论文标题
在与Navier-Stokes方程相关的玩具模型上
On a Toy-Model related to the Navier-Stokes Equations
论文作者
论文摘要
在本文中，我们考虑了不可压缩的Navier-Stokes系统的抛物线玩具模型。正如我们将在下面看到的那样，该模型与不可压缩的模型共享许多类似的功能。其中的能量不平等，缩放对称性，并且在$ 3 $ d中也是超临界的。我们的目标是为此玩具模型建立一些规律性的结果，以便在可能的情况下更好地了解标准的Navier-Stokes系统。我们还以直接的方式证明了我们模型的caffarelli-kohn-nirenberg类型结果。最后，利用缺乏无差异约束的优势，我们能够在径向对称的环境中研究该模型，我们能够建立完全规律性。
In this paper we consider a parabolic toy-model for the incompressible Navier-Stokes system. This model, as we shall see below, shares a lot of similar features with the incompressible model; among which the energy inequality, the scaling symmetry, and it is also supercritical in $3$D. Our goal is to establish some regularity results for this toy-model in order to get, if possible, better insight to the standard Navier-Stokes system. We also prove here, in a direct manner, a Caffarelli-Kohn-Nirenberg type result for our model. Finally, taking advantage of the absence of the divergence-free constraint, we are able to study this model in the radially symmetric setting for which we are able to establish full regularity.
