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Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-\(\alpha\)) equations on bounded domains. (English) Zbl 1006.35074

The Lagrangian averaged Navier-Stokes (LANS-\(\alpha\)) equations for a fluid moving in a bounded region \(\Omega\subset \mathbb{R}^3\) with smooth boundary \(\partial\Omega\) are given by \[ \begin{aligned} &\frac{\partial v}{\partial t}+U^{\alpha}(v)+\nu A v+ v\cdot\nabla v +(1-\alpha^2\Delta)^{-1}\nabla p=0,\\ &\quad \nabla\cdot v=0\qquad \text{in} \Omega\times(0,T),\\&v=0 \quad\text{on}\quad \partial\Omega\times(0,T),\qquad v(x,0)=v_0(x), \end{aligned} \] where \(U^{\alpha}(v)=\alpha^2(1-\alpha^2\Delta)^{-1}\text{div}\left[\nabla v \cdot\nabla v^T+\nabla v\cdot \nabla v-\nabla v^T\cdot \nabla v\right]\) and \(A=-P\Delta\) is the Stokes operator.
The global well-posedness and regularity to the problem is proved for initial data in the set \(\{v\in H^s\cap H^1_0:\;Av=0\;\text{on} \partial\Omega,\nabla\cdot v=0\}\), \(s\in[3,5)\). The equations converge to the Navier-Stokes equations as \(\alpha\to 0\). It is shown also that classical solutions of the LANS-\(\alpha\) equations converge almost all in \(H^s\) for \(s\in(5/2,3)\) to solutions of the inviscid equation \((\nu=0)\) called the Lagrangian averaged Euler equations.
The history of the problem and related mathematical models are discussed in the paper.

MSC:

35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35Q05 Euler-Poisson-Darboux equations
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