Dynamics of the \(g\)-Navier--Stokes equations
The author considers a variant of the Navier-Stokes equations in dimension \(d=2\) which differs from the standard version in that it involves a weight \(g(x_1,x_2)\) which is \(1\)-periodic in \(x_1\), \(x_2\), smooth and close to \(g= 1\). The equation considered is \[ \begin{gathered} u_t- \Delta_gu+ g^{-1}(\nabla g\cdot\nabla) u+ (u\cdot\nabla)u+ \nabla p= f,\;x\in\Omega= (0,1)\times (0,1),\\ \text{div}(gu)= 0,\quad\Delta u+ g^{-1}(\nabla g\cdot\nabla) u= \Delta_g u.\end{gathered}\tag{1} \] \(u(x_1,u_2)\) is \(1\)-periodic in \(x_1\), \(x_2\). On system (1) a functional \(L^2_g\)-setting is imposed which differs from the usual \(L^2\)-setting in that the weight \(g\) enters in an obvious way. E.g. one considers the space \(H_g\) which is the \(L^2_g\)-closure of the fields \(u\in C^\infty(\Omega)^2\) which satisfy \(\text{div}(gu)= 0\) and \(\int_\Omega u\,dx= 0\). An orthogonal projection \(P_g\) which projects from \(L^2_g\) onto \(H_g\) is then introduced accordingly, leading to an abstract evolution equation \[ u_t+ A_g u+ B_g(u,u)= q\tag{2} \] with \(A_g\), \(B_g\) the \(g\)-versions of the Stokes operator and the nonlinearity. If \(g=1\) then the evolution semigroup \(S_1(t)\) associated with (2) for \(g=1\) admits a global attractor \({\mathcal A}_1\), as proved by Foias, Temam, Constantin. The main result of the author is that if \(g\) is sufficiently close to \(g= 1\), then the evolution semigroup \(S_g(t)\) associated with (2), admits a global attractor \({\mathcal A}_g\) which is close to \({\mathcal A}_1\) in some sense. The paper contains a large number of propositions and lemmas without proof; these can be found in the author's thesis.
scientific article; zbMATH DE number 2164782