35B30 Dependence of solutions on initial and boundary data, parameters [See also 37Cxx]
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Keywords
- Evolutionary Γ-convergence, gradient systems, homogeniza- tion, Cahn–Hilliard equation, evolutionary variational inequality, energy- dissipation principle, two-scale convergence. (1)
- Fredholm alternative (1)
- continuous dependence (1)
- data-to-solution map (1)
- nonlinear differential equations (1)
- nonstandard phase field system (1)
- small denominators (1)
- uniqueness (1)
In these notes we discuss two approaches to evolutionary Γ- convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ Γ- convergence methods. In the first approach we consider families of uni- formly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savar ́e 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004.
We apply both approaches to rigorously derive homogenization limits for Cahn–Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals Γ-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.
The present note deals with a nonstandard systems of differential equations describing a two-species phase segregation. This system naturally arises in the asymptotic analysis carried out recently by the same authors,
as the diffusion coefficient in the equation governing
the evolution of the order parameter tends to zero. In particular, an existence result has been proved for the limit system in a very general framework. On the contrary, uniqueness was shown by assuming a constant mobility coefficient. Here, we generalize this result and prove
a continuous dependence property in the case that the mobility coefficient suitably depends on the chemical potential.
This paper concerns $n\times n$ linear one-dimensional hyperbolic systems of the type
$$
\om\partial_tu_j + a_j(x)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x)u_k = f_j(x,t),\; j=1,\ldots,n,
$$
with periodicity conditions in time and reflection boundary conditions in space.
We state sufficient conditions on the data $\om, a_j$ and $b_{jk}$
such that the system has a Fredholm like solvability behavior.
Moreover, we state sufficient conditions on the data such that for any right hand side there exists exactly
one solution, that the solution survives under small perturbations of the data, and that the corresponding
data-to-solution-map
is smooth with respect to appropriate function space norms. In particular, those sufficient conditions
imply that no small denominator effects occur. Moreover, we show that perturbations of the coefficients $\om$ and $a_j$ lead to
essentially different results than perturbations of the coefficients $b_{jk}$, in general.