The Euler equations in a critical case of the generalized Campanato space

Abstract

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.