Unicité des solutions des équations de Navier–Stokes dans les espaces de Morrey–Campanato

Abstract

In a recent work [P.G. Lemarié-Rieusset, Uniqueness for the Navier–Stokes problem: Remarks on a theorem of Jean-Yves Chemin, Nonlinearity 20 (2007) 1475–1490], P.G. Lemarié-Rieusset proved the uniqueness of solution to the Navier–Stokes equations in the space L p ( [ 0 , T ] , L q ( R d ) ) ∩ C ( [ 0 , T ] , B ∞ − 1 , ∞ ) provided that p > 2 and q > d . In this paper, we prove a local version of this result which covers the limit case q = d . Precisely, we prove the uniqueness of solution to the Navier–Stokes equations in the space L p ( [ 0 , T ] , M ˜ r , d ( R d ) ) ∩ C ( [ 0 , T ] , B ∞ − 1 , ∞ ) for every p > 2 and r > 2 where M ˜ r , d ( R d ) is the closure of the test functions in the Morrey–Campanato space M r , d ( R d ) . The prove of our result relies on an extension of the Comparison Theorem of P.G. Lemarié-Rieusset (Theorem 21.1 in [P.G. Lemarié-Rieusset, Recent developments in the Navier–Stokes problem, Chapman & Hall/CRC, 2002]). Moreover, this extension allows us to prove the uniqueness of solution to the Navier–Stokes equations in a functional space closed to the critical space C ( [ 0 , T ] , M 2 , d ( R d ) ) .