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Register a new userFinite Dimensionality of the attractor for the hyperbolic Cahn-Hilliard-Oono Equation in R^3
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In this paper, we continue the study of the hyperbolic relaxation of the Cahn-Hilliard-Oono equation with the sub-quintic non-linearity in the whole space $R^3$ started in our previous paper and verify that under the natural assumptions on the non-linearity and the external force, the fractal dimension of the associated global attractor in the natural energy space is finite.
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We prove the global well-posedness of the so-called hyperbolic relaxation of the Cahn-Hilliard-Oono equation in the whole space R^3 with the non-linearity of the sub-quintic growth rate. Moreover, the dissipativity and the existence of a smooth global attractor in the naturally defined energy space is also verified. The result is crucially based on the Strichartz estimates for the linear Scroedinger equation in R^3.
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.
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.
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.
P. Galenko et al. proposed a modified Cahn-Hilliard equation to model rapid spinodal decomposition in non-equilibrium phase separation processes. This equation contains an inertial term which causes the loss of any regularizing effect on the solutions. Here we consider an initial and boundary value problem for this equation in a two-dimensional bounded domain. We prove a number of results related to well-posedness and large time behavior of solutions. In particular, we analyze the existence of bounded absorbing sets in two different phase spaces and, correspondingly, we establish the existence of the global attractor. We also demonstrate the existence of an exponential attractor.
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