TY - JOUR
T1 - On the Cahn-Hilliard-Brinkman Equations in $\mathbb{R}^4$: Global Well-Posedness
AU - Li , Bing
AU - Fang Wang , 
AU - Ling Xue , 
AU - Kai Yang , 
AU - Kun Zhao , 
JO - Annals of Applied Mathematics
VL - 4
SP - 513
EP - 536
PY - 2021
DA - 2021/12
SN - 37
DO - http://doi.org/10.4208/aam.OA-2021-0011
UR - https://global-sci.org/intro/article_detail/aam/20093.html
KW - Cahn-Hilliard-Brinkman equations, energy criticality, Cauchy problem, classical solution, global well-posedness.
AB - <p style="text-align: justify;">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.</p>