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On the global existence of mild solutions to the Boltzmann equation for small data in \(L^D\). (English) Zbl 1209.35094

Summary: We develop a new theory of existence of global solutions to the Boltzmann equation for small initial data. These new mild solutions are analogous to the mild solutions for the Navier-Stokes equations. The existence comes as a result of the study of the competing phenomena of dispersion, due to the transport operator, and of singularity formation, due to the nonlinear Boltzmann collision operator. It is the joint use of the so-called dispersive estimates with new convolution inequalities on the gain term of the collision operator that allows to obtain uniform bounds on the solutions and thus demonstrate the existence of solutions.

MSC:

35Q20 Boltzmann equations
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
35B45 A priori estimates in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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