degenerate variational inequalities of evolution on etudie des inegalites variationnelles de la forme re 0 pour tout v k by 0 bu 0 ou b est lineaire mais peut s annuler a est non lineaire et k est un ensemble convexe ferme

cauchy problems of semilinear pseudo parabolic equations abstract this paper concerns with the cauchy problems of semilinear pseudo parabolic equations after establishing the necessary existence uniqueness and comparison principle for mild solutions which are also classical ones provided that the initial data are appropriately smooth we investigate large time behavior of solutions it is shown that there still exist the critical global existence exponent and the critical fujita exponent for pseudo parabolic equations and that these two critical exponents are consistent with the corresponding semilinear heat equations

homogenization of some degenerate pseudoparabolic variational inequalities abstract multiscale analysis of a degenerate pseudoparabolic variational inequality modelling the two phase flow with dynamical capillary pressure in a perforated domain is the main topic of this work regularisation and penalty operator methods are applied to show the existence of a solution of the nonlinear degenerate pseudoparabolic variational inequality defined in a domain with microscopic perforations as well as to derive a priori estimates for solutions of the microscopic problem the main challenge is the derivation of a priori estimates for solutions of the variational inequality uniformly with respect to the regularisation parameter and to the small parameter defining the scale of the microstructure the method of two scale convergence is used to derive the corresponding macroscopic obstacle problem

a pseudo parabolic variational inequality and stefan problem abstract we study existence and uniqueness questions concerning non linear variational inequalities of the type of 1 1a 1 1b in the introduction applications to free boundary problems are also given author

simulating finger phenomena in porous media with a moving finite element method the non equilibrium richards equation is solved using a moving finite element method in this paper the governing equation is discretized spatially with a standard finite element method and temporally with second order runge kutta schemes a strategy of the mesh movement is based on the work by li et al r li t tang p w zhang a moving mesh finite element algorithm for singular problems in two and three space dimensions journal of computational physics 177 2002 365 393 a beckett and mackenzie type monitor function is adopted to obtain high quality meshes around the wetting front a smoothing method which is based on the diffusive mechanism is used with the moving mesh technique high mesh quality and high numerical accuracy are obtained successfully the numerical convergence and the advantage of the algorithm are demonstrated by a series of numerical experiments

pseudo parabolic regularization of forward backward parabolic equations power type nonlinearities we study a quasilinear parabolic equation of forward backward type under assumptions on the nonlinearity which hold for a wide class of mathematical models using a pseudo parabolic regularization of power type we prove existence and uniqueness of positive solutions of the regularized problem in a space of radon measures it is shown that these solutions satisfy suitable entropy inequalities we also study their qualitative properties in particular proving that the singular part of the solution with respect to the lebesgue measure is constant in time

the global existence and time decay for the solutions of the fractional pseudo parabolic equation abstract we consider the cauchy problem of fractional pseudo parabolic equation on the whole space r n n 1 here the fractional order is related to the diffusion type source term behaving as the usual diffusion term on the high frequency part it has a feature of regularity gain and regularity loss for 1 and 0 1 respectively we establish the global existence and time decay rates for small amplitude solutions to the cauchy problem for 0 in the case that 0 1 we introduce the time weighted energy method to overcome the weakly dissipative property of the equation

analysis of a darcy flow model with a dynamic pressure saturation relation we consider a new model for groundwater flow the model differs from previous models in that the saturation pressure relation is extended with a dynamic term namely the time derivative of the saturation the resulting model equation is of nonlinear degenerate pseudoparabolic type we give a rigorous analysis of travelling wave solutions and the local behavior near fronts part of this analysis is then used in a formal asymptotic analysis of solutions with initial support in a half line as an illustrative special case we also consider heaviside initial data

small and waiting time behavior of a darcy flow model with a dynamic pressure saturation relation we address the small time evolution of interfaces fronts for the pseudoparabolic generalization partial u over partial t partial over partial x left u alpha partial u over partial x u beta partial 2 u over partial x partial t right of the porous medium equation identifying regimes in which the local behavior remains fixed for some finite time and others in which it changes instantaneously a number of phenomena beyond those exhibited by the porous medium equation are elucidated including retreating fronts and novel types of local behavior related results for the important limit case partial u over partial t partial over partial x left u beta partial 2 u over partial x partial t right are also described

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

ward type identities for the two dimensional anderson model at weak disorder using the particular momentum conservation laws in dimension d 2 we rewrite the anderson model in terms of low momentum long range fields at the price of introducing electron loops the corresponding loops satisfy a ward type identity hence are much smaller than expected this fact should be useful for the study of the weak coupling model in the middle of the spectrum of the free hamiltonian

boundary control for a class of pseudo parabolic differential equations abstract the boundary stabilization problem for a class of linear and nonlinear pseudo parabolic differential equations is considered the proposed control laws are used to achieve global exponential stability for the linear system and semi global exponential stability for the nonlinear system in the h 1 sense an h 2 bound of the solution for the nonlinear system is also derived a numerical example is included to illustrate the application of the proposed control laws

random matrix theory and the anderson model this paper is devoted to a discussion of possible strategies to prove rigorously the existence of a metal insulator anderson transition for the anderson model in dimension d 3 the possible criterions used to define such a transition are presented it is argued that at low disorder the lowest order in perturbation theory is described by a random matrix model various simplified versions for which rigorous results have been obtained in the past are discussed it includes a free probability approach the wegner n orbital model and a class of models proposed by disertori pinson and spencer comm math phys 232 83 124 2002 at last a recent work by magnen rivasseau and the author markov process and related fields 9 261 278 2003 is summarized it gives a toy modeldescribing the lowest order approximation of anderson model and it is proved that for d 2 its density of states is given by the semicircle distribution a short discussion of its extension to d 3 follows

on a class of forward backward parabolic equations existence of solutions abstract we study the initial boundary value problem u t u x x u t x x in 0 t u u t 0 in 0 t u u 0 in 0 where is an interval and u 0 is a nonnegative radon measure on the map is increasing in 0 and decreasing in for some 0 and satisfies 0 0 the regularizing map is increasing and bounded we prove existence of suitably defined nonnegative radon measure valued solutions the solution class is natural since smooth initial data may generate solutions which become measure valued after finite time

unstable atomicity and loop spaces on lie groups a k 1 connected space x with hk x zp z p has been defined to be patomic if any map f x x which induces an isomorphism f hk x zp hk x zp must induce an isomorphism f h x zp h x zp we describe such x as being c p p atomic cohen peterson and consider some variants of the definition c p p atomicity is an indecomposability condition on a space implying for example that the space has no non trivial rood p homotopy retracts we define a space x to be weakly p atomic if any map f x x is either a mod p homotopy equivalence or for each n f lq x zp h x zp is nilpotent under composition if x is a k 1 connected cw complex of finite type hk x zp zp k 1 and x is either an h space or a co h space in particular a suspension then x is c p p atomic if and only if it is weakly p atomic see comment 2 following proposition 5 1 the spaces considered satisfy a stronger condition a space x is defined to be p atomic if any map f x x is either a m o d p homotopy equivalence or f x zp h x zv is nilpotent not every weakly p atomic space is p atomic for example bsu is weakly 2 atomic but is not 2 atomic see comment 2 following lemma 3 3 a p atomic space x is defined to be p atomic of degree t if whenever f h x zv h x zv is nilpotent f t 0 but there exists some such f with f t1 0 t 0 the author does not know if every p atomic space is p atomic of finite degree the significance of the degree t will become apparent below we consider the 2 atomicity of o g where g is a 1 connected simple lie group

existence of weak solutions for a nonlocal pseudo parabolic model for brinkman two phase flow in asymptotically flat porous media abstract we study a nonlocal evolution equation that involves a pseudo parabolic third order term the equation models almost uni directional two phase flow in brinkman regimes we prove the existence of weak solutions for this equation we also give a series of numerical examples that demonstrate the ability of the equation to support overshooting like in 26 and explore the behavior of solutions in various limit regimes

the richards equation with hysteresis and degenerate capillary pressure abstract we study the richards equation with a dynamic capillary pressure including hysteresis we provide existence and approximation results for degenerate capillary pressure curves p c treating two cases in the first case the permeability function k can be degenerate but the initial saturation does not take the critical values in the second case the permeability function k is strictly positive but the capillary pressure function can be multi valued in both cases the degenerate behavior of p c leads to the physically desired uniform bounds for the saturation variable our approach exploits maximum principles and relies on the corresponding uniform bounds for pressure and saturation a new compactness result for the saturation variable allows to take limits in nonlinear terms the solution concept uses tools of convex analysis

numerical analysis of a finite volume scheme for two incompressible phase flow with dynamic capillary pressure abstract in this paper we propose and analyze the convergence of a tpfa two points flux approximation finite volume scheme to approximate the two incompressible phase flow with dynamic capillary pressure in porous media the fully implicit scheme is based on nonstandard approximation on mobilities and capillary pressure on the dual mesh we derive a discrete variational formulation and we present a new result of convergence in a two or three dimensional porous medium in comparison with static capillary pressure the non equilibrium capillary model requires more powerful techniques especially the discrete energy estimates are not standard

remarks on the stability of shock profiles for conservation laws with dissipation two remarks are made the first is to establish the stability of monotone shock profiles of the kdv burgers equation based on an energy method of goodman the second remark illustrates specifically in burgers equation that uniform rates of decay are not to be expected for perturbations of shock profiles in typical norms introduction we will prove two results the first is to establish the orbital stability of monotone shock profile solutions 4 x st of the kdv burgers equation 1 1 ut uux uxx uxxx 0 0 the second is to show that for shock profile solutions of viscous conservation laws and specifically for burgers equation 1 2 ut uux uxx one cannot expect perturbations to decay at any uniform rate algebraic or otherwise in ordinary norms such as lp the first result is related to goodman s conditional stability result in 5 for small amplitude shock profiles x st in systems of genuinely nonlinear conservation laws with viscosity u f u x buxx u e rm the second is intended to be relevant to the question of full stability for shock profiles of such systems the nonlinear stability to small perturbations of constant state solutions u0 x c of the system u f u x b u ux x where b c is a strictly stable viscosity matrix see 9 can be demonstrated based on a uniform rate of decay of solutions to the linearized equation the method used is similar to that developed in 8 and 10 for systems in n 1 space dimensions faster decay rates in higher space dimensions make the method easier to apply when n is large but it does just work for conservation laws with stable viscosities in one space dimension see 13 thus for example assuming c 0 if iiuo iih6 ll is sufficiently small then iiu t llh4 ct 1 4iiuoii h4 n la in this note we want to point out that such uniform decay rates cannot be valid for perturbations of shock profiles in particular for the shock profile solution of burgers equations we construct a family of perturbations which decay to zero but do so at no uniform rate in most ordinary norms including all lp norms thus for received by the editors october 15 1984 and in revised form december 19 1984 1980 mathematics subject classification primary 35q20 35l65 1985 american mathematical society 0002 9947 85 1 00 25 per page

degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities abstract the existence of solutions of degenerate quasilinear pseudoparabolic equations where the term t u is replace by t b u with memory terms and quasilinear variational inequalities is shown the existence of solutions of equations is proved under the assumption that the nonlinear function b is monotone and a gradient of a convex continuously differentiable function the uniqueness is proved for lipschitz continuous elliptic parts the existence of solutions of quasilinear variational inequalities is proved under stronger assumptions namely the nonlinear function defining the elliptic part is assumed to be a gradient and the function b to be lipschitz continuous

two phase entropy solutions of a forward backward parabolic equation this article deals with the cauchy problem for a forward backward parabolic equation which is of interest in physical and biological models considering such an equation as the singular limit of an appropriate pseudoparabolic third order regularization we consider the framework of entropy solutions namely weak solutions satisfying an additional entropy inequality inherited by the higher order equation moreover we restrict the attention to two phase solutions that is solutions taking values in the intervals where the parabolic equation iswell posed proving existence and uniqueness of such solutions

the riemann problem for a forward backward parabolic equation abstract the riemann problem for a forward backward parabolic equation of interest in physical and biological models is studied a complete classification of suitably defined entropy solutions is provided thereafter the existence and uniqueness of a solution is proven and its type is identified

practical aspects of formulation and solution of moving mesh partial differential equations moving mesh partial differential equations mmpdes are used in the mmpde moving mesh method to generate adaptive moving meshes for the numerical solution of time dependent problems how mmpdes are formulated and solved is crucial to the efficiency and robustness of the method in this paper several practical aspects of formulating and solving mmpdes are studied they include spatial balance scaling invariance effective control of mesh concentration bounds on time steps multiple sub steps and two level mesh movement numerical results are also given c 2001 academic press

the buckley leverett equation with dynamic capillary pressure the buckley leverett equation for two phase flow in a porous medium is modified by including dependence of the capillary pressure on the rate of change of saturation this model due to hassanizadeh and gray results in a nonlinear pseudoparabolic partial differential equation phase plane analysis including a separation function to measure the distance between invariant manifolds is used to determine when the equation supports traveling waves corresponding to undercompressive shocks the riemann problem for the underlying conservation law is solved and the structures of the various solutions are confirmed with numerical simulations of the partial differential equation

stable patterns in a viscous diffusion equation we consider a pseudoparabolic regularization of a forward backward nonlinear diffusion equation ut a f u vut motivated by the problem of phase separation in a viscous binary mixture the function f is nonmonotone so there are discontinuous steady state solutions corresponding to arbitrary arrangements of phases we find that any bounded measurable steady state solution u x satisfying f u constant f u x 0 a e is dynamically stable to perturbations in the sense of convergence in measure in particular smooth solutions may achieve discontinuous asymptotic states furthermore stable states need not correspond to absolute minimizers of free energy thus violating gibbs principle of stability for phase mixtures

a model problem for groundwater flow with dynamic capillary pressure stability of travelling waves modelling vertical nonsteady groundwater flow with dynamic capillary pressure leads to a nonlinear diffusion convection equation extended with a nonlinear third order term with two space derivatives and one time derivative we consider the special case in which the original equation is the burgers equation and the third order term is linear we investigate stability of travelling wave solutions of the resulting pseudo parabolic burgers equation for which we first obtain well posedness results

finite element galerkin method for 2d sobolev equations with burgers type nonlinearity abstract in this article the global existence of a unique strong solution to the 2d sobolev equation with burgers type nonlinearity is established using weak or weak compactness type arguments when the forcing function f 0 is in l l2 new a priori bounds are derived which are valid uniformly in time as t and with respect to the dispersion coefficient as 0 it is further shown that the solution of the sobolev equation converges to the solution of the 2d burgers equation with order o a finite element method is applied to approximate the solution in the spatial direction and the existence of a global attractor is derived for the semidiscrete scheme further using a priori bounds and an integral operator optimal error estimates are derived in l l2 norm which hold uniformly with respect to as 0 since the constants in the error estimates have exponential growth in time therefore under a certain uniqueness condition the error bounds are derived which are uniformly in time more importantly all the above results remain valid as tends to zero finally this paper concludes with some numerical examples

oscillatory instability of traveling waves for a kdv burgers equation the stability of traveling wave solutions of a generalization of the kdv burgers equation o u u p o u 03u a o u is studied as the parameters p and a are varied the eigenvalue problem for the linearized evolution of perturbations is analyzed by numerically computing evans function d a an analytic function whose zeros correspond to discrete eigenvalues in particular the number of unstable eigenvalues in the complex plane is evaluated by computing the winding number of d a analytical and numerical evidence suggests that a hopf bifurcation occurs for oscillatory traveling wave profiles in certain parameter ranges dynamic simulations suggest that the bifurcation is subcritical no stable time periodic solution is found

a global existence result for the equations describing unsaturated flow in porous media with dynamic capillary pressure in this paper we investigate the pseudoparabolic equation tc div k c pc c c tc g en where is a positive constant c is the moisture content k is the hydraulic conductivity and pc is the static capillary pressure this equation describes unsaturated flows in porous media with dynamic capillary pressure saturation relationship in general such models arise in a number of cases when non equilibrium thermodynamics or extended non equilibrium thermodynamics are used to compute the flux for this equation existence of the traveling wave type solutions was extensively studied nevertheless the existence seems to be known only for the non degenerate case when k is strictly positive we use the approach from statistical hydrodynamics and construct the corresponding entropy functional for the regularized problem such approach permits to get existence for any time interval of an appropriate weak solution with square integrable first derivatives in x and in t and square integrable time derivative of the gradient negative part of such weak solution is small in l2 norm with respect to x uniformly in time as the square root of the relative permeability at the value of the regularization parameter next we control the regularized entropy a fine balance between the regularized entropy and the degeneracy of the capillary pressure permits to get an lq uniform bound for the time derivative of the gradient these estimates permit passing to the limit when the regularization parameter tends to zero and obtaining the existence of global nonnegative weak solution

blow up phenomena for a pseudo parabolic equation with p laplacian and logarithmic nonlinearity terms abstract this paper deals with a pseudo parabolic equation with p laplacian and logarithmic nonlinearity terms under homogeneous dirichlet boundary condition in a smooth bounded domain which was studied in 12 and the global existence and finite time blow up of the weak solution were studied under different energy levels we generalize and extend those results by discussing the asymptotic behavior of the weak solution and proving that the weak solution converges to the corresponding stationary solution as time tends to infinity moreover a lower and an upper bound estimation for blow up time and rate are obtained for the blow up weak solution in different initial energy cases furthermore we establish the weak solution with high initial energy that is global bounded or blow up under some sufficient conditions as well as a nonblow up criterion and algebraic decay result in other assumptions

linking anisotropic sharp and diffuse surface motion laws via gradient flows we compare four surface motion laws for sharp surfaces with their diffuse interface counterparts by means of gradient flows on corresponding energy functionals the energy functionals can be defined to give the same dependence on normal direction for the energy of sharp plane surfaces as for their diffuse counterparts the anisotropy of the kinetics can be incorporated into the inner product without affecting the energy functional

generalized nonequilibrium capillary relations for two phase flow through heterogeneous media for two phase flow in porous media the natural medium heterogeneity necessarily gives rise to capillary nonequilibrium effects the relaxation to the equilibrium is a slow process which should be introduced in macroscopic flow models many nonequilibrium models are based on a phenomenological approach at the same time there exists a rigorous mathematical way to develop the nonequilibrium equations its formalism developed by bourgeat and panfilov computational geosciences 2 191 1998 is based on the homogenization of the microscale flow equations over medium heterogeneities in contrast with the mentioned paper in which the case of a sufficiently fast relaxation was analyzed we consider the case of long relaxation which leads to the appearance of long term memory on the macroscale due to coupling between the nonlinearity and nonlocality in time the macroscopic model remains however incompletely homogenized in the general case at the same time frequently only the relationship for the nonequilibrium capillary pressure is of interest for applications in the present paper we obtain such an exact relationship in two different independent forms for the case of long term memory this relationship is more general than that obtained by bourgeat and panfilov in addition we prove the comparison theorem which determines the upper and lower bounds for the macroscopic model these bounds represent linear flow models which are completely homogenized the results obtained are illustrated by numerical simulations

a filtration of the loops on su n by schubert varieties il y a une filtration f n k de l espace su n qui realise la filtration r n k en homologie de plus il y a un diagramme commutatif entre su n bu bu k f n k par des applications j et h usuelles et le quotient f n k de la filtration est homeomorphe au complexe de thorn de u 1 k sur f u 1 k

non uniqueness results for entropy two phase solutions of forward backward parabolic problems with unstable phase abstract this paper studies the well posedness of the entropy formulation given by plotnikov 1994 in 23 for forward backward parabolic problem obtained as singular limit of a proper pseudoparabolic approximation it was proved in mascia et al 2009 20 that such a formulation gives uniqueness when the solution takes values in the stable phases here we consider the situation in which unstable phase is taken in account proving that in general uniqueness does not hold

the unsaturated flow in porous media with dynamic capillary pressure abstract in this paper we consider a degenerate pseudoparabolic equation for the wetting saturation of an unsaturated two phase flow in porous media with dynamic capillary pressure saturation relationship where the relaxation parameter depends on the saturation following the approach given in 13 the existence of a weak solution is proved using galerkin approximation and regularization techniques a priori estimates needed for passing to the limit when the regularization parameter goes to zero are obtained by using appropriate test functions motivated by the fact that considered pde allows a natural generalization of the classical kullback entropy finally a special care was given in obtaining an estimate of the mixed derivative term by combining the information from the capillary pressure with the obtained a priori estimates on the saturation

two phase porous media flows with dynamic capillary effects and hysteresis in this paper we obtain the uniqueness of weak solutions for a two phase flow model in a porous medium a particularity of the model is that the dynamic effects and hysteresis are included in the capillary pressure to prove the uniqueness we define an auxiliary elliptic system

numerical investigations of two phase flow with dynamic capillary pressure in porous media via a moving mesh method abstract motivated by observations of saturation overshoot this paper investigates numerical modeling of two phase flow in porous media incorporating dynamic capillary pressure the effects of the dynamic capillary coefficient the infiltrating flux rate and the initial and boundary values are systematically studied using a traveling wave ansatz and efficient numerical methods the traveling wave solutions may exhibit monotonic non monotonic or plateau shaped behavior special attention is paid to the non monotonic profiles the traveling wave results are confirmed by numerically solving the partial differential equation using an accurate adaptive moving mesh solver comparisons between the computed solutions using the brooks corey model and the laboratory measurements of saturation overshoot verify the effectiveness of our approach

global solution and blow up for a class of pseudo p laplacian evolution equations with logarithmic nonlinearity the main goal of this work is to study an initialboundary value problem for a nonlinear pseudoparabolic equation with logarithmic nonlinearity by using the potential well method and a logarithmic sobolev inequality we obtain results of existence or nonexistence of global weak solutions in addition we also provide sufficient conditions for the large time decay of global weak solutions and the finite time blow up of weak solutions

error estimates for a mixed finite element discretization of a two phase porous media flow model with dynamic capillarity abstract we analyze a fully discrete numerical scheme for the model describing two phase immiscible flow in porous media with dynamic effects in the capillary pressure we employ the euler implicit method for the time discretization the spatial discretization is based on the mixed finite element method mfem specifically the lowest order raviart thomas elements are applied in this paper the error estimates for the saturation fluxes and phase pressures in l 0 t l 2 are derived for the temporal and spatial discretization to show the convergence of the scheme finally we present some numerical results to support the theoretical findings

effect of aggregation on population recovery modeled by a forward backward pseudoparabolic equation in this paper we study the equation u t u f u u t f u in a bounded domain of r d d 1 with homogeneous boundary conditions of the neumann type as a model of aggregating population with a migration rate determined by and total birth and mortality rates characterized by f we will show that the aggregating mechanism induced by u allows the survival of a species in danger of extinction numerical simulations suggest that the solutions stabilize asymptotically in time to a not necessarily homogeneous stationary solution this is shown to be the case for a particular version of the function u

nonuniqueness of solutions for a class of forward backward parabolic equations abstract we study the initial boundary value problem u t u x x e u t x x in 0 u e u t 0 in 0 u u 0 0 in 0 with measure valued initial data here is a bounded open interval 0 0 is increasing in 0 and decreasing in and the regularising term is increasing but bounded it is natural to study measure valued solutions since singularities may appear spontaneously in finite time nonnegative radon measure valued solutions are known to exist and their construction is based on an approximation procedure until now nothing was known about their uniqueness in this note we construct some nontrivial examples of solutions which do not satisfy all properties of the constructed solutions whence uniqueness fails in addition we classify the steady state solutions

constructive field theory and applications perspectives and open problems in this paper we review many interesting open problems in mathematical physics which may be attacked with the help of tools from constructive field theory they could give work for future mathematical physicists trained with constructive methods well into the 21st century

analysis of an interior penalty discontinuous galerkin scheme for two phase flow in porous media with dynamic capillary effects we present an interior penalty discontinuous galerkin scheme for a two phase porous media flow model that incorporates dynamic effects in the capillary pressure the approximation of the mass conservation laws is performed in their original formulation without introducing a global pressure we prove the existence of a solution to the emerging fully discrete systems and the convergence of the scheme error estimates are obtained for sufficiently smooth data

degenerate regularization of forward backward parabolic equations the regularized problem we study a quasilinear parabolic equation of forward backward type in one space dimension under assumptions on the nonlinearity which hold for a number of important mathematical models for example the one dimensional perona malik equation using a degenerate pseudoparabolic regularization proposed in barenblatt et al siam j math anal 24 1414 1439 1993 which takes time delay effects into account we prove existence and uniqueness of positive solutions of the regularized problem in a space of radon measures we also study qualitative properties of such solutions in particular concerning their decomposition into an absolutely continuous part and a singular part with respect to the lebesgue measure in this respect the existence of a family of viscous entropy inequalities plays an important role

qualitative analysis of a nonlinear viscous diffusion equation abstract we study the cauchy problem of multidimensional viscous diffusion equations by using an equivalent integral equation we first give the existence of local solutions then under some assumptions we obtain the global existence of solutions

the characterization of reaction convection diffusion processes by travelling waves it has long been known that the heat equation displays infinite speed of propagation this is to say that if the initial data are nonnegative and have nonempty compact support the solution of an initial value problem is positive everywhere after any infinitesimal time however since the nineteen fifties it has also been known that certain nonlinear diffusion equations of degenerate parabolic type do not display this phenomenon for these equations the generalized solution of an initial value problem with compactly supported initial data will have bounded support with respect to the spatial variable at all times in this paper the necessary and sufficient criterion for finite speed of propagation for the general nonlinear reaction convection diffusion equation formula is determined the assumptions on the coefficientsa bandcare such that the classification unifies and generalizes previously known results the technique employed is comparision of an arbitrary solution of the equation with suitably constructed travelling wave solutions and subsolutions basically the central conclusion is that the equation exhibits finite speed of propagation if and only if it admits a travelling wave solution with bounded support concurrently the search for a travelling wave solution with bounded support can be reduced to the study of a singular nonlinear integral equation whose solution must satisfy a certain constraint

passage to the limit over small parameters in the viscous cahn hilliard equation abstract we study singular passage to the limit over different small parameters for the viscous cahn hilliard equation under weak growth assumptions on the nonlinearity a rigorous proof of convergence to solutions of either the cahn hilliard equation or of the allen cahn equation or of the sobolev equation depending on the choice of the parameter is provided we also study the singular limit of the cahn hilliard equation as the parameter in the fourth order term goes to zero in particular we show that a radon measure valued solution of the limiting ill posed problem can arise depending on the behavior of the nonlinearity at infinity

on convergence of a discrete problem describing transport processes in the pressing section of a paper machine including dynamic capillary effects one dimensional case this work presents a proof of convergence of a discrete solution to a continuous one at first the continuous problem is stated as a system of equations which describe the filtration process in the pressing section of a paper machine two flow regimes appear in the modeling of this problem the model for the saturated flow is presented by the darcy s law and the mass conservation the second regime is described by the richards approach together with a dynamic capillary pressure model the finite volume method is used to approximate the system of pdes then the existence of a discrete solution to the proposed finite difference scheme is proven compactness of the set of all discrete solutions for different mesh sizes is proven the main theorem shows that the discrete solution converges to the solution of the continuous problem at the end we present numerical studies for the rate of convergence

dynamics as a mechanism preventing the formation of finer and finer microstructure we study the dynamics of pattern formation in the one dimensional partial differential equation u u w u x x u xxt u 0 text u u x t text x in 0 1 text t 0 proposed recently by ball holmes james pego swart bhjps as a mathematical cartoon for the dynamic formation of microstructures observed in various crystalline solids here w is a double well potential like 1 4 ux 2 1 2 what makes this equation interesting and unusual is that it possesses as a lyapunov function a free energy consisting of kinetic energy plus a nonconvex elastic energy but no interfacial energy contribution which does not attain a minimum but favours the formation of finer and finer phase mixtures e u u t int limits 0 1 frac u t 2 2 w u x frac u 2 2 dx our analysis of the dynamics confirms the following surprising and striking difference between statics and dynamics conjectured in bhjps on the basis of numerical simulations of swart holmes sh while minimizing the above energy predicts infinitely fine patterns mathematically weak but not strong convergence of all minimizing sequences unvn of e u v in the sobolev space w1p 0 1 l2 0 1 solutions to the evolution equation of ball et al typically develop patterns of small but finite length scale mathematically strong convergence in w1p 0 1 l2 0 1 of all solutions u t ut t with low initial energy as time t

numerical schemes for a pseudo parabolic burgers equation discontinuous data and long time behaviour we consider a simplified model for vertical non stationary groundwater flow which includes dynamic capillary pressure effects specifically we consider a viscous burgers type equation that is extended with a third order term containing mixed derivatives in space and time we analyse the one dimensional boundary value problem and investigate numerically its long time behaviour the numerical schemes discussed here take into account possible discontinuities of the solution

localization for some continuous random hamiltonians in d dimensions abstract we prove the existence with probability one of an interval of pure point spectrum for some families of continuous random schrodinger operators in d dimensions for anderson like models with positive short range single site potentials we also prove that the corresponding eigenfunctions decay exponentially and that the integrated density of states is lipschitz continuous for the other families of random potentials that we study we show that the corresponding eigenfunctions decay faster than an inverse power of x which depends upon the decay rate of the single site potential to obtain these results we develop an extension of the classical aronszajn donoghue theory for a class of relatively compact perturbations and a spectral averaging method which extends kotani s trick to these more general families of operators

sobolev regularization of a class of forward backward parabolic equations we address existence and asymptotic behaviour for large time of young measure solutions of the dirichlet initial boundary value problem for the equation ut u where the function need not satisfy monotonicity conditions under suitable growth conditions on these solutions are obtained by a vanishing viscosity method from solutions of the corresponding problem for the regularized equation ut u ut the asymptotic behaviour as t of young measure solutions of the original problem is studied by limit set techniques relying on the tightness of sequences of time translates of the limiting young measure when n 1 this measure is characterized as a linear combination of dirac measures with support on the branches of the graph of

existence of infinitely many solutions for a forward backward heat equation let 4 be a piecewise linear function which satisfies the condition s40 s 2 cs2 c 0 s e r and which is monotone decreasing on an interval a b c r it is shown that for f e c2 0 11 with max f a there exists a t 0 such that the initial boundary value problem ut 00 x x ux os t u ls t u 0 f has infinitely many solutions u satisfying 11ul u x u x u 1l 2 6 c f 4 on 0 1i x 0 t 0 introduction consider the initial boundary value problem ut u x x t e 0 1 x 0 t 1 u t uj l t 0 t e 0 t u x 0 f x x e 0 1 if 4 is strictly monotone increasing with 4 c 0 1 has a unique solution which is roughly speaking as smooth as the function o on the other hand if 4 c 0 this assumption allows 4 to have monotone decreasing parts e g the model cubic et s 1s3 s2 2s 4 s 3 42 2s a natural and interesting question is whether problem 1 has a solution if the usual assumption 4 0 is replaced by the weaker coercivity condition c under this hypothesis j bona j nohel and l wahlbin bnw hn obtained several a received by the editors april 2 1982 and in revised form may 14 1982 1980 mathematics subject classification primary 35k55 35k65

variational mesh adaptation ii error estimates and monitor functions the key to the success of a variational mesh adaptation method is to define a proper monitor function which controls mesh adaptation in this paper we study the choice of the monitor function for the variational adaptive mesh method developed in the previous work j comput phys 174 2001 924 two types of monitor functions scalar matrix and non scalar matrix ones are defined based on asymptotic estimates of interpolation error obtained using the interpolation theory of finite element methods the choice of the adaptation intensity parameter is also discussed for each of these monitor functions asymptotic bounds on interpolation error are obtained for adaptive meshes that satisfy the regularity and equidistribution conditions two dimensional numerical results are given to verify the theoretical findings

equivalent formulations and numerical schemes for a class of pseudo parabolic equations this paper investigates three different formulations for a class of pseudo parabolic equations such equations are encountered for example as a model for two phase porous media flows when dynamic effects in the capillary pressure are included we first show the equivalence of the three different formulations for the original equation on the basis of this we further investigate the corresponding discretization in time and give some numerical examples

on the well posedness of a spectral fractional forward backward pseudo parabolic equation abstract we prove existence uniqueness and stability of solution of a forward backward pseudo parabolic equation with spectral fractional laplacian then we define and prove existence of young measure valued solution for fractional forward backward problem by the vanishing viscosity method

degenerate two phase porous media flow model with dynamic capillarity abstract in this paper we investigate a degenerate elliptic parabolic system which describes the flow of two incompressible immiscible fluids in porous media and including dynamic effects in the phase pressure difference first for the regularized case the existence and uniqueness of the weak solution are obtained then we let the regularization parameter go to zero to show the existence of weak solutions under degenerate case

on a pseudoparabolic regularization of a forward backward forward equation abstract we consider an initial boundary value problem for a degenerate pseudoparabolic regularization of a nonlinear forward backward forward parabolic equation with a bounded nonlinearity which is increasing at infinity we prove existence of suitably defined nonnegative solutions of the problem in a space of radon measures solutions satisfy several monotonicity and regularization properties in particular their singular part is nonincreasing and may disappear in finite time the problem is of intrinsic mathematical interest but also arises naturally when studying by time reversal the spontaneous appearance of singularities in a specific application

localization lengths and boltzmann limit for the anderson model at small disorders in dimension 3 we prove lower bounds on the localization length of eigenfunctions in the three dimensional anderson model at weak disorders our results are similar to those obtained by schlag shubin and wolff j anal math 88 2002 for dimensions one and two we prove that with probability one most eigenfunctions have localization lengths bounded from below by o left lambda 2 over log 1 over lambda right where is the disorder strength this is achieved by time dependent methods which generalize those developed by erdos and yau commun pure appl math liii 667 753 2003 to the lattice and non gaussian case in addition we show that the macroscopic limit of the corresponding lattice random schrodinger dynamics is governed by a linear boltzmann equation

similarity solutions of the porous medium equation with sign changes with m 1 here u is a function of x and t for nonnegative u eq 1 1 arises in the theory of gas flow through a one dimensional porous medium and an extensive theory has been developed over the last decades see e g l 3 151 more recently 4 5 12 131 applied and pure mathematicians have also become interested in solutions of 1 1 with sign changes the most striking property of 1 1 is that solutions may have compact support whose free boundaries or interfaces move outwards with a finite possibly zero speed proportional to the one sided x derivative of 1 1 at the free boundary taken from the interior of the support this is usually called the interface condition the expression between brackets in l l i e 1 1 ux l m iu u which we shall think of as theflux because of the divergence form of l l is continuous with respect to x in zeros of u if it is nonzero we shall call this the sign change condition we observe that for u bounded away from zero and infinity eq 1 1 is uniformly parabolic and local smoothness follows from standard regularity theory for the development of the theory of 1 1 similarity solutions play an important role essentially these are solutions whose profiles remain the same as t varies in the nonnegative case they were studied extensively in 7 91 the most familiar of them are of course the burenblatt pattle solutions given by

convergence analysis of p1 finite element method for free boundary problems on non overlapping subdomains abstract the paper applies a conforming p1 finite element method to general variational inequalities derived from free boundary problems the domain of the free boundary problem can be properly split into two non overlapping subdomains where the free boundary is located in only one subdomain the variational inequality is reduced to a partial differential equation in the subdomain which does not contain the free boundary but still keeps its original form in the other subdomain therefore the original variational inequality can be discretized separately with p1 finite element in different subdomains a non overlapping domain decomposition method is introduced to solve these two discretized sub problems by p1 finite element method iteratively while a robin type boundary condition is utilized for the data transfer on the common boundary we show that the sequence of such finite element solutions converges to the discretized solution of the original problem application to a free seepage problem verifies the theory

on a class of forward backward parabolic equations formation of singularities abstract we study the formation of singularities for the problem u t u x x e u t x x in 0 t u e u t 0 in 0 t u u 0 0 in 0 where and t are positive constants a bounded interval u 0 a nonnegative radon measure on a nonmonotone and nonnegative function with 0 0 and an increasing bounded function we show that if u 0 is a bounded or continuous function singularities may appear spontaneously the class of singularities which can arise in finite time is remarkably large and includes infinitely many dirac masses and singular continuous measures

degenerate regularization of forward backward parabolic equations the vanishing viscosity limit we address the vanishing viscosity limit of the regularized problem studied in smarrazzo and tesei arch rat mech anal 2012 in press we show that the limiting points in a suitable topology of the family of solutions of the regularized problem can be regarded as suitably defined weak measure valued solutions of the original problem in general these solutions are the sum of a regular term which is absolutely continuous with respect to the lebesgue measure and a singular term which is a radon measure singular with respect to the other by using a family of entropy inequalities we prove that the singular term is nondecreasing in time we also characterize the disintegration of the young measure associated with the regular term proving that it is a superposition of two dirac masses with support on the branches of the graph of the nonlinearity varphi

asymptotic behaviour and changes of phase in one dimensional nonlinear viscoelasticity in this paper we study the asymptotic behaviour as t co of solutions u x t to the nonlinear partial differential equation ut m4 a o o 1 1 with initial conditions u 0 q x u x u x o 0 1 3a

universal scaling of spontaneous imbibition for water wet systems 1 spontaneous counter current imbibition si is a key mechanism in many multiphase flow processes such as cleanup of nonaqueous phase liquids napls bioremediation or co2 storage for interpreting and upscaling laboratory si data and modeling and prediction purposes scaling groups are an essential tool the question of how to formulate a general scaling group has been debated for over 90 years here we propose the first scaling group that incorporates the influence of all parameters on si that are present in the two phase darcy model the group is derived rigorously from the only known exact analytical solution for spontaneous imbibition by relating the cumulative water phase imbibed to the normalized pore volume we show the validity of the group by applying it to 42 published si studies for water oil and water air experiments for a wide range of viscosity ratios different materials different initial water saturations and different length scales in all cases water was the wetting phase our group serves as a master equation whose generality allows the rigorous prediction of the validity of a large number of specialized scaling groups proposed during the last 90 years furthermore our results give strong evidence that the darcy model is suitable for describing si and that including dynamic effects in capillary pressure is not necessary for counter current si contrary to what has been hypothesized two key applications of the group are discussed first the group can serve as the long sought after general transfer rate for imbibition used in dual porosity models second it is the so far missing proportionality constant in imbibition germination models for plant seeds