Weak solutions and attractors for three-dimensional Navier-Stokes equations with nonregular force
The authors consider a viscous incompressible Newtonian fluid in a bounded open domain \(D\subset\mathbb{R}^3\) with smooth boundary \(\partial D\), described by the classical Navier-Stokes equations \[ {\partial u\over\partial t}+(u\cdot\nabla)u+\nabla p=\nu\Delta u+f+{\partial g\over \partial t}\text{ in }[0,\infty)\times D, \] \[ \begin{aligned} \text{div} & u=0\text{ in }[0, \infty)\times D,\\ & u=0\text{ in }[0,\infty)\times\partial D,\\ & u(0,x)=u_0(x),\;x\in D,\end{aligned} \] where \(u\) is the velocity field, \(p\) the pressure field, \(\nu>0\) the kinematic viscosity, and \(f+(\partial g/\partial t)\) the body force. Let \(X=L^2_{\text{loc}}[0,\infty;H)\), and \(W(g)\subset X\) be the space of generalized weak solutions. Let \(G\) be the space of all \(g\in C_0(\mathbb{R};V)\) that have polynomial growth at \(-\infty\) and satisfy the property \[ \liminf_{\alpha\to\infty}\limsup_{t\to\infty}{1\over t}\int^0_{-t} \bigl|z_\alpha(s,g)\bigr|^8_{L^4}ds=0, \] where \(z_\alpha(t,g)=g(t)-\int^t_{-\infty}(A+\alpha)e^{-(t-s)(A+\alpha)}g(s)ds\). Let \[ B(M_0,g)= \left\{ u\in W(g): \int^1_0\bigl|u(s)\bigr|^2 ds\leq M_0\right\} \] be defined for all constants \(M_0>0\) and all \(g\in G\). Let \(M(\cdot)\) be a function, defined on the backward orbit of \(g\) (the set \(\{\theta_{-\tau}g;\tau\geq 0\})\) with values in \((0,\infty)\), so that the subexponential growth condition is fulfilled, \[ \lim_{\tau\to+\infty}{\log^+M(\theta-\tau g)\over\tau}=0.\tag{1} \] For \(g\in G\) and a function \(M(\cdot)\) satisfying the assumption (1) let us define the set \[ {\mathcal A}\bigl(g,M(\cdot)\bigr)=\bigcap_{\tau\geq 0}\overline {\bigcup_{t\geq\tau}\Phi(t,\theta_{-t}g)B\bigl(M(\theta_{-t}g), \theta_{-t}g \bigr)} \] and let for \(g\in G\) \[ {\mathcal A}(g)=\overline{\bigcup_{M(\cdot)}{\mathcal A}(g,M(\cdot))}.\tag{2} \] Let \({\mathcal D}\) be the set of all measurable multifunctions contained in a ball \(B(M(g),g)\), so that (1) is fulfilled with \(P_Q\)-probability 1. There exists an \(\alpha>0\) and a \(\{\theta_t\}_{t \in\mathbb{R}} \)-invariant set \(G_\alpha\subset C_0(\mathbb{R};V)\) of full \(P_Q\)-measure consisting of functions \(g\) of polynomial growth so that \[ 2C^*\lim_{\tau \to\pm \infty} {1 \over\tau}\int^0_{-\tau} |z_\alpha(0, \theta_sg) |^8_{L^4} ds=\beta_\alpha. \] The main theorem of this paper says that for sufficiently large \(\alpha>0\) we have the following: (i) The attractor \({\mathcal A}(g)\) defined by (2) for \(g\in G_\alpha\) (for \(g\in C_0(\mathbb{R},V)\setminus G_\alpha\) we set \({\mathcal A}(g)= \{0\})\) defines a \(\overline{\mathcal B}^{P_Q}_{C_0(-\infty,\infty;V)}\)-measurable multifunction. (ii) \({\mathcal A}(g)\) is the unique attractor in \({\mathcal D}\). (iii) For any \(D\in{\mathcal D}\), \(\varepsilon>0\), \[ P_Q\biggl\{d\bigl(\overline {\Phi (t,g)D(g)},{\mathcal A}(\theta_tg)\bigr)> \varepsilon\biggr\}\to 0\quad\text{for}\quad t\to\infty. \]
scientific article; zbMATH DE number 1333046