Extensions for Systems of Conservation Laws

Helge Kristian Jenssen, Irina A. Kogan

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)1096-1140
Number of pages45
JournalCommunications in Partial Differential Equations
Volume37
Issue number6
DOIs
StatePublished - Jun 1 2012

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Systems of Conservation Laws
Jacobian matrices
Eigenvalues and eigenfunctions
Conservation
Trivial
Differential-algebraic Systems
Conservative System
Second-order Systems
Jacobian matrix
General Solution
Scalar, inner or dot product
Conservation Laws
Eigenvector
Integrability
Weak Solution
Breakdown
First-order
Theorem

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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Extensions for Systems of Conservation Laws. / Jenssen, Helge Kristian; Kogan, Irina A.

In: Communications in Partial Differential Equations, Vol. 37, No. 6, 01.06.2012, p. 1096-1140.

Research output: Contribution to journalArticle

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