On a bilinear estimate in weak-Morrey spaces and uniqueness for Navier-Stokes equations
The Navier-Stokes equations are studied, where the bilinear term \(B(u,u)\) is given in terms of the heat semigroup and Leray projector -- see formula (5.1). The main point is to give a new estimate of \(B(u,u)\) in the functional space \(L^{\infty}((0,T);X)\), where \(X\) is a Morrey space. The Morrey spaces are used in order to improve the Sobolev embedding theorem -- see the useful paper [\textit{M. Rupflin}, ``What is a\dots Morrey space? What is a\dots Campanato space?''], available on Google. The new estimate of \(B(u,u)\) is obtained by using the Lorentz spaces (see Section 2.1) and block spaces (see Section 3.1). The block spaces are defined in terms of Lorentz spaces and contain a weight \(\rho\). On this way, the author can avoid the Besov spaces. It is proved that the block space is the predual of the Morrey space. The estimate of the bilinear term is obtained by using Holder type inequalities, some estimates of the heat semigroup on the block spaces, interpolation of operators and duality.
scientific article; zbMATH DE number 6524293
scientific article