Approximate inertial manifolds to the generalized symmetric regularized long wave equations with damping term (Q1879126)

The authors study a generalized long wave equation, i.e. \[ \begin{gathered} u_t- \nu u_{xx}+ \rho_x+ f(u)_x- u_{xxt}= g_1(x),\quad (x,t)\in \mathbb{R}\times \mathbb{R}_+,\\ \rho_t+ u_x+ h(\rho)= g_2(x);\;u(x+ D,t)= u(x- D,t),\quad \rho(x+ D,t)= \rho(x- D,t),\\ u(x,0)= u_0(x),\quad \rho(x, 0)= \rho_0(x),\quad x\in\mathbb{R}.\end{gathered}\tag{1} \] Here \(D\), \(\nu> 0\), \(f,h\in C^\infty(\mathbb{R})\), \(g_j(x)\in L^2(\Omega)\), with \(\Omega= (-D,D)\). The goal is to establish (under some assumptions) the existence of approximate inertial manifolds, that is of finite-dimensional manifolds \(\Sigma\) which have the property that every solution remains asymptotically in a thin neighbourhood of \(\Sigma\). First, some preparatory steps are provided. Theorem 1, stated without proof, asserts that if \(u_0\in H^k(\Omega)\), \(\rho_0\in H^{k-1}(\Omega)\), \(g_1\in H^{k-1}(\Omega)\), \(g_2\in H^{k-2}(\Omega)\) and \(h'(\rho)\geq \gamma> 0\) (for some \(\gamma\)) then (1) possesses a unique global solution \(u\in H^k(\Omega)\), \(\rho\in H^{k-1}(\Omega)\). Here \(H^k(\Omega)\) denote the standard 2D-periodic Sobolev spaces. Assuming \(\int_\Omega g_1\,dx= 0\), Lemma 1 asserts the existence of an absorbing set for solutions, that is of \(E_1> 0\) such that \[ \| u(t)\|_{H^1}\leq E_1\text{ and }\|\rho(t)\|\leq E_1,\quad \forall t\geq t_1,\text{ for some }t_1.\tag{2} \] The proof is based on energy type estimates. Lemma 2 extends (2) in that it provides similar bounds for the \(\|\;\|_{H^k}\)-norm. The proof of Lemma 2 requires a lengthy induction step from \(k-1\) to \(k\). Lemma 3 extends this estimates to the derivatives \(u_t\), \(\rho_t\). In a last step, the existence of approximate inertial manifolds is proved. To this end, system (1) is cast into an abstract form. The existence of two types of manifolds is then proved, following the lines of Fojas, Temam. The first is a flat manifold of the form \(\Sigma_0= P_m L^2 (\Omega)\times P_m L^2 (\Omega)\), with \(P_m\) a projection operator of finite-dimensional range, obtained by spectral methods. The second manifold, while still finite-dimensional, is nonlinear. Finally, bounds for the asymptotic distance of trajectories from the respective manifolds are given.