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We investigate a diffuse-interface model that describes the dynamics of viscous incompressible two-phase flows with surfactant. The resulting system of partial differential equations consists of a sixth-order Cahn–Hilliard equation for the difference of local concentrations of the binary fluid mixture coupled with a fourth-order Cahn–Hilliard equation for the local concentration of the surfactant. The former has a smooth potential, while the latter has a singular potential. Both equations are coupled with a Navier–Stokes system for the (volume averaged) fluid velocity. The evolution system is endowed with suitable initial conditions, a no-slip boundary condition for the velocity field and homogeneous Neumann boundary conditions for the phase functions as well as for the chemical potentials. We first prove the existence of a global weak solution, which turns out to be unique in two dimensions. Stronger regularity assumptions on the initial data allow us to prove the existence of a unique global (respectively, local) strong solution in two (respectively, three) dimensions. In the two-dimensional case, we derive a continuous dependence estimate with respect to the norms controlled by the total energy. Moreover, we establish instantaneous regularization properties of global weak solutions for [Formula: see text]. In particular, we show that the surfactant concentration stays uniformly away from the pure states 0 and 1 after some positive time.

A phase field model which describes the formation of protein-RNA complexes subject to phase segregation is analyzed. A single protein, two RNA species, and two complexes are considered. Protein and RNA species are governed by coupled reaction-diffusion equations which also depend on the two complexes. The latter ones are driven by two Cahn-Hilliard equations with Flory-Huggins potential and reaction terms depending on the solution variables. The resulting nonlinear coupled system is equipped with no-flux boundary conditions and suitable initial conditions. The former ones entail some mass conservation constraints which are also due to the nature of the reaction terms. The existence of global weak solutions in a bounded (two- or) three-dimensional domain is established. In dimension two, some weighted-in-time regularity properties are shown. In particular, the complexes instantaneously get uniformly separated from the pure phases. Taking advantage of this result, a unique continuation property is proven. Among the many technical difficulties, the most significant one arises from the fact that the two complexes are initially nonexistent, so their initial conditions are zero i.e., they start from a pure phase. Thus we must solve, in particular, a system of two coupled Cahn-Hilliard equations with singular potential and nonlinear sources without the usual assumption on the initial datum, i.e., the initial phase cannot be pure. This novelty requires a new approach to estimate the chemical potential in a suitable $L^p(L^2)$-space with $p\in(1,2)$. This technique can be extended to other models like, for instance, the well-known Cahn-Hilliard-Oono equation., 56 pages

We consider a system which consists of a Cahn-Hilliard equation coupled with a Cahn-Hilliard-Oono equation in a bounded domain of $\mathbb{R}^d$, $d = 2, 3$. This system accounts for macrophase and microphase separation in a polymer mixture through two order parameters $u$ and $v$. The free energy of this system is a bivariate interaction potential which contains the mixing entropy of the two order parameters and suitable coupling terms. The equations are endowed with initial conditions and homogeneous Neumann boundary conditions both for $u,v$ and for the corresponding chemical potentials. We first prove that the resulting problem is well posed in a weak sense. Then, in the conserved case, we establish that the weak solution regularizes instantaneously. Furthermore, in two spatial dimensions, we show the strict separation property for $u$ and $v$, namely, they both stay uniformly away from the pure phases $\pm 1$ in finite time. Finally, we investigate the long-time behavior of a finite energy solution showing, in particular, that it converges to a single stationary state., Comment: to appear in Nonlinear Analysis: Real World Applications

We study a Cahn-Hilliard-Hele-Shaw (or Cahn-Hilliard-Darcy) system for an incompressible mixture of two fluids. The relative concentration difference $\varphi$ is governed by a convective nonlocal Cahn-Hilliard equation with degenerate mobility and logarithmic potential. The volume averaged fluid velocity $\mathbf{u}$ obeys a Darcy's law depending on the so-called Korteweg force $\mu\nabla \varphi$, where $\mu$ is the nonlocal chemical potential. In addition, the kinematic viscosity $\eta$ may depend on $\varphi$. We establish first the existence of a global weak solution which satisfies the energy identity. Then we prove the existence of a strong solution. Further regularity results on the pressure and on $\mathbf{u}$ are also obtained. Weak-strong uniqueness is demonstrated in the two dimensional case. In the three-dimensional case, uniqueness of weak solutions holds if $\eta$ is constant. Otherwise, weak-strong uniqueness is shown by assuming that the pressure of the strong solution is $\alpha$-H\"{o}lder continuous in space for $\alpha\in (1/5,1)$., Comment: 70 pages

We consider a nonlinear system which consists of the incompressible Navier–Stokes equations coupled with a convective nonlocal Cahn–Hilliard equation. This is a diffuse interface model which describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluids having the same density. We assume that both the viscosity and mobility functions depend smoothly on the order parameter. Moreover, we assume that the mobility degenerates at the pure phases and that the potential is singular (e.g. of logarithmic type). This system is endowed with a no-slip boundary condition for the (average) velocity and a homogeneous Neumann boundary condition for the chemical potential. Thus the total mass is conserved. In the two-dimensional case, this problem was already analyzed in some joint papers of the first three authors. However, in the present general case, only the existence of a global weak solution, the (conditional) weak–strong uniqueness and the existence of the global attractor were proven. Here we are able to establish the existence of a (unique) strong solution through an approximation procedure based on time discretization. As a consequence, we can prove suitable uniform estimates which allow us to show some smoothness of the global attractor. Finally, we discuss the existence of strong solutions for the convective nonlocal Cahn–Hilliard equation, with a given velocity field, in the three dimensional case as well.

In this paper, we consider a two-dimensional diffuse interface model for the phase separation of an incompressible and isothermal binary fluid mixture with matched densities. This model consists of the Navier–Stokes equations, nonlinearly coupled with a convective nonlocal Cahn–Hilliard equation. The system rules the evolution of the (volume-averaged) velocity $$\varvec{u}$$ of the mixture and the (relative) concentration difference $$\varphi $$ of the two phases. The aim of this work is to study an optimal control problem for such a system, the control being a time-dependent external force $$\varvec{v}$$ acting on the fluid. We first prove the existence of an optimal control for a given tracking type cost functional. Then we study the differentiability properties of the control-to-state map $$\varvec{v}\mapsto [\varvec{u},\varphi ]$$, and we establish first-order necessary optimality conditions. These results generalize the ones obtained by the first and the third authors jointly with Rocca (SIAM J Control Optim 54:221–250, 2016). There the authors assumed a constant mobility and a regular potential with polynomially controlled growth. Here, we analyze the physically more relevant case of a degenerate mobility and a singular (e.g., logarithmic) potential. This is made possible by the existence of a unique strong solution which was recently proved by the authors and Gal (WIAS preprint series No. 2309, Berlin, 2016).

We study a diffuse interface model of the two-component Hele–Shaw flow. This is an advective Cahn–Hilliard equation for the relative concentration , where the incompressible velocity field is determined by the Darcy's law depending on the Korteweg force . Here μ is the derivative of a nonlocal non-convex free energy characterized by a logarithmic potential. The system is subject to no-flux boundary conditions for and μ along with an initial condition for . First of all, we prove that the corresponding problem is globally well posed with respect to a natural notion of weak solution. Also, we establish the existence of global strong solutions. In dimension two, we show the validity of the so-called instantaneous separation property. This means that any solution, which is not a pure phase initially, stays away from the pure phases, uniformly with respect to the initial energy and total mass. Finally we prove the existence of the global attractor for the corresponding dynamical system as well as the convergence to a single equilibrium of any weak solution in the two-dimensional case.

We consider a diffuse interface model for the phase separation of an incompressible and isothermal non-Newtonian binary fluid mixture in three dimensions. The averaged velocity u is governed by a Navier–Stokes system with a shear dependent viscosity controlled by a power p > 2 . This system is nonlinearly coupled through the Korteweg force with a convective nonlocal Cahn–Hilliard equation for the order parameter φ, that is, the (relative) concentration difference of the two components. The resulting equations are endowed with the no-slip boundary condition for u and the no-flux boundary condition for the chemical potential μ. The latter variable is the functional derivative of a nonlocal and nonconvex Ginzburg–Landau type functional which accounts for the presence of two phases. We first prove the existence of a weak solution in the case p ≥ 11 / 5 . Then we extend some previous results on time regularity and uniqueness if p > 11 / 5 .

We consider the nonlocal Cahn–Hilliard equation with singular potential and constant mobility. Well-posedness and regularity of weak solutions are studied. Then we establish the validity of the strict separation property in dimension two. Further regularity results as well as the existence of regular finite dimensional attractors and the convergence of a weak solution to a single equilibrium are also provided. Finally, regularity results and the strict separation property are also proven for the two-dimensional nonlocal Cahn–Hilliard–Navier–Stokes system with singular potential.

We consider the so-called Cahn–Hilliard–Oono equation with singular (e.g. logarithmic) potential in a bounded domain of [Formula: see text], [Formula: see text]. The equation is subject to an initial condition and Neumann homogeneous boundary conditions for the order parameter as well as for the chemical potential. However, contrary to the Cahn–Hilliard equation, the total mass might not be conserved. The existence of a global finite energy solution to such a problem was proven by Miranville and Temam. We first establish some regularization properties in finite time of the (unique) solution. Then, in dimension two, we prove the so-called strict separation property, namely, we show that any finite energy solution stays away from pure phases, uniformly with respect to the initial energy and the total mass. Taking advantage of these results, we study the long-time behavior of solutions. More precisely, we establish the existence of the global attractor in both two and three dimensions. Due to the strict separation property in dimension two, we also prove the existence of exponential attractors and we show that a finite energy solution always converges to a single equilibrium even though the mass is not conserved.

We propose a space semi-discrete and a fully discrete finite element scheme for the modified phase field crystal equation (MPFC). The space discretization is based on a splitting method and consists in a Galerkin approximation which allows low order (piecewise linear) finite elements. The time discretization is a second-order scheme which has been introduced by Gomez and Hughes for the Cahn-Hilliard equation. The fully discrete scheme is shown to be unconditionally energy stable and uniquely solvable for small time steps, with a smallness condition independent of the space step. Using energy estimates, we prove that in both cases, the discrete solution converges to the unique energy solution of the MPFC equation as the discretization parameters tend to 0. Using a Lojasiewicz inequality, we also establish that the discrete solution tends to a stationary solution as time goes to infinity. Numerical simulations illustrate the theoretical results.

We consider a diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids. This model consists of the Navier–Stokes equations coupled with a convective nonlocal Cahn–Hilliard equation. Several results were already proven by two of the present authors. However, in the two-dimensional case, the uniqueness of weak solutions was still open. Here we establish such a result even in the case of degenerate mobility and singular potential. Moreover, we show the weak–strong uniqueness in the case of viscosity depending on the order parameter, provided that either the mobility is constant and the potential is regular or the mobility is degenerate and the potential is singular. In the case of constant viscosity, on account of the uniqueness results, we can deduce the connectedness of the global attractor whose existence was obtained in a previous paper. The uniqueness technique can be adapted to show the validity of a smoothing property for the difference of two trajectories which is crucial to establish the existence of an exponential attractor. The latter is established even in the case of variable viscosity, constant mobility and regular potential.

We devise a rst-order in time convex splitting scheme for a nonlocal Cahn HilliardOono type equation with a transport term and subject to homogeneous Neu-mann boundary conditions. The presence of the transport term is not a minor modi-cation, since, for instance, we lose the unconditional unique solvability and stability. However, we prove the stability of our scheme when the time step is suciently small. Furthermore, we prove the consistency of this scheme and the convergence to the exact solution. Finally, we give some numerical simulations which conrm our theoretical results and demonstrate the performance of our scheme not only for phase separation, but also for crystal nucleation, for several choices of the interaction kernel.

The Cahn–Hilliard–Hele–Shaw system is a fundamental diffuse-interface model for an incompressible binary fluid confined in a Hele–Shaw cell. It consists of a convective Cahn–Hilliard equation in which the velocity u is subject to a Korteweg force through Darcy's equation. In this paper, we aim to investigate the system with a physically relevant potential (i.e., of logarithmic type). This choice ensures that the (relative) concentration difference φ takes values within the admissible range. To the best of our knowledge, essentially all the available contributions in the literature are concerned with a regular approximation of the singular potential. Here we first prove the existence of a global weak solution with finite energy that satisfies an energy dissipative property. Then, in dimension two, we further obtain the uniqueness and regularity of global weak solutions. In particular, we show that any two-dimensional weak solution satisfies the so-called strict separation property, namely, if φ is not a pure state at some initial time, then it stays instantaneously away from the pure states. When the spatial dimension is three, we prove the existence of a unique global strong solution, provided that the initial datum is regular enough and sufficiently close to any local minimizer of the free energy. This also yields the local Lyapunov stability of the local minimizer itself. Finally, we prove that under suitable assumptions any global solution converges to a single equilibrium as time goes to infinity.

We consider the nonlocal Cahn-Hilliard equation with singular potential and degenerate mobility in a bounded domain Ω ⊂ R d , d ≤ 3 . We first prove the existence of maximal strong solutions in weighted (in time) L p spaces. Then we establish further regularity properties of the solution through maximal regularity theory. Finally, we revisit the separation property in an appendix.

In this note, we want to highlight and correct an error in the paper "On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems" [Comm. Pure Appl. Anal. 15 (2016), 299-317] written by the authors.

We consider the modified phase-field crystal (MPFC) equation that has recently been proposed by P. Stefanovic et al. This is a variant of the phase-field crystal (PFC) equation, introduced by K.-R. Elder et al., which is characterized by the presence of an inertial term $\beta\phi_{tt}$. Here $\phi$ is the phase function standing for the number density of atoms and $\beta\geq 0$ is a relaxation time. The associated dynamical system for the MPFC equation with respect to the parameter $\beta$ is analyzed. More precisely, we establish the existence of a family of exponential attractors $\mathcal{M}_\beta$ that are H\"older continuous with respect to $\beta$., Comment: arXiv admin note: text overlap with arXiv:1306.5857

We consider a diffuse interface model for phase separation of an isothermal incompressible binary fluid in a Brinkman porous medium. The coupled system consists of a convective Cahn-Hilliard equation for the phase field $\phi$, i.e., the difference of the (relative) concentrations of the two phases, coupled with a modified Darcy equation proposed by H.C. Brinkman in 1947 for the fluid velocity $\mathbf{u}$. This equation incorporates a diffuse interface surface force proportional to $\phi \nabla \mu$, where $\mu$ is the so-called chemical potential. We analyze the well-posedness of the resulting Cahn-Hilliard-Brinkman (CHB) system for $(\phi,\mathbf{u})$. Then we establish the existence of a global attractor and the convergence of a given (weak) solution to a single equilibrium via {\L}ojasiewicz-Simon inequality. Furthermore, we study the behavior of the solutions as the viscosity goes to zero, that is, when the CHB system approaches the Cahn-Hilliard-Hele-Shaw (CHHS) system. We first prove the existence of a weak solution to the CHHS system as limit of CHB solutions. Then, in dimension two, we estimate the difference of the solutions to CHB and CHHS systems in terms of the viscosity constant appearing in CHB.

We consider an evolution system describing the phenomenon of marble sulphation of a monument, accounting of the surface rugosity. We first prove a local in time well posedness result. Then, stronger assumptions on the data allow us to establish the existence of a global in time solution. Finally, we perform some numerical simulations that illustrate the main feature of the proposed model.

We consider a modification of the so-called phase-field crystal (PFC) equation introduced by K. R. Elder et al. This variant has recently been proposed by P. Stefanovic et al. to distinguish between elastic relaxation and diffusion time scales. It consists of adding an inertial term (i.e. a second-order time derivative) into the PFC equation. The mathematical analysis of the resulting equation is more challenging with respect to the PFC equation, even at the well-posedness level. Moreover, its solutions do not regularize in finite time as in the case of PFC equation. Here we analyze the modified PFC (MPFC) equation endowed with periodic boundary conditions. We first prove the global existence and uniqueness of a solution with initial data in a bounded energy space. This solution satisfies some uniform dissipative estimates which allow us to study the long-time behavior of the corresponding dynamical system. In particular, we establish the existence of the global attractor as well as an exponential attractor. Then we demonstrate that any trajectory originating from the bounded energy phase space converges to a single equilibrium. This is done by means of a suitable version of the Łojasiewicz–Simon inequality. An estimate on the convergence rate is also given.

A well-known diffuse interface model for incompressible isothermal mixtures of two immiscible fluids consists of the Navier–Stokes system coupled with a convective Cahn–Hilliard equation. In some recent contributions the standard Cahn–Hilliard equation has been replaced by its nonlocal version. The corresponding system is physically more relevant and mathematically more challenging. Indeed, the only known results are essentially the existence of a global weak solution and the existence of a suitable notion of global attractor for the corresponding dynamical system defined without uniqueness. In fact, even in the two-dimensional case, uniqueness of weak solutions is still an open problem. Here we take a step forward in the case of regular potentials. First we prove the existence of a (unique) strong solution in two dimensions. Then we show that any weak solution regularizes in finite time uniformly with respect to bounded sets of initial data. This result allows us to deduce that the global attractor is the union of all the bounded complete trajectories which are strong solutions. We also demonstrate that each trajectory converges to a single equilibrium, provided that the potential is real analytic and the external forces vanish.

We consider a phase-field model of Caginalp type where the free energy depends on the order parameter $\chi$ in a nonlocal way. Therefore, the resulting system consists of the energy balance equation coupled with a nonlinear and nonlocal ODE for $\chi$. Such system has been analyzed by several authors, in particular when the configuration potential is a smooth double-well function. More recently, the first author has established the existence of a finite-dimensional global attractor in the case of a potential defined on $(-1,1)$ and singular at the endpoints. Here we examine both the case of regular potentials as well as the case of physically more relevant singular potentials (e.g., logarithmic). We prove well-posedness results and the eventual global boundedness of solutions uniformly with respect to the initial data. In addition, we show that the separation property holds in the case of singular potentials. Thanks to these results, we are able to demonstrate the existence of a finite-dimensional global attractor in the present cases as well.

We consider a modified Cahn-Hiliard equation where the velocity of the order parameter $u$ depends on the past history of $\Delta \mu $, $\mu $ being the chemical potential with an additional viscous term $ \alpha u_{t},$ $\alpha >0.$ In addition, the usual no-flux boundary condition for $u$ is replaced by a nonlinear dynamic boundary condition which accounts for possible interactions with the boundary. The aim of this work is to analyze the passage to the singular limit when the memory kernel collapses into a Dirac mass. In particular, we discuss the convergence of solutions on finite time-intervals and we also establish stability results for global and exponential attractors.

We consider a well-known diffuse interface model for the study of the evolution of an incompressible binary fluid flow in a two or three-dimensional bounded domain. This model consists of a system of two evolution equations, namely, the incompressible Navier-Stokes equations for the average fluid velocity u coupled with a convective Cahn–Hilliard equation for an order parameter $$\phi $$ . The novelty is that the system is endowed with boundary conditions which account for a moving contact line slip velocity. The existence of a suitable global energy solution is proven and the convergence of any such solution to a single equilibrium is also established.

The phase separation of an isothermal incompressible binary fluid in a porous medium can be described by the so-called Brinkman equation coupled with a convective Cahn-Hilliard (CH) equation. The former governs the average fluid velocity $\mathbf{u}$, while the latter rules evolution of $\varphi$, the difference of the (relative) concentrations of the two phases. The two equations are known as the Cahn-Hilliard-Brinkman (CHB) system. In particular, the Brinkman equation is a Stokes-like equation with a forcing term (Korteweg force) which is proportional to $\mu\nabla\varphi$, where $\mu$ is the chemical potential. When the viscosity vanishes, then the system becomes the Cahn-Hilliard-Hele-Shaw (CHHS) system. Both systems have been studied from the theoretical and the numerical viewpoints. However, theoretical results on the CHHS system are still rather incomplete. For instance, uniqueness of weak solutions is unknown even in 2D. Here we replace the usual CH equation with its physically more relevant nonlocal version. This choice allows us to prove more about the corresponding nonlocal CHHS system. More precisely, we first study well-posedness for the CHB system, endowed with no-slip and no-flux boundary conditions. Then, existence of a weak solution to the CHHS system is obtained as a limit of solutions to the CHB system. Stronger assumptions on the initial datum allow us to prove uniqueness for the CHHS system. Further regularity properties are obtained by assuming additional, though reasonable, assumptions on the interaction kernel. By exploiting these properties, we provide an estimate for the difference between the solution to the CHB system and the one to the CHHS system with respect to viscosity.

Phase-field crystal models are used to describe several pattern formation phenomena like crystallization of liquid, diffusion defects and glass formation. The prototypical equation is obtained as the conserved gradient flow associated with a free-energy functional of Swift–Hohenberg type. Here we consider a variant of the phase-field crystal equation proposed by P. Galenko et al. to account for fast dynamics. This version is characterized by the fact that the gradient flow depends on the past history of the particle density through a memory kernel k e , e > 0 being a relaxation time. Therefore the resulting nonlinear evolution equation is an integro-differential equation of sixth order which is equipped with the physically relevant periodic boundary conditions. We show that this problem generates a dissipative dynamical system on a suitable phase space. Moreover, we prove the existence of a robust family of exponential attractors which is Holder continuous with respect to e.

In this paper we propose and analyze a Discontinuous Galerkin method for a linear parabolic problem with dynamic boundary conditions. We present the formulation and prove stability and optimal a priori error estimates for the fully discrete scheme. More precisely, using polynomials of degree $$p\ge 1$$p?1 on meshes with granularity h along with a backward Euler time-stepping scheme with time-step $$\Delta t$$Δt, we prove that the fully-discrete solution is bounded by the data and it converges, in a suitable (mesh-dependent) energy norm, to the exact solution with optimal order $$h^p + \Delta t$$hp+Δt. The sharpness of the theoretical estimates are verified through several numerical experiments.

The aim of this paper is to study the well-posedness and the existence of global attractors for a family of Cahn–Hilliard equations with a mobility depending on the chemical potential. Such models ...

Two different models for the evolution of incompressible binary fluid mixtures in a three-dimensional bounded domain are considered. They consist of the 3D incompressible Navier-Stokes equations, subject to time-dependent external forces and coupled with either a convective Allen-Cahn or Cahn-Hilliard equation. Such systems can be viewed as generalizations of the Navier-Stokes equations to two-phase fluids. Using the trajectory approach, the authors prove the existence of the trajectory attractor for both systems.

P. Galenko et al. proposed a Cahn–Hilliard model with inertial term in order to model spinodal decomposition caused by deep supercooling in certain glasses. Here we analyze a finite element space semidiscretization of their model, based on a scheme introduced by C. M. Elliott et al. for the Cahn–Hilliard equation. We prove that the semidiscrete solution converges weakly to the continuous solution as the discretization parameter tends to 0. We obtain optimal a priori error estimates in energy norm and related norms, assuming enough regularity on the solution. We also show that the semidiscrete solution converges to an equilibrium as time goes to infinity and we give a simple finite difference version of the scheme.

We consider a singular perturbation of the one-dimensional Cahn–Hilliard equation subject to periodic boundary conditions. We construct a family of exponential attractors \({\{{\mathcal M}_\epsilon\}, \epsilon\geq 0}\) being the perturbation parameter, such that the map \({\epsilon \mapsto {\mathcal M}_\epsilon}\) is Holder continuous. Besides, the continuity at \({\epsilon=0}\) is obtained with respect to a metric independent of \({\epsilon.}\) Continuity properties of global attractors and inertial manifolds are also examined.

We consider a model for the flow of a mixture of two homogeneous and incompressible fluids in a two-dimensional bounded domain. The model consists of a Navier–Stokes equation governing the fluid velocity coupled with a convective Cahn–Hilliard equation for the relative density of atoms of one of the fluids. Endowing the system with suitable boundary and initial conditions, we analyze the asymptotic behavior of its solutions. First, we prove that the initial and boundary value problem generates a strongly continuous semigroup on a suitable phase-space which possesses the global attractor A . Then we establish the existence of an exponential attractors E . Thus A has finite fractal dimension. This dimension is then estimated from above in terms of the physical parameters. Moreover, assuming the potential to be real analytic and in absence of volume forces, we demonstrate that each trajectory converges to a single equilibrium. We also obtain a convergence rate estimate in the phase-space metric.

We study in this paper the well-posedness and the asymptotic behavior, in terms of global attractors, of the Caginalp system with coupled dynamic boundary conditions and possibly singular potentials (e.g., of logarithmic type).

Here we consider a singular perturbation of the Hodgkin–Huxley system which is derived from the Lieberstein's model. We study the associated dynamical system on a suitable bounded phase space, when the perturbation parameter e (i.e., the axon specific inductance) is sufficiently small. We prove the existence of bounded absorbing sets as well as of smooth attracting sets. We deduce the existence of a smooth global attractor A e . Finally we prove the main result, that is, the existence of a family of exponential attractors { E e } which is Holder continuous with respect to e.

We consider a singularly perturbed phase-field model of Caginalp type which is thermally isolated and whose order parameter φ is subject to a dynamic boundary condition. More precisely, we indicate by e a (small) coefficient multiplying ∂tu in the heat equation, u being the temperature, and we construct a family of exponential attractors which is robust as e goes to 0. This is physically meaningful since the limiting problem is the viscous Cahn-Hilliard equation for the sole φ with a dynamic boundary condition. The upper semicontinuity of the global attractor is also analyzed. The paper extends and revisits some results previously obtained by A. Miranville et al. Mathematics Subject Classification (2000). Primary 35B41, 35K55, 37L30; Sec

We analyze a system of nonlinear parabolic equations which describes the evolution of an order parameter f f and the relative temperature v v in a superconducting material occupying a bounded domain Ω ⊂ R n \Omega \subset {\mathbb R}^n , n ≤ 3 n\leq 3 . Here the dependent variable f f is subject to the homogeneous Neumann boundary condition, while v v is equal to a time-dependent Dirichlet datum u ~ \tilde u on the boundary. Therefore, the corresponding dynamical system is nonautonomous. Our main goal is to analyze the asymptotic behavior of its solutions. We first show that the system has a bounded absorbing set and a global attractor which are uniform with respect to a sufficiently general class of u ~ \tilde u . Then, we give sufficient conditions on u ~ \tilde u which ensure the convergence of a given trajectory to a single stationary state and we estimate the convergence rate. Finally, we demonstrate the existence of an exponential attractor of finite fractal dimension for quasi-periodic or stabilizing boundary data u ~ \tilde u .

We consider a model of nonisothermal phase transitions taking place in a bounded spatial region. The order parameter $\psi$ is governed by an Allen-Cahn type equation which is coupled with the equation for the temperature $\theta$. The former is subject to a dynamic boundary condition recently proposed by some physicists to account for interactions with the walls. The latter is endowed with a boundary condition which can be a standard one (Dirichlet, Neumann or Robin) or a dynamic one of Wentzell type. We thus formulate a class of initial and boundary value problems whose local existence and uniqueness is proven by means of a fixed point argument. The local solution becomes global owing to suitable a priori estimates. Then we analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dynamical systems. In particular, we demonstrate the existence of the global attractor as well as of an exponential attractor.

We consider an integrodifferential reaction-diffusion system on a multidimensional spatial domain, subject to homogeneous Neumann boundary conditions. This system finds applications in population dynamics and it is characterized by nonlocal delay terms depending on both the temporal and the spatial variables. The distributed time delay effects are represented by memory kernels which decay exponentially but they are not necessarily monotonically decreasing. We first show how to construct a (dissipative) dynamical system on a suitable phase-space. Then we discuss the existence of the global attractor as well as of an exponential attractor.

We consider a reaction-diffusion system subject to homogeneous Neumann boundary conditions on a given bounded domain. The reaction term depends on the population densities as well as on their past histories in a very general way. This class of systems is widely used in population dynamics modelling. Due to its generality, the longtime behavior of the solutions can display a certain complexity. Here we prove a qualitative result which can be considered as a common denominator of a large family of specific models. More precisely, we demonstrate the existence of an exponential attractor, provided that a bounded invariant region exists and the past history decays exponentially fast. This result will be achieved by means of a suitable adaptation of the l-trajectory method coming back to the seminal paper of Malek and Necas.

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn–Hilliard equation characterized by the presence of an inertial term χtt, χ being the order parameter, which is linearly coupled with an evolution equation for the (relative) temperature ϑ. The latter can be of hyperbolic type if the Cattaneo–Maxwell heat conduction law is assumed. The state variables and the chemical potential are subject to the homogeneous Neumann boundary conditions. We first provide conditions which ensure the well-posedness of the initial and boundary value problem. Then, we prove that the corresponding dynamical system is dissipative and possesses a global attractor. Moreover, assuming that the nonlinear potential is real analytic, we establish that each trajectory converges to a single steady state by using a suitable version of the Łojasiewicz–Simon inequality. We also obtain an estimate of the decay rate to equilibrium.

We study a phase-field system where the energy balance equation has the standard (parabolic) form, while the kinetic equation ruling the evolution of the order parameter X is a nonlocal and nonlinear second-order ODE. The main features of the latter equation are a space convolution term which models long-range interactions of particles and a singular configuration potential that forces X to take values in (-1,1). We first prove the global existence and uniqueness of a regular solution to a suitable initial and boundary value problem associated with the system. Then, we investigate its long time behavior from the point of view of ω-limits. In particular, using a nonsmooth version of the Lojasiewicz-Simon inequality, we show that the ω-limit of any trajectory contains one and only one stationary solution, provided that the configuration potential in the kinetic equation is convex and analytic.

This paper is concerned with the well-posedness and the asymptotic behavior of solutions to the following parabolic–hyperbolic phase-field system (0.1) { ( θ + χ ) t − Δ θ = 0 , χ t t + χ t − Δ χ + ϕ ( χ ) − θ = 0 , in Ω × ( 0 , + ∞ ) , subject to the Neumann boundary condition for θ (0.2) ∂ ν θ = 0 , on Γ × ( 0 , + ∞ ) , the dynamical boundary condition for χ (0.3) ∂ ν χ + χ + χ t = 0 , on Γ × ( 0 , + ∞ ) , and the initial conditions (0.4) θ ( 0 ) = θ 0 , χ ( 0 ) = χ 0 , χ t ( 0 ) = χ 1 , in Ω , where Ω is a bounded domain in R 3 with smooth boundary Γ, ν is the outward normal direction to the boundary and ϕ is a real analytic function. In this paper we first establish the existence and uniqueness of a global strong solution to (0.1)–(0.4). Then, we prove its convergence to an equilibrium as time goes to infinity and we provide an estimate of the convergence rate.

We consider a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities. This model consists of the Navier–Stokes system coupled with a convective non-local Cahn–Hilliard equation with non-constant mobility. We first prove the existence of a global weak solution in the case of non-degenerate mobilities and regular potentials of polynomial growth. Then we extend the result to degenerate mobilities and singular (e.g. logarithmic) potentials. In the latter case we also establish the existence of a global attractor in dimension two. Using a similar technique, we show that there is a global attractor for the convective non-local Cahn–Hilliard equation with degenerate mobility and singular potential in dimension three.

We consider a diffuse interface model of tumour growth proposed by A. Hawkins-Daruudet al.((2013)J. Math. Biol.671457–1485). This model consists of the Cahn–Hilliard equation for the tumour cell fraction ϕ nonlinearly coupled with a reaction–diffusion equation for ψ, which represents the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation functionp(ϕ) multiplied by the differences of the chemical potentials for ϕ and ψ. The system is equipped with no-flux boundary conditions which give the conservation of the total mass, that is, the spatial average of ϕ + ψ. Here, we prove the existence of a weak solution to the associated Cauchy problem, provided that the potentialFandpsatisfy sufficiently general conditions. Then we show that the weak solution is unique and continuously depends on the initial data, provided thatpsatisfies slightly stronger growth restrictions. Also, we demonstrate the existence of a strong solution and that any weak solution regularizes in finite time. Finally, we prove the existence of the global attractor in a phase space characterized by ana prioribounded energy.

We consider a nonlinear parabolic system which governs the evolution of the (relative) temperature θ and of an order parameter χ. This system describes phase transition phenomena like, e.g., melting-solidification processes. The equation ruling χ is characterized by a singular potential W which forces χ to take values in the interval [−1, 1]. We provide reasonable conditions on W which ensure that, from a certain time on, χ stays uniformly away from the pure phases 1 and −1. Combining this separation property with the Lojasiewicz-Simon inequality, we show that any smooth and bounded trajectory uniformly converges to a stationary state and we give an estimate of the decay rate. ∗This work was partially supported by the Italian MIUR PRIN Research Projects Modellizzazione Matematica ed Analisi dei Problemi a Frontiera Libera and Aspetti Teorici e Applicativi di Equazioni a Derivate Parziali, and by the Italian MIUR FIRB Research Project Analisi di Equazioni a Derivate Parziali, Lineari e Non Lineari: Aspetti Metodologici, Modellistica, Applicazioni †The work of H.P. was supported by the Grant A1019302 of GA AV CR ‡The work of G.S. was partially supported by the HYKE Research Training Network

We consider a one-dimensional reaction-diffusion type equation with memory, originally proposed by W.E. Olmstead et al. to model the velocity $u$ of certain viscoelastic fluids. More precisely, the usual diffusion term $u_{x x}$ is replaced by a convolution integral of the form $\int_0^\infty k(s) u_{x x}(t-s)ds$, whereas the reaction term is the derivative of a double-well potential. We first reformulate the equation, endowed with homogeneous Dirichlet boundary conditions, by introducing the integrated past history of $u$. Then we replace $k$ with a time-rescaled kernel $k_\varepsilon$, where $\varepsilon>0$ is the relaxation time. The obtained initial and boundary value problem generates a strongly continuous semigroup $S_\varepsilon(t)$ on a suitable phase-space. The main result of this work is the existence of the global attractor for $S_\varepsilon(t)$, provided that $\varepsilon$ is small enough.

Our aim in this note is to analyze the relation between two notions of attractors for the study of the long time behavior of equations with memory, namely, the global attractor in the so-called past history approach, and the more recently proposed notion of trajectory attractor.