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

2014/02/10 by Marin Misur, Darko Mitrovic, Misur, Marin +1 · 1 citation
Mathematics · #35A27 #35K55 #42B15 #42B30 #46G10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #math.AP #math.FA #msc:35A27 #msc:35K55 #msc:42B15 #msc:42B30 #msc:46G10

paper · pdf · doi:10.48550/arxiv.1402.2259

arxiv created 2014/10/31 · arxiv updated 2014/11/03

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 ≤ 1, a quadratic form q(x;u_r,v_r)=∑_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.

Cited by

• Global L^p Second Commutation Lemma

Related

• The Malliavin derivative and compactness: application to a degenerate PDE-SDE coupling
• L^p-parabolic regularity and non-degenerate Ornstein-Uhlenbeck type operators
• On a nonlinear compactness lemma in L^p(0,T;B)
• Some variants of the classical Aubin-Lions Lemma
• L^p estimates for the Laplacian via blow-up