The Boltzmann-BGK model with nontrivial collision frequency near vacuum
We study the Cauchy problem for the Boltzmann-BGK model in the three-dimensional whole space with nontrivial collision frequency $ν(f)=ρ^αT^β$, where $α\in(1/3,1]$, $β\in[0,1]$, and $3α+2β\geq 2$. Existing well-posedness results for such models have largely been confined to near-equilibrium regimes and stationary problems. For nonnegative initial data with finite mass and energy that are sufficiently small in suitable polynomial-weighted $L^{\infty}$ norms, we establish global existence and uniqueness of mild solutions near vacuum. No uniform positive lower bound on the macroscopic density or temperature is imposed. The analysis relies on two complementary mechanisms. First, a phase-space weight invariant along free-transport characteristics yields dispersive estimates for the collision frequency and the gain term. The resulting time-integrable decay controls the nonlinear growth and closes the global weighted estimates. Second, we establish a Lipschitz estimate for the relaxation operator in a weighted $L^1$ space controlling mass and energy, with a Lipschitz constant depending only on weighted upper bounds for the distribution functions. The relaxation operator $ν\mathcal{M}$ has a more favorable structure for Lipschitz estimates than the local Maxwellian $\mathcal{M}$ alone: the collision frequency compensates for the singular dependence on the macroscopic fields, allowing us to establish Lipschitz continuity without uniform positive lower bounds for the density or temperature. Together, these estimates yield a unique global mild solution with uniform weighted bounds.