Null Lagrangian Measures in Subspaces, Compensated Compactness and Conservation Laws
Null Lagrangian Measures in Subspaces, Compensated Compactness and Conservation Laws
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DOI：
10.1007/s00205-019-01403-7
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发表时间：
2019-11-01
影响因子：
2.5
通讯作者：
Peng, Guanying
中科院分区：
文献类型：
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作者：
Lorent, Andrew;Peng, Guanying
补偿紧性是求解非线性偏微分方程的重要方法，特别是在双曲守恒定律的研究中。补偿紧性问题的最简单的表述之一是要求紧集 K. Mmxn 上的条件，使得 lim j.8 dist(Du j, K) L p = 0 且 sup j u j W1, p < 8。{Du j} j 是 L p 中的预紧。 (1) 设M1，M2，...，Mq 表示Mmxn 的所有次数的集合。 (1) 的充分条件是，对于所有 k (2)，K 满足 Mk (X) d mu(X) = Mk Xd mu(X) 的任何概率测度 mu 都是狄拉克测度。我们将满足 (2) 空拉格朗日测度的测度称为空拉格朗日测度，并根据[21]，我们将 K 上支持的空拉格朗日测度集表示为 Mpc(K)。对于一般的 m、n，即使在 K 是 Mmxn 的线性子空间的情况下，Mpc(K) 的平凡性的充分必要条件也是一个悬而未决的问题。我们回答这个问题并提供任意线性子空间K.Mmxn的充要条件。这些想法还允许我们证明对于任何 d。 {1, 2, 3}，d 维子空间 K。当且仅当 K 具有 Rank-1 连接时，Mmxn 支持非平凡的空拉格朗日测度。从 [5] 可知，对于 d = 4，这是错误的。进一步利用所开发的想法，我们能够回答 Kirchheim 等人的问题。 [18]。对于某个函数 a 及其原语 F，设 P1(u, v) :=.. u v a(v) u ua(v) 1 2 u2 + F(v).. 和 K1 := {P1(u, v) : u, v. IR} 。集合 K1 出现在研究 2 x 2 守恒定律系统 ut = a(v) x 和 vt = ux 的熵解的过程中。在[18]中，作者询问函数 a 的条件是什么，使得 Mpc(K1 n U) 由狄拉克测度组成，其中 U 是 K1 中任意矩阵的开邻域。给定 a = (a1, a2)。 IR2，如果 a (a2) > 0，那么我们对于任何 d > 0 构造非平凡测度 inMpc(K1 n Bd (P1(a)))。另一方面，如果 a (a2) < 0 那么对于足够小的 d > 0，我们表明 Mpc(K1 n Bd (P1(a))) 由狄拉克测度组成。
Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the study of hyperbolic conservation laws. One of the simplest formulations of a compensated compactness problem is to ask for conditions on a compact set K. Mmxn such that lim j.8 dist(Du j, K) L p = 0 and sup j u j W1, p < 8. {Du j} j is precompact in L p. (1) Let M1, M2,..., Mq denote the set of all minors of Mmxn. A sufficient condition for (1) is that any probability measure mu supported on K satisfying Mk (X) d mu(X) = Mk Xd mu(X) for all k (2) is a Dirac measure. We call measures that satisfy (2) Null Lagrangian Measures and following [ 21], we denote the set of Null Lagrangian Measures supported on K byMpc(K). For general m, n, a necessary and sufficient condition for triviality ofMpc(K) was an open question even in the case where K is a linear subspace of Mmxn. We answer this question and provide a necessary and sufficient condition for any linear subspace K. Mmxn. The ideas also allow us to show that for any d. {1, 2, 3}, d- dimensional subspaces K. Mmxn support non- trivial Null Lagrangian Measures if and only if K has Rank- 1 connections. This is known to be false for d = 4 from [ 5]. Further using the ideas developed we are able to answer a question of Kirchheim et al. [ 18]. Let P1(u, v) :=.. u v a(v) u ua(v) 1 2 u2 + F(v).. and K1 := {P1(u, v) : u, v. IR} for some function a and its primitive F. The set K1 arises in the study of entropy solutions to the 2 x 2 system of conservation laws ut = a(v) x and vt = ux. In [ 18], the authors asked what are the conditions on the function a such that Mpc(K1 n U) consists of Dirac measures, where U is an open neighborhood of an arbitrary matrix in K1. Given a = (a1, a2). IR2, if a (a2) > 0 then we construct non-trivial measures inMpc(K1 n Bd (P1(a))) for anyd > 0. On the other hand if a (a2) < 0 then for sufficiently smalld > 0, we show thatMpc(K1 n Bd (P1(a))) consists of Dirac measures.