## Dynamics of the $$g$$-Navier–Stokes equations.(English)Zbl 1072.35145

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.

### MSC:

 35Q30 Navier-Stokes equations 35B41 Attractors 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
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### References:

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