Quantitative Hydrodynamic Limit of the Chern--Simons--Higgs System

We study the hydrodynamic limit of the Chern–Simons–Higgs system, a relativistic gauge field model involving the Chern–Simons interaction. We introduce a single scaling parameter capturing both the non-relativistic (infinite speed of light) and semi-classical (vanishing Planck constant) regimes. This unified scaling allows us to justify the simultaneous non-relativistic and semi-classical limit, while retaining the nontrivial influence of the Chern–Simons gauge structure. Using a modulated energy method, we establish quantitative convergence rates toward the corresponding compressible Euler–Chern–Simons system as the scaling parameter tends to zero.