phase flow have been developed the generated clusters which

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

piecewise constant subsolutions for the muskat problem we show the existence of infinitely many admissible weak solutions for the incompressible porous media equations for all muskat type initial data with c 3 alpha regularity of the interface in the unstable regime and for all non horizontal data with c 3 alpha regularity in the stable regime our approach involves constructing admissible subsolutions with piecewise constant densities this allows us to give a rather short proof where it suffices to calculate the velocity and acceleration at time zero thus emphasizing the instantaneous nature of non uniqueness due to discontinuities in the initial data

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

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

on the solvability of a transmission problem for the laplace operator with a dynamic boundary condition on a nonregular interface abstract in this paper we consider a transmission problem for the laplace operator when an interface contains angular points and time derivatives of the unknown functions enter in the transmission condition we prove the existence of a unique solution in the weighted holder classes

on a nonclassical fractional boundary value problem for the laplace operator abstract in this paper we consider a boundary value problem for the poisson equation with a boundary condition comprising the fractional derivative in time and the right hand sides dependent on time we prove the one valued solvability of this problem and provide the coercive estimates of the solution

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

on well posedness of the muskat problem with surface tension abstract we consider the muskat problem with surface tension for one fluid or two fluids with or without viscosity jump with infinite depth or lipschitz rigid boundaries and in arbitrary dimension d of the interface the problem is nonlocal quasilinear and to leading order is scaling invariant in the sobolev space h s c r d with s c 1 d 2 we prove local well posedness for large data in all subcritical sobolev spaces h s r d s s c allowing for initial interfaces whose curvatures are unbounded and furthermore when d 1 not locally square integrable to the best of our knowledge this is the first large data well posedness result that covers all subcritical sobolev spaces for the muskat problem with surface tension we reformulate the problem in terms of the dirichlet neumann operator and use a paradifferential approach to reduce the problem to an explicit parabolic equation which is of independent interest

analytical behavior of two dimensional incompressible flow in porous media in this paper we study the analytic structure of a two dimensional mass balance equation of an incompressible fluid in a porous medium given by darcy s law we obtain local existence and uniqueness by the particle trajectory method and we present different global existence criterions these analytical results with numerical simulations are used to indicate nonformation of singularities moreover blow up results are shown in a class of solutions with infinite energy

the hele shaw problem and area preserving curve shortening motions we prove existence locally in time of a solution of the following hele shaw problem given a simply connected curve contained in a smooth bounded domain find the motion of the curve such that its normal velocity equals the jump of the normal derivatives of a function which is harmonic in the complement of the curve in and whose boundary value on the curve equals its curvature we show that this motion is a curve shortening motion which does not change the area of the region enclosed by the curve in case is the whole plane 2 we also show that if the initial curve is close to an equilibrium curve i e to a circle then there exists a global solution and the global solution tends to some circle exponentially fast as time tends to infinity

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

rational landen transformations on ℝ the landen transformation a b a b 2 ab preserves the value of an elliptic integral and its iteration produces the classical arithmetic geometric mean agm a b we present analogous transformations for rational functions integrated over the whole real line

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

local and global existence results for anisotropic hele shaw flows in this paper we study a moving boundary problem for an anisotropic two phase hele shaw flow using a regularization technique we prove existence of a local solution under suitable conditions on the initial free boundary we obtain a global solution and study its asymptotic behaviour

motion of elastic thin films by anisotropic surface diffusion with curvature regularization in this paper we prove short time existence uniqueness and regularity for a surface diffusion evolution equation with curvature regularization in the context of epitaxially strained two dimensional films this is achieved by using the h 1 gradient flow structure of the evolution law via de giorgi s minimizing movements this seems to be the first short time existence result for a surface diffusion type geometric evolution equation in the presence of elasticity

the transmission problem in domains with a corner point for the laplace operator in weighted holder spaces abstract we prove the existence and uniqueness of a solution to the elliptic transmission problem in nonsmooth domains in the weighted holder space the coercive estimates of the solution are given

spectral operators generated by timoshenko beam model we announce a series of results on the spectral analysis for a class of nonselfadjoint operators which are the dynamics generators for the systems governed by the equations of the timoshenko beam model with a 2 parameter family of dissipative boundary conditions our results split into three groups 1 we present asymptotic formulas for the spectra of the aforementioned operators the spectrum of each operator consists of two branches of discrete complex eigenvalues and for their generalized eigenvectors 2 we show that these operators are riesz spectral this result follows from the fact that the systems of generalized eigenvectors form riesz bases in the corresponding energy spaces 3 we give the asymptotics of the spectra and the eigenfunctions for the nonselfadjoint polynomial operator pencils associated with these operators our results on one hand provide a class of nontrivial examples of spectral operators nonselfadjoint operators which admit an analog of spectral decomposition on the other hand these results give a key to the solutions of various control and stabilization problems for the timoshenko beam model using the spectral decomposition method

lyapunov functions identities and the cauchy problem for the hele shaw equation this article is devoted to the study of the hele shaw equation we introduce an approach inspired by the water wave theory starting from a reduction to the boundary introducing the dirichlet to neumann operator and exploiting various cancellations we exhibit parabolic evolution equations for the horizontal and vertical traces of the velocity on the free surface this allows to quasi linearize the equations in a natural way by combining these exact identities with convexity inequalities we prove the existence of hidden lyapunov functions of different natures we also deduce from these identities and previous works on the water wave problem a simple proof of the well posedness of the cauchy problem the analysis contains two side results of independent interest firstly we give a principle to derive estimates for the modulus of continuity of a pde under general assumptions on the flow secondly we prove and give applications of a convexity inequality for the dirichlet to neumann operator

global classical solution of muskat free boundary problem abstract in this paper the muskat problem which describes a two phase flow of two fluids for example oil and water in porous media is discussed the problem involves in seeking two time dependent harmonic functions u1 x y t and u2 x y t in oil and water regions respectively and the interface between oil and water i e the free boundary y x t such that on the free boundary u 1 u 2 v n k 1 u 1 n k 2 u 2 n where n the unit normal vector on the free boundary toward oil region vn is the normal velocity of the free boundary k1 and k2 are positive constants satisfying k1 k2 we prove the existence of classical solution globally in time under some reasonable assumptions the argument developed in this paper can be used in any multidimensional case