Global solvability and blow up for the convective Cahn-Hilliard equations with concave potentials

We study initial boundary value problems for the convective Cahn-Hilliard equation \Dt u +\px^4u +u\px u+\px^2(|u|^pu)=0. It is well-known that without the convective term, the solutions of this equation may blow up in finite time for any $p>0$. In contrast to that, we show that the presence of the convective term u\px u in the Cahn-Hilliard equation prevents blow up at least for $0<p<\frac49$. We also show that the blowing up solutions still exist if $p$ is large enough ($p\ge2$). The related equations like Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard equation, are also considered

Coalescence in the 1D Cahn-Hilliard model

We present an approximate analytical solution of the Cahn-Hilliard equation describing the coalescence during a first order phase transition. We have identified all the intermediate profiles, stationary solutions of the noiseless Cahn-Hilliard equation. Using properties of the soliton lattices, periodic solutions of the Ginzburg-Landau equation, we have construct a family of ansatz describing continuously the processus of destabilization and period doubling predicted in Langer's self similar scenario

Weak Solutions to the Degenerate Viscous Cahn-Hilliard Equation

The Cahn--Hilliard equation is a common model to describe phase separation processes of a mixture of two components. In this paper, we study the viscous Cahn--Hilliard equation with degenerate phase-dependent mobility. We define a notion of weak solutions and prove the existence of such weak solutions by considering the limits of the viscous Cahn--Hilliard equation with positive mobility. Also, we prove that such weak solutions satisfy an energy dissipation inequality under some additional conditions

Optimal distributed control of a stochastic Cahn-Hilliard equation

We study an optimal distributed control problem associated to a stochastic Cahn-Hilliard equation with a classical double-well potential and Wiener multiplicative noise, where the control is represented by a source-term in the definition of the chemical potential. By means of probabilistic and analytical compactness arguments, existence of an optimal control is proved. Then the linearized system and the corresponding backward adjoint system are analysed through monotonicity and compactness arguments, and first-order necessary conditions for optimality are proved.Comment: Key words and phrases: stochastic Cahn-Hilliard equation, phase separation, optimal control, linearized state system, adjoint state system, first-order optimality condition

Degenerate Mobilities in Phase Field Models are Insufficient to Capture Surface Diffusion

Phase field models frequently provide insight to phase transitions, and are robust numerical tools to solve free boundary problems corresponding to the motion of interfaces. A body of prior literature suggests that interface motion via surface diffusion is the long-time, sharp interface limit of microscopic phase field models such as the Cahn-Hilliard equation with a degenerate mobility function. Contrary to this conventional wisdom, we show that the long-time behaviour of degenerate Cahn-Hilliard equation with a polynomial free energy undergoes coarsening, reflecting the presence of bulk diffusion, rather than pure surface diffusion. This reveals an important limitation of phase field models that are frequently used to model surface diffusion