Hyperbolic versus parabolic asymptotics in kinetic theory toward fluid dynamic models (Q2862268)

Interests in this paper are first to analyze the hyperbolic asymptotic behavior of a general family of kinetic equations with a linear interaction kernel using a nonstandard scaling, and second to adapt these kinds of techniques to the nonlinear setting. Another objective of this paper is to connect the compressible or incompressible character of the fluid limit with the parabolic or hyperbolic scalings in kinetic theory.NEWLINENEWLINEThe general one-dimensional kinetic model is considered NEWLINE\[NEWLINE\varepsilon^{1+\gamma}\partial_t f_\varepsilon+\varepsilon v\partial_x f_\varepsilon - \varepsilon^\gamma \partial_x\phi_\varepsilon\partial_v f_\varepsilon=L[f_\varepsilon],\tag{1}NEWLINE\]NEWLINE where \(f_\varepsilon(t,x,v)\) represents the distribution of the particles, \(-\partial_x\phi(t,x)\) stands for a conservative field acting on the particles, and \(L[f_\varepsilon]\) is a operator representing the particular interaction phenomena. The Poisson law for the field is added NEWLINE\[NEWLINE\partial_{xx}^2\phi_\varepsilon=\rho_\varepsilon, \,\,\,\, \rho_\varepsilon(t,x)=\int_\mathbb{R}f_\varepsilon(t,x,v)dv.NEWLINE\]NEWLINE The first part of the article is devoted to the rigorous passage to the limit in the system (1) as \(\varepsilon\) goes to 0. The particular cases of the Fokker-Planck and relaxation models are presented as examples; this will be used to show the real need for intermediate scalings, which arise in some particular situations in a natural way. This part is also devoted to the study of the Chapman-Enskog expansions and the connection between the compressibility and incompressibility characters of the macroscopic fluid regimes with intermediate hyperbolic limits.NEWLINENEWLINEA nonlinear example consisting in a kinetic Kac model with a field term is considered in the second part of the article. At first the model is linearized around a global steady state to finally analyze its behavior under intermediate scaling. A new difficulty appears because the conservation of the kinetic energy produces a lack of compactness.