Uniqueness for some Leray-Hopf solutions to the Navier-Stokes equations.
The author first considers three formulations of solutions to the Navier-Stokes equations, namely distributional solutions defined in a standard way (NSE), the formulation with initial data incorporated as an inhomogeneous term (NSE)\('\), and Fujita-Kato's integral formulation (NSI). Equivalence of (NSE) and (NSI) was previously shown by \textit{G. Furiola, P. G. Lemarié-Rieusset} and \textit{E. Terraneo} [Rev. Mat. Iberoam. 16, No. 3, 605--667 (2000; Zbl 0970.35101)], and here the equivalence of those to (NSE)\('\) is proved. Next, under the assumptions of the trilinear estimates in abstract function space, \(X\), the author proves the weak-strong uniqueness following Foias and Serrin [\textit{C. Foias}, Bull, Soc. Math. Fr. 89, 1--8 (1961; Zbl 0107.07602); \textit{J. Serrin}, Nonlinear Probl., Proc. Sympos. Madison 1962, 69--98 (1963; Zbl 0115.08502)]. Third, the above trilinear estimates are established when the space \(X\) is one of the Lorentz spaces \(L^{p,\infty}(\mathbb{R}^3)\), Morrey spaces \(\dot M^q_p(\mathbb{R}^3)\) and Besov spaces based on Morrey spaces \(\dot B^{s,\infty}_{\dot M^q_p}(\mathbb{R}^3)\), thus proving the weak-strong uniqueness in these spaces. This implies previous results due to Serrin, Ribaud, and Gallagher-Planchon respectively, as corollaries. Finally, motivated by C. Foias (loc. cit.), for the solution of (NSE)\('\) in the Leray-Hopf class, the author exhibits new uniqueness classes, which are not weak-strong ones.
scientific article; zbMATH DE number 1901943