a semi linear elliptic equation in a strip arising in a two dimensional flame propagation model on montre l existence d une solution stationnaire pour un probleme a 2 dimensions pour l equation elliptique semilineaire de la forme u v y u x g u dans la bande s x y r 2 x r 0 0

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

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

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

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

some global dynamical properties of the kuramoto sivashinsky equations nonlinear stability and attractors abstract the kuramoto sivashinsky equations model pattern formations on unstable flame fronts and thin hydrodynamic films they are characterized by the coexistence of coherent spatial structures with temporal chaos we investigate some global dynamical properties including nonlinear stability we demonstrate their low modal behavior in terms of determining modes and that the fractal dimension of all attractors is bounded by a universal constant times l 13 8 where l is a dimensionless pattern cell size in the one dimensional case such equations are indeed a paradigm of low dimensional behavior for infinite dimensional systems

many hard examples for resolution for every choice of positive integers c and k such that k 3 and c 2 k 0 7 there is a positive number e such that with probability tending to 1 as n tends to a randomly chosen family of cn clauses of size k over n variables is unsatisfiable but every resolution proof of its unsatisfiability must generate at least 1 e n clauses

on sufficient conditions for unsatisfiability of random formulas a descriptive complexity approach to random 3 sat is initiated we show that unsatisfiability of any significant fraction of random 3 cnf formulas cannot be certified by any property that is expressible in datalog combined with the known relationship between the complexity of constraint satisfaction problems and expressibility in datalog our result implies that any constraint propagation algorithm working with small constraints will fail to certify unsatisfiability almost always our result is a consequence of designing a winning strategy for one of the players in the existential pebble game the winning strategy makes use of certain extension axioms that we introduce and hold almost surely on a random 3 cnf formula the second contribution of our work is the connection between finite model theory and propositional proof complexity to make this connection explicit we establish a tight relationship between the number of pebbles needed to win the game and the width of the resolution refutations as a consequence to our result and the known size width relationship in resolution we obtain new proofs of the exponential lower bounds for resolution refutations of random 3 cnf formulas and the pigeonhole principle

stability of receding traveling waves for a fourth order degenerate parabolic free boundary problem abstract consider the thin film equation h t h h y y y y 0 with a zero contact angle at the free boundary that is at the triple junction where liquid gas and solid meet previous results on stability and well posedness of this equation have focused on perturbations of equilibrium stationary or self similar profiles the latter eventually wetting the whole surface these solutions have their counterparts for the second order porous medium equation h t h m y y 0 where m 1 is a free parameter both porous medium and thin film equation degenerate as h 0 but the porous medium equation additionally fulfills a comparison principle while the thin film equation does not in this note we consider traveling waves h v 6 x 3 x 2 for x 0 where x y v t and v 0 are free parameters these traveling waves are receding and therefore describe de wetting a phenomenon genuinely linked to the fourth order nature of the thin film equation and not encountered in the porous medium case as it violates the comparison principle the linear stability analysis leads to a linear fourth order degenerate parabolic operator for which we prove maximal regularity estimates to arbitrary orders of the expansion in x in a right neighborhood of the contact line x 0 this leads to a well posedness and stability result for the corresponding nonlinear equation as the linearized evolution has different scaling as x 0 and x the analysis is more intricate than in related previous works we anticipate that our approach is a natural step towards investigating other situations in which the comparison principle is violated such as droplet rupture

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

a general approach to stability in free boundary problems abstract every solution of a linear elliptic equation on a bounded domain may be considered as an equilibrium of a free boundary problem the free boundary problem consists of the corresponding parabolic equation on a variable unknown domain with free boundary conditions prescribing both dirichlet and neumann data we establish a rigorous stability analysis of such equilibria including the construction of stable and unstable manifolds for this purpose we transform the free boundary problem to a fully nonlinear and nonlocal parabolic problem on a fixed domain with fully nonlinear lateral boundary conditions and we develop the general theory for such problems as an illustration we give two examples the second being the focussing flame problem in combustion theory

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

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

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

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

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

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

lower bounds for the weak pigeonhole principle and random formulas beyond resolution we work with an extension of resolution called res 2 that allows clauses with conjunctions of two literals in this system there are rules to introduce and eliminate such conjunctions we prove that the weak pigeonhole principle phpncn and random unsatisfiable cnf formulas require exponential size proofs in this system this is the strongest system beyond resolution for which such lower bounds are known as a consequence to the result about the weak pigeonhole principle res log is exponentially more powerful than res 2 also we prove that resolution cannot polynomially simulate res 2 and that res 2 does not have feasible monotone interpolation solving an open problem posed by krajicek

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

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

existence of solutions and separation from singularities for a class of fourth order degenerate parabolic equations a nonlinear parabolic equation of the fourth order is analyzed the equation is characterized by a mobility coefficient that degenerates at 0 existence of at least one weak solution is proved by using a regularization procedure and deducing suitable a priori estimates if a viscosity term is added and additional conditions on the nonlinear terms are assumed then it is proved that any weak solution becomes instantaneously strictly positive this in particular implies uniqueness for strictly positive times and further time regularization properties the long time behavior of the problem is also investigated and the existence of trajectory attractors and under more restrictive conditions of strong global attractors is shown

on ks type equations describing the evolution and rupture of a liquid interface abstract study of core annular flow of two liquids of different viscosity generates two new amplitude equations of the kuramoto sivashinsky ks type in the first an integral operator of dispersive nature appends the ks it modifies the bifurcation structure of the ks and delays parametrically the appearances of a specific pattern the second modification specific to cylindrical geometry is due to a sensitive part in the curvature dispensed within a conventional expansion when retained it may cause a linearly unstable interface to rupture and to form experimentally observed bubbles

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

on the structure of positive solutions to an elliptic problem arising in thin film equations we consider the structure of positive radial solutions of u u u 0in br u 0on br where br is a ball in rn with radius r when 0 0 such that when r r the dirichlet problem has exactly two radial solutions when r r the solution is unique and there is no solution for r 0 the radial solution exists and is unique

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

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

positive solutions of a semilinear elliptic equation with singular nonlinearity abstract this paper is devoted to the study of positive solutions of the semilinear elliptic equation u k x u p 0 x r n with n 3 and p 0 asymptotic behaviours of sky states and uniqueness of singular sky states are obtained via invariant manifold theory of dynamical systems the dirichlet problem in exterior domains is also studied it is proved that this problem has infinitely many positive solutions with fast growth

small world properties emerge in highly compartmentalized networks with intermediate group sizes and numbers many recent studies have focused on two statistical properties observed in diverse real world networks the small world property and compartmentalization d j watts and s h strogatz nature 393 440 1998 m girvan and m e j newman proc natl acad sci 99 7821 2002 models that include group affiliations have been shown to produce networks with high clustering coefficients a necessary condition for small world properties m e j newman phys rev e 68 026121 2003 m e j newman and j park phys rev e 68 036122 2003 however the consequences of varying the number and size of groups in a network are not well understood in order to investigate the consequences of group organization we examined sets of networks that varied simultaneously in the size and number of groups while maintaining the same overall size and average degree here we show that the small world property arises in maximally compartmentalized and clustered networks that occur in the intermediate region between few very large groups and many very small groups

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

preconditioning a class of fourth order problems by operator splitting we develop preconditioners for systems arising from finite element discretizations of parabolic problems which are fourth order in space we consider boundary conditions which yield a natural splitting of the discretized fourth order operator into two discrete linear second order elliptic operators and exploit this property in designing the preconditioners the underlying idea is that efficient methods and software to solve second order problems with optimal computational effort are widely available we propose symmetric and non symmetric preconditioners along with theory and numerical experiments they both document crucial properties of the preconditioners as well as their practical performance it is important to note that we neither need h s regularity s 1 of the continuous problem nor quasi uniform grids

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

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

positive solutions to singular second and third order differential equations for quantum fluids we analyze a quantum trajectory model given by a steady state hydrodynamic system for quantum fluids with positive constant temperature in bounded domains for arbitrary large data the momentum equation can be written as a dispersive third order equation for the particle density where viscous effects are incorporated the phenomena that admit positivity of the solutions are studied the cases one space dimensional dispersive or non dispersive viscous or non viscous are thoroughly analyzed with respect to positivity and existence or non existence of solutions all depending on the constitutive relation for the pressure law we distinguish between isothermal linear and isentropic power law pressure functions of the density it is proved that in the dispersive non viscous model a classical positive solution only exists for small positive particle current densities both for the isentropic and isothermal case uniqueness is also shown in the isentropic subsonic case when the pressure law is strictly convex however we prove that no weak isentropic solution can exist for large current densities the dispersive viscous problem admits a classical positive solution for all current densities both for the isentropic and isothermal case with an ultra diffusion condition

resolution proofs of generalized pigeonhole principles we extend results of haken to give an exponential lower bound on the size of resolution proofs for propositional formulas encoding a generalized pigeonhole principle these propositional formulas express the fact that there is no one one mapping from c n objects to n objects when c 1 as a corollary resolution proof systems do not p simulate constant formula depth frege proof systems

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

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

a free boundary problem for the heat equation arising in flame propagation we introduce a new free boundary problem for the heat equation of interest in combustion theory it is obtained in the description of laminar flames as an asymptotic limit for high activation energy the problem asks for the determination of a domain in space time q c rn x 0 t and a function u x t 0 defined in q such that ut au in q u takes certain initial conditions u x 0 uo x for x e qo oqfn t 0 and two conditions are satisfied on the free boundary r a2 n t 0 u o and u1 1 where u1 denotes the derivative of u along the spatial exterior normal to r we approximate this problem by means of a certain regularization on the boundary and prove the existence of a weak solution under suitable assumptions on the initial data

singularities in a modified kuramoto sivashinsky equation describing interface motion for phase transition abstract phase transitions can be modeled by the motion of an interface between two locally stable phases a modified kuramoto sivashinsky equation h t 2 h 4 h 1 h 2 2 h 2 h xx h yy h xy 2 describes near planar interfaces which are marginally long wave unstable we study the question of finite time singularity formation in this equation in one and two space dimensions on a periodic domain such singularity formation does not occur in the kuramoto sivashinsky equation 0 for all 1 0 we provide sufficient conditions on the initial data and size of the domain to guarantee a finite time blow up in which a second derivative of h becomes unbounded using a bifurcation theory analysis we show a parallel between the stability of steady periodic solutions and the question of finite time blow up in one dimension finally we consider the local structure of the blow up in the one dimensional case via similarity solutions and numerical simulations that employ a dynamically adaptive self similar grid the simulations resolve the singularity to over 25 decades in h xx l and indicate that the singularities are all locally described by a unique self similar profile in h xx we discuss the relevance of these observations to the full intrinsic equations of motion and the associated physics

a free boundary problem arising in flame propagation abstract we study a free boundary problem describing the propagation of laminar flames the problem arises as the limit of a singular perturbation problem we introduce the notion of viscosity solutions for the problem to show the maximum principle type property of the solutions using this property we show the uniform convergence of the approximating solutions and the uniqueness of the viscosity solution under several geometric conditions on the initial data

the efficiency of resolution and davis putnam procedures we consider several problems related to the use of resolution based methods for determining whether a given boolean formula in conjunctive normal form is satisfiable first building on the work of clegg edmonds and impagliazzo in proceedings of the twenty eighth annual acm symposium on theory of computing philadelphia pa 1996 acm new york 1996 pp 174 183 we give an algorithm for unsatisfiability that when given an unsatisfiable formula of f finds a resolution proof of f the runtime of our algorithm is subexponential in the size of the shortest resolution proof of f next we investigate a class of backtrack search algorithms for producing resolution refutations of unsatisfiability commonly known as davis putnam procedures and provide the first asymptotically tight average case complexity analysis for their behavior on random formulas in particular for a simple algorithm in this class called ordered dll we prove that the running time of the algorithm on a randomly generated k cnf formula with n variables and m clauses is 2 theta n n m 1 k 2 with probability 1 o 1 finally we give new lower bounds on mbox res f the size of the smallest resolution refutation of f for a class of formulas representing the pigeonhole principle and for randomly generated formulas for random formulas chvatal and szemeredi j acm 35 1988 pp 759 768 had shown that random 3 cnf formulas with a linear number of clauses require exponential size resolution proofs and fu on the complexity of proof systems ph d thesis university of toronto toronto on canada 1995 extended their results to k cnf formulas these proofs apply only when the number of clauses is omega n log n we show that a lower bound of the form 2 n gamma holds with high probability even when the number of clauses is n k 2 4 epsilon

on the automatizability of resolution and related propositional proof systems a propositional proof system is automatizable if there is an algorithm that given a tautology produces a proof in time polynomial in the size of its smallest proof this notion can be weakened if we allow the algorithm to produce a proof in a stronger system within the same time bound this new notion is called weak automatizability among other characterizations we prove that a system is weakly automatizable exactly when a weak form of the satisfiability problem is solvable in polynomial time after studying the robustness of the definition we prove the equivalence between i resolution is weakly automatizable ii res k is weakly automatizable and iii res k has feasible interpolation when k 1 in order to prove this result we show that res 2 has polynomial size proofs of the reflection principle of resolution which is a version of consistency we also show that res k for every k 1 proves its consistency in polynomial size while resolution does not in fact we show that resolution proofs of its own consistency require almost exponential size this gives a better lower bound for the monotone interpolation of res 2 and a separation from resolution as a byproduct our techniques also give us a way to obtain a large class of examples that have small resolution refutations but require relatively large width this answers a question of alekhnovich and razborov related to whether resolution is automatizable in quasipolynomial time

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

on the equation of a curved flame front abstract a strongly nonlinear spatially invariant equation for the dynamics of premixed flame is derived on the assumption that the curvature of the flame front is small the equation generalizes the corresponding weakly nonlinear equation obtained previously near the stability threshold

stability of cylindrical bodies in the theory of surface diffusion abstract perturbation arguments and finite element calculations are employed to study the nonlinear partial differential equations governing morphological changes induced by curvature driven diffusion of mass in the surface of an axially symmetric body isotropy of surface properties is assumed second and higher order perturbation analyses indicate that the familiar result of the linear theory of small amplitude longitudinal perturbations of a cylinder to the effect that a long cylinder is stable against all perturbations with spatial fourier spectra containing only wavelengths less than the circumference of the cylinder does not hold in the full nonlinear theory the perturbation analyses yield criteria for determining when longitudinal perturbations with high wave number spectra grow in amplitude after an initial decay followed by an incubation time and result in break up of the body into a necklace of beads the principal conclusions of the formal perturbation analyses are found to be in good accord with numerical solutions obtained by finite element methods

a switching lemma for small restrictions and lower bounds for k dnf resolution we prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to formulas in disjunctive normal form dnfs with small terms we use this to prove lower bounds for the res k propositional proof system an extension of resolution which works with k dnfs instead of clauses we also obtain an exponential separation between depth d circuits of bottom fan in k and depth d circuits of bottom fan in k 1 our results for res k are as follows the 2n to n weak pigeonhole principle requires exponential size to refute in res k for k leq sqrt log n log log n for each constant k there exists a constant w k so that random w cnfs require exponential size to refute in res k for each constant k there are sets of clauses which have polynomial size res k 1 refutations but which require exponential size res k refutations

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

properties of highly clustered networks we propose and solve exactly a model of a network that has both a tunable degree distribution and a tunable clustering coefficient among other things our results indicate that increased clustering leads to a decrease in the size of the giant component of the network we also study susceptible infective recovered type epidemic processes within the model and find that clustering decreases the size of epidemics but also decreases the epidemic threshold making it easier for diseases to spread in addition clustering causes epidemics to saturate sooner meaning that they infect a near maximal fraction of the network for quite low transmission rates

mean payoff games and propositional proofs we associate a cnf formula to every instance of the mean payoff game problem in such a way that if the value of the game is non negative the formula is satisfiable and if the value of the game is negative the formula has a polynomial size refutation in s 2 frege i e dnf resolution this reduces mean payoff games to the weak automatizability of s 2 frege and to the interpolation problem for s 2 2 frege since the interpolation problem for s 1 frege i e resolution is solvable in polynomial time our result is close to optimal up to the computational complexity of solving mean payoff games the proof of the main result requires building low depth formulas that compute the bits of the sum of a constant number of integers in binary notation and low complexity proofs of the required arithmetic properties

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

symmetries and solutions to the thin film equations we study symmetries and solutions of the generalized fourth order nonlinear partial differential equations which arise from studies of thin liquid films it is shown that the equations admit extended scaling and rotation symmetries and a class of generalized conditional symmetries for certain coefficient functions exact solutions associated to the symmetries are obtained

pseudorandom generators hard for k dnf resolution and polynomial calculus resolution a pseudorandom generator gn f0 1g n f0 1g m is hard for a propositional proof system p if roughly speaking p cannot eciently

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

the intractability of resolution abstract we prove that for infinitely many disjunctive normal form propositional calculus tautologies the length of the shortest resolution proof of cannot be bounded by any polynomial of the length of the tautologies we use were introduced by cook and reckhow 1979 and encode the pigeonhole principle extended resolution can furnish polynomial length proofs of these formulas

a global attracting set for the kuramoto sivashinsky equation new bounds are given for the l2 norm of the solution of the kuramoto sivashinsky equation partial t u x t partial x 2 partial x 4 u x t u x t partial x u x t for initial data which are periodic with periodl there is no requirement on the antisymmetry of the initial data the result is mathop lim sup limits t to infty left u cdot t right 2 leqslant const l 8 5

self adjoint sub classes of generalized thin film equations in this work we consider a class of fourth order nonlinear partial differential equation containing several un specified coefficient functions of the dependent variable which encapsulates various mathematical models used e g for describing the dynamics of thin liquid films we determine the subclasses of these equations which are self adjoint by using a general theorem on conservation laws proved by one of the authors nhi we find conservation laws for some of these partial differential equations without classical lagrangians

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

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

on the complexity of resolution with bounded conjunctions we analyze size and space complexity of res k a family of propositional proof systems introduced by krajicek in fund math 170 1 3 2001 123 which extend resolution by allowing disjunctions of conjunctions of up to k 1 literals we show that the treelike res k proof systems form a strict hierarchy with respect to proof size and also with respect to space moreover resolution while simulating treelike res k is almost exponentially separated from treelike res k to study space complexity for general res k we introduce the concept of dynamical satisfiability which allows us to prove in a unified way all known space lower bounds for resolution and to extend them to res k

why social networks are different from other types of networks we argue that social networks differ from most other types of networks including technological and biological networks in two important ways first they have nontrivial clustering or network transitivity and second they show positive correlations also called assortative mixing between the degrees of adjacent vertices social networks are often divided into groups or communities and it has recently been suggested that this division could account for the observed clustering we demonstrate that group structure in networks can also account for degree correlations we show using a simple model that we should expect assortative mixing in such networks whenever there is variation in the sizes of the groups and that the predicted level of assortative mixing compares well with that observed in real world networks

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

short proofs are narrow resolution made simple the width of a resolution proof is defined to be the maximal number of literals in any clause of the proof in this paper we relate proof width to proof length size in both general resolution and its tree like variant the following consequences of these relations reveal width as a crucial resource of resolution proofs in one direction the relations allow us to give simple unified proofs for almost all known exponential lower bounds on size of resolution proofs as well as several interesting new ones they all follow from width lower bounds and we show how these follow from natural expansion property of clauses of the input tautology in the other direction the width size relations naturally suggest a simple dynamic programming procedure for automated theorem proving one which simply searches for small width proofs this relation guarantees that the runnuing time and thus the size of the produced proof is at most quasi polynomial in the smallest tree like proof this algorithm is never much worse than any of the recursive automated provers such as dll used in practice in contrast we present a family of tautologies on which it is exponentially faster