The generalized incompressible Navier-Stokes equations in Besov spaces

This paper is concerned with global solutions of the generalizedNavier-Stokes equations. The generalized Navier-Stokes equationshere refer to the equations obtained by replacing the Laplacian inthe Navier-Stokes equations by the more general operator$(-\Delta)^\alpha$ with $\alpha>0$. It has previously been shownthat any classical solution of the $d$-dimensional generalizedNavier-Stokes equations with $\alpha\ge \frac{1}{2}+\frac{d}{4}$ isalways global in time. Thus, attention here is solely focused on thecase when $\alpha<\frac{1}{2}+\frac{d}{4}$. We consider solutionsemanating from initial data in several Besov spaces and establishthe global existence and uniqueness of the solutions when thecorresponding initial data are comparable to the diffusioncoefficient in these Besov spaces.

Keywords

generalized Navier-Stokes equations, global solutions, Besov spaces

2010 Mathematics Subject Classification

Primary 35Q30. Secondary 76D03.

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Published 1 January 2004