On the Cahn-Hilliard-Brinkman Equations in $\mathbb{R}^4$: Global Well-Posedness

Abstract

We study the global well-posedness of large-data solutions to the Cauchy problem of the energy critical Cahn-Hilliard-Brinkman equations in $\mathbb{R}^4$. By developing delicate energy estimates, we show that for any given initial datum in $H^5(\mathbb{R}^4)$, there exists a unique global-in-time classical solution to the Cauchy problem. As a special consequence of the result, the global well-posedness of large-data solutions to the energy critical Cahn-Hilliard equation in $\mathbb{R}^4$ follows, which has not been established since the model was first developed over 60 years ago. The proof is constructed based on extensive applications of Gagliardo-Nirenberg type interpolation inequalities, which provides a unified approach for establishing the global well-posedness of large-data solutions to the energy critical Cahn-Hilliard and Cahn-Hilliard-Brinkman equations for spatial dimension up to four.