### Abstract

We prove that isentropic gas flow does not admit non-degenerate TVD fields on any invariant set ℳ(r_{0}, s_{0}) = {r_{0} < r < s < s_{0}}, where r, s are Riemann coordinates. A TVD field refers to a scalar field whose spatial variation Var_{X}(φ{symbol}(τ(t, X), u(t, X))) is non-increasing in time along entropic solutions. The result is established under the assumption that the Riemann problem defined by an overtaking shock-rarefaction interaction gives the asymptotic states in the exact solution.Little is known about global existence of large-variation solutions to hyperbolic systems of conservation laws u_{t} + f(u)_{x} = 0. In particular it is not known if isentropic gas flow admits a priori BV bounds which apply to all BV data. In the few cases where such results are available (scalar case, Temple class, systems satisfying Bakhvalov's condition, isothermal gas dynamics) there are TVD fields which play a key role for existence. Our results show that the same approach cannot work for isentropic flow.

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

Pages (from-to) | 629-657 |

Number of pages | 29 |

Journal | Communications in Partial Differential Equations |

Volume | 38 |

Issue number | 4 |

DOIs | |

State | Published - Apr 1 2013 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

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*Communications in Partial Differential Equations*, vol. 38, no. 4, pp. 629-657. https://doi.org/10.1080/03605302.2012.755543

**No TVD Fields for 1-D Isentropic Gas Flow.** / Chen, Geng; Jenssen, Helge Kristian.

Research output: Contribution to journal › Article

TY - JOUR

T1 - No TVD Fields for 1-D Isentropic Gas Flow

AU - Chen, Geng

AU - Jenssen, Helge Kristian

PY - 2013/4/1

Y1 - 2013/4/1

N2 - We prove that isentropic gas flow does not admit non-degenerate TVD fields on any invariant set ℳ(r0, s0) = {r0 < r < s < s0}, where r, s are Riemann coordinates. A TVD field refers to a scalar field whose spatial variation VarX(φ{symbol}(τ(t, X), u(t, X))) is non-increasing in time along entropic solutions. The result is established under the assumption that the Riemann problem defined by an overtaking shock-rarefaction interaction gives the asymptotic states in the exact solution.Little is known about global existence of large-variation solutions to hyperbolic systems of conservation laws ut + f(u)x = 0. In particular it is not known if isentropic gas flow admits a priori BV bounds which apply to all BV data. In the few cases where such results are available (scalar case, Temple class, systems satisfying Bakhvalov's condition, isothermal gas dynamics) there are TVD fields which play a key role for existence. Our results show that the same approach cannot work for isentropic flow.

AB - We prove that isentropic gas flow does not admit non-degenerate TVD fields on any invariant set ℳ(r0, s0) = {r0 < r < s < s0}, where r, s are Riemann coordinates. A TVD field refers to a scalar field whose spatial variation VarX(φ{symbol}(τ(t, X), u(t, X))) is non-increasing in time along entropic solutions. The result is established under the assumption that the Riemann problem defined by an overtaking shock-rarefaction interaction gives the asymptotic states in the exact solution.Little is known about global existence of large-variation solutions to hyperbolic systems of conservation laws ut + f(u)x = 0. In particular it is not known if isentropic gas flow admits a priori BV bounds which apply to all BV data. In the few cases where such results are available (scalar case, Temple class, systems satisfying Bakhvalov's condition, isothermal gas dynamics) there are TVD fields which play a key role for existence. Our results show that the same approach cannot work for isentropic flow.

UR - http://www.scopus.com/inward/record.url?scp=84875191527&partnerID=8YFLogxK

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U2 - 10.1080/03605302.2012.755543

DO - 10.1080/03605302.2012.755543

M3 - Article

AN - SCOPUS:84875191527

VL - 38

SP - 629

EP - 657

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 4

ER -