Stability and Decay for the 2D Anisotropic Navier-Stokes Equations with Fractional Horizontal Dissipation on $\mathbb{R}^2$

The stability problem for the 2D Navier-Stokes equations with dissipation in only one direction on $\mathbb R^2$ is not fully understood. This dissipation is in the intermediate regime between the fully dissipative Navier-Stokes and the inviscid Euler. Navier-Stokes solutions in $\mathbb R^2$ decay algebraically in time while Euler solutions can grow rather rapidly in time. This paper solves the fundamental stability and large-time behavior problem on the anisotropic Navier-Stokes with fractional dissipation $\Lambda_1^{2s}$ for all $0\leq s<1$. The case $s=1$ corresponds to the standard one directional dissipation $\partial_1^2$. Different techniques are developed to treat different ranges of fractional exponents: $0\leq s\leq \frac34$, $\frac34<s<\frac{11}{12}$, and $\frac{11}{12} \leq s <1$. The final range is the most difficult case, for which we introduce the spatial polynomial $A_2$ weights and exploit the boundedness of Riesz transforms on weighted $L^2$-spaces.