No TVD Fields for 1-D Isentropic Gas Flow

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Abstract

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.

Original languageEnglish (US)
Pages (from-to)629-657
Number of pages29
JournalCommunications in Partial Differential Equations
Volume38
Issue number4
DOIs
StatePublished - Apr 1 2013

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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