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The Euler equations in a critical case of the generalized Campanato space
Annales de l'Institut Henri Poincaré C, Analyse non linéaire ( IF 1.9 ) Pub Date : 2020-07-13 , DOI: 10.1016/j.anihpc.2020.06.006
Dongho Chae 1, 2 , Jörg Wolf 1
Affiliation  

In this paper we prove local in time well-posedness for the incompressible Euler equations in Rn for the initial data in L1(1)1(Rn), which corresponds to a critical case of the generalized Campanato spaces Lq(N)s(Rn). The space is studied extensively in our companion paper [9], and in the critical case we have embeddings B,11(Rn)L1(1)1(Rn)C0,1(Rn), where B,11(Rn) and C0,1(Rn) are the Besov space and the Lipschitz space respectively. In particular L1(1)1(Rn) contains non-C1(Rn) functions as well as linearly growing functions at spatial infinity. We can also construct a class of simple initial velocity belonging to L1(1)1(Rn), for which the solution to the Euler equations blows up in finite time.



中文翻译:

广义Campanato空间临界情况下的欧拉方程。

本文证明了不可压缩的Euler方程在时间上的局部适定性 [Rñ 对于中的初始数据 大号1个1个1个[Rñ,它对应于广义Campanato空间的一个临界情况 大号qñs[Rñ。在我们的配套论文[9]中对空间进行了广泛的研究,在关键情况下,我们有嵌入方法。1个1个[Rñ大号1个1个1个[RñC01个[Rñ,在哪里 1个1个[RñC01个[Rñ分别是Besov空间和Lipschitz空间。特别是大号1个1个1个[Rñ 包含非C1个[Rñ空间无穷大处的线性函数和线性增长函数。我们还可以构造一类属于大号1个1个1个[Rñ,对此,欧拉方程的解在有限的时间内爆炸了。

更新日期:2020-07-13
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