Strain, Robert M.; Yun, Seok-Bae Spatially homogeneous Boltzmann equation for relativistic particles. (English) Zbl 1321.35127 SIAM J. Math. Anal. 46, No. 1, 917-938 (2014). 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 \( L^{1}\). Reviewer: Alain Brillard (Riedisheim) Cited in 13 Documents 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 Keywords:spatially homogeneous Boltzmann equation; relativistic particles; reduction; Povzner inequality; Cauchy problem; existence and uniqueness; differential inequality Citations:Zbl 0776.45008 PDFBibTeX XMLCite \textit{R. M. Strain} and \textit{S.-B. Yun}, SIAM J. Math. Anal. 46, No. 1, 917--938 (2014; Zbl 1321.35127) Full Text: DOI