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

TL;DR

This paper extends compensated compactness theory to the $L^p-L^q$ setting involving fractional derivatives and variable coefficients, using $H$-distributions, and applies it to nonlinear degenerate parabolic equations.

Contribution

It introduces a generalized framework for compensated compactness in the $L^p-L^q$ setting with fractional derivatives and variable coefficients, expanding the applicability of the theory.

Findings

Established new conditions for quadratic form convergence in distribution.

Extended compensated compactness to fractional derivatives and variable coefficients.

Applied the theoretical results to nonlinear degenerate parabolic equations.

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, $1/p+1/q \leq 1$, a quadratic form $q(x;u_r,v_r)=\sum\limits_{j,m=1}^N q_{j m}(x)u_{j r} v_{m r}$ 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.