About Lanford's theorem in the half-space with specular reflection

TL;DR

This work extends the rigorous derivation of the Boltzmann equation to the half-space with specular boundary conditions, establishing a Lanford-type convergence for hard spheres under the wall. It introduces a rigorous collision-operator definition and a Gaussian-weighted functional framework for the BBGKY and Boltzmann hierarchies, and adapts the recollision-control strategy to boundary geometry. The main technical contributions include explicit Duhamel-type representations of solutions, a geometric pseudo-trajectory interpretation, and a sequence of cut-offs (in adjunctions, velocity, and wall-proximity) that tame recollisions and grazing events. Consequently, under Boltzmann-Grad scaling $N\varepsilon^{d-1}=1$ and suitable initial data, the first marginal of the particle density converges locally uniformly to the Boltzmann solution with specular reflection, with a quantified rate, thereby extending Lanford's theorem to domains with boundaries and illustrating the boundary’s influence on the convergence rate and time horizon.

Abstract

The present article proposes a rigorous derivation of the Boltzmann equation in the half-space. We show an analog of the Lanford's theorem in this domain, with specular reflection boundary condition, stating the convergence in the low density limit of the first marginal of the density function of a system of N hard spheres towards the solution of the Boltzmann equation associated to the initial data corresponding to the initial state of the one-particle-density function. The original contributions of this work consist in two main points: the rigorous definition of the collision operator and of the functional space in which the BBGKY hierarchy is solved in a strong sense; and the adaptation to the case of the half-space of the control of the recollisions performed by Gallagher, Saint-Raymond and Texier, which is a crucial step to obtain the Lanford's theorem.

Figures (8)

• Figure 1: A representation of the specular reflection against an obstacle $\Omega$.
• Figure 2: A representation of an elastic collision between two hard spheres.
• Figure 3: Construction of a pseudo-trajectory for the BBGKY hierarchy.
• Figure 4: Comparison of the pseudo-trajectories of the two hierarchies.
• Figure 5: Case of a recollision in the dynamics of the hard spheres, and divergence from the dynamics of particles of radius zero, following the free flow.

Theorems & Definitions (52)

• Theorem 1: Lanford's theorem
• Proposition 1: Rigorous definition of the hard sphere dynamics almost everywhere, globally in time
• Remark 1
• Definition 1: Hard sphere transport
• Definition 2: Boundary condition for the hard sphere dynamics
• Theorem 2: Definition of the collision operator of the BBGKY hierarchy for functions of $\mathcal{C}([0,T],L^\infty(\mathcal{D}^\varepsilon_{s+1}))$ decaying sufficiently fast at infinity in the velocity variables
• proof : Sketch of proof of Theorem \ref{['SECT2TheorDefinOperaColliBBGKY']}
• Definition 3: Norms $|\cdot|_{\varepsilon,s,\beta}$ and $|\cdot|_{0,s,\beta}$, spaces $X_{\varepsilon,s,\beta}$ and $X_{0,s,\beta}$
• Definition 4: Norms ${\left\vert\left\vert \cdot \right\vert\right\vert}_{N,\varepsilon,\beta,\mu^\alpha}$ and ${\left\vert\left\vert \cdot \right\vert\right\vert}_{0,\beta,\mu^\alpha}$, spaces $\textbf{X}_{N,\varepsilon,\beta,\mu^\alpha}$ and $\textbf{X}_{0,\beta,\mu^\alpha}$
• Proposition 2: Embeddings of the spaces $X_{\varepsilon,s,\beta}$, $X_{0,s,\beta}$, $\textbf{X}_{N,\varepsilon,\beta,\mu^\alpha}$ and $\textbf{X}_{0,\beta,\mu^\alpha}$