On a generalization of compensated compactness in the $L^p-L^q$ setting

Abstract

We investigate conditions under which, for two sequences $(u_r)$ and $(v_r)$ weakly converging to $u$ and $v$ in $L^p(R^d;R^N)$ and $L^q(R^d;R^N)$, respectively, $1/p+1/q\le 1$, a quadratic form $q(x;u_r,v_r)=\sum _{j,m=1}^N{q}_{jm}(x)u_{jr}{v}_{mr}$ converges toward $q(x;u,v)$ in the sense of distributions. The conditions involve fractional derivatives and variable coefficients, and they represent a generalization of the known compensated compactness theory. The proofs are accomplished using a recently introduced $H$-distribution concept. We apply the developed techniques to a nonlinear (degenerate) parabolic equation.

References in corpus (1)

• On the velocity averaging for equations with optimal heterogeneous rough coefficients