In this paper we prove local in time well-posedness for the incompressible Euler equations in for the initial data in , which corresponds to a critical case of the generalized Campanato spaces . The space is studied extensively in our companion paper , and in the critical case we have embeddings , where and are the Besov space and the Lipschitz space respectively. In particular contains non- functions as well as linearly growing functions at spatial infinity. We can also construct a class of simple initial velocity belonging to , for which the solution to the Euler equations blows up in finite time.
Cite this article
Dongho Chae, Jörg Wolf, The Euler equations in a critical case of the generalized Campanato space. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 2, pp. 201–241DOI 10.1016/J.ANIHPC.2020.06.006