论文标题
对称性和刚性：只有一种对称性允许非零的真实对称解
Symmetry and rigidity: Only one kind of symmetry allow non-zero real symmetric solution
论文作者
论文摘要
Leray猜测，爆炸解决方案应具有与初始数据相似的结构，并提议考虑自相似解决方案。但是Necas-Ruzicka-Sverak在1996年证明了这种解决方案应为零。也就是说，Navier-Stokes方程对于自相似结构具有刚性。最近，杨 - 旺发现对称性能在不适合性结果的证明中起着重要作用。此外，杨应用傅立叶变换来考虑对称溶液。他表明，对称解决方案的一方应为零，并且存在某些对称属性可能导致对称解决方案。在本文中，我们考虑了与初始数据自变量相关的对称性，并分析了非线性项的对称结构。 （i）我们已经发现哪些类型的对称属性可以生成对称解，我们还证明了其余的对称属性在某种意义上仅允许零解决方案。对于真实的初始数据，我们证明只有一种对称性可以生成非零的对称解。 （ii）此外，要了解$ b（u，v）$的结构，我们表明足以考虑所有对称案例。 （iii）第三，我们为某些大初始值建立了良好的性质。 （iv）最后，我们将这种对称结果应用于域上的Navier-Stokes方程，并证明具有能量保护的平滑溶液。
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.
