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

Brandolese, Lorenzo

C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 2, 125-130 (2001).

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

Keywords:

space-time decay of solutions of the Navier-Stokes equations; localization condition; symmetries of small initial data