Global strong well-posedness and unconditional large-data dynamics for the inhomogeneous Boltzmann equation

Establish global well-posedness in a strong or mild solution class, uniqueness, and unconditional convergence to equilibrium for arbitrary large data in the spatially inhomogeneous cutoff Boltzmann equation on the three-dimensional torus, and derive the coercivity and weighted pointwise bounds needed to determine these global dynamics from weak or macroscopic information.

Background

The paper contrasts the relatively complete large-data theory for the spatially homogeneous Boltzmann equation with the substantially less understood spatially inhomogeneous problem. Although DiPerna–Lions renormalized solutions provide global existence for arbitrary physical data, they do not generally provide uniqueness or the strong pointwise estimates required for quantitative dynamical conclusions.

The unresolved issues identified here motivate the paper’s conditional theory. The authors impose a uniform velocity-tail condition on a global mild solution and show that this condition generates macroscopic coercivity, weighted pointwise control, uniqueness in a dissipative mild class, and exponential convergence to the global Maxwellian. These results therefore address a conditional intermediate regime rather than resolving the stated unconditional large-data problem.