Resumen de Transport equation in generalized Campanato spaces

Dongho Chae, Jörg Wolf

• In this paper we study the transport equation in Rn .0; T /, T > 0, n 2, @tf C v rf D g; f . ; 0/ D f0 in R n ; in generalized Campanato spaces L s q.p;N /.Rn/. 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 D q D N D 1, we have the embeddings B 1 1;1.Rn/ ,! L 1 1.p;1/.Rn/ ,! C 0;1.Rn/, where B 1 1;1.Rn/ and C 0;1.Rn/ are the Besov and Lipschitz spaces, respectively. For f0 2 L 1 1.p;1/.Rn/, v 2 L1 .0; T IL 1 1.p;1/.Rn/// and g 2 L1 .0; T IL 1 1.p;1/.Rn///, we prove the existence and uniqueness of solutions to the transport equation in L1.0; T I L 1 1.p;1/.Rn// such that kf kL1.0;T IL1 1.p;1/.Rn/// C kvkL1.0;T IL1 1.p;1/.Rn///; kgkL1.0;T IL1 1.p;1/.Rn/// : Similar results for the other cases are also proved.