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AUTHOR Chae, Dongho
TITLE The Euler equations in a critical case of the generalized Campanato space
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JOURNAL ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE, 2021
ABSTRACT 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) (Rn). The space is studied extensively in our companion paper [9], and in the critical case we have embeddings B-infinity,1(1)(R-n) hooked right arrow L-1(1)(1)(R-n) hooked right arrow C-0,C-1(R-n), where B-infinity,1(1) (R-n) and C-0,C-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. (C) 2020 L'Association Publications de l'Institut Henri Poincare. Published by Elsevier B.V. All rights reserved.
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