Initial layer analysis of relaxation-time limit of the collisional QHD
We study the structure of the initial layer arising in the relaxation-time limit of the collisional quantum hydrodynamic (QHD) system. When the initial data are not well prepared, a fast transient regime appears near the initial time, which prevents the uniform-in-time convergence of the momentum density to its limiting value. Using the method of matched asymptotic expansions, we derive a systematic asymptotic description of the solution with respect to the relaxation-time parameter $\tau$. In particular, we identify the fast time scale $t/\tau^{2}$ governing the initial layer and explicitly construct the corresponding inner expansion for the momentum density together with the outer expansion describing the slow dynamics. The leading-order outer dynamics are shown to coincide with the quantum drift-diffusion equation. The asymptotic expansion is rigorously justified by establishing uniform in $\tau$ energy estimates for the remainder terms under suitable regularity assumptions on the solutions. As a consequence, we prove the strong convergence of the momentum density in $L^\infty$ in time after subtracting the leading initial-layer correction. The analysis further shows that the convergence rate of order $\tau$ is optimal for general initial data and explains the improved rate in the well-prepared case.