Trajectory attractors for the 2D Navier-Stokes system and some generalizations
The authors extend known results on global attractors of 2D-Navier-Stokes: \[ \partial_tu= P\Delta u- P(u\nabla)u+ g(x, t),\quad\text{div }u= 0,\quad u= 0\quad\text{on }\partial\Omega.\tag{1} \] Here \(\Omega\) is a smooth bounded domain. Much is known about attractors of (1). The authors study global attractors of (1) in an extended setting, adapted to the nonautonomous case. Let \(U\), \(V\), \(H_2\) be the closures in \(L^2(\Omega)^2\), \(H^1(\Omega)^2\), \(H^2(\Omega)^2\) respectively of all \(v\in C^\infty_0(\Omega)^2\) with \(\text{div }v= 0\). The intention is to study (1) not in the usual phase space, say \(H\), but rather in a space of trajectories. To this end spaces \(L^{\text{loc}}_2= L^{\text{loc}}_2(R_+, H)\) and \(L^{\text{loc}}_{2,w}= L^{\text{loc}}_{2,w}(R_+, H)\) are needed; \(L^{\text{loc}}_2\) is a Fréchet space under the family of seminorms \(\int^{t_2}_{t_1} \int_\Omega| u(x, t)|^2dx dt\), \(0< t_1<t_2\), and \(L^{\text{loc}}_{2,w}\) is endowed with a corresponding weak topology. A \(g\in L^{\text{loc}}_{2,w}\) is translation compact (t-c for short) if the set of translates \(g(\cdot+h)\), \(h\geq 0\) is precompact in \(L^{\text{loc}}_{2,w}\); \(H_+(g)\) is the closure of the hull of translates \(g(\cdot+ h)\), \(h\geq 0\) in \(L^{\text{loc}}_{2,w}\). With \(g_0\) t-c in \(L^{\text{loc}}_{2,w}\) and \(g\in H_+(g_0)\), let \(K_g\) be the set of solutions of (1) with \(g\) as force, set \(K_+= \bigcup K_g\), \(g\in H_+g_0\). The action of the translation semigroup \((T_hg)(t)=g(t+ h)\) now extends in a natural way from \(H_+(g_0)\) to \(K_+\); moreover \(T_hK_+\subseteq K_+\subseteq L^{\text{loc}}_{2,w}\) follows from the definitions. The authors now endow \(K_+\) with a Fréchet topology which is adapted to the notion of strong solution of (1). Based on this notions, the concept of trajectory attractor in \(K_+\) with respect to the translation semigroup \(T_h\), \(h\geq 0\) is defined. The main result states among others that if \(g_0\in L^{\text{loc}}_2\) is t-c in \(L^{\text{loc}}_{2,w}\) then \(K_+\) has a trajectory attractor with respect to the translation semigroup \(T_h\), \(h\geq 0\), which is endowed with various properties. The proof is based on a number of lengthy estimates. The paper concludes with further material on an extension of these concepts to 3D-Navier-Stokes.
scientific article; zbMATH DE number 1127696