不可压缩欧拉方程Hölder解的尖锐能量正则性与典型性结果
Sharp energy regularity and typicality results for Hölder solutions of incompressible Euler equations
AI总结：
本文在Baire范畴意义下证明，对于 $θ<1/3$ 的欧拉方程 $C^θ$ 弱解，具有尖锐动能正则性 $e_v\in C^{\frac{2θ}{1-θ}}$ 且不具备更高可积正则性的解构成剩余集，同时光滑解构成无处稠密集，部分解决了相关猜想。
摘要：
本文致力于证明欧拉方程弱解 $v\in C^θ$ 在 $θ<1/3$ 情形下的若干典型性结果。已知凸积分方案会产生展现出动能 $e_v$ 反常耗散的野生弱解。我们证明这些解在Baire范畴意义下是典型的。由文献[8]可知，欧拉方程的 $θ$-Hölder连续弱解 $v$ 的动能 $e_v$ 满足 $e_v\in C^{\frac{2θ}{1-θ}}$。作为第一个结果，我们证明具有该行为的解在合适的完备度量空间 $X_θ$ 中构成剩余集，该空间包含于所有 $C^θ$ 弱解构成的空间中，其选取方式在本文末尾讨论。更精确地说，我们证明满足 $e_v \in C^{\frac{2θ}{1-θ}}$ 但不属于任何开区间 $I \subset [0,T]$ 上的 $\bigcup_{p\ge 1,\varepsilon>0}W^{\frac{2θ}{1-θ} + \varepsilon,p}(I)$ 的解 $v\in X_θ$ 的集合是 $X_θ$ 中的剩余集。这尤其部分解决了文献[9, Conjecture 1]。我们还证明光滑解在所有 $C^θ$ 弱解构成的空间中构成无处稠密集。所用技术相同，而真正区分这两种情形的是后者无需相对于自然度量引入不同的完备度量空间。
英文摘要：
This paper is devoted to show a couple of typicality results for weak solutions $v\in C^θ$ of the Euler equations, in the case $θ<1/3$. It is known that convex integration schemes produce wild weak solutions that exhibit anomalous dissipation of the kinetic energy $e_v$. We show that those solutions are typical in the Baire category sense. From [8], it is know that the kinetic energy $e_v$ of $θ$-Hölder continuous weak solution $v$ of the Euler equations satisfy $ e_v\in C^{\frac{2θ}{1-θ}}$. As a first result we prove that solutions with that behavior are a residual set in suitable complete metric space $X_θ$, that is contained in the space of all $C^θ$ weak solutions, whose choice is discussed at the end of the paper. More precisely we show that the set of solutions $v\in X_θ$ with $e_v \in C^{\frac{2θ}{1-θ}}$ but not to $\bigcup_{p\ge 1,\varepsilon>0}W^{\frac{2θ}{1-θ} + \varepsilon,p}(I)$ for any open $I \subset [0,T]$, are a residual set in $X_θ$. This, in particular, partially solves [9, Conjecture 1]. We also show that smooth solutions form a nowhere dense set in the space of all the $C^θ$ weak solutions. The technique is the same and what really distinguishes the two cases is that in the latter there is no need to introduce a different complete metric space with respect to the natural one.
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