On a Toy-Model Related to the Navier–Stokes Equations

Abstract

A parabolic toy-model for the incompressible Navier-Stokes system is considered. This model shares a lot of similar features with the incompressible model, including the energy inequality, the scaling symmetry, and it is also supercritical in 3D. A goal is to establish some regularity results for this toy-model in order to get, if possible, better insight to the standard Navier-Stokes system. A Caffarelli-Kohn-Nirenberg type result for the model is also proved in a direct manner. Finally, the absence of divergence-free constraint allows us to study this model in the radially symmetric setting for which full regularity is established. Bibliography: 16 titles.

Decay Estimate for some Toy-models related to the Navier-Stokes system

“…The particular case a = b = 0 was tackled in [12] (see Theorem 7.1). Finally, let us point out that the case κ = 0 and a = b = 0 was tackled in [4] using a more direct method which could also be extended to the general case (2.1) upon some minor changes.…”