Continuous dependence on initial data for the solutions of 3-D anisotropic Navier-Stokes equations

Abstract

In this paper, we prove the continuous dependence on the initial data for the solutions of 3-D incompressible anisotropic Navier-Stokes equations in the functional space $X_s\left(T\right)\stackrel{\mathrm{def}}{=}\{u:u\in C([0,T];H^{0,s})\text{with}{\nabla }_{h}u\in L^2(]0,T[;H^{0,s}\left)\right\}$ for $s>\frac{1}{2}$. We also show the non-uniform continuity of the data-to-solution map in $C\left(\right[0,T];H^{0,s})$ for $s>\frac{1}{2}$, which makes sharp contrast with the corresponding result for the classical 3-D Navier-Stokes equations.