Symmetric and uniform analytic solutions in phase space for Navier–Stokes equations

For incompressible Navier–Stokes equations, Leray believed that, for a blow up solution, the initial data and the solution should both have the same special structure for different time and proposed to consider self-similar solutions. Necas–Ruzicka–Sverak proved that self-similar solution has to be zero in 1996. In the study of ill-posedness, Yang–Yang–Wu find symmetry property plays an important role. In this paper, we consider two categories of symmetry properties. On the one hand, we found some kinds of symmetric solution has to be zero as self-similar solution. On the other hand, we prove that three kinds of symmetry initial data can have uniform analytic and symmetric solution in the general Fourier–Herz spaces. We use symmetric and uniform analytic functions to approximate the solution. We take two steps: (i) For these kinds of symmetry of initial data, we prove that the solution has also the same symmetric structure. (ii) We prove that the uniform analyticity is equivalent to the convolution inequality on Herz spaces.

Keywords

Navier–Stokes equations, symmetry about the component of velocity field, symmetry and anti-symmetry for the independent variable of velocity field, uniform analyticity, Fourier–Herz space

2010 Mathematics Subject Classification

Primary 35Q30, 42B35. Secondary 76D03.

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Received 16 July 2019

Published 18 February 2020