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Acta Mathematica Vietnamica

On the Global Attractor for the 3D Viscous Camassa-Holm Equations with Damping Term in the Whole Space

Nguyen Hai Ha Giang , Trinh Dang Duong , icon-email Vu Manh Toi

Abstract

We consider the 3D viscous Camassa-Holm equations with damping term in the whole space. Firstly, we prove the existence, uniqueness and regularity of global weak solutions to the equations. Then we prove the existence of a compact global attractor for the associated continuous semigroup. To overcome the essential difficulty when proving the asymptotic compactness of the semigroup, which arises due to the lack of compactness of the Sobolev embeddings, we exploit the energy equation method. Next, by using the regularity of weak solutions and inductive arguments, we study the Sobolev regularity of the global attractor. Finally, we give an explicit upper bound for the fractal dimension of the global attractor.

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Viện Hàn lâm Khoa học và Công nghệ Việt Nam Hội Toán học Việt Nam Viện Nghiên cứu Cao cấp về toán MathScinet Arxiv