2019/04/18 by Chae, Dongho, Wolf, Joerg · 2 citations
#35Q30 #76D03 #76D05 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1904.08676
In this paper we prove local in time well-posedness for the incompressible Euler equations in \Bbb Rn for the initial data in \mathscr L 1 1(1)(\mathbb Rn) , which corresponds to a critical case of the generalized Campanato spaces \mathscr L s q(N)(\mathbb Rn). The space is studied extensively in our companion paper\citetrans, and in the critical case we have embeddings B1∞, 1 (\Bbb Rn) \hookrightarrow \mathscr L 1 1(1)(\mathbb Rn) \hookrightarrow C0, 1 (\Bbb Rn), where B1∞, 1 (\Bbb Rn) and C0, 1 (\Bbb Rn) are the Besov space and the Lipschitz space respectively. In particular \mathscr L 1 1(1)(\mathbb Rn) contains non-C1(\Bbb Rn) functions as well as linearly growing functions at spatial infinity. We can also construct a class of simple initial velocity belonging to \mathscr L 1 1(1)(\mathbb Rn), for which the solution to the Euler equations blows up in finite time.