### Abstract

The paper studies the possible blowup of the total variation for entropy weak solutions of the p-system, modeling isentropic gas dynamics. It is assumed that the density remains uniformly positive, while the initial data can have arbitrarily large total variation (measured in terms of Riemann invariants). Two main results are proved. (I) If the total variation blows up in finite time, then the solution must contain an infinite number of large shocks in a neighborhood of some point in the t-x plane. (II) Piecewise smooth approximate solutions can be constructed whose total variation blows up in finite time. For these solutions the strength of waves emerging from each interaction is exact, while rarefaction waves satisfy the natural decay estimates stemming from the assumption of genuine nonlinearity.

Original language | English (US) |
---|---|

Pages (from-to) | 1242-1280 |

Number of pages | 39 |

Journal | Communications in Partial Differential Equations |

Volume | 43 |

Issue number | 8 |

DOIs | |

State | Published - Aug 3 2018 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

*Communications in Partial Differential Equations*,

*43*(8), 1242-1280. https://doi.org/10.1080/03605302.2018.1499115

}

*Communications in Partial Differential Equations*, vol. 43, no. 8, pp. 1242-1280. https://doi.org/10.1080/03605302.2018.1499115

**On finite time BV blow-up for the p-system.** / Bressan, Alberto; Chen, Geng; Zhang, Qingtian.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On finite time BV blow-up for the p-system

AU - Bressan, Alberto

AU - Chen, Geng

AU - Zhang, Qingtian

PY - 2018/8/3

Y1 - 2018/8/3

N2 - The paper studies the possible blowup of the total variation for entropy weak solutions of the p-system, modeling isentropic gas dynamics. It is assumed that the density remains uniformly positive, while the initial data can have arbitrarily large total variation (measured in terms of Riemann invariants). Two main results are proved. (I) If the total variation blows up in finite time, then the solution must contain an infinite number of large shocks in a neighborhood of some point in the t-x plane. (II) Piecewise smooth approximate solutions can be constructed whose total variation blows up in finite time. For these solutions the strength of waves emerging from each interaction is exact, while rarefaction waves satisfy the natural decay estimates stemming from the assumption of genuine nonlinearity.

AB - The paper studies the possible blowup of the total variation for entropy weak solutions of the p-system, modeling isentropic gas dynamics. It is assumed that the density remains uniformly positive, while the initial data can have arbitrarily large total variation (measured in terms of Riemann invariants). Two main results are proved. (I) If the total variation blows up in finite time, then the solution must contain an infinite number of large shocks in a neighborhood of some point in the t-x plane. (II) Piecewise smooth approximate solutions can be constructed whose total variation blows up in finite time. For these solutions the strength of waves emerging from each interaction is exact, while rarefaction waves satisfy the natural decay estimates stemming from the assumption of genuine nonlinearity.

UR - http://www.scopus.com/inward/record.url?scp=85061636612&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85061636612&partnerID=8YFLogxK

U2 - 10.1080/03605302.2018.1499115

DO - 10.1080/03605302.2018.1499115

M3 - Article

AN - SCOPUS:85061636612

VL - 43

SP - 1242

EP - 1280

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 8

ER -