### 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 language | English (US) |
---|---|

Pages (from-to) | 1096-1140 |

Number of pages | 45 |

Journal | Communications in Partial Differential Equations |

Volume | 37 |

Issue number | 6 |

DOIs | |

State | Published - Jun 1 2012 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

*Communications in Partial Differential Equations*,

*37*(6), 1096-1140. https://doi.org/10.1080/03605302.2011.626827

}

*Communications in Partial Differential Equations*, vol. 37, no. 6, pp. 1096-1140. https://doi.org/10.1080/03605302.2011.626827

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - Extensions for Systems of Conservation Laws

AU - Jenssen, Helge Kristian

AU - Kogan, Irina A.

PY - 2012/6/1

Y1 - 2012/6/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84861547404&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84861547404&partnerID=8YFLogxK

U2 - 10.1080/03605302.2011.626827

DO - 10.1080/03605302.2011.626827

M3 - Article

VL - 37

SP - 1096

EP - 1140

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 6

ER -