Transport equation theory in the Triebel-Lizorkin spaces and its applications to the ideal fluid flows

In this paper, we develop a general theory for the transport equation within the framework of Triebel-Lizorkin spaces. We first derive commutator estimates in these spaces, dispensing with the conventional divergence-free condition, via the Bony paraproduct decomposition and vector-valued maximal function inequalities. Building on these estimates and combining the method of characteristics with a compactness argument, we then obtain the new a priori estimates and prove local well-posedness for the transport equation in Triebel-Lizorkin spaces. The resulting theory is applicable to a wide range of evolution equations, including models for incompressible and compressible ideal fluid flows, shallow water waves, among others. As an illustration, we consider the incompressible ideal magnetohydrodynamics (MHD) system. Employing the general transport theory developed here yields a complete local well-posedness result in the sense of Hadamard, covering both sub-critical and critical regularity regimes, and provides corresponding blow-up criteria for the ideal MHD equations in Triebel-Lizorkin spaces. Our results refine and substantially extend earlier work in this direction.