Well Posedness and Global Attractors for the 3D Periodic BBM Equation Below the Energy Space.

This work is devoted to the long time behavior for the damped BBM equation on three dimensional torus below the energy space. First, it is shown that the BBM equation is globally well-posed in H s (T 3) with 1 2 ≤ s ≤ 1 . The result can not be improved by our method since the bilinear estimate used in the proof fails to hold for smaller s. Second, it is proved that there exists a global attractor in H s (T 3) for every 3 4 < s ≤ 1 . The main difficulty lies in proving the existence of a bounded absorbing set in H s (T 3) , since no conservation law is available below H 1 (T 3) . It is overcome by Bourgain's high-low frequency decomposition approach. Finally, some quantitative regularities of the global attractor in terms of the Fourier projection are derived. [ABSTRACT FROM AUTHOR]