## On the solutions to the normal form of the Navier-Stokes equations.(English)Zbl 1246.76019

Let $$u(t)=u(\cdot,t)$$ is the regular solution to the Navier-Stokes equations \begin{aligned} \frac{\partial u}{\partial t}-\nu\Delta u+(u\cdot\nabla)u+\nabla p&=-\nabla F,\quad \text{div} u=0 \quad \text{in}\quad x\in \mathbb{R}^3,\;t>0 \end{aligned}\tag{1} satisfying to the spatial periodicity condition. $$u(t)$$ has the asymptotic expansion $u(t)\,\sim\, q_1(t)e^{-t}+q_2(t)e^{-2t}+q_3(t)e^{-3t}+\cdots ,\tag{2}$ where $$q_j(t)$$ is the polynomial in $$t$$ of degree at most $$j-1$$. A normal form of the Navier-Stokes equations is the system of differential equations associated with the asymptotic expansion (2). The sufficient conditions for the convergence of the expansion (2) are given. The authors give examples of normed spaces in which the normal form of the Navier-Stokes equations constitutes a well-behaved infinite-dimensional system of ordinary differential equations. Some open problems in the studied sphere are formulated in the paper.

### MSC:

 76D05 Navier-Stokes equations for incompressible viscous fluids 35Q30 Navier-Stokes equations

### Keywords:

Navier-Stokes equations; asymptotic expansion
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