On the Initial Boundary Value Problems for the Enskog Equation in Irregular Domains

Abstract

The paper is concerned with the Enskog equation with a constant high density factor for large initial data in L1(Rn). The initial boundary value problem is investigated for bounded domains with irregular boundaries. The proof of an H-theorem for the case of general domains and boundary conditions is given. The main result guarantees the existence of global solutions of boundary value problems for large initial data with all v-moments initially finite and domains having boundary with finite Hausdorff measure and satisfying a cone condition. Existence and uniqueness are first proved for the case of bounded velocities. The solution has finite norm \(\int {(\sup _{0 \leqslant t \leqslant T} f(t_0 + t,x + vt))\sqrt {1 + v^2 } dq{\text{ }}dv,} \) where q = (t0, x) is taken on all possible n-dimensional planes Q(v) in Rn+l intersecting a fixed point and orthogonal to vectors (1, v), v ∈ Rn.