On the Cahn-Hilliard-Brinkman Equations in R^(4):Global Well-Posedness

摘要

We study the global well-posedness of large-data solutions to the Cauchy problem of the energy critical Cahn-Hilliard-Brinkman equations in R^(4).By developing delicate energy estimates,we show that for any given initial datum in H^(5)(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 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.