Abstract We provide a rigorous mathematical framework to establish the hydrodynamic limit of Vlasov model introduced in Takata and Noguchi (J. Stat. Phys. 172:880-903, 2018) by order describe phase transition fluids kinetic equations. prove that, when scale parameter tends 0, this converges nonlocal Cahn-Hilliard equation with degenerate mobility. For our analysis, we introduce apropriate forms...

Our aim in this paper is to study generalizations of the Allen-Cahn equation based on a modification of the Ginzburg-Landau free energy proposed in [25]. In particular, the free energy contains an additional term called Willmore regularization. We prove the existence, uniqueness and regularity of solutions, as well as the existence of the global attractor. Furthermore, we study the convergence ...

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation ut + ∆ ` ε∆u− ε−1f(u) ́ = 0. It is shown that the a posteriori error bounds depends on ε−1 only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct an adaptive algorithm f...

Formation of modulated phase patterns can be modelized by a modi…ed Cahn-Hilliard equation which includes a non local term preventing the formation of macroscopic domains. Using stationnary solutions of the original Cahn-Hilliard equation as analytical ansatzs, we compute the thermodynamically stable period of a 1D modulated phase pattern. We …nd that the period scales like the power -1/3 of th...

We study finite element approximations of stochastic partial differential equations of Ginzburg-Landau type and the main paradigm considered in this paper is the stochastic Allen-Cahn model. We first demonstrate that the constructed stochastic finite element approximations are within an arbitrary level of tolerance from the corresponding one-dimensional stochastic partial differential equation;...

Abstract-Consider the motion of a curve in the plane under its mean curvature. It is a very interesting problem to investigate what happens when there are intersection points on the curve at which the mean curvature is singular. In this paper, we study this issue numerically by solving the Allen-Cahn equation and the nonlocal evolution equation with Kac potential. The Allen-Cahn equation is dis...

We consider a generalisation of the Cahn–Hilliard equation that incorporates an elastic energy density which, being quasiconvex, incorporates microstructure formation on smaller length scales. We prove global existence of weak solutions in certain microstructural regimes in (one and) two dimensions and present sufficient conditions for uniqueness. Preliminary numerical computations to illustrat...