Extensions for Systems of Conservation Laws
Extensions for Systems of Conservation Laws
复制标题
守恒定律体系的扩展
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
I. Kogan
中科院分区：
文献类型：
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作者：
H. Jenssen;I. Kogan
延拓是守恒律理论的核心，它为弱解提供了内在的选择标准。在给定系统ut+f(U)x=0的情况下，延拓求解通常超定的某些二阶偏微分方程组。确定扩展集的大小可能是一项具有挑战性的任务。我们不是直接分析这个二阶系统，而是考虑由雅可比矩阵df的特征向量ri的长度βi满足的方程，用由扩张定义的内积来测量。对于给定的特征框架{ri}，扩张由这些长度唯一地确定，直到平凡的仿射部分。βI求解一阶代数-微分系统(β-系统)，标准的可积性定理可以应用于该系统。扩展的数量通过确定β系统的一般解中的自由常量和函数的数量来确定。我们给出了3×3-系的完全分解，以及β-系具有平凡代数部分的富框架的完全分解。我们的框架是受工作[16]的启发，其中作者考虑了具有指定特征框架的保守系统。人们很自然地会问，最终的系统有多少扩展，而答案在很大程度上取决于规定的框架。
Extensions are central in the theory of conservation laws by providing intrinsic selection criteria for weak solutions. Given a system u t + f(u) x = 0 the extensions solve certain second order PDEs which are typically overdetermined. Determining the size of the set of extensions can be a challenging task. Instead of analyzing this second order system directly, we consider the equations satisfied by the lengths β i of the eigenvectors r i of the Jacobian matrix Df, measured with the inner product defined by an extension. For a given eigen-frame {r i } the extensions are determined uniquely, up to trivial affine parts, by these lengths. The β i solve a first order algebraic-differential system (the β-system) to which standard integrability theorems can be applied. The number of extensions is determined by determining the number of free constants and functions in the general solution to the β-system. We provide a complete breakdown for 3 × 3-systems, and for rich frames whose β-system has trivial algebraic part. Our framework is motivated by the work [16] where the authors consider conservative systems with prescribed eigen-frames. It is natural to ask how many extensions the resulting systems have, and the answer depends in an essential manner on the prescribed frame.
影响因子：
2.5
作者：
De Lellis, Camillo;Szekelyhidi, Laszlo, Jr.
通讯作者：
Szekelyhidi, Laszlo, Jr.