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Long time behavior of solutions for a damped Benjamin-Ono equation (2010.05520v1)
Published 12 Oct 2020 in math.AP
Abstract: We consider the Benjamin-Ono equation on the torus with an additional damping term on the smallest Fourier modes (cos and sin). We first prove global well-posedness of this equation in $L2_{r,0}(\mathbb{T})$. Then, we describe the weak limit points of the trajectories in $L2_{r,0}(\mathbb{T})$ when time goes to infinity, and show that these weak limit points are strong limit points. Finally, we prove the boundedness of higher-order Sobolev norms for this equation. Our key tool is the Birkhoff map for the Benjamin-Ono equation, that we use as an adapted nonlinear Fourier transform.