A real space method for averaging lemmas
The authors study the kinetic equation \[ v\cdot \nabla_x f (x,v)= \Delta^{\alpha/2} g (x,v)\, , \] where \(x\in {\mathbb R}^d\), \(v\in {\mathbb R}^d\) and \(\alpha\in [0,1]\). In particular they prove several results about the regularity of ``velocity averages'', i.e. of quantities like \[ \rho (x)= \int f(x,v)\phi (v)\, dv, \qquad \phi\in C^\infty_c ({\mathbb R}^d)\, . \] It was first observed in [\textit{F. Golse, B. Perthame} and \textit{R. Sentis}, C. R. Acad. Sci., Paris, Sér. I 301, 341--344 (1985; Zbl 0591.45007)] that \(\rho\) is more regular than \(f\), motivating a large number of subsequent works. The authors' results improve the known regularity of \(\rho\) when the integrability of the solutions is different in space and velocity. Moreover their method of proof completely avoids the use of Fourier transform or wavelet decomposition, in contrast with all the methods known so far.
scientific article; zbMATH DE number 2164949
scientific article