Anisotropic Mixed Fractional Landau Inequalities for Rotating Compressible Flows

We develop a rigorous theory of anisotropic mixed fractional Landau inequalities for rotating compressible fluid flows at high Mach numbers, incorporating Coriolis and centrifugal forces. We introduce rotating fractional Sobolev spaces $\mathcal{W}^{\nu,p}_{\alpha,\Omega}(\mathbb{R}^k)$, which encode directional scaling, fractional dissipation and rotational effects. We prove sharp fractional Landau inequalities with explicit dependence on the Mach number, the rotation rate $|\Omega|$ and the anisotropy vector $\alpha$. Key tools are a rotating Littlewood--Paley decomposition, anisotropic maximal estimates with rotational corrections, and commutator estimates for the Coriolis term. As applications, we establish stability bounds for neural operators approximating rotating compressible flows and derive optimal approximation rates of order $N^{-\nu/d_{\alpha,\Omega}}$, where $d_{\alpha,\Omega}=\sum_i\alpha_i^{-1}+\kappa|\Omega|^{2/\nu}$ is the effective anisotropic--rotational dimension. Our results provide a mathematical foundation for analysing high--Mach rotating flows and for designing physically informed neural architectures.