We study the three-dimensional many-particle quantum dynamics in mean-field setting. We forge together the hierarchy method and the modulated energy method. We prove rigorously that the compressible Euler equation is the limit as the particle number tends to infinity and the Planck’s constant tends to zero. We improve the previous sufficient small time hierarchy argument to any finite time via a new iteration scheme and Strichartz bounds first raised by Klainerman and Machedon in this context. We establish strong and quantitative microscopic to macroscopic convergence of mass and momentum densities up to the 1st blow up time of the limiting Euler equation. We justify that the macroscopic pressure emerges from the space-time averages of microscopic interactions via the Strichartz-type bounds. We have hence found a physical meaning for Strichartz-type bounds.

In this paper, we study the isotropic–nematic phase transition for the nematic liquid crystal based on the Landau–de Gennes $\mathbf{Q}$-tensor theory. We justify the limit from the Landau–de Gennes flow to a sharp interface model: in the isotropic region, $\mathbf{Q}\equiv 0$; in the nematic region, the $\mathbf{Q}$-tensor is constrained on the manifolds $\mathcal{N}=\{s_{+}(\mathbf{n}\otimes \mathbf{n}-\frac{1}{3}\mathbf{I}),\mathbf{n}\in {\mathbb{S}}^2\}$ with $s_{+}$ a positive constant, and the evolution of alignment vector field $\mathbf{n}$ obeys the harmonic map heat flow, while the interface separating the isotropic and nematic regions evolves by the mean curvature flow. This problem can be viewed as a concrete but representative case of the Rubinstein–Sternberg–Keller problem introduced in Rubinstein et al. (SIAM J. Appl. Math. 49:116–133, 1989; SIAM J. Appl. Math. 49:1722–1733, 1989).