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| On the Cahn-Hilliard-Brinkman Equations in R4：Global Well-Posedness |
| Bing Li,Fang Wang,Ling Xue,Kai Yang,Kun Zhao |
| (School of Mathematical Sciences, Tiangong University, Tianjin 300387,
China;School of Mathematics and Statistics, Changsha University of Science
and Technology, Changsha, Hunan 410114, China;College ofMathematical Sciences, HarbinEngineering University, Harbin,
Heilongjiang 150001, China;School of Mathematics, Southeast University, Nanjing, Jiangsu 211189,
China;Department of Mathematics, Tulane University, New Orleans, LA 70118,
USA) |
| DOI： |
| 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. |
| Key words: Cahn-Hilliard-Brinkman equations, energy criticality, Cauchy problem, classical solution, global well-posedness. |