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In this paper we prove local in time well-posedness for the incompressible Euler equations in R n for the initial data in L 1 (1) 1 (R n) , which corresponds to a critical case of the generalized Campanato spaces L q (N) s (R n). The space is studied extensively in our companion paper [9] , and in the critical case we have embeddings B ∞ , 1 1 (R n) ↪ L 1 (1) 1 (R n) ↪ C 0 , 1 (R n) , where B ∞ , 1 1 (R n) and C 0 , 1 (R n) are the Besov space and the Lipschitz space respectively. In particular L 1 (1) 1 (R n) contains non- C 1 (R n) functions as well as linearly growing functions at spatial infinity. We can also construct a class of simple initial velocity belonging to L 1 (1) 1 (R n) , for which the solution to the Euler equations blows up in finite time. [ABSTRACT FROM AUTHOR]

We prove the existence of a forward discretely self-similar solutions to the Navier–Stokes equations in R 3 × ( 0 , + ∞ ) for a discretely self-similar initial velocity belonging to L l o c 2 ( R 3 ) . [ABSTRACT FROM AUTHOR]