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Asymptotic analysis of the Navier-Stokes equations in thin domains. (English) Zbl 0957.35108

From the introduction: We are engaged with the Navier-Stokes equations of viscous incompressible fluids in three dimensional thin domains \(\Omega_\varepsilon= \omega\times (0,\varepsilon)\), where \(\omega\) is a suitable domain in \(\mathbb{R}^2\). We derive an asymptotic expansion of the strong solution \(u^\varepsilon\) of the Navier-Stokes equations in the thin domain \(\Omega_\varepsilon\) when \(\varepsilon\) is small, which is valid uniformly in time. We consider two types of boundary conditions: the Dirichlet-periodic boundary condition and the purely periodic condition. For the first type of boundary condition we prove that the solution can be written, for \(\varepsilon\) small, as \[ u^\varepsilon(t)= w^\varepsilon +\overline u^\varepsilon \exp\left(- {\nu t\over 2\varepsilon^2} \right),\quad \forall t> 0, \] where \(w^\varepsilon\) is the solution of the associated Stokes problem and \(\overline u^\varepsilon\) is a bounded (in time) function depending on the initial data. For the purely periodic boundary condition case, we prove that the solution can be written, as: \[ u^\varepsilon(t) =w^\varepsilon +u^\varepsilon_{2 D} (t)+\overline u^\varepsilon \exp\left(- {\nu t\over 2 \varepsilon^2} \right), \quad \forall t>0 \text{ and }\varepsilon \text{ small}, \] where \(w^\varepsilon\) is the solution of the auxiliary Stokes problem, \(u^\varepsilon_{2\text{D}} (t)\) is the solution of the 2D-Navier-Stokes equations with three components and \(\overline u^\varepsilon\) is a bounded (in time) function depending on the initial data.

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35C20 Asymptotic expansions of solutions to PDEs
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