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Spatially homogeneous Boltzmann equation for relativistic particles. (English) Zbl 1321.35127

The authors prove some properties of the spatially homogeneous Boltzmann equation for relativistic particles written as \(\partial _{t}g+\widehat{p} \cdot \nabla _{x}g=Q(g,g)\) where \(g(x,p,t)\) is the number density of particles at position \((x,p)\in \mathbb{R}_{x}^{3}\times \mathbb{R}_{p}^{3}\) and at time \(t\in \mathbb{R}_{+}\). \(Q\) is the collision operator which can be decomposed in a gain term \(Q^{+}\) and a loss term \(Q^{-}\). The authors start with two reductions of the collision operator: the Glassey-Strauss and the center of momentum reductions. The first main result of the paper presents the relativistic version of the Povzner inequality (see the paper by R. T. Glassey and W. A. Strauss [Publ. Res. Inst. Math. Sci. 29, No. 2, 301–347 (1993; Zbl 0776.45008)] for the classical Newtonian one), using the center of momentum reduction. Then the authors prove an existence and uniqueness result for the Cauchy problem \(\frac{\partial g}{ \partial t}=Q(g,g)\), \(g(0)=g_{0}\), assuming appropriate hypotheses on the initial data \(g_{0}\). Finally the authors prove that the \(k\)th polynomial moment \(m_{k}(t)=\int g(p,t)(p^{0})^{k}dp\) satisfies a differential inequality. They also study how the exponential moment is propagated in \( L^{1}\).

MSC:

35Q20 Boltzmann equations
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness

Citations:

Zbl 0776.45008
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