Global well-posedness of solutions to the Cauchy problem of convective Cahn–Hilliard equation

Abstract

In this paper, we consider the global well-posedness and time-decay rates of solution to the Cauchy problem for 3D convective Cahn-Hilliard equation with doublewell potential via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained, theḢ −s (0 < s ≤ 1 2 ) negative Sobolev norms is shown to be preserved along time evolution and enhance the decay rates.2010 Mathematics Subject Classification. 35Q35, 35B65, 76N10, 76D05.

Well-posedness of the classical solution for the Kuramto–Sivashinsky equation with anisotropy effects

“…Observe that Eq. (1.5) is has been studied in the multidimensional case in the papers [7,18,66] and their references.…”

On global dynamics of 2D convective Cahn–Hilliard equation

“…The convective Cahn-Hilliard equation [10][11][12][13][14][15][16], which arises naturally as a continuous model for the formation of facets and corners in crystal growth, is a typical fourth order nonlinear parabolic equation. Let = [0, L] × [0, L], where L > 0, γ is a positive constant, β is a vector.…”

On the Cauchy problem of 3D incompressible Navier-Stokes-Cahn-Hilliard system

“…The authors established some results on global existence and global attractor. There are also some papers related to the Cauchy problem of Cahn-Hilliard equation, please refer to [20,56] and the reference therein. It is worth pointing out that the assumptions imposed on the nonlinear function f (u) = u 0 F (s)ds in [10,43,17,20,56] are too strict.…”

“…There are also some papers related to the Cauchy problem of Cahn-Hilliard equation, please refer to [20,56] and the reference therein. It is worth pointing out that the assumptions imposed on the nonlinear function f (u) = u 0 F (s)ds in [10,43,17,20,56] are too strict. One of the most nature assumption is f (u) = u 3 − u, which is the derivative of a double-well potential (the other is logarithmic potential).…”