Global well-posedness of solutions to the Cauchy problem of convective Cahn-Hilliard equation. Cauchy problem of convective Cahn-Hilliard equation (Q1756411)

The author proves the existence of a unique global smooth solution \(u=u(x,t)\) of the Cauchy problem associated with the \(N\)-dimensional convective Cahn-Hilliard equation, \[ \begin{cases} \partial_tu+\Delta^2u-\Delta f(u)-\nabla \cdot g(u)=0,&\\ u(x,0)=u_0(x), \;\;x\in \mathbb{R}^N, \;t\geq 0, \end{cases} \] where \(f\) and \(g\) are smooth functions satisfying certain local growth condition at \(u=\bar{u}\) for some \(\bar{u}\in \mathbb{R}\) and the \(L^{N(l-1)/3}(\mathbb{R})\)-norm of \(u_0\) is sufficiently small. A similar result is obatained for the \(N\)-dimensional Cahn-Hilliard equation without convective term.