No TVD Fields for 1-D Isentropic Gas Flow

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 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.