A Kinetic Concentration criterion for Large-data Boltzmann Dynamics

Abstract: We establish a conditional theory of genuinely non-perturbative global dynamics for the cutoff Boltzmann equation with hard potentials on the three-dimensional torus. We prove that any global mild solution satisfying [ \sup_{t\ge0}\int_{\mathbb R^3}W(v)\sup_{x\in\mathbb T^3}F(t,x,v)\,dv<\infty, ] for a suitable stretched-exponential weight (W), enjoys uniform weighted (L^\infty) control, is unique in the dissipative mild class, and converges exponentially to the global Maxwellian. In particular, these conclusions hold for arbitrary-amplitude solutions admitting a uniform stretched-exponential velocity upper bound. The key mechanism is a tail-to-coercivity principle: uniform velocity-tail control generates positive lower bounds for the local density and nonlinear collision frequency, without imposing macroscopic coercivity a priori. A double-Duhamel positivity argument also generates a Gaussian lower bound at arbitrarily short positive times, allowing initial data with local vacuum. We establish a polynomially weighted counterpart under an additional spatially uniform lower velocity profile, obtaining analogous weighted (L^\infty) bounds, uniqueness, and exponential relaxation. Our results provide a low-regularity alternative to conditional theories requiring smoothness and pointwise macroscopic bounds. They reveal how a single velocity-tail condition controls large-data Boltzmann dynamics and suggest a route toward problems with physical boundary conditions.