Fractional Navier-Stokes equations
In this paper, the authors investigate some fractional generalization of the Navier-Stokes equations. The flow domain \(\Omega \subset\mathbb{R}^N\), \(N \geq 2\) is supposed to be smooth and bounded. The authors present a detailed analysis of the existence, uniqueness, regularization and continuation properties of the solution by using the geometric theory of abstract parabolic problems. Concerning the original Navier-Stokes problem next global solutions of the Leray-Hopf type are constructed which satisfy Duhamel's integral formula, too. Finally, studying the 3-D model with zero external force they estimate a time after which the latter solution regularizes to strong solution. Furthermore, it was possible to estimate a time such that if a local strong solution exists until that time, then it exists forever. \par The paper is rather comprehensive. The proofs are mainly presented in detail and they use various modern tools from functional analysis and others like the elliptic regularity theory. The bibliography which gives a very good overview on the topic contains 52 items.
initial-boundary value problems for pseudodifferential operators
scientific article; zbMATH DE number 6996817
scientific article