Symmetry and rigidity: Only one kind of symmetry allow non-zero real symmetric solution

Abstract

Leray guessed that, a blow-up solution should have similar structure as its initial data and proposed to consider self-similar solution. But Necas-Ruzicka-Sverak proved in 1996 that such solution should be zero. That is to say, Navier-Stokes equations have rigidity for self-similar structure. Recently, Yang-Yang-Wu found that the symmetry property plays an important role in the proof of ill-posedness result. Further, Yang applied Fourier transformation to consider symmetric solutions. He has shown that a party of symmetric solution should be zero and there exists some symmetric property can result in symmetric solution. In this paper, we consider the symmetry related to the independent variables of initial data and we analyze the symmetric structure of non-linear term. (i) We have found out what kinds of symmetric properties can generate symmetric solutions and we have also proved that the rest symmetric properties allow only zero solutions in some sense. For real initial data, we prove there exists only one kind of symmetry can generate non-zero symmetric solution. (ii) Further, to understand the structure of $B(u,v)$, we show it is sufficient to consider all the symmetric cases. (iii) Thirdly, we establish the well-posedness for some big initial values. (iv) Lastly, we apply such symmetric result to the Navier-Stokes equations on the domain and we prove the existence of smooth solution with energy conservation.