36 research outputs found
Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction
In this paper we study the long-time behavior of a nonlocal Cahn-Hilliard
system with singular potential, degenerate mobility, and a reaction term. In
particular, we prove the existence of a global attractor with finite fractal
dimension, the existence of an exponential attractor, and convergence to
equilibria for two physically relevant classes of reaction terms
The weighted energy-dissipation principle and evolutionary Gamma-convergence for doubly nonlinear problems
We consider a family of doubly nonlinear evolution equations that is given by families of convex dissipation potentials, nonconvex energy functionals, and external forces parametrized by a small parameter ε. For each of these problems, we introduce the so-called weighted energy-dissipation (WED) functional, whose minimizer correspond to solutions of an elliptic-in-time regularization of the target problems with regularization parameter δ. We investigate the relation between the Γ-convergence of the WED functionals and evolutionary Γ-convergence of the associated systems. More precisely, we deal with the limits δ→0, ε→0, as well as δ+ ε→0 either in the sense of Γ-convergence of functionals or in the sense of evolutionary Γ-convergence of functional-driven evolution problems, or both. Additionally, we provide some quantitative estimates on the rate of convergence for the limit ε→0, in the case of quadratic dissipation potentials and uniformly λ-convex energy functionals. Finally, we discuss a homogenization problem as an example of application
The weighted energy-dissipation principle and evolutionary [Gamma]-convergence for doubly nonlinear problems
We consider a family of doubly nonlinear evolution equations that is given by families of convex dissipation potentials, nonconvex energy functionals, and external forces parametrized by a small parameter ε. For each of these problems, we introduce the so-called weighted energy-dissipation (WED) functional, whose minimizer correspond to solutions of an elliptic-in-time regularization of the target problems with regularization parameter δ. We investigate the relation between the Γ-convergence of the WED functionals and evolutionary Γ-convergence of the associated systems. More precisely, we deal with the limits δ→0, ε→0, as well as δ+ ε→0 either in the sense of Γ-convergence of functionals or in the sense of evolutionary Γ-convergence of functional-driven evolution problems, or both. Additionally, we provide some quantitative estimates on the rate of convergence for the limit ε→0, in the case of quadratic dissipation potentials and uniformly λ-convex energy functionals. Finally, we discuss a homogenization problem as an example of application
On a nonlocal Cahn--Hilliard equation with a reaction term
We prove existence, uniqueness, regularity and separation properties for a nonlocal Cahn- Hilliard equation with a reaction term. We deal here with the case of logarithmic potential and degenerate mobility as well an uniformly lipschitz in u reaction term g(x, t, u)
From nonlocal to local Cahn-Hilliard equation
In this paper we prove the convergence of a nonlocal version of the
Cahn-Hilliard equation to its local counterpart as the nonlocal convolution
kernel is scaled using suitable approximations of a Dirac delta in a periodic
boundary conditions setting. This convergence result strongly relies on the
dynamics of the problem. More precisely, the -gradient flow structure
of the equation allows to deduce uniform estimates for solutions of the
nonlocal Cahn-Hilliard equation and, together with a Poincar\'e type inequality
by Ponce, provides the compactness argument that allows to prove the
convergence result.Comment: 13 page
The weighted energy-dissipation principle and evolutionary Γ-convergence for doubly nonlinear problems
We consider a family of doubly nonlinear evolution equations that is given by families of convex dissipation potentials, nonconvex energy functionals, and external forces parametrized by a small parameter ε. For each of these problems, we introduce the so-called weighted energy-dissipation (WED) functional, whose minimizers correspond to solutions of an elliptic-in-time regularization of the target problems with regularization parameter δ. We investigate the relation between the Γ-convergence of the WED functionals and evolutionary Γ-convergence of the associated systems. More precisely, we deal with the limits δ → 0, ε → 0, as well as δ + ε → 0 either in the sense of Γ-convergence of functionals or in the sense of evolutionary Γ-convergence of functional-driven evolution problems, or both. Additionally, we provide some quantitative estimates on the rate of convergence for the limit ε → 0, in the case of quadratic dissipation potentials and uniformly λ-convex energy functionals. Finally, we discuss a homogenization and a dimension reduction problem as examples of application