Pullback \(\mathcal{D}\)-attractors for three-dimensional Navier-Stokes equations with nonlinear damping

When \(\alpha=0\), problem (1) becomes the classical 3D Navier-Stokes equation with external force, which has been studied by many authors (see [1–8]). The damping arises from the resistance to the motion of the flow and describes various physical phenomena, such as porous media flow, drag or friction effects, and some dissipative mechanisms (see [9–11]). For the autonomous case, in [12], Cai and Jiu proved that Cauchy problem (1) possesses global strong solutions when \(\beta\geq\frac{7}{2}\), and the global strong solution is unique when \(\frac{7}{2}\leq\beta\leq5\). In [13], Zhang, Wu and Lu also investigated the uniqueness of strong solution of problem (1). They established that the strong solution exists when \(\beta> 3\), and the global strong solution is unique when \(3<\beta\leq5\). This improved the earlier results in [12]. In [14, 15], some authors discussed the \(L^{2}\)-decay rate of solutions of problem (1). In [16, 17], we studied the existence of global attractors and uniform attractors of strong solution of problem (1) when \(\frac{7}{2}\leq\beta \leq5\).

In this paper, our aim is to study the long-time behavior of strong solution of problem (1) by the theory of pullback attractors. Pullback attractor theory is a natural generalization of the theory of global attractors developed to study autonomous dynamical systems, and it is well suited to study the non-autonomous dynamical systems. We shall prove the existence of pullback attractors in \((H_{0}^{1}(\Omega))^{3}\) and \((H^{2}(\Omega))^{3}\) under the assumption of an external force \(f(x,t)\) satisfying a certain integrability condition. To attain our goal we use the methods introduced in [18–20], which will be explained in more detail in Section 2. Before formulating the main results of the paper, we shall introduce some function spaces and some notations.

In this paper, the letter C is a generic positive constant, which may change its value from line to line, even in the same line.

In the next section, we provide basic definitions and results we shall use in this paper. In Section 3 we give some prior estimates of solutions. Based on these uniform estimation, in Section 4 we prove the existence of pullback attractors.

In this section, we will recall some basic definitions and abstract results about pullback attractor and state the theorems about the existence and uniqueness of global solutions of problem (1).

Let \(\mathcal{D}\) be a nonempty class of families \(\hat{D}=\{D(t):t\in \mathbb {R}\}\) of nonempty subsets of X.

Such a family \(\hat{\mathcal{A}}\) is called minimal if \(A(t)\subset C(t)\) for any family \(\hat{C}=\{C(t):t\in \mathbb {R}\}\) of closed subsets of X such that \(\lim_{\tau\rightarrow-\infty}\operatorname{dist}(U(t,\tau)B(\tau),C(t))=0\) for any \(\hat{B}\in\mathcal{D}\).

Obviously, a continuous process is a norm-to-weak continuous process. The following result is very useful to check that the process is norm-to-weak continuous.

Now, we recall the existence and uniqueness theorem for strong solution of problem (1).

Because \(\Omega\subset R^{3}\) is sufficiently regular, so \(V\hookrightarrow\mathbf{L}^{6}(\Omega)\), and because \(\mathbf {L}^{6}(\Omega)\hookrightarrow\mathbf{L}^{\beta+1}(\Omega)\) (\(3<\beta\leq 5\)), so \(V\cap\mathbf{L}^{\beta+1}(\Omega)=V\). In the following, we use \(u_{\tau}\in V\) to replace \(u_{\tau}\in V\cap\mathbf{L}^{\beta+1}(\Omega)\).

In [16], we have proved that the strong solution \(u(x,t)\) is continuous with respect to the initial value condition \(u_{0}\) in the space V when \(\frac{7}{2}\leq\beta\leq5\) (Proposition 7). From the proof, we can easily deduce that, when \(3<\beta\leq5\), for the non-autonomous case, the strong solution \(u(x,t)\) is also continuous with respect to the initial data \(u_{\tau}\) in V.

In order to construct a process \(\{U(t,\tau)\}_{t\geq\tau}\) for problem (1), we define \(U(t,\tau): V\rightarrow V\) by \(U(t,\tau)u_{\tau}=u(t)\), \(t\geq\tau\). Obviously, the process \(\{U(t,\tau)\}_{t\geq\tau}\) is a continuous process in V, so it is also a norm-to-weak continuous process in V.

In this section, we derive uniform estimates on solutions of problem (1) when \(\tau\rightarrow-\infty\). These estimates are necessary to prove the existence of pullback \(\mathcal{D}\)-absorbing sets and the pullback asymptotic compactness of process \(\{U(t,\tau)\}_{t\geq\tau}\) associated with the system.

In Section 2, we have known that the process \(\{U(t,\tau)\}_{t\geq\tau }\) associated with (1) is norm-to-weak continuous in V. In this section, we prove the existence of pullback attractors in V and \(\mathbf{H}^{2}(\Omega)\) for the non-autonomous Navier-Stokes equation with nonlinear damping.

In this paper, we consider the 3D Navier-Stokes equations with nonlinear damping \(\alpha|u|^{\beta-1}u\) with initial and Dirichlet boundary conditions which arises in the fluid dynamics. Under suitable assumptions on the external force function f, we obtain the existence of pullback \(\mathcal{D}\)-attractors of solutions in V and \(\mathbf {H}^{2}(\Omega)\), respectively. In [16] and [17], we have discussed the existence of global attractors and uniform attractors of such 3D NSEs in V and \(\mathbf{H}^{2}(\Omega)\) with \(\frac{7}{2}\leq \beta\leq5\). From this paper, we find that the pullback \(\mathcal {D}\)-attractors can exist in V and \(\mathbf{H}^{2}(\Omega)\) with \(\beta \in(3,5]\). Obviously, this improves the results in [16] and [17].