Steady Solutions to the Relativistic Boltzmann Equation in a Slab

Abstract

We study steady solutions to the relativistic Boltzmann equation with hard-sphere interactions in a slab geometry. Under a spatial symmetry assumption in the transverse variables $x_2$ and $x_3$, the problem reduces to a one-dimensional spatial slab $x_1 \in [0,1]$ while retaining full three-dimensional momentum dependence. For non-negative inflow boundary conditions prescribed at $x_1=0$ and $x_1=1$, we prove the existence and uniqueness of a stationary solution in a weighted $L^1_p L^\infty_{x_1}$ framework, together with exponential decay in momentum. Our analysis treats the full slab domain and does not rely on any smallness assumption on the slab width. We establish sharp coercivity and continuity estimates for the collision frequency, together with weighted convolution and pointwise bounds for the nonlinear gain term. These estimates generate and propagate a $(-Δ_p)^{-1}$-type regularity within the solution framework, which plays a crucial role in the existence and uniqueness argument. In addition, we obtain uniform weighted integrability of the solution over arbitrary two-dimensional hyperplanes through the origin. This hyperplane estimate is derived as a genuinely a posteriori regularity property, without imposing any a priori hyperplane bounds, and follows from a Lorentz-invariant geometric reduction.

Theorems & Definitions (26)

• Theorem 1.1: Existence, uniqueness, and intrinsic regularity
• Theorem 1.2: Hyperplane integrability of the steady state
• Remark 1.3
• Lemma 1.4: Coercive inequality for the relative momentum
• Remark 1.5
• Lemma 1.6
• proof
• Proposition 2.1: Upper- and lower-bound estimates for $\mathcal{L}f$
• proof
• Lemma 2.2