Asymptotic behaviour of a phase-field model with three coupled equations without uniqueness

We prove the existence of weak solutions for a phase-field model with three coupled equations with unknown uniqueness, and state several dynamical systems depending on the regularity of the initial data. Then, the existence of families of global attractors (level-set depending) for the corresponding multi-valued semiflows is established, applying an energy method. Finally, using the regularizing effect of the problem, we prove that these attractors are in fact the same.     

Universal Attractor for a Penrose-Fife System with Special Heat Flux Law

A Penrose-Fife system for non isothermal phase transitions with non conserved order parameter is introduced. A linear growth of the latent heat density with respect to the phase field is allowed. Continous dependence on data and the existence of the universal attractor for the associated nonlinear semigroup are shown. These properties hold with respect to a strong metric accounting for the nonlinear and even singular terms characterizing the system. The present analysis extends a former result by the same authors, holding in the case of a constant latent heat.     

On a nonlocal diffusion model with Neumann boundary conditions

We study a nonlocal diffusion model analogous to heat equation with Neumann boundary conditions. We prove the existence and uniqueness of solutions and a comparison principle. Furthermore, we analyze the asymptotic behavior of the solutions as the temporal variable goes to infinity and the boundary datum depends only on a spacial variable.     

Asymptotic equipartition and long time behavior of solutions of a thin-film equation

We investigate the large-time behavior of classical solutions to the thin-film type equation ut=−x(uuxxx). It was shown in previous work of Carrillo and Toscani that for non-negative initial data u₀ that belongs to H¹(R) and also has a finite mass and second moment, the strong solutions relax in the L¹(R) norm at an explicit rate to the unique self-similar source type solution with the same mass. The equation itself is gradient flow for an energy functional that controls the H¹(R) norm, and so it is natural to expect that one should also have convergence in this norm. Carrillo and Toscani raised this question, but their methods, using a different Lyapunov functions that arises in the theory of the porous medium equation, do not directly address this since their Lyapunov functional does not involve derivatives of u. Here we show that the solutions do indeed converge in the H¹(R) norm at an explicit, but slow, rate. The key to establishing this convergence is an asymptotic equipartition of the excess energy. Roughly speaking, the energy functional whose dissipation drives the evolution through gradient flow consists of two parts: one involving derivatives of u, and one that does not. We show that these must decay at related rates—due to the asymptotic equipartition—and then use the results of Carrillo and Toscani to control the rate for the part that does not depend on derivatives. From this, one gets a rate on the dissipation for all of the excess energy.     

带非局部源的退化半线性抛物方程的解的爆破性质

This paper deals with the blow-up properties of the positive solutions to the nonlocal degenerate semilinear parabolic equation u _t − (x ^a u _x ) _x =∫ ₀ ^a f(u)dx in (0,a) × (0,T) under homogeneous Dirichlet conditions. The local existence and uniqueness of classical solution are established. Under appropriate hypotheses, the global existence and blow-up in finite time of positve solutions are obtained. It is also proved that the blow-up set is almost the whole domain. This differs from the local case. Furthermore, the blow-up rate is precisely determined for the special case: f(u)=u ^p , p>1.     

Existence of a universal attractor for a fully hyperbolic phase-field system

Here we study a nonlinear hyperbolic integrodifferential system which was proposed by H.G. Rotstein et al. to describe certain peculiar phase transition phenomena. This system governs the evolution of the (relative) temperature and the order parameter (or phase-field) . We first consider an initial and boundary value problem associated with the system and we frame it in a history space setting. This is done by introducing two additional variables accounting for the histories of and . Then we show that the reformulated problem generates a dissipative dynamical system in a suitable infinite-dimensional phase space. Finally, we prove the existence of a universal attractor.     

The viscous Cahn-Hilliard equation with inertial term

We consider a hyperbolic relaxation of the viscous Cahn-Hilliard equation. This equation describes the early stages of spinodal decomposition in certain glasses. We establish the existence of families of exponential attractors and inertial manifolds which are continuous at any parameter of viscosity ?≥0. Continuity properties of the global attractors are also examined.     

On some nonlocal evolution equations in Banach spaces

This note is concerned with the initial value problem for the abstract nonlocal equation where A is a maximal monotone operator from a reflexive Banach space E to its dual E^*, while B is a nonlocal maximal monotone operator from . Under proper boundedness and coercivity assumptions on the operators, a solution is achieved by means of a discretization argument. Uniqueness and continuous dependence are also discussed and we prove some estimates for the discretization error. Finally, we deal with the approximation of linear Volterra integrodifferential operators.     

Phase-field systems with nonlinear coupling and dynamic boundary conditions

We consider phase-field systems of Caginalp type on a three-dimensional bounded domain. The order parameter fulfills a dynamic boundary condition, while the (relative) temperature is subject to a homogeneous boundary condition of Dirichlet, Neumann or Robin type. Moreover, the two equations are nonlinearly coupled through a quadratic growth function. Here we extend several results which have been proven by some of the authors for the linear coupling. More precisely, we demonstrate the existence and uniqueness of global solutions. Then we analyze the associated dynamical system and we establish the existence of global as well as exponential attractors. We also discuss the convergence of given solutions to a single equilibrium.     

Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system

In this paper, we investigate the spatial dynamics of a nonlocal and time-delayed reaction-diffusion system, which is motivated by an age-structured population model with distributed maturation delay. The spreading speed c^*, the existence of traveling waves with the wave speed c?c^*, and the nonexistence of traveling waves with c<c^* are obtained. It turns out that the spreading speed coincides with the minimal wave speed for monotone traveling waves.     

Weighted iterative solutions of linear differential equations and heat conduction of ideal gases in moment theory

An iterative method is proposed to find a particular solution of a system of linear differential equations, in the form of a fixed-point problem, with no boundary conditions. To circumvent the unboundedness of differential operators, iterative approximation with gradually decreasing weight is used. Conditions for convergence that can easily be checked in numerical iterations are established. Furthermore, for the numerical iterative scheme, uniqueness and stability theorems are proved. These results are applied to heat conduction of ideal gases in moment theory.     

L -decay rates of planar viscous rarefaction wave for scalar conservation law with degenerate viscosity in n -dimensions

We investigate the Cauchy problem and the initial-boundary value problem for multi-dimensional conservation laws with degenerate viscosity in the whole space and in the half-space respectively. We give the optimal decay estimates in the W^1,p(1≤p≤∞)$W^{1,p}(1\le p\le \infty )$ norm for the perturbation from the planar viscous rarefaction wave. The analysis based on the new L^p$L^p$-energy method and L¹$L^1$-estimates.     

Weak solutions for the fully hyperbolic phase-field system of conserved type

We consider a conserved phase-field system coupling two nonlinear hyperbolic integro-differential equations. The model results from the assumption that the material undergoing phase transition exhibits some thermal memory effects (cf. [15]) and that the response of the order parameter to the variation of the free-energy functional is delayed (cf. [10, 23]). We prove the existence of the solution to the corresponding initial-boundary value problem associated with the resulting PDE system and a (conditioned) continuous dependence estimate of the solution with respect to the data of the problem. This work is partially supported by the Italian Ministero dell’Istruzione, dell’Università e della Ricerca, PRIN no. 2004011204, Project Analisi Matematica nei Problemi Inversi     

On the discretization of a partial differential equation in the neighborhood of a periodic orbit

Sumamry This article is concerned with the comparison of the dynamic of a partial differential equation and its time discretization. We restrict our attention to the neighborhood of a hyperbolic periodic orbit. We show that the discretization possesses an invariant closed curve near the periodic orbit and that the trajectories of the semigroups defined by the partial differential equations and its approximation are close in a sense to be precised provided that different data are allowed. This answers partly an open problem posed in [4]. Examples of application to dissipative partial equations are provided.     

Decay rates to the planar rarefaction waves for a model system of the radiating gas in n dimensions

In this paper, we study the asymptotic decay rates to the planar rarefaction waves to the Cauchy problem for a hyperbolic-elliptic coupled system called as a model system of the radiating gas in Rn (n=3,4,5) if the initial perturbations corresponding to the planar rarefaction waves are sufficiently small in (H²∩L¹∩W2,6) (Rn). The analysis is based on the Lp-energy method and several special interpolation inequalities.     

Subelliptic operators on Lie groups: Variable coefficients

Let G be a Lie group with Lie algebra g and a _i,...,a _d and algebraic basic of g. Futher, if A _i=dL(a_i) are the corresponding generators of left translations by G on one of the usual function spaces over G, let% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbciab-Heaijaab2dadaaeqbqa% aiaadogadaWgaaWcbaqedmvETj2BSbacgmGae4xSdegabeaakiaadg% eadaahaaWcbeqaaiab+f7aHbaaaeaacqGFXoqycaGG6aGaaiiFaiab% +f7aHjaacYhatuuDJXwAK1uy0HMmaeXbfv3ySLgzG0uy0HgiuD3BaG% Wbbiab9rMiekaaikdaaeqaniabggHiLdaaaa!5EC1!\[H{\rm{ = }}\sum\limits_{\alpha :|\alpha | \le 2} {c_\alpha A^\alpha } \] be a second-order differential operator with real bounded coefficients c . The operator is defined to be subelliptic if% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiGacMgacaGGUbGaaiOzamXvP5wqonvsaeHbfv3ySLgzaGqbaKaz% aasacqWF7bWEcqWFTaqlkmaaqafabaGaam4yamaaBaaaleaarmWu51% MyVXgaiyWacqGFXoqyaeqaaaqaaiab+f7aHjaacQdacaGG8bGae4xS% deMaaiiFaiabg2da9iaaikdaaeqaniabggHiLdGccqWFOaakiuGacq% qFNbWzcqWFPaqkcqaH+oaEdaahaaWcbeqaamaaBaaameaacqGFXoqy% aeqaaaaakiaacUdacqqFNbWzcqGHiiIZcqqFhbWrcqqFSaalcqqFGa% aicqaH+oaEcqGHiiIZrqqtubsr4rNCHbachaGaeWxhHe6aaWbaaSqa% beaacqqFKbazcqqFNaWjcqaFaC-jaaGccaGGSaGaaiiFaiabe67a4j% aacYhacqGH9aqpjqgaGeGae8xFa0NccqGH+aGpcaaIWaGaaiOlaaaa% !7884!\[\inf \{ - \sum\limits_{\alpha :|\alpha | = 2} {c_\alpha } (g)\xi ^{_\alpha } ;g \in G, \xi \in ^{d'} ,|\xi | = \} > 0.\]We prove that if the principal coefficients {c ; ||=2} of the subelliptic operator are once left differentiable in the directions a ₁,...,a _d with bounded derivatives, then the operator has a family of semigroup generator extensions on the L _p-spaces with respect to left Haar measure dg, or right Haar measure d, and the corresponding semigroups S are given by a positive integral kernel,% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbaiab-HcaOGqbciab+nfatnaa% BaaaleaacaWG0baabeaaruqqYLwySbacgiGccaqFgpGae8xkaKIae8% hkaGIae43zaCMae8xkaKIae8xpa0Zaa8qeaeaacaqGKbaaleaacqGF% hbWraeqaniabgUIiYdGcceWGObGbaKaacaWGlbWaaSbaaSqaaiaads% haaeqaaOGae8hkaGIae43zaCMae43oaSJae4hAaGMae8xkaKIaa0NX% diab-HcaOiab+HgaOjab-LcaPiab-5caUaaa!5DFA!\[(S_t \phi )(g) = \int_G {\rm{d}} \hat hK_t (g;h)\phi (h).\]The semigroups are holomorphic and the kernel satisfies Gaussian upper bounds. If in addition the coefficients with ||=2 are three times differentiable and those with ||=1 are once differentiable, then the kernel also satisfies Gaussian lower bounds.Some original features of this article are the use of the following: a priori inequalities on L in Section 3, fractional operator expansions for resolvent estimates in Section 4, a parametrix method based on reduction to constant coefficient operators on the Lie group rather than the usual Euclidean space in Section 5, approximation theory of semigroups in Section 11 and time dependent perturbation theory to treat the lower order terms of H in Sections 11 and 12.     

Iterative solution of a singular convection-diffusion perturbation problem

An iterative domain decomposition method is developed to solve a singular perturbation problem. The problem consists of a convection-diffusion equation with a discontinuous (piecewise-constant) diffusion coefficient, and the problem domain is decomposed into two subdomains, on each of which the coefficient is constant. After showing that the boundary value problem is well posed, we indicate a specific numerical implementation of the iterative technique that combines the finite element method on one subdomain with the method of matched asymptotic expansions on the other subdomain. This procedure extends work by Carlenzoli and Quarteroni, which was originally intended for a boundary layer problem with an outer region and an inner region. Our extension carries over to a problem where the domain consists of the outer and inner boundary layer regions plus a region in which the diffusion coefficient is constant and significant in magnitude. An unexpected benefit of our new implementation is its efficiency, which is due to the fact that at each iteration the problem needs to be solved explicitly only on one subdomain. It is only when the final approximation on the entire domain is desired that the matched asymptotic expansions approximation need be computed on the second subdomain. Two-dimensional convergence results and numerical results illustrating the method for a two-dimensional test problem are given.Received: February 12, 2004     

Robust exponential attractors for a phase-field system with memory

H.G. Rotstein et al. proposed a nonconserved phase-field system characterized by the presence of memory terms both in the heat conduction and in the order parameter dynamics. These hereditary effects are represented by time convolution integrals whose relaxation kernels k and h are nonnegative, smooth and decreasing. Rescaling k and h properly, we obtain a system of coupled partial integrodifferential equations depending on two relaxation times ɛ and σ. When ɛ and σ tend to 0, the formal limiting system is the well-known nonconserved phase-field model proposed by G. Caginalp. Assuming the exponential decay of the relaxation kernels, the rescaled system, endowed with homogeneous Neumann boundary conditions, generates a dissipative strongly continuous semigroup S_{ɛ, σ}(t) on a suitable phase space, which accounts for the past histories of the temperature as well as of the order parameter. Our main result consists in proving the existence of a family of exponential attractors for S_{ɛ, σ}(t), with ɛ, σ ∈ [0, 1], whose symmetric Hausdorff distance from tends to 0 in an explicitly controlled way.