Entropy methods for nonstrictly hyperbolic systems. (English summary) 10 (1997), no. 4, 333–346.

In this paper we examine the nonstrictly hyperbolic wave equation ytt−σ(yx)x=0 under the following conditions: (H1) σ∈C3(R,R), σ(0)=σ′(0)=σ′′(0)=0; (H2) σ′(ξ)>0 for ξ≠0; (H3) ξσ′′(ξ)>0 for ξ≠0; (H4) for every ε>0 there exists a constant c>0 such that |σ′′(ξ)|≤cσ′(ξ) when |ξ|>ε. This problem is a model example for nonstrictly hyperbolic systems with degeneracy along a sonic line. The analysis consists of investigating the convergence of approximating solutions (in the sense of DiPerna and Tartar) satisfying the Kruzhkov-Lax entropy inequality. An examination of the classical Lax entropies, carried out by studying the asymptotics of a related Fuchs-type equation, shows these to become singular when yx=0, and the coefficients of the asymptotic series for the progressing waves then diverge at the sonic line. Then we are led to consider instead energy estimates based on the mechanical entropy (which are not equivalent to L2 norms because of the condition that σ′(0)=0). By constructing an appropriate antisymmetric form, and using in part third-order derivatives of the entropy-entropy flux pairs to ensure coercivity, the related Young measure is shown to reduce to a Dirac mass if it is contained in {(x,t): yx≥0}. The Young mass cannot be expected to be Dirac in general due to the existence of time-periodic solutions.