The global attractor of the damped, forced generalized Korteweg de Vries-Benjamin-Ono equation in \(L^2\) (Q862085)

The goal of this paper is to study existence and regularity of the global attractor of the damped, forced generalized KdV-Benjamin-Ono equation in \(L^2(\mathbb{R})\), \[ \partial_t u+ \gamma\cdot{\mathcal H}\partial^2_x u+ \beta\partial^3_x u+\mu\partial_x u+\lambda u+ u\partial_x u= f,\tag{1} \] \[ u(x,0)= u_0(x)\in L^2(\mathbb{R}),\quad (x,t)\in \mathbb{R}\times\mathbb{R},\tag{2} \] where \(\gamma\), \(\beta\), \(\mu\), \(\lambda\) are real constants and \(\gamma\cdot\beta\neq 0\), \(\lambda> 0\). The time-independent function \(f\in L^2(\mathbb{R})\), and \({\mathcal H}\) denotes the Hilbert transform \[ {\mathcal H}\theta(x)= \text{p.v. }{1\over\pi} \int {\theta(x- y)\over y}\,dy, \] where p.v. denotes the Cauchy principal value. Moreover, the authors show that the global attractor of (1)--(2) is actually a compact set in \(H^3(\mathbb{R})\).