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 fractionalderivatives and variable coefficients, and they represent a generalization ofthe known compensated compactness theory. The proofs are accomplished using arecently introduced $H$-distribution concept. We apply the developed techniquesto a nonlinear (degenerate) parabolic equation.
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机译:我们研究在两个序列$(u_r)$和$(v_r)$分别收敛到$ L ^ p(R ^ d; R ^ N)$和$ L ^ { q}(R ^ d; R ^ N)$分别为$ 1 / p + 1 / q \ leq 1 $,二次形式$ q(x; u_r,v_r)= \ sum \ limits_ {j,m = 1 } ^ N q_ {jm}(x)u_ {jr} v_ {mr} $在分布意义上收敛于$ q(x; u,v)$。这些条件包括分数导数和可变系数,它们代表了已知的补偿紧度理论的一般化。证明是使用最近引入的$ H $-分布概念来完成的。我们将开发的技术应用于非线性(简并)抛物线方程。
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