calculation of various diffuser flows with inlet swirl and inlet distortion effects a method of predicting the properties of various turbulent flows in planar conical and annular diffusers with inlet swirl and inlet distortion effects has been developed and appraised the numerical scheme is based on the fully conservative control volume representation of governing conservation equations the method is capable of carrying out the calculation when the flowfield contains stalled or recirculation flow regions three different discretization schemes are compared to analyze the numerical diffusion an algebraic reynolds stress model is used to represent the large streamline curvature the appraisal is achieved by comparing predicted results with various experimental data the results show that the numerical scheme with the utilized turbulence closure model predicts various diffuser flows with the effects of inlet swirl and inlet distortion within measurement accuracy and is able to provide various guiding information for many engineering applications

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

a level set approach for simulations of flows with multiple moving contact lines with hysteresis the level set method of sussman et al j comput phys 148 1999 81 is extended such that flows with multiple moving contact lines can be simulated accounting for inertia a relation between contact line speed and contact angle slip and contact line hysteresis the convergence properties of the method are investigated with particular attention on the resolution of the contact line stress singularity results are compared with a lubrication theory for spreading droplets

existence and features of similarity solutions for non boussinesq gravity currents in this paper the flow dynamics of gravity currents on a horizontal plane is investigated from a theoretical point of view by seeking similarity solutions the current is generated by unleashing a varying volume of heavy fluid within an ambient fluid of much lower density unlike earlier investigators we assume that the ambient fluid exerts no significant resisting action on the current and therefore the flow depth is expected to drop to zero at the front in the absence of friction in this context the shallow water equations are highly appropriate for computing the mean velocity and flow depth of the current the boundary condition imposed at the front leads to technical mathematical difficulties indeed unlike in the boussinesq case no regular solution to the shallow water equations satisfies the downstream condition but when the flow is supercritical at the channel inlet it is possible to construct a piecewise solution by patching a regular solution to an exceptional solution which represents the head behavior to better understand this result and make sure that the result is physically relevant we consider the navier stokes equations within the high reynolds number limit approximate similarity solutions can be worked out which support our earlier analysis on the shallow water equations while the flow body is self similar and weakly rotational the head is not self similar but tends toward a self similarity shape at long times it is characterized by a strong vorticity a straight free surface and a nonuniform velocity profile which becomes flatter and flatter with time

on some third order nonlinear boundary value problems existence location and multiplicity results abstract we prove an ambrosetti prodi type result for the third order fully nonlinear equation u t f t u t u t u t s p t with f 0 1 r 3 r and p 0 1 r continuous functions s r under several two point separated boundary conditions from a nagumo type growth condition an a priori estimate on u is obtained an existence and location result will be proved by degree theory for s r such that there exist lower and upper solutions the location part can be used to prove the existence of positive solutions if a non negative lower solution is considered the existence nonexistence and multiplicity of solutions will be discussed as s varies

travelling waves in dilatant non newtonian thin films abstract we prove the existence of a travelling wave solution for a gravity driven thin film of a viscous and incompressible dilatant fluid coated with an insoluble surfactant the governing system of second order partial differential equations for the film s height h and the surfactant s concentration are derived by means of lubrication theory applied to the non newtonian navier stokes equations

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

cancellation over commutative rings of dimension one and two all rings in this paper are commutative and noetherian and all modules are finitely generated we will study the question if a c z b c is a z b here a b and c are modules over a ring r in 1962 chase 4 gave an affirmative answer for r k x y the polynomial ring in two variables over an algebraically closed field k provided a and b are torsionfree and char k trank a in the first section of this paper we extend chase s result to include all two dimensional regular affine k domains most of the paper deals with one dimensional rings in section 2 we prove some general results which we hope will eventally lead to some sort of structure theory for torsionfree modules over one dimensional reduced rings with finite normalization in this context we show that one can cancel from projectives that is the answer is yes when a and b are projective if and only if pit r pit z where g is the normalization when a and b are assumed to be merely torsionfree the problem is more subtle but we have some partial results the last three sections deal with three naturally occurring examples of one dimensional rings coordinate rings of curves quadratic orders and integral group rings of finite abelian groups in each of these cases we are able to give a fairly complete answer to the cancellation problem for torsionfree modules many of the ideas in this paper are the direct or indirect results of innumerable conversations with raymond heitmann and lawrence levy in particular levy discovered a serious error in the proof of the main theorem of an earlier version and heitmann showed by example that the statement itself was incorrect i am extremely grateful to both of them for their interest and insight

phase plane analysis and traveling wave solution of third order nonlinear singular problems arising in thin film evolution a nonlinear third order singular ordinary differential equation is a model for a viscous fluid which drains over a wet surface as a thin film down a vertical wall here we discuss the physical processes of a thin liquid film evolution the classical thin film equation is being derived and the series solution is obtained by employing the adomian decomposition method adm in this paper a new way for the phase planes analysis and traveling wave solution by adm is presented the proposed algorithm is implemented using mathematical software mathematica 6 0

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

on the structure of positive solutions to an elliptic problem with a singular nonlinearity abstract we consider the following boundary value problem u u p u q in u 0 on where p q 0 and is a bounded smooth domain in r n n 2 it is shown that for 1 there is at least one solution u of this problem satisfying u in for any 0 the profiles of u for 0 and are analyzed for 1 it is clear that u 1 is a constant solution to the problem we also present the branch of solutions satisfying 0 u 1 in for the equation with 1 we obtain more information on the structure of positive radial solutions of the problem when is an annulus

linear stability of steady states for thin film and cahn hilliard type equations we study the linear stability of smooth steady states of the evolution equation h t f h xxx x g h h x x ah under both periodic and neumann boundary conditions if a 0 we assume f 1 in particular we consider positive periodic steady states of thin film equations where a 0 and f g might have degeneracies such as f 0 0 as well as singularities like g 0

on the determination of the steady film profile for a non newtonian thin droplet abstract a shooting method is used to determine a solution to a third order ode modeling the steady profile of a non newtonian thin droplet we compare a direct approach to an iterative approach using a secant method we obtain a nonlinear relationship between the contact angle and the position of the contact line r from this nonlinear relationship we use curve fitting to obtain an empirical law of the form tan r f k where k is the power law coefficient and f is a nonlinear function of k

traveling wave solutions to thin film equations thin films can be effectively described by the lubrication approximation in which the equation of motion is mathit h mathit t mathit h mathit n mathit h mathit x mathit x mathit x mathit x 0 here h is a necessarily positive quantity which represents the height or thickness of the film different values of n especially 1 2 and 3 correspond to different physical situations this equation permits solutions in the form of traveling disturbances with a fixed form if u is the propagation velocity the resulting equation for the disturbance is mathit uh mathit x mathit h mathit n mathit h mathit x mathit x mathit x mathit x here quantitative and qualitative solutions to the equation are presented the study has been limited to the intervals in x where the solutions are positive it is found that transitions between different qualitative behaviors occur at n 3 2 3 2 and 1 2 for example if u is not zero solitonlike solutions defined on a finite interval are only possible for n3 more specific results can be obtained in the case in which the velocity is zero solitons occur for n2 for n 1 the region 3 2n is characterized by the presence of advancing front solutions with support on ensuremath infty t for ng1 2 single minimum solutions diverging at ifmmode pm else textpm fi ensuremath infty are possible the generic solution present for all positive values of n is a receding front which diverges at finite x for n0

cubic splines method for a system of third order boundary value problems we use uniform cubic polynomial splines to develop a numerical method for computing approximations to the solution of a system of third order boundary value problem associated with odd order obstacle problems such problems arise in physical oceanography and can be studied in the framework of variational inequality theory the convergence analysis of the new method is studied and an upper bound for the error is derived a numerical example is given to illustrate the efficiency of the new method

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

low reynolds number modeling of turbulent flows with and without wall transpiration a full reynolds stress closure that is capable of describing the flow all the way to the wall is formulated the closure is based on the conventional high reynolds number form of the redistribution model the inclusion of molecular diffusion and a modified dissipation model to account for viscous effects near a wall two dissipation models are investigated along with two gradient diffusion and two redistribution models their respective effects on the calculated flow properties are assessed by comparing them with the data of fully developed turbulent flows and a developing pipe flow with wall transpiration the near wall behavior is very well predicted however the wall correction to the redistribution modeling is found to have little effect on the calculated results the overall behavior of the fully developed turbulent flows is best described by a nonisotropic gradient diffusion model a return to isotropy redistribution model and a dissipation model that accounts for viscous behavior near a wall this same closure also gives the best prediction of the axial pressure drop behavior along a pipe with a uniform wall suction furthermore the near wall behavior of such a flow is very well predicted by this closure

stability of cahn hilliard fronts we prove stability of the kink solution of the cahn hilliard equation partial derivative t u partial derivative x 2 partial derivative x 2 u u 2 u 3 2 x is an element of r the proof is based on an inductive renormalization group method and we obtain detailed asymptotics of the solution as t infinity we prove stability of the kink solution of the cahn hilliard equation partial derivative t u partial derivative x 2 partial derivative x 2 u u 2 u 3 2 x is an element of r the proof is based on an inductive renormalization group method and we obtain detailed asymptotics of the solution as t infinity c 1999 john wiley sons inc

spectral comparison principles for the cahn hilliard and phase field equations and time scales for coarsening we initiate a study of the instability properties of steady solutions of the cahn hilliard ch equation and the phase field pf system in terms of the coarseness of those solutions other spectral results are obtained as well steady state solutions for the ch equation with no flux or periodic boundary conditions are also stationary states for the bistable reaction diffusion equation and the pf system with constant temperature thus it is meaningful to linearize each of these equations about a common stationary solution and discuss the stability of this state in each dynamical process converting the linearized operators to self adjoint form and expressing eigenvalues in terms of modified rayleigh quotients enables us to relate the spectra of the three operators in particular we relate the indices of instability with respect to the three processes we also discuss the change in the spectrum of the linearized ch operator in one dimension when we change the stationary solution at which it is linearized we conjecture that the eigenvalues increase with the number of oscillations of the steady state for a given mass and prove that this is so in certain cases this lends credence to the idea that motion of these dynamical systems near coarse solutions takes place on longer time scales

on the roughness induced effective boundary conditions for an incompressible viscous flow abstract we consider the laminar viscous channel flow with the lateral surface of the channel containing surface irregularities it is supposed that a uniform pressure gradient is maintained in the longitudinal direction of the channel after studying the corresponding boundary layers we obtain rigorously the navier friction condition it is valid when the size and amplitude of the imperfections tend to zero furthermore the coefficient in the law is determined through an auxiliary boundary layer type problem and the tangential drag force and the effective mass flow are determined up to order o e3 2 the value of the effective coefficient is shown to be independent with respect to the position of the mean surface in the range of o e

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

traveling waves for a model of gravity driven film flows in cylindrical domains abstract traveling wave solutions are studied for a recently derived model of a falling viscous film on the interior of a vertical rigid tube by identifying a hopf bifurcation and using numerical continuation software families of non trivial traveling wave solutions may be traced out in parameter space these families all contain a single solution at a turnaround point with larger film thickness than all others in the family in an earlier paper it was conjectured that this turnaround point may represent a critical thickness separating two distinct flow regimes observed in physical experiments as well as two distinct types of behavior in transient solutions to the model here these hypotheses are verified over a range of parameter values using a combination of numerical and analytical techniques the linear stability of these solutions is also discussed both large and small amplitude solutions are shown to be unstable though the instability mechanisms are different for each wave type specifically for small amplitude waves the region of relatively flat film away from the localized wave crest is subject to the same instability that makes the trivial flat film solution unstable for large amplitude waves this mechanism is present but dwarfed by a much stronger tendency to relax to a regime close to that followed by small amplitude waves

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

instabilities in gravity driven flow of thin fluid films this paper presents theoretical computational and experimental aspects of the instability development in the flow of thin fluid films the theoretical part involves basic fluid me chanics and presents derivation of the thin film equation using lubrication approximation a simplified version of this equation is then analyzed analytically using linear stability analysis and also numerically the results are then compared directly to experiments the experimental part outlines the setup as well as data acquisition and analysis this immediate comparison to experiments is very useful for gaining better insight into the interpretation of various theoretical and computational results

droplet spreading intermediate scaling law by pde methods we consider the spreading of a thin droplet of viscous liquid on a plane surface driven by capillarity the standard lubrication approximation leads to an evolution equation for the film height h that is ill posed when the spreading is limited by the no slip boundary condition at the liquid solid interface due to a singularity at the moving contact line the most common relaxation of the no slip boundary condition removes this singularity but introduces a new physical length scale the slippage length b it is believed that this microscopic length scale only enters logarithmically in the effective that is macroscopic spreading behavior in this paper we rigorously show that the naively expected spreading rate is indeed only altered by a logarithmic term involving b more precisely we prove a scaling law for the diameter of the apparent that is macroscopic support of the droplet in time this is an intermediate scaling law it takes an initial layer to forget the initial droplet shape whereas after a long time the droplet is so thin that its spreading is governed by the physics on the scale b our proof works by deriving suitable estimates for physically relevant integral quantities the free energy the length of the apparent support and their respective rates of change as opposed to matched asymptotic methods this pde approach closely mimics a simple heuristic argument based on the gradient flow structure 2002 john wiley sons inc

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

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

a simple proof of stability of fronts for the cahn hilliard equation we apply a method developed in our earlier work on a non local phase kinetics equation to give a simple proof of the non linear stability of fronts for the cahn hilliard equation

on the cauchy problem for a reaction diffusion equation with a singular nonlinearity abstract we consider the following cauchy problem with a singular nonlinearity p u t u u x r n t 0 0 u t 0 c lb r n with n 3 and having a positive lower bound we find some conditions on the initial value such that the local solutions of p vanish in finite time meanwhile we obtain optimal conditions on for global existence and study the large time behavior of those global solutions in particular we prove that if 0 and n 3 x u s x 2 1 n 2 2 1 1 1 x 2 1 where u s is a singular equilibrium of p and 1 then p has a unique global classical solution u with u u s and u x t 1 1 1 1 1 1 1 t 1 1 on the other hand the structure of positive radial solutions of the steady state of p is studied and some interesting properties of the positive solutions are obtained moreover the stability and weakly asymptotic stability of the positive radial solutions of the steady state of p are also discussed

existence and uniqueness results for third order nonlinear differential systems abstract sufficient conditions are given for the existence of solutions to third order boundary value problems for nonlinear differential systems some appropriate conditions are given to guarantee that the nagumo condition is satisfied by constructing appropriate a lower solution upper solution pair a concept that is defined in this paper the uniqueness result of the problem is also established the emphasis here is that the differential systems has nonlinear dependence on all over order derivatives and the boundary conditions are nonlinear

analysis of adomian decomposition applied to a third order ordinary differential equation from thin film flow abstract the adomian decomposition method is applied to the third order ordinary differential equation y y k obtained by considering a travelling wave solution admitted by a generalized thin film equation or for thin film flow down a vertical wall the adomian decomposition method leads to a power series approximation to the solution of y y k we show that the domain of convergence of the adomian decomposition solution is dependent on k we truncate the adomian decomposition solution at a suitable order to ensure that the truncated solution satisfies the contact line condition y 0 at some point x x we then determine the contact angle where tan d y d x at x x and plot the variation of contact angle with increasing k

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

thin film equations with partial wetting energy existence of weak solutions abstract the capillarity driven evolution with slip of a thin liquid film over a dry surface is considered in the regime of partial wetting the focus is on the simplest model case of a constant non zero dynamic contact angle in the lubrication approximation for the analytical treatment of the corresponding free boundary problem a new strategy is proposed based on the introduction of an ad hoc class of disjoining pressures which tend to concentrate at triple junctions a first investigation of this approach yields the existence of weak solutions which satisfy the dissipation relation as an inequality and which are different from those with zero contact angle a heuristic argument is also presented in order to clarify the connection between contact angle and dissipation relation it shows that moving droplets which satisfy the dissipation relation as an equality are forced to have the prescribed contact angle

numerical solutions of third order system of boundary value problems we develop a two stage numerical method for computing approximate solutions of a system of third order boundary value problems associated with odd order obstacle problems such a problem arise in physical oceanography and can be studied in the framework of variational inequality theory we show that the present method is of order two and gives numerical results which are better than the other available results numerical example is presented to illustrate the applicability of the new method our results can be viewed as an important and significant refinement of the previously known results

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

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

on finite generation of kernels of locally nilpotent r derivations of r x y z let r be a noetherian domain containing the field of rationals we show that if r is dedekind then the kernel of any locally nilpotent r derivation of r x y z is a finitely generated r algebra conversely we show that if r is neither a field nor a dedekind domain then there exists a locally nilpotent r derivation of r x y z whose kernel is not finitely generated over r

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

non polynomial spline approach to the solution of a system of third order boundary value problems we use a quartic spline equivalent nonpolynomial spline functions to develop a numerical method for computing approximations to the solution of a system of third order boundary value problems associated with obstacle unilateral and contact problems we show that the present method gives approximations which are better than those produced by other collocation finite difference and spline methods a numerical example is given to illustrate the applicability and efficiency of the new method

existence results for doubly nonlinear higher order parabolic equations on unbounded domains on demontre un resultat d existence pour le probleme aux valeurs limites et initiales u q 1 sign u t a 1 u f sur 0 t q u satisfait des conditions de dirichlet sur 0 t u u 0 sur pour t 0 a 1 u 1 m m d d u p 1 sign d u pour p q 1 m n 1 et r n un ensemble ouvert

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

the asymptotic solution of a class of third order boundary value problems arising in the theory of thin film flows this paper studies boundary and interior layer phenomena exhibited by solutions of certain singularly perturbed third order boundary value problems which govern the motion of thin liquid films subject to viscous capillary and gravitational forces precise conditions specifying where and when the third order derivative terms in the differential equations can be neglected are derived and improved estimates for the actual solutions in terms of solutions of the lower order models are constructed the paper also contains a technique for replacing a third order problem with an asymptotically equivalent second order one that may have wider applicability

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

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

on nontrivial traveling waves in thin film flows including contact lines we discuss dynamics of thin liquid films spreading down an inclined plane the fronts of these films are known to be unstable with respect to formation of finger like and triangular patterns in this work we concentrate on one particular aspect of these flows and that is the existence of nonlinear traveling waves we find evidence for presence of these waves for all inclination angles less than 90 to understand better the relevant pattern formation mechanism we explore via numerical simulations the bifurcation structure of the stability diagram close to the critical wavenumber the recovered structure is consistent with the existence of a subcritical bifurcation we discuss the connection between the bifurcation diagram and the existence of nontrivial traveling waves

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

mathematical analysis to a nonlinear fourth order partial differential equation abstract the paper first study the steady state thin film type equation u n u q 2 u u m u f x u with navier boundary conditions in multidimensional space by the truncation method a fixed point argument and some energy estimates the existence and asymptotic limit 0 for the positive weak solutions are given second the parabolic equation u t u n u x x x q 2 u x x x x u m u x x 0 with a navier boundary in one dimensional space is researched the existence is obtained by applying a semi discrete method for the time variable and solving the corresponding elliptic problem the uniqueness is shown for q 2 depending on an energy estimate in addition the iteration relation of the semi discrete problem gives an exponential decay result for the time t the thin film equation which is usually used to describe the motion of a very thin layer of viscous in compressible fluids along an inclined plane is a class of nonlinear fourth order parabolic equations and the maximum principle does not hold directly for applying the classic theory of partial differential equation the paper transforms the fourth order problem into a second order elliptic elliptic system or a second order parabolic elliptic system

on center subspace behavior in thin film equations the large time behavior of weak nonnegative solutions of the thin film equation tfe with absorption u t nabla cdot u n nabla delta u u p 1 u with parameters n in 0 3 and p 1 is studied the standard free boundary problem fbp with zero height zero contact angle and zero flux conditions at the interface and bounded compactly supported initial data is considered it is shown that there exists the critical absorption exponent p 0 1 n frac 4 n such that for p p 0 the asymptotic behavior of solutions u x t for t gg1 is represented by the well known source type solution of the pure tfe absorption u s x t t beta n f y y x t beta with the exponent beta frac 1 4 nn which is perturbed by a couple of ln t factors for n 1 this behavior is associated with the center subspace for the rescaled linearized thin film operator and is given by u x t sim nobreak t ln t beta n f x t beta ln t beta n 4 with beta frac 1 4 n where f y fr

calculation of planar and conical diffuser flows etude numerique de l ecoulement turbulent a proximite de la paroi on utilise une version modifiee du modele k e aux faibles nombres de reynolds tenant compte des effets de gradients adverses rencontres dans les ecoulements en diffuseur

singular integral equations arising in draining and coating flows abstract we present a very general existence theorem for a nonlinear singular integral equation using the schauder tychonoff fixed point theorem our result automatically guarantees the existence of a positive solution to a problem arising in draining and coating flows

on the cancellation problem for the affine space mathbb a 3 in characteristic p we show that the cancellation conjecture does not hold for the affine space mathbb a 3 k over any field k of positive characteristic we prove that an example of t asanuma provides a three dimensional k algebra a for which a is not isomorphic to k x1 x2 x3 although a t is isomorphic to k x1 x2 x3 x4

stability theorems for overrings of polynomial rings let r be a polynomial ring over a field projective modules over rings of the type r ix y r x y where r is a non zero element of r have been considered by murthy swan weibel see mu and w these rings lie between r x and r x x 1 and even when they are regular projective modules over them need not be free mu example 6 2 this leads us to study stability properties of projective modules over rings a which lie between r x and r x x 1 when r is any commutative noetherian ring we prove stability theorems for projective modules over such rings a in w these results theorems 4 2 and 4 3 have been proved for a r x by plumstead p and for a r x x 1 by mandal ma in w we prove similar results for the allied class of rings d of the type r x y xy stability theorems for gl a and gl d are proved in w when a r x or r x x 1 our theorems 6 2 i and 6 4 i had already been proved by suslin s finally in w 7 we study pic r x y r xy when r is a pid theorem 7 1 extends a result of murthy mu corollary 5 3 it would be interesting to know whether theorems 4 1 and 4 2 can be extended to the polynomial ring a t 1 tm where a lies between r x and r x x 1 more precisely for a projective a t 1 t module p of rank dim a we would like to know if the following statements are true i p has a unimodular element ii p has the cancellation property we have been able to show that when a r x the statement i is true b r theorem 3 1

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

a numerical and asymptotic study of some third order ordinary differential equations relevant to draining and coating flows some draining or coating fluid flow problems in which surface tension forces are important can be described by third order ordinary differential equations accurate computations are provided here