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Creators/Authors contains: "Schrecker, Matthew"

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  1. ; ; (, Archive for Rational Mechanics and Analysis)
    Abstract In this paper, we rigorously prove the existence of self-similar converging shock wave solutions for the non-isentropic Euler equations for$$\gamma \in (1,3]$$ γ ∈ ( 1 , 3 ] . These solutions are analytic away from the shock interface before collapse, and the shock wave reaches the origin at the time of collapse. The region behind the shock undergoes a sonic degeneracy, which causes numerous difficulties for smoothness of the flow and the analytic construction of the solution. The proof is based on continuity arguments, nonlinear invariances, and barrier functions. 
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  2. ; ; (, SIAM Journal on Mathematical Analysis)
    Free, publicly-accessible full text available February 28, 2026
  3. ; ; ; (, Archive for Rational Mechanics and Analysis)
    Abstract In the supercritical range of the polytropic indices$$\gamma \in (1,\frac{4}{3})$$ γ ∈ ( 1 , 4 3 ) we show the existence of smooth radially symmetric self-similar solutions to the gravitational Euler–Poisson system. These solutions exhibit gravitational collapse in the sense that the density blows up in finite time. Some of these solutions were numerically found by Yahil in 1983 and they can be thought of as polytropic analogues of the Larson–Penston collapsing solutions in the isothermal case$$\gamma =1$$ γ = 1 . They each contain a sonic point, which leads to numerous mathematical difficulties in the existence proof. 
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