Singular kernels and compactness in nonlinear conservation laws
A nonlinear equation of the form \[ {\partial\over\partial t}(K*u)+\sum^N_{i=1}a_i(u){\partial u\over\partial x_i}=0,\qquad u=u(t,x),\quad t\in(0,T),\;x\in\mathbb{R}^N \] where \(K*u(t,x)\equiv\int^t_{-\infty}K(t-s)u(s,x)ds\), and it is supposed \(K\in L^1(\mathbb{R}^+)\); \(K\) nonnegative, nonincreasing and convex on \(\mathbb{R}^+\); \(0<k_1\leq\lim_{t\to 0_+}\inf t^\alpha K(t)\leq\lim_{t\to 0_+}\sup t^\alpha K(t)\leq k_2<\infty\), where \(\alpha\in (0,1)\) and \(k_1,k_2\) are suitable constants; \(a_i\in C^2(\mathbb{R}^1)\), meas \(\{v\in[-Y,Y]\left|\sum^N_{i=1}a_i(v)\xi_i\right.=0\}=0\) for all \(|\xi|\neq 0\). The authors proved that the set of all entropy solutions \(u\) satisfying \(-Y\leq u(t,x)\leq Y\) for a.e. \(t\in(-\infty,T)\), \(x\in\mathbb{R}^N\) is compact in \(L^1_{\text{loc}}([0,T]\times\mathbb{R}^N)\). This compactness result may be considered as a generalization of the velocity average lemmata of Di Perna, Lions and Meyer to anisotropic problems containing fractional derivatives.
scientific article; zbMATH DE number 1212875