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On the dimension of the attractor for the non-homogeneous Navier-Stokes equations in non-smooth domains. (English) Zbl 0969.35105
The authors consider the inhomogeneous Navier-Stokes equations in two dimensions: \[ \partial_t u=\nu\Delta u- (u\nabla)u+\nabla p+f,\quad \text{div }u= 0,\quad u=\varphi\quad\text{on }\partial\Omega.\tag{1} \] Here \(f\in L^2(\Omega)\), \(\varphi\in L^\infty(\partial\Omega)\) are time independent while \(\Omega\) is a bounded Lipschitz domain. They prove that there is an attractor \(A\) which attracts asymptotically all solutions. They obtain the following estimates, both for the Hausdorff and fractal dimension of \(A\): \[ \dim(A)\leq c_1 G+ c_2\text{Re}^{3/2},\tag{2} \] where \(G\) is the Grashoff number and Re the Reynolds number, while \(c_1\), \(c_2\) are dimensionless constants. A major part of the technical difficulties are related to the question of finding a field \(\psi\) (a background flow) such that \[ \text{div }\psi= 0\quad\text{in }\Omega\quad\text{and}\quad \psi=\varphi\quad\text{on }\partial\Omega.\tag{3} \] With the aid of such \(\psi\), and by setting \(v=u-\psi\), (1) transform into \[ \partial_t v=\nu\Delta v+ (v\nabla)v+ (v\nabla)\psi+ (\psi\nabla)v+ \nabla p+ f+\nu\Delta\psi- (\psi\nabla) \psi\tag{4} \]
\[ \text{with}\quad \text{div }v= 0\quad\text{in }\Omega,\quad v= 0\quad\text{on }\partial\Omega. \] An appropriate notion of weak solutions of (4) is given and shown that weak solutions exist, are unique and provided with certain regularity properties. A major result (Theorem 1.11) states that the dynamical system, associated with (4), has a universal attractor \(A\), and that the Hausdorff and fractional dimension of \(A\) satisfy an estimate of type (2). The proof requires the solution of the Stokes system: \[ -\Delta u+\nabla q=0\quad\text{in }\Omega,\quad \text{div }u= 0\quad\text{in }\Omega\quad\text{and}\quad u=\varphi\quad\text{a.e. on }\partial\Omega\tag{5} \] in the sense of nontangential convergence.
The background flow \(\psi\) in (3) is now defined in terms of \(u\) and a suitable smooth function \(\eta_\varepsilon\). Due to the singular behaviour of \(\psi\) in the vicinity of \(\partial\Omega\), standard estimates (e.g. relating to the Galerkin method) have now to be treated with considerably more care than usual. The proof of the main theorem splits into proofs of several auxiliary theorems and an appendix devoted to a discussion of (5).

35Q30 Navier-Stokes equations
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35D05 Existence of generalized solutions of PDE (MSC2000)
35D10 Regularity of generalized solutions of PDE (MSC2000)
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