Cahn–Hilliard equation with nonlocal singular free energies

Annali di Matematica Pura ed Applicata (1923 -) - We consider a Cahn–Hilliard equation which is the conserved gradient flow of a nonlocal total free energy functional. This functional is...     

Sixth-order Cahn–Hilliard equations with singular nonlinear terms

Our aim in this paper is to study the well-posedness for a class of sixth-order Cahn–Hilliard equations with singular nonlinear terms. More precisely, we prove the existence and uniqueness of variational solutions, based on a variational inequality.     

Strong solutions for two-dimensional nonlocal Cahn–Hilliard–Navier–Stokes systems

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 In some recent contributions the standard Cahn–Hilliard equation has been replaced by its nonlocal version.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.     

Doubly nonlocal reaction–diffusion equations and the emergence of species

The paper is devoted to a reaction–diffusion equation with doubly nonlocal nonlinearity arising in various applications in population dynamics. One of the integral terms corresponds to the nonlocal consumption of resources while another one describes reproduction with different phenotypes. Linear stability analysis of the homogeneous in space stationary solution is carried out. Existence of traveling waves is proved in the case of narrow kernels of the integrals. Periodic traveling waves are observed in numerical simulations. Existence of stationary solutions in the form of pulses is shown, and transition from periodic waves to pulses is studied. In the applications to the speciation theory, the results of this work signify that new species can emerge only if they do not have common offsprings. Thus, it is shown how Darwin’s definition of species as groups of morphologically similar individuals is related to Mayr’s definition as groups of individuals that can breed only among themselves.     

Global existence of weak solutions to a nonlocal Cahn–Hilliard–Navier–Stokes system

A well-known diffuse interface model consists of the Navier–Stokes equations nonlinearly coupled with a convective Cahn–Hilliard type equation. This system describes the evolution of an incompressible isothermal mixture of binary fluids and it has been investigated by many authors. Here we consider a variant of this model where the standard Cahn–Hilliard equation is replaced by its nonlocal version. More precisely, the gradient term in the free energy functional is replaced by a spatial convolution operator acting on the order parameter φ, while the potential F may have any polynomial growth. Therefore the coupling with the Navier–Stokes equations is difficult to handle even in two spatial dimensions because of the lack of regularity of φ. We establish the global existence of a weak solution. In the two-dimensional case we also prove that such a solution satisfies the energy identity and a dissipative estimate, provided that F fulfills a suitable coercivity condition.     

Error control for the approximation of Allen–Cahn and Cahn–Hilliard equations with a logarithmic potential

A fully computable upper bound for the finite element approximation error of Allen–Cahn and Cahn–Hilliard equations with logarithmic potentials is derived. Numerical experiments show that for the sharp interface limit this bound is robust past topological changes. Modifications of the abstract results to derive quasi-optimal error estimates in different norms for lowest order finite element methods are discussed and lead to weaker conditions on the residuals under which the conditional error estimates hold.     

Global attractor for the Cahn–Hilliard/Allen–Cahn system

We study the coupled Cahn–Hilliard/Allen–Cahn problem with constraints, which describes the isothermal diffusion-driven phase transition phenomena in binary systems. Our aim is to show the existence–uniqueness result and to construct the global attractor for the related dynamical system.     

On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework

We provide a thermodynamic basis for the development of models that are usually referred to as ??phase-field models?? for compressible, incompressible, and quasi-incompressible fluids. Using the theory of mixtures as a starting point, we develop a framework within which we can derive ??phase-field models?? both for mixtures of two constituents and for mixtures of arbitrarily many fluids. In order to obtain the constitutive equations, we appeal to the requirement that among all admissible constitutive relations that which is appropriate maximizes the rate of entropy production (see Rajagopal and Srinivasa in Proc R Soc Lond A 460:631?C651, 2004). The procedure has the advantage that the theory is based on prescribing the constitutive equations for only two scalars: the entropy and the entropy production. Unlike the assumption made in the case of the Navier?CStokes?CFourier fluids, we suppose that the entropy is not only a function of the internal energy and the density but also of gradients of the partial densities or the concentration gradients. The form for the rate of entropy production is the same as that for the Navier?CStokes?CFourier fluid. As observed earlier in Heida and Málek (Int J Eng Sci 48(11):1313?C1324, 2010), it turns out that the dependence of the rate of entropy production on the thermodynamical fluxes is crucial. The resulting equations are of the Cahn?CHilliard?CNavier?CStokes type and can be expressed both in terms of density gradients or concentration gradients. As particular cases, we will obtain the Cahn?CHilliard?CNavier?CStokes system as well as the Korteweg equation. Compared to earlier approaches, our methodology has the advantage that it directly takes into account the rate of entropy production and can take into consideration any constitutive assumption for the internal energy (or entropy).     

The Cahn–Hilliard equation with time-dependent constraint

We propose an abstract variational inequality formulation of the Cahn–Hilliard equation with a time-dependent constraint. We introduce notions of strong and weak solutions, and prove that a strong solution, if it exists, is a weak solution, and that the existence of a unique weak solution holds under an appropriate time-dependence condition on the constraint. We also show that the weak solution is a strong solution under appropriate assumptions on the data. Our abstract results can be applied to various concrete problems.     

Phase Separation in the Advective Cahn–Hilliard Equation

The Cahn–Hilliard equation is a classic model of phase separation in binary mixtures that exhibits spontaneous coarsening of the phases. We study the Cahn–Hilliard equation with an imposed advection term in order to model the stirring and eventual mixing of the phases. The main result is that if the imposed advection is sufficiently mixing, then no phase separation occurs, and the solution instead converges exponentially to a homogeneous mixed state. The mixing effectiveness of the imposed drift is quantified in terms of the dissipation time of the associated advection–hyperdiffusion equation, and we produce examples of velocity fields with a small dissipation time. We also study the relationship between this quantity and the dissipation time of the standard advection–diffusion equation.

     

A Gamma-convergence approach to the Cahn–Hilliard equation

We study the asymptotic dynamics of the Cahn–Hilliard equation via the “Gamma-convergence” of gradient flows scheme initiated by Sandier and Serfaty. This gives rise to an H ¹-version of a conjecture by De Giorgi, namely, the slope of the Allen–Cahn functional with respect to the H ⁻¹-structure Gamma-converges to a homogeneous Sobolev norm of the scalar mean curvature of the limiting interface. We confirm this conjecture in the case of constant multiplicity of the limiting interface. Finally, under suitable conditions for which the conjecture is true, we prove that the limiting dynamics for the Cahn–Hilliard equation is motion by Mullins–Sekerka law. Partially supported by a Vietnam Education Foundation graduate fellowship.     

Cahn–Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics

In this paper, we establish a novel approach to proving existence of non-negative weak solutions for degenerate parabolic equations of fourth order, like the Cahn–Hilliard and certain thin film equations. The considered evolution equations are in the form of a gradient flow for a perturbed Dirichlet energy with respect to a Wasserstein-like transport metric, and weak solutions are obtained as curves of maximal slope. Our main assumption is that the mobility of the particles is a concave function of their spatial density. A qualitative difference of our approach to previous ones is that essential properties of the solution – non-negativity, conservation of the total mass and dissipation of the energy – are automatically guaranteed by the construction from minimizing movements in the energy landscape.     

Multi-component Cahn–Hilliard systems with dynamic boundary conditions

Our aim in this paper is to study the well-posedness and the asymptotic behavior, in terms of finite-dimensional attractors, of Cahn–Hilliard systems describing phase separation processes in multi-component alloys, endowed with dynamic boundary conditions. Such boundary conditions take into account the interactions with the walls when considering confined systems.     

Optimal control of the convective Cahn–Hilliard equation

This article studies the problem for optimal control of the convective Cahn–Hilliard equation in one-space dimension. The optimal control under boundary condition is given, the existence of optimal solution to the equation is proved and the optimality system is established.     

Singularly perturbed 1D Cahn–Hilliard equation revisited

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}We consider a singular perturbation of the one-dimensional Cahn–Hilliard equation subject to periodic boundary conditions. We construct a family of exponential attractors {M_e}, e 3 0{\{{\mathcal M}_\epsilon\}, \epsilon\geq 0} being the perturbation parameter, such that the map e? M_e{\epsilon \mapsto {\mathcal M}_\epsilon} is H?lder continuous. Besides, the continuity at e = 0{\epsilon=0} is obtained with respect to a metric independent of e.{\epsilon.} Continuity properties of global attractors and inertial manifolds are also examined.     

The Cahn–Hilliard–Hele–Shaw system with singular potential

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.