On a generalization of compensated compactness in the Lp−LqLp−Lq setting

We investigate conditions under which, for two sequences (ur)(ur) and (vr)(vr) weakly converging to u and v in Lp(Rd;RN)Lp(Rd;RN) and Lq(Rd;RN)Lq(Rd;RN), respectively, 1/p+1/q≤11/p+1/q≤1, a quadratic form q(x;ur,vr)=∑j,m=1Nqjm(x)ujrvmr converges toward q(x;u,v)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.