不可压Euler方程的奇异Hölder解集的维数
Dimension of the singular set of wild Hölder solutions of the incompressible Euler equations
摘要：
对于$β<\frac13$，我们考虑了不可压缩Euler方程的不守恒动能的$C^β(\mathbb{T}^3\times [0,T])$弱解。我们证明了对于这样的解，奇异次$\mathcal{B}$的闭非空集满足$\dim_{\mathcal{H}}(\mathcal{B})\geq \frac{2β}{1-β}$。这个时间奇异集的豪斯多夫维数的下界与动能的赫尔德正则性有内在联系，我们猜想它是尖锐的。作为在这个方向上的第一步，对于每个$β<β'<\frac{1}{3}$，我们能够构造，通过凸积分计划，非保守的$C^β(\mathbb{T}^3\times [0,T])$弱解的不可压缩欧拉系统，使$\dim_{\mathcal{H}}(\mathcal{B})\leq \frac{1}{2}+\frac{1}{2}\frac{2β'}{1-β'}$。此外，我们建立的野生解决方案的结构允许推导出非唯一性的$C^β(\mathbb{T}^3\times [0,T])$弱解的柯西问题的欧拉从每个光滑的初始数据。
英文摘要：
For $β<\frac13$, we consider $C^β(\mathbb{T}^3\times [0,T])$ weak solutions of the incompressible Euler equations that do not conserve the kinetic energy. We prove that for such solutions the closed and non-empty set of singular times $\mathcal{B}$ satisfies $\dim_{\mathcal{H}}(\mathcal{B})\geq \frac{2β}{1-β}$. This lower bound on the Hausdorff dimension of the singular set in time is intrinsically linked to the Hölder regularity of the kinetic energy and we conjecture it to be sharp. As a first step in this direction, for every $β<β'<\frac{1}{3}$ we are able to construct, via a convex integration scheme, non-conservative $C^β(\mathbb{T}^3\times [0,T])$ weak solutions of the incompressible Euler system such that $\dim_{\mathcal{H}}(\mathcal{B})\leq \frac{1}{2}+\frac{1}{2}\frac{2β'}{1-β'}$. The structure of the wild solutions that we build allows moreover to deduce non-uniqueness of $C^β(\mathbb{T}^3\times [0,T])$ weak solutions of the Cauchy problem for Euler from every smooth initial datum.
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