Existence and regularity of the global attractor for a weakly damped forced shallow water equation in \(H^1(\mathbb R)\). (English) Zbl 1176.35158
Summary: We prove the existence of the global attractor for a weakly damped forced shallow water equation in \(H^1(\mathbb R)\). Moreover, we show that the global attractor is actually compact in \(H^4(\mathbb R)\). The main method is the combination of the energy equation, a suitable splitting of the solutions, and bilinear estimates in Bourgain spaces.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 37L50 | Noncompact semigroups, dispersive equations, perturbations of infinite-dimensional dissipative dynamical systems |
Keywords:
shallow water equation; global attractor; asymptotic smoothing effect; Bourgain spaces; bilinear estimatesReferences:
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