The derivation of fluid equations from the Boltzmann equation is one of the most important problems in kinetic theory. In this talk, we investigate several results concerning the diffusive limit of the Boltzmann equation within the L²–L^∞ framework. Based on this framework, we present two results: (1) a global diffusive expansion in an exterior domain, and (2) the global hydrodynamic limit with Maxwell boundary conditions in a bounded domain.

We consider a nonlocal semilinear elliptic equation in a bounded smooth domain with the inhomogeneous Dirichlet boundary condition, which arises as the stationary problem of the Keller-Segel system with physical boundary conditions describing the boundary-layer formation driven by chemotaxis. This problem has a unique steady-state solution which possesses a boundary-layer profile as the nutrient diffucion coefficient tends to zero. Using the Fermi coordinates and delicate analysis with subtle estimates, we also rigorously derive the asymptotic expansion of the boundary-layer profile and thickness in terms of the small diffusion rate with coefficients explicitly expressed by the domain geometric properties including mean curvature, volume and surface area. By these expansions, one can explicitly find the joint impact of the mean curvature, surface area and volume of the spatial domain on the boundary-layer steepness and thickness.

We study the gradient theory of phase transitions through the asymptotic analysis of variational problems introduced by Modica (1987). As the perturbation parameter tends to zero, minimizers converge to two-phase functions whose interfaces minimize area. The proof uses techniques from the theory of functions of bounded variation and Γ-convergence. This framework has applications in materials science and the study of minimal surfaces. 4참고자료: L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal. 98 (1987), 123–142.