\(L^ p\) regularity of velocity averages
The paper is concerned with the regularity of velocity averages for solutions of transport equations: \[ v.\nabla_ xf=g\quad\text{for } x\in\mathbb{R}^ N,\quad v\in\mathbb{R}^ N,\quad\text{or} \tag{1} \] \[ \partial f/\partial t+v.\nabla_ xf=g\quad\text{for } x\in\mathbb{R}^ N,\quad v\in\mathbb{R}^ N,\quad t\in\mathbb{R}. \tag{2} \] In the time-independent case (equation (1)), the authors prove that, if \(f\in L^ p(\mathbb{R}^ N\times\mathbb{R}^ N)\) and \(g\in L^ p(\mathbb{R}^ N\times\mathbb{R}^ N)\), \(1<p\leq 2\), then for every \(\psi\in D(\mathbb{R}^ N)\), the velocity average \(\bar f(x)=\int_{\mathbb{R}^ N}f(x,v)\psi(v)dv\) belongs to the Besov space \(B_ 2^{s,p}(\mathbb{R}^ N)\) where \(s=1/p'\). In the time dependent case (equation (2)), they prove similar results when \(g\) admits the following decomposition: \(g=(I-\Delta_ x)^{\tau/2}(I-\Delta_ v)^{m/2}G\), \(\;\tau\in[0,1)\), \(\;m\geq 0\), \(\;G\in L^ p(\mathbb{R}^ N\times\mathbb{R}^ N\times\mathbb{R})\). Some applications to Vlasov-Maxwell systems and to other models are also given. These results extend many previous results [see for example \textit{F. Golse}, \textit{P. L. Lions}, \textit{B. Perthame}, \textit{R. Sentis}, J. Funct. Anal. 76, No. 1, 110-125 (1988; Zbl 0652.47031)].
scientific article; zbMATH DE number 21236
scientific article