new dissipated energy and partial regularity for thin film equations abstract we study the thin film equation h t h n h x x x x and prove that the energy h p h x 2 d x dissipates for suitable weak solutions when n p lies in the whole domain e defined by laugesen which extends partially the corresponding results of laugesen chugunova and taranets and then improve some previous results of partial regularity by chou and the first author

cahn hilliard and thin film equations with nonlinear mobility as gradient flows in weighted wasserstein metrics abstract in this paper we establish a novel approach to proving existence of non negative weak solutions for degenerate parabolic equations of fourth order like the cahn hilliard and certain thin film equations the considered evolution equations are in the form of a gradient flow for a perturbed dirichlet energy with respect to a wasserstein like transport metric and weak solutions are obtained as curves of maximal slope our main assumption is that the mobility of the particles is a concave function of their spatial density a qualitative difference of our approach to previous ones is that essential properties of the solution non negativity conservation of the total mass and dissipation of the energy are automatically guaranteed by the construction from minimizing movements in the energy landscape

finite element approximation of a sixth order nonlinear degenerate parabolic equation we consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation equation here where generically equation here for any given equation here in addition to showing well posedness of our approximation we prove convergence in space dimensions d leq 3 furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed finally some numerical experiments in one and two space dimensions are presented

wavenumber based filtering of marine point source data in seismic processing plane wave decomposition has played a fundamental role serving as a basis for developing sophisticated processing techniques valid for depth dependent models by comparing analytical expressions for the decomposed wavefields we review several processing algorithms of interest for the geophysicist the algorithms may be applied to marine point source data acquired over a horizontally layered viscoelastic and anisotropic medium the plane wave decomposition is based on the fourier transform integral for line source data and the hankel transform integral for point source data the involved wavenumber integrals of the cosine or bessel function type are difficult to evaluate accurately and efficiently however a number of the processing techniques can easily be run as a filtering operation in the spatial domain without transforming to the wavenumber domain the mathematical expressions for the spatial filters are derived using plane wave analysis with numerical examples we demonstra

higher order nonlinear degenerate parabolic equations this paper is concerned with nonlinear degenerate parabolic equations of the form ut 1 m 1 d f u d2m 1u 0 with f u u n n 1 near u 0 and d x under appropriate boundary conditions it is shown that there exists a weak solution u some of the main results of the paper are that u 0 if u0 0 and that the support of u t when u0 0 increases with t for the last property we require that n 2 and m 1

doubly nonlinear thin film equations in one space dimension we consider a free boundary problem for a class of fourth order nonlinear parabolic equations which are degenerate both with respect to the unknown and to its third derivative the problem is relevant in the description of the surface tension driven spreading of a non newtonian liquid over a solid surface in the complete wetting regime relying solely on global and local energy estimates and on bernis inequalities we prove existence of solutions to this problem and obtain sharp upper bounds for the propagation of their support a necessary condition for the occurrence of waiting time phenomena is also derived

on a thin film model with insoluble surfactant abstract this paper studies the existence and asymptotic behavior of global weak solutions for a thin film equation with insoluble surfactant under the influence of gravitational capillary and van der waals forces we prove the existence of global weak solutions for medium sized initial data in large function spaces moreover exponential decay towards the flat equilibrium state is established where an estimate on the decay rate can be computed explicitly

reverse draining of a magnetic soap film analysis and simulation of a thin film equation with non uniform forcing we analyze and classify equilibrium solutions of the one dimensional thin film equation with no flux boundary conditions and in the presence of a spatially dependent external forcing we prove theorems that shed light on the nature of these equilibrium solutions guarantee their validity and describe how they depend on the properties of the external forcing we then apply these results to the reverse draining of a one dimensional magnetic soap film subject to an external non uniform magnetic field numerical simulations illustrate the convergence of the solutions towards equilibrium configurations we then present bifurcation diagrams for steady state solutions we find that multiple stable equilibrium solutions exist for fixed parameters and uncover a rich bifurcation structure to these solutions demonstrating the complexity hidden in a relatively simple looking evolution equation finally we provide a simulation describing how numerical solutions traverse the bifurcation diagram as the amplitude of the forcing is slowly increased and then decreased

a generalized thin film equation in multidimensional space abstract this paper deals with the existence and nonnegativity of the weak periodic domain solution for a degenerate fourth order parabolic equation modeling the evolution of thin films this study in the multidimensional domain follows the recent results related to the resolution in one dimensional space j e rakotoson j m rakotoson c verbeke generalized lubrification models blow up and global existence racsam 99 2 2005 235 241 c verbeke quelques modeles d equations d evolution de surfaces explosion en temps fini et diverses proprietes qualitatives these de doctorat de l universite de poitiers 15 decembre 2005 j e rakotoson j m rakotoson c verbeke higher order equations related to thin film blow up and global existence the influence of the initial data submitted for publication

on the periodic intermediate long wave equation the authors discuss some properties of a physically interesting nonlinear integro differential equation with periodic boundary conditions it is the natural periodic analogue of the intermediate long wave equation and it provides a periodic analogue of the benjamin ono equation in the appropriate limit due to the speciality of the integral operator the equation admits a backlund transformation an infinity of motion constants etc two simple periodic solutions are exhibited the equation may be transformed into more than one kind of bilinear equation

wave equation processing using finite difference propagators part 1 wavefield dissection and imaging of marine multicomponent seismic data abstractmethods for wavefield injection are used in for instance reverse time extrapolation of shot gathers in reverse time migration for correct injection of recorded data without any ambiguity of the propagation direction the wavefield injection methodology requires pressure and particle velocity data such as multicomponent towed marine or seabed seismic recordings we discovered that by carefully considering the models medium parameters and boundary conditions for injection wavefield injection of multicomponent data can also be used to solve several long standing challenges in marine seismic data processing by means of conventional time space domain finite difference propagators we outlined and demonstrated several of these important applications including up down separation of wavefields deghosting direct wave removal source signature estimation multiple removal and imaging using primaries and multiples only acoustic models are considered but the concepts are straightforward to general

periodic solutions of the intermediate long wave equation a nonlinear superposition principle periodic stationary wave solutions of the intermediate long wave ilw equation are derived using the bilinear transformation method and a new expression for the dispersion relation is obtained the class of physically important real valued solutions is identified these solutions may be represented as an infinite superposition of solitary wave profiles a property shared by the related korteweg de vries kdv and benjamin ono bo equation this nonlinear superposition principle which has been the subject of various interpretations in the literature is discussed the ilw periodic solution approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes respectively for intermediate amplitudes the solution can be well approximated by either a sine wave or solitary wave in the shallow water kdv limit the ilw periodic solution leads to the familiar cnoidal wave whereas the deep water bo limit yields benjamin s periodic wave a previously unknown expression for the cnoidal wave dispersion relation in terms of theta functions is obtained

equations of the korteweg de vries type with non trivial conserved quantities the authors study a family of equations of the korteweg de vries type different elements of this family are characterised by their linear dispersion relation w k they prove that there are only three members of this family which possess a non trivial polynomial conserved quantity and have a linear dispersion relation analytic in a neighbourhood of the origin these correspond to the korteweg de vries equation itself the benjamin ono equation and the intermediate long wave equation as is well known these equations have indeed infinitely many conserved quantities

well posedness for the navier slip thin film equation in the case of partial wetting we are interested in the thin lm equation with zero contact angle and quadratic mobility modeling the spreading of a thin liquid lm driven by capillarity and limited by viscosity in conjunction with a navier slip condition at the substrate this degenerate fourthorder parabolic equation has the contact line as a free boundary from the analysis of the self similar source type solution one expects that the solution is smooth only as a function of two variables x x where x denotes the distance from the contact line with p 13 1 4 0 6514 irrational therefore the well posedness theory is more subtle than in case of linear mobility coming from darcy dynamics or in case of the second order counterpart the porous medium equation here we prove global existence and uniqueness for one dimensional initial data that are close to traveling waves the main ingredients are maximal regularity estimates in weighted l 2 spaces for the linearized evolution after suitable subtraction of a t b t x terms

source deghosting by depth apparition abstractmarine seismic data are distorted by ghosts as waves propagating upward reflect downward from the sea surface ghosts appear on the source side and the receiver side however whereas the receiver side ghost problem has been studied in detail and many different solutions have been proposed and implemented commercially the source side ghost problem has remained largely unsolved with few satisfactory solutions available we have developed a new and simple method to remove sea surface ghosts that is related to the recently introduced concept of signal apparition as opposed to the temporal spatial source signature modulation functions used in the original signal apparition theory our source deghosting method relies on using sources at different depths but not at the same lateral positions the new method promises to be particularly suitable for 3d applications on sparse or incomplete acquisition geometries

on a functional equation related to the intermediate long wave equation we resolve an open problem stated by ablowitz et al 1982 j phys a math gen 15 781 concerning the integral operator appearing in the intermediate long wave equation we explain how this is resolved using the perturbative symmetry approach introduced by one of us with mikhailov by solving a certain functional equation we prove that the intermediate long wave equation and the benjamin ono equation are the unique integrable cases within a particular class of integro differential equations furthermore we explain how the perturbative symmetry approach is naturally extended to treat equations on a periodic domain

higher order equations related to thin films blow up and global existence the influence of the initial data we are interested in the following class of equations ksdv ht hhxx 12 hx 2 xx hx 3h x hx 6 h t periodic on 1 1 t 0 h 0 h0 with r 0 0 when 0 the model was established by j r king j r king two generalizations of the thin film equation math comput modelling 34 2001 737 756 here we show that if the initial data h0 0 23 then any admissible weak local solution h is necessarily nonnegative moreover there is no global weak solution on r of ksdv 0 and the blow up time must occur before t0 2h 0h02 h 0 2 provided that h0 is nonconstant h 0 is the average of h0 over 1 1 on the other hand if h0 0 then we have a value c 56 such that if c for all t 0 there is a nonpositive global weak solution h on 0 t being in particular in l2 0 t hper2 1 1 and if c we show that there exists a weak solution if is greater than 1 2 moreover adapting the energy method used by bernis f bernis finite speed of propagation and continuity of the interface for thin viscous flows adv differential equations 1 3 1996 337 368 in the case 1 and 0 we can show that weak solutions have a finite speed of propagation when 0 0 the model was established by spencer davis and voorhees b j spencer s h davis p w voorhees morphological instability in epitaxially strained dislocation free solid films nonlinear evolution appl math technical report 9201 dept of engineering sci and appl math mccormick school of eng and appl sci northwestern university evanston il 60208 september 1992 if 0 h0 0 then we show that there exists a global solution h 0 provided that h0 belongs to a certain class of functions if h0 0 blow up should occur in this case we show this fact under the dirichlet boundary conditions for the weak solution of ksdv 00 the blow up time tmax 1 13 ln 1 1 h0 1 t assuming h0 1 1 limt tmax 0t h l1 1 1 6d

finite element approximation of the cahn hilliard equation with degenerate mobility we consider a fully practical finite element approximation of the cahn hilliard equation with degenerate mobility textstyle frac partial u partial t del b u del gamma lap u psi u where b cdot geq 0 is a diffusional mobility and psi cdot is a homogeneous free energy in addition to showing well posedness and stability bounds for our approximation we prove convergence in one space dimension furthermore an iterative scheme for solving the resulting nonlinear discrete system is analyzed we also discuss how our approximation has to be modified in order to be applicable to a logarithmic homogeneous free energy finally some numerical experiments are presented

well posedness for a class of thin film equations with general mobility in the regime of partial wetting we establish well posedness for the family of thin film equations left begin array ll h t h n h xxx x 0 quad rm in h 0 h 0 h x 1 quad quad rm on partial h 0 end array right 1 with n in 0 frac 14 5 backslash 1 2 the model 1 with n in 0 3 has been used to describe the evolution of a capillary driven thin liquid droplet on a solid substrate in terms of its height profile h geqq 0 the family of thin film equations 1 provides a model problem to investigate contact line propagation in fluid dynamics under relaxed slip conditions the parabolicity of the fourth order parabolic problem degenerates at the free boundary which leads to a loss of regularity at the moving contact point our solutions are regular in terms of the two variables d x and d x 3 n where d x is the distance to the free boundary the main technical difficulty in the analysis of 1 is related to the loss of regularity at the contact points

on seismic deghosting using green s theorem abstractwe have examined theoretically how receiver side deghosting of pressure measurements can be derived from the green s theorem method we split the green s function that obeys dirichlet boundary conditions on the sea surface and at the receiver plane into two contributions the first emitting energy downward only from its source location and the other emitting energy only upward using the normal derivative of the source side downgoing green s function in the green s theorem evaluation over the receiver plane the upgoing part of the pressure field is predicted this is the receiver side deghosted field by inserting the source side upgoing normal derivative green s function in green s theorem its evaluation over the receiver plane predicts the downgoing part of the pressure field for a plane horizontal receiver surface the required green s function can be derived using the image series expansion method to display the fundamental frequencies of this green s function we have applied a fourier se

periodic ilw equation with discrete laplacian we study an integro differential equation which generalizes the periodic intermediate long wave ilw equation the kernel of the singular integral involved is an elliptic function written as a second order difference of the weierstrass function using sato s formulation we show the integrability and construct some special solutions an elliptic solution is also obtained we present a conjecture based on a poisson structure that gives an alternative description of this integrable hierarchy we note that this poisson algebra in turn is related to a quantum algebra related to the family of macdonald difference operators

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

the navier slip thin film equation for 3d fluid films existence and uniqueness abstract we consider the thin film equation t h h 2 h 0 in physical space dimensions i e one dimension in time t and two lateral dimensions with h denoting the height of the film in the third spatial dimension which corresponds to the lubrication approximation of the navier stokes equations of a three dimensional viscous thin fluid film with navier slip at the substrate this equation can have a free boundary the contact line moving with finite speed at which we assume a zero contact angle condition complete wetting regime previous results have focused on the 1 1 dimensional version where it has been found that solutions are not smooth as a function of the distance to the free boundary in particular a well posedness and regularity theory is more intricate than for the second order counterpart the porous medium equation or the thin film equation with linear mobility corresponding to darcy dynamics in the hele shaw cell here we prove existence and uniqueness of classical solutions that are perturbations of an asymptotically stable traveling wave profile this leads to control on the free boundary and in particular its velocity

analysis of a thin film approximation for two fluid taylor couette flows abstract in this work we study the evolution of the interface between two different fluids in two concentric cylinders when the velocity is given by the navier stokes equation and one of the fluids is thin we present a formal asymptotic derivation of the evolution equation for the interface under different scaling assumptions for the surface tension we then study the different types of the stationary solutions and travelling waves for the resulting equation in particular we state a global well posedness result and using center manifold theory we obtain detailed information about the long time asymptotics of the solutions of the problem

the lubrication approximation for thin viscous films regularity and long time behavior of weak solutions we consider the fourth order degenerate diffusion equation in one space dimension this equation derived from a lubrication approximation models the surface tension dominated motion of thin viscous films and spreading droplets 15 the equation with f h h also models a thin neck of fluid in the hele shaw cell 10 11 23 in such problems h x t is the local thickness of the the film or neck this paper considers the properties of weak solutions that are more relevant to the droplet problem than to hele shaw for simplicity we consider periodic boundary conditions with the interpretation of modeling a periodic array of droplets we consider two problems the first has initial data h0 0 and f h h n 0 n 3 we show that there exists a weak nonnegative solution for all time also we show that this solution becomes a strong positive solution after some finite time t and asymptotically approaches its means as t the weak solution is in the classical sense of distributions for 3 8 n 3 and in a weaker sense introduced in 1 for the remaining 0 n 3 8 furthermore the solutions have high enough regularity to just include the unique source type solutions 2 with zero slope at the edge of the support they do not include any of the less regular solutions with positive slope at the edge of the support second we consider strictly positive initial data h0 m 0 and f h h n 0 n for this problem we show that even if a finite time singularity of the form h 0 does occur there exists a weak nonnegative solution for all time t this weak solution becomes strong and positive again after some critical time t as in the first problem we show that the solution approaches its mean as t the main technical idea is to introduce new classes of dissipative entropies to prove existence and higher regularity we show that these entropies are related to norms of the difference between the solution and its mean to prove the relaxation result 1996 john wiley sons inc

time rescaling and asymptotic behavior of some fourth order degenerate diffusion equations in this paper we investigate the large times behavior of weak solutions to the fourth order degenerate parabolic equation ut u nuxxx x modeling the evolution of thin films in particular for all n 0 we prove exponential decay of u x t towards its mean value 1 u dx in l1 norm for long times and we give the explicit n dependent rate of decay the result is based on classical entropy estimates and on detailed lower bounds for the entropy production

counting the stationary states and the convergence to equilibrium for the 1 d thin film equation abstract this paper is concerned with the one dimensional thin film equation u t x m u x 2 u x 2 p u 0 p u 1 u n m n u m 0 n m 0 in 0 l r with the homogeneous neumann boundary conditions u x x p u x x 0 l 0 u x x 0 l 0 for all t 0 we prove that for any given positive initial datum the number of positive stationary states is at most infinitely countable furthermore we prove that the solution of the evolution problem converges to an equilibrium as time tends to infinity

non negative global weak solutions for a degenerated parabolic system approximating the two phase stokes problem abstract we establish the existence of non negative global weak solutions for a strongly coupled degenerated parabolic system which was obtained as an approximation of the two phase stokes problem driven solely by capillary forces moreover the system under consideration may be viewed as a two phase generalization of the classical thin film equation

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

optimal lower bounds on asymptotic support propagation rates for the thin film equation abstract we derive lower bounds on asymptotic support propagation rates for strong solutions of the cauchy problem for the thin film equation the bounds coincide up to a constant factor with the previously known upper bounds and thus are sharp our results hold in case of at most three spatial dimensions and n 1 2 92 the result is established using weighted backward entropy inequalities with singular weight functions to yield a differential inequality combined with some entropy production estimates the optimal rate of propagation is obtained to the best of our knowledge these are the first lower bounds on asymptotic support propagation rates for higher order nonnegativity preserving parabolic equations

well posedness for the navier slip thin film equation in the case of complete wetting we are interested in the thin film equation with zero contact angle and quadratic mobility modeling the spreading of a thin liquid film driven by capillarity and limited by viscosity in conjunction with a navier slip condition at the substrate this degenerate fourth order parabolic equation has the contact line as a free boundary from the analysis of the self similar source type solution one expects that the solution is smooth only as a function of two variables x x where x denotes the distance from the contact line with 13 14 0 6514 irrational therefore the well posedness theory is more subtle than in case of linear mobility coming from darcy dynamics or in case of the second order counterpart the porous medium equation here we prove global existence and uniqueness for one dimensional initial data that are close to traveling waves the main ingredients are maximal regularity estimates in weighted l2 spaces for the linearized evolution after suitable subtraction of a t b t x terms

on a fourth order degenerate parabolic equation in higher space dimensions we are concerned with the existence of non negative solutions to the following initial boundary value problem of a fourth order degenerate parabolic equation in higher space dimensions ut div u n u 0 in 0 t u u 0 on 0 t and u x 0 u0 x in

approximate solutions of acoustic 3d integral equation and their application to seismic modeling and full waveform inversion abstract over the recent decades a number of fast approximate solutions of lippmann schwinger equation which are more accurate than classic born and rytov approximations were proposed in the field of electromagnetic modeling those developments could be naturally extended to acoustic and elastic fields however until recently they were almost unknown in seismology this paper presents several solutions of this kind applied to acoustic modeling for both lossy and lossless media we evaluated the numerical merits of those methods and provide an estimation of their numerical complexity in our numerical realization we use the matrix free implementation of the corresponding integral operator we study the accuracy of those approximate solutions and demonstrate that the quasi analytical approximation is more accurate than the born approximation further we apply the quasi analytical approximation to the solution of the inverse problem it is demonstrated that this approach improves the estimation of the data gradient comparing to the born approximation the developed inversion algorithm is based on the conjugate gradient type optimization numerical model study demonstrates that the quasi analytical solution significantly reduces computation time of the seismic full waveform inversion we also show how the quasi analytical approximation can be extended to the case of elastic wavefield

an analytical solution for benchmark problem 1 the ideal wedge results from an exact analytical solution for the field in a wedge with pressure release boundaries benchmark problem 1 are presented in the form of transmission loss as a function of horizontal range from the source these computed curves contain no significant error and thus may be used as a primary benchmark for establishing the accuracy of range dependent numerical propagation models

a thin film equation with a singular diffusion in this paper a thin film equation with a singular diffusion term u t u n u a u u 0 is studied with periodic boundary conditions in multidimensional space it has a lot of applications in fluids theory such as draining of foams and the movement of contact lenses in order to overcome the difficulty of singular diffusion term the approximation problems need to be constructed and the entropy functional method is applied for the limit process moreover a compactness argument shows the existence of nonnegative weak solutions

on the convergence of entropy consistent schemes for lubrication type equations in multiple space dimensions we present nonnegativity preserving finite element schemes for a general class of thin film equations in multiple space dimensions the equations are fourth order degenerate parabolic and may contain singular terms of second order which are to model van der waals interactions a subtle discretization of the arising nonlinearities allows us to prove discrete counterparts of the essential estimates found in the continuous setting by use of the entropy estimate strong convergence results for discrete solutions are obtained in particular the limit of discrete fluxes mh uh ph will be identified with the flux m u w u u in the continuous setting as a by product first results on existence and positivity almost everywhere of solutions to equations with singular lower order terms can be established in the continuous setting

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

towards optimal regularity for the fourth order thin film equation in rn graveleau type focusing self similarity abstract an approach to some optimal more precisely non improvable regularity of solutions of the thin film equation u t u n u in r n r u x 0 u 0 x in r n where n 0 2 is a fixed exponent with smooth compactly supported initial data u 0 x in dimensions n 2 is discussed namely a precise exponent for the holder continuity with respect to the spatial radial variable x is obtained by construction of a graveleau type focusing self similar solution as a consequence optimal regularity of the gradient u in certain l p spaces as well as a holder continuity property of solutions with respect to x and t are derived which cannot be obtained by classic standard methods of integral identities inequalities several profiles for the solutions in the cases n 0 and n 0 are also plotted in general we claim that even for arbitrarily small n 0 and positive analytic initial data u 0 x the solutions u x t cannot be better than c x 2 e smooth where e n o n as n 0

reductions of pdes to second order odes and symbolic computation a new method to obtain second order reductions for ordinary differential equations which are polynomial in the derivatives of the dependent variable is presented the method is applied to obtain reductions and new solutions to several well known equations of mathematical physics a lubrication equation a thin film equation the zoomeron equation and a family of 5 th order partial differential equations which includes the caudrey dodd gibbon sawada kotera kaup kupershmidt ito and lax equations some pieces of computer algebra code to derive the reductions are also included

positivity preserving numerical schemes for lubrication type equations lubrication equations are fourth order degenerate diffusion equations of the form h t nabla cdot f h nabla delta h 0 describing thin films or liquid layers driven by surface tension recent studies of singularities in which h to 0 at a point describing rupture of the fluid layer show that such equations exhibit complex dynamics which can be difficult to simulate accurately in particular one must ensure that the numerical approximation of the interface does not show a false premature rupture generic finite difference schemes have the potential to manifest such instabilities especially when underresolved we present new numerical methods in one and two space dimensions that preserve positivity of the solution regardless of the spatial resolution whenever the pde has such a property we also show that the schemes can preserve positivity even when the pde itself is only known to be nonnegativity preserving we prove that positivity preserving finite difference schemes have unique positive solutions at all times we prove stability and convergence of both positivity preserving and generic methods in one and two space dimensions to positive solutions of the pde showing that the generic methods also preserve positivity and have global solutions for sufficiently fine meshes we generalize the positivity preserving property to a finite element framework and show via concrete examples how this leads to the design of other positivity preserving schemes

nonnegative solutions of a fourth order nonlinear degenerate parabolic equation elena beretta michiel bertsch roberta dal passo communicated by m guatin o introduction in this paper we study nonnegative solutions of the fourth order equation 0 1 ut u uxxx x o where n is a real positive constant equation 0 1 is a nonlinear parabolic partial differential equation which is degenerate at points at which u vanishes it arises in several applications see the references in 2 4 5 6 in which u x t stands for a nonnegative quantity for example in the study of the dynamics of thin liquid films u denotes the thickness of the film or another relevant nonnegative quantity the case n 1 models the flow in a hele shaw cell while n 3 corresponds to viscous flow on a solid surface without slip driven by surface tension see also 6 7 8 for a related equation it is well known that fourth order equations do not satisfy a maximum principle in particular classical solutions of the linear equation ut u 0 may become negative in finite time bernis 8 friedman 2 have observed that if n is positive equation 0 1 has generalized solutions which remain nonnegative for all times in addition they show that the qualitative positivity of these solutions strongly depends on the value of n in more recent papers by kadanoff et al a combination of numerical calculations and matched asymptotics 4 and the analysis of travelling wave solutions 5 leads to results which suggest a very rich structure of the solutions of 0 1 it is the purpose of the present paper to clarify several aspects of the nonlinear equation 0 1 and its dependence on the parameter n our results lead to several conjectures and open problems as was pointed out in 4 for sufficiently small values of n solutions of 0 1 which are initially strictly positive may vanish at some point x 0 after a finite time to but to the best of our knowledge there is no rigorous proof of this phenomenon in the literature on the other hand it is of particular interest for applications for example if u denotes the thickness of a liquid film it implies that at time to the film breaks at xo

movcol4 a moving mesh code for fourth order time dependent partial differential equations in this paper we develop and analyze a moving mesh code for the adaptive simulation of fourth order pdes based on collocation the scheme is shown to enforce discrete conservation for problems written in a generalized conservation form to demonstrate the breadth of applicability we present examples from both cahn hilliard and thin film type equations exhibiting metastable behavior finite time solution blow up finite time extinction and moving interfaces

the linear limit of thin film flows as an obstacle type free boundary problem we study the limit as n to 0 of the nonnegative self similar source type solutions of the thin film equationut unuxxx x 0 we obtain a unique limiting function u which is a solution of an obstacle type free boundary problem with the constraint u geq 0 associated with the linear equationut uxxxx 0 the function u is c1 for t 0 and has finite speed of propagation the positivity set um 0 being bounded by two contact lines x pm a t 1 4 a constant the function u has a dirac mass as initial condition and satisfies the linear equation in the positivity set but not across the free boundaries or contact lines also known as moving boundaries we give an integral representation of u in the positivity set we set up a precise definition of the general non self similar obstacle type free boundary problem which is different from a standard parabolic variational inequality and compare it with the cauchy problem we also consider source type solutions for negative values of n which are so

on seismic deghosting by spatial deconvolution abstractreceiver side deghosting can be derived and implemented in the frequency domain as spatial deterministic deconvolution of marine pressure recordings the deghosting deconvolution operator is found analytically as the inverse fourier transform of the wavenumber domain wave equation deghosting function for a sea surface reflection coefficient of 1 the wavenumber domain deghosting function has well known poles at fundamental frequencies equal to an integer multiple of a function of the receiver depth and the plane wave dip angle relative to the depth axis the first singularity is always at 0 hz the spatial deghosting operator has singularities at fundamental frequencies equal to an integer multiple of a function of the receiver depth independent of its lateral coordinates the first singularity is again at 0 hz in addition the deghosting operator that is applied to 3d data has singularity when its lateral coordinate is zero a simple numerical example demonstrates the method

blow up of solutions to a degenerate parabolic equation not in divergence form abstract we study nonglobal positive solutions to the dirichlet problem for ut up u u in bounded domains where 0 t t 1 p if either the space dimension is one or p t t 1 p u t consists of continuous functions solving w w 1 p w 1 p in one space dimension it is shown that actually t t 1 p u t w as t t where w coincides with an element of a one parameter family of functions inside each component of its positivity set furthermore we study the size of the components of w 0 with the result that this size is uniquely determined by in the case p 1 the positivity set can have the maximum possible size 2 p for certain initial data but it may also be arbitrarily close to the minimal length

a fourth order parabolic equation modeling epitaxial thin film growth abstract we study the continuum model for epitaxial thin film growth from phys d 132 1999 520 542 which is known to simulate experimentally observed dynamics very well we show existence uniqueness and regularity of solutions in an appropriate function space and we characterize the existence of nontrivial equilibria in terms of the size of the underlying domain in an investigation of asymptotical behavior we give a weak assumption under which the limit set of the dynamical system consists only of steady states in the one dimensional setting we can characterize the set of steady states and determine its unique asymptotically stable element the article closes with some illustrative numerical examples

asymmetric travelling waves for the thin film equation abstract we study the existence uniqueness and asymptotic behaviour of travelling wave solutions to the thin film equation with non zero contact angle at one interface we obtain explicit lower and upper bounds for these solutions moreover we analyse in an appendix the stability of a related class of stationary travelling wave solutions

finite element approximation of soluble surfactant spreading on a thin film we consider a fully practical finite element approximation of the following system of nonlinear degenerate parabolic equations begin align frac partial u partial t frac12 nabla cdot u 2 nabla sigma v frac13 nabla cdot u 3 nabla w 0 w c delta u delta u nu a u 3 hspace 0 1cm frac partial v partial t nabla cdot u v nabla sigma v frac12 nabla cdot u 2 v nabla w rho s delta v k left psi v right 0 frac partial psi partial t frac12 u nabla sigma v cdot nabla psi frac13 u 2 nabla w cdot nabla psi rho b u 1 nabla cdot left u nabla psi right beta k u 1 psi v 0 hspace 2cm nonumber end align the above equations model a marangoni driven thin film laden with a soluble surfactant in which the bulk surfactant concentration has been vertically averaged the model accounts for the presence of both attractive a geq 0 and repulsive delta 0 with nu 3 van der waals forces here u is the height of the film v is the concentration of the interfacial surfactant species psi is the concentration of the surfactant species within the bulk phase and sigma v 1 v is the typical surface tension moreover rhos geq 0 rhob 0 and c 0 are the inverses of the surface peclet number the bulk peclet number and the modified capillary number respectively finally beta 0 and k 0 are parameters that characterize the solubility and the rate of interfacial adsorption in addition to showing stability bounds for our approximation we prove convergence and hence existence of a solution to this nonlinear degenerate parabolic system i in one space dimension when rhos 0 and moreover ii in two space dimensions if in addition nu geq 7 furthermore iterative schemes for solving the resulting nonlinear discrete system are discussed finally some numerical experiments are presented

nonnegativity preserving convergent schemes for the thin film equation we present numerical schemes for fourth order degenerate parabolic equations that arise e g in lubrication theory for time evolution of thin films of viscous fluids we prove convergence and nonnegativity results in arbitrary space dimensions a proper choice of the discrete mobility enables us to establish discrete counterparts of the essential integral estimates known from the continuous setting hence the numerical cost in each time step reduces to the solution of a linear system involving a sparse matrix furthermore by introducing a time step control that makes use of an explicit formula for the normal velocity of the free boundary we keep the numerical cost for tracing the free boundary low

thin films with high surface tension this paper is a review of work on thin fluid films where surface tension is a driving mechanism its aim is to highlight the substantial amount of literature dealing with relevant physical models and also analytic work on the resultant equations in general the introduction of surface tension into standard lubrication theory leads to a fourth order nonlinear parabolic equation pad h t pad x left c frac h 3 3 frac partial 3 h partial x 3 f h h x h xx right 0 label abeq1 where h h x t is the fluid film height for steady situations this equation may be integrated once and a third order ordinary differential equation is obtained appropriate forms of this equation have been used to model fluid flows in physical situations such as coating draining of foams and the movement of contact lenses in the introduction a form of the above equation is derived for flow driven by surface tension surface tension gradients gravity and long range molecular forces modifications to the equation due to slip the effect of two free surfaces two phase fluids and higher dimensional forms are also discussed the second section of this paper describes physical situations where surface tension driven lubrication models apply and the governing equations are given the third section reviews analytical work on the model equations as well as the generalized lubrication equation pad h t pad x left h n h xxx right 0 in particular the discussion focusses on asymptotic results travelling waves stability and similarity solutions numerical work is also discussed while for analytical results the reader is directed to existing literature

integrated receiver deghosting and closed loop surface multiple elimination abstractaccurate surface related multiple removal is an important step in conventional seismic processing and more recently primaries and surface multiples are separated such that each of them is available for imaging algorithms current developments in the field of surface multiple removal aim at estimating primaries in a large scale inversion process using such a so called closed loop process in each iteration primaries and surface multiples will be updated until they fit the measured data the advantage of redefining surface multiple removal as a closed loop process is that certain preprocessing steps can be included which can lead to an improved multiple removal in principle the surface related multiple elimination process requires deghosted data as input thus the source and receiver ghost must be removed we have focused on the receiver ghost effect and assume that the source is towed close to the sea surface such that the source ghost effect is well represented by a dipole source the rece

upper bounds on waiting times for the thin film equation the case of weak slippage we derive upper bounds on the waiting time of solutions to the thin film equation in the regime of weak slippage n in 2 frac 32 11 in particular we give sufficient conditions on the initial data for instantaneous forward motion of the free boundary for n in 2 frac 32 11 our estimates are sharp for n 2 they are sharp up to a logarithmic correction term note that the case n 2 corresponds with a grain of salt to the assumption of the navier slip condition at the fluid solid interface we also obtain results in the regime of strong slippage n in 1 2 however in this regime we expect them not to be optimal our method is based on weighted backward entropy estimates hardy s inequality and singular weight functions we deduce a differential inequality which would enforce blowup of the weighted entropy if the contact line were to remain stationary for too long

smooth zero contact angle solutions to a thin film equation around the steady state in the simplest case of a linearly degenerate mobility we view the thin film equation as a classical free boundary problem our focus is on the regularity of solutions and of their free boundary in the complete wetting regime which prescribes zero slope at the free boundary in order to rule out of the analysis possible changes in the topology of the positivity set we zoom into the free boundary by looking at perturbations of the stationary solution our strategy is based on a priori energy type estimates which provide minimal conditions on the initial datum under which a unique global solution exists in fact this solution turns out to be smooth for positive times and to converge to the stationary solution for large times as a consequence we obtain smoothness and large time behavior of the free boundary

on seismic deghosting using integral representation for the wave equation use of green s functions with neumann or dirichlet boundary conditions abstractthe recent interest in broadband seismic technology has spurred research into new and improved seismic deghosting solutions one starting point for deriving deghosting methods is the representation theorem which is an integral representation for the wave equation recent research results show that by using green s functions with dirichlet boundary conditions in the representation theorem source side deghosting of already receiver side deghosted wavefields can be achieved we found that the choice of green s functions with neumann boundary conditions on the sea surface and the plane that contains the sources leads to an identical but simpler solution with fewer processing steps in addition we found that pressure data can be receiver side deghosted by introducing green s functions with dirichlet boundary conditions on the sea surface and the plane containing the receivers into a modified representation theorem the deghosting methods derived from the representation theorem are wave theoretic alg

nonlocal models for nonlinear dispersive waves abstract considered herein are model equations for the unidirectional propagation of small amplitude nonlinear dispersive long waves such as those governed by the classical korteweg de vries equation of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial so that the operator modelling dispersive effect is nonlocal particular cases in view here are the benjamin ono equation and the intermediate long wave equation which arise in internal wave theory and smith s equation which governs certain types of continental shelf waves the initial value problem for these equations is shown to be globally well posed in the classical sense including continuous dependence upon the initial data and in certain cases upon the modelling of nonlinear and dispersive effects whilst the results are stated for the specific equations listed above the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear dispersive evolution equations particularly worthy of note is our theorem implying that solutions of the intermediate long wave equation converge strongly to solutions of the korteweg de vries equation or to solutions of the benjamin ono equation in appropriate asymptotic limits

a family of fourth order q logarithmic equations we prove the existence of global in time weak nonnegative solutions to a family of nonlinear fourth order evolution equations parametrized by a real parameter q 0 1 which includes the well known thin film q 1 2 and the derrida lebowitz speer spohn dlss equation q 1 subject to periodic boundary conditions in one spatial dimension in contrast to the gradient flow approach in 25 our method relies on dissipation property of the corresponding entropy functionals tsallis entropies resulting in required a priori estimates and extends the existence result from 25 to a wider range of the family members namely to 0 q 1 2 generalized beckner type functional inequalities yield an exponential decay rate of relative entropies which in further implies the exponential stability in the l1 norm of the constant steady state finally we provide illustrative numerical examples supporting the analytical results

wave equation processing using finite difference propagators part 2 deghosting of marine hydrophone seismic data abstractwe have developed a new and simple method for deghosting of conventional hydrophone streamer data towed at arbitrary variable depths the method uses a time space domain finite difference fd solution to the wave equation with pressure field boundary conditions to predict and remove ghosts because it operates in the time domain our method is unaffected by any number of notches in the frequency spectrum of the data and therefore will deghost through notches apart from the acquired hydrophone data the method relies on the depth profile of the streamer recording the data beneath a sea surface with a known reflection coefficient as well as the propagation velocity in the water above the streamer the method was applied to simple and more complex synthetic data which illustrated its ability to deal with complex data and any acquisition geometry

existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order abstract this paper is devoted to studying the existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order u t u p 2 u f u in r n with boundary condition u u 0 and initial data u 0 the substantial difficulty is that the general maximum principle does not hold for it the solutions are obtained for both the steady state case and the developing case by the fixed point theorem and the semi discretization method unlike the general procedures used in the previous papers on the subject we introduce two families of approximate solutions with determining the uniform bounds of derivatives with respect to the time and space variables respectively by a compactness argument with necessary estimates we show that the two approximation sequences converge to the same limit i e the solution to be determined in addition the decays of solutions towards the constant steady states are established via the entropy method finally it is interesting to observe that the solutions just tend to the initial data u 0 as p

coupled mode solutions in generalized ocean environments a two way coupled mode model has been developed to solve propagation problems in fluid layered range dependent ocean environments the model incorporates piecewise linear depth dependent range varying sound speed profiles svps a fast and efficient normal mode model which is based on a propagator matrix method is used to obtain the eigenvalues and eigenfunctions the coupling coefficients are expressed in terms of depth integrals of products of airy functions and are solved analytically the coupled equations are solved exactly by an approach referred to as the u k method which incorporates the lanczos method previously used in solving similar coupled channel equations in nuclear theory and successfully applied to nuclear response problems in heavy nuclei knobles solutions of coupled mode equations with a large dimension in underwater acoustics j acoust soc am 96 1741 1747 1994 highlights of the formalism will be presented which expand on the overview given previously stotts et al solutions of coupled mode equations with a large dimension in underwater acoustics j acoust soc am 95 2909 a 1994 several benchmark comparisons will be presented including propagation over a hill upslope wedge propagation and an analytic depth and range dependent svp in a flat waveguide

positivity and the attractor dimension in a fourth order reaction diffusion equation in this paper we investigate the semilinear partial differential equation ut fuxxxx uxx u 1 u2 with a view particularly to obtaining some insight into how one might establish positivity preservation results for equations containing fourth order spatial derivatives the maximum principle cannot be applied to such equations however progress can be made by employing some very recent best possible interpolation inequalities due to the third named author in which the interpolation constants are both explicitly known and sharp these are used to estimate the lx distance between u and 1 during the evolution a positivity preservation result can be obtained under certain restrictions on the initial datum we also establish an explicit two sided estimate for the fractal dimension of the attractor which is sharp in terms of the physical parameters

a positivity preserving numerical scheme for a nonlinear fourth order parabolic system noindent a positivity preserving numerical scheme for a fourth order nonlinear parabolic system arising in quantum semiconductor modeling is studied the system is numerically treated by introducing an additional nonlinear potential and a subsequent semidiscretization in time the resulting sequence of nonlinear second order elliptic systems admits at each time level strictly positive solutions which is proved by an exponential transformation of variables the stability of the scheme is shown and convergence is proved in one space dimension numerical results concerning the switching behavior of a resonant tunneling diode are presented

on genus two solutions for the ilw equation the existence of theta function solutions of genus two for the intermediate long wave equation is established a numerical example is also presented the method basically goes along with krichever s construction of theta function solutions for soliton equations such as the kronecker product equation this idea leads us to a question whether a riemann surface exists which allows a peculiar abelian integral of the third kind the answer is affirmative at least for genus two curves

the linear limit of the dipole problem for the thin film equation we investigate self similar solutions of the dipole problem for the one dimensional thin film equation on the half line x ge 0 we study compactly supported solutions of the linear moving boundary problem and show how they relate to solutions of the nonlinear problem the similarity solutions are generally of the second kind given by the solution of a nonlinear eigenvalue problem although there are some notable cases where first kind solutions also arise we examine the conserved quantities connected to these first kind solutions difficulties associated with the lack of a maximum principle and the non self adjointness of the fundamental linear problem are also considered seeking similarity solutions that include sign changes yields a surprisingly rich set of coexisting stable solutions for the intermediate asymptotics of this problem our results include analysis of limiting cases and comparisons with numerical computations

long time asymptotics for strong solutions of the thin film equation in this paper we investigate the large time behavior of strong solutions to the one dimensional fourth order degenerate parabolic equation u t u u xxx x modeling the evolution of the interface of a spreading droplet for nonnegative initial values u 0 x h 1 both compactly supported or of finite second moment we prove explicit and universal algebraic decay in the l 1 norm of the strong solution u x t towards the unique among source type solutions strong source type solution of the equation with the same mass the method we use is based on the study of the time decay of the entropy introduced in 13 for the porous medium equation and uses analogies between the thin film equation and the porous medium equation

noise attenuation by a hard wedge shaped barrier this paper is concerned with the problem of sound screening by a wedge like barrier the sound source is assumed to be point like and the receiver is located in the shadow of the source sound field so that according to geometrical optics only the field diffracted by the edge of the barrier is considered first for the hard wedge in space three models are used for calculating the amplitude of the edge diffracted field these are the uniform theory of diffraction utd the hadden pierce model both in the frequency domain and the biot tolstoy theory of diffraction which is a time domain formulation it is first shown that even at relatively low frequencies the frequency domain models perform quite satisfactorily as compared to the exact time domain theory hence and due to its relative simplicity the utd is proposed as an accurate calculation scheme for solving problems with edge diffraction by hard wedges it is also proved from theoretical calculations that the amplitude of the edge diffracted field increases for an increasing angle of the wedge and consequently the hard half plane gives the lowest field amplitude in the shadow zone some applications are then considered for evaluating the performance of a barrier on a flat ground either completely hard or with mixed homogeneous boundary conditions an improvement of the scheme for calculating the sound field in the all hard case is achieved through considering the multiple diffraction in this case only to the second order between the top of the wedge barrier and its base the results show that for usually occurring situations increasing the angle of the hard wedge barrier affects negatively its efficiency through diminishing its insertion loss these conclusions are also supported by the results of some experimental measurements conducted at a scale model level

a dynamical systems approach for the contact line singularity in thin film flows abstract we are interested in a complete characterization of the contact line singularity of thin film flows for zero and nonzero contact angles by treating the model problem of source type self similar solutions we demonstrate that this singularity can be understood by the study of invariant manifolds of a suitable dynamical system in particular we prove regularity results for singular expansions near the contact line for a wide class of mobility exponents and for zero and nonzero dynamic contact angles key points are the reduction to center manifolds and identifying resonance conditions at equilibrium points the results are extended to radially symmetric source type solutions in higher dimensions furthermore we give dynamical systems proofs for the existence and uniqueness of self similar droplet solutions in the nonzero dynamic contact angle case

on loglog blow up in higher order reaction diffusion and wave equations a formal geometric approach abstract in 1934 petrovskii s boundary regularity study of the heat equation gave the first appearance of the ln ln t t blow up factor in pde theory we discuss the origin of analogous non self similar blow up in higher order reaction diffusion parabolic or wave hyperbolic equations of the form u t u 2 u x x x x u or u t t u 2 u x x x x u in l l 0 t with zero dirichlet boundary conditions at x l where l l 0 2 we present formal arguments that the standard similarity blow up rate 1 t t acquires an extra universal ln ln t t factor the explanation is based on a geometric matching with the so called logarithmic travelling waves as group invariant solutions of the pdes we also discuss connections with log log blow up factors occurring in earlier studies of plasma physics parabolic equations and the nonlinear critical schrodinger equation

thin film free boundary problems for partial wetting we present a novel framework to solve thin film equations with an explicit non zero contact angle where the support of the solution is treated as an unknown the algorithm uses a finite element method based on a gradient formulation of the thin film equations coupled to an arbitrary lagrangian eulerian method for the motion of the support features of this algorithm are its simplicity and robustness we apply this algorithm in 1d and 2d to problems with surface tension contact angles and gravity

well posedness of the cauchy problem for a fourth order thin film equation via regularization approaches abstract this paper is devoted to some aspects of well posedness of the cauchy problem the cp for short for a quasilinear degenerate fourth order parabolic thin film equation the tfe 4 0 1 u t u n u in r n r u x 0 u 0 x in r n where n 0 is a fixed exponent with bounded smooth compactly supported initial data dealing with the cp for at least n 0 3 2 requires introducing classes of infinitely changing sign solutions that are oscillatory close to finite interfaces the main goal of the paper is to detect proper solutions of the cp for the degenerate tfe 4 by uniformly parabolic analytic e regularizations at least for values of the parameter n sufficiently close to 0 firstly we study an analytic homotopy approach based on a priori estimates for solutions of uniformly parabolic analytic e regularization problems of the form u t e u u in r n r where e u for e 0 1 is an analytic e regularization of the problem 0 1 such that 0 u u n and 1 u 1 using a more standard classic technique of passing to the limit in integral identities for weak solutions however this argument has been demonstrated to be non conclusive basically due to the lack of a complete optimal estimate regularity theory for these types of problems secondly to resolve that issue more successfully we study a more general similar analytic homotopy transformation in both the parameters as e 0 and n 0 and describe branching of solutions of the tfe 4 from the solutions of the notorious bi harmonic equation u t 2 u in r n r u x 0 u 0 x in r n which describes some qualitative oscillatory properties of cp solutions of 0 1 for small n 0 providing us with the uniqueness of solutions for the problem 0 1 when n is close to 0 finally riemann like problems occurring in a boundary layer construction that occur close to nodal sets of the solutions as e 0 are discussed in other to get uniqueness results for the tfe 4 0 1

removable singularity of the polyharmonic equation abstract let x 0 r n n 2 be a domain and let m 2 we will prove that a solution u of the polyharmonic equation m u 0 in x 0 has a removable singularity at x 0 if and only if k u x o x x 0 2 n k 0 1 2 m 1 as x x 0 0 for n 3 and o log x x 0 1 k 0 1 2 m 1 as x x 0 0 for n 2 for m 2 we will also prove that u has a removable singularity at x 0 if u x o x x 0 2 m n as x x 0 0 for n 3 and u x o x x 0 2 m 2 log x x 0 1 as x x 0 0 for n 2

on the convective cahn hilliard equation with degenerate mobility abstract in this paper we study the existence of weak solutions for the convective cahn hilliard equation with degenerate mobility based on the schauder type estimates we establish the global existence of classical solutions for regularized problems after establishing some necessary uniform estimates on the approximate solutions we prove the existence of weak solutions the nonnegativity and the finite speed of propagation of perturbations of solutions are also discussed

the cutoff method for the numerical computation of nonnegative solutions of parabolic pdes with application to anisotropic diffusion and lubrication type equations the cutoff method which cuts off the values of a function less than a given number is studied for the numerical computation of nonnegative solutions of parabolic partial differential equations a convergence analysis is given for a broad class of finite difference methods combined with cutoff for linear parabolic equations two applications are investigated linear anisotropic diffusion problems satisfying the setting of the convergence analysis and nonlinear lubrication type equations for which it is unclear if the convergence analysis applies the numerical results are shown to be consistent with the theory and in good agreement with existing results in the literature the convergence analysis and applications demonstrate that the cutoff method is an effective tool for use in the computation of nonnegative solutions cutoff can also be used with other discretization methods such as collocation finite volume finite element and spectral methods and for the computation of positive solutions