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Register a new userEnergy barrier and $Gamma$-convergence in the $d$-dimensional Cahn-Hilliard equation
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We study the d-dimensional Cahn-Hilliard equation on the flat torus in a parameter regime in which the system size is large and the mean value is close---but not too close---to -1. We are particularly interested in a quantitative description of the energy landscape in the case in which the uniform state is a local but not global energy minimizer. In this setting, we derive a sharp leading order estimate of the size of the energy barrier surrounding the uniform state. A sharp interface version of the proof leads to a $Gamma$-limit of the rescaled energy gap between a given function and the uniform state.
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We give a detailed study of the infinite-energy solutions of the Cahn-Hilliard equation in the 3D cylindrical domains in uniformly local phase space. In particular, we establish the well-posedness and dissipativity for the case of regular potentials of arbitrary polynomial growth as well as for the case of sufficiently strong singular potentials. For these cases, we prove the further regularity of solutions and the existence of a global attractor. For the cases where we have failed to prove the uniqueness (e.g., for the logarithmic potentials), we establish the existence of the trajectory attractor and study its properties.
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.
This paper deals with the Cauchy-Dirichlet problem for the fractional Cahn-Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In particular, to prove the convergence result, a variant of the so-called L ojasiewicz-Simon inequality is provided for the fractional Dirichlet Laplacian and (possibly) non-analytic (but $C^1$) nonlinearities.
In this paper, we consider the advective Cahn-Hilliard equation in 2D with shear flow: $$ begin{cases} u_t+v_1(y) partial_x u+gamma Delta^2 u=gamma Delta(u^3-u) quad & quad textrm{on} quad mathbb T^2; u textrm{periodic} quad & quad textrm{on} quad partial mathbb T^2, end{cases} $$ where $mathbb T^2$ is the two-dimensional torus. Under the assumption that the shear has a finite number of critical points and there are linearly growing modes only in the direction of the shear, we show the global existence of solutions with arbitrary initial $H^2$ data. The main difficulty of this paper is to handle the high-regularity and non-linearity underlying the term $Delta(u^3)$ in a proper way. For such a purpose, we modify the methods by Iyer, Xu, and Zlatov{s} in 2021 under a shear flow setting.
Experiments with diblock co-polymer melts display undulated bilayers that emanate from defects such as triple junctions and endcaps, cite{batesjain_2004}. Undulated bilayers are characterized by oscillatory perturbations of the bilayer width, which decay on a spatial length scale that is long compared to the bilayer width. We mimic defects within the functionalized Cahn-Hillard free energy by introducing spatially localized inhomogeneities within its parameters. For length parameter $varepsilonll1$, we show that this induces undulated bilayer solutions whose width perturbations decay on an $O!left(varepsilon^{-1/2}right)$ inner length scale that is long in comparison to the $O(1)$ scale that characterizes the bilayer width.
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