## On the localization of symmetric and asymmetric solutions of the Navier-Stokes equations in $$\mathbb{R}^n$$.(English. Abridged French version)Zbl 0973.35149

Summary: The aim of this note is to present some results on the space-time decay of solutions of the Navier-Stokes equations in $$\mathbb{R}^n$$, with data $$u(0)= a$$. We show that the localization condition $$L^1(\mathbb{R}^n,(1+|x|) dx)$$ is instantaneously lost, during the Navier-Stokes evolution, if the data has non-orthogonal components with respect to the $$L^2$$ inner product. We also show that some supplementary symmetries of small initial data allow us to obtain global strong solutions of the Navier-Stokes equations with an over-critical decay, both pointwise and of the energy norm.

### MSC:

 35Q30 Navier-Stokes equations 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids
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