Transport equation in generalized Campanato spaces.

In this paper we study the transport equation in generalized Campanato spaces Ls The critical case is particularly interesting, and is applied to the local well-posedness problem for the incompressible Euler equations in a space close to the Lipschitz space in our companion paper [Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 2, 201-241]. In the critical case s Dq DN D1, we have the embeddings B1 1;1.Rn/,!L1 1.p;1/.Rn/,!C0;1.Rn/, where B11; 1.Rn/and C0;1.Rn/are the Besov and Lipschitz spaces, respectively. For f0 2L1 1.p;1/.Rn/, v 2L1.0;T IL1 1.p;1/.Rn///and g 2L1.0;T IL1 1.p;1/.Rn///, we prove the existence and uniqueness of solutions to the transport equation in L1.0; T IL1 1.p;1/.Rn//such that Similar results for the other cases are also proved. [ABSTRACT FROM AUTHOR]

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