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A new type of stable shock formation in gas dynamics

  • *Corresponding author: Isaac Neal

    *Corresponding author: Isaac Neal 

The third author is supported by NSF grant DMS-2007606, the NSF grant DMS-200760, and the Department of Energy Advanced Simulation and Computing (ASC) Program. The fourth author is supported by the NSF CAREER grant DMS-1911413, and the NSF grant DMS-230768

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  • From an open set of initial data, we construct a family of classical solutions to the 1D nonisentropic compressible Euler equations which form $ C^{0,\nu} $ cusps as a first singularity, for any $ \nu \in [\frac{1}{2}, 1) $. For this range of $ \nu $, this is the first result demonstrating the stable formation of such $ C^{0,\nu} $ cusp-type singularities, also known as pre-shocks. The proof uses a new formulation of the differentiated Euler equations along the fast acoustic characteristic, and relies on a novel set of $ L^p $ energy estimates for all $ 1<p<\infty $, which may be of independent interest.

    Mathematics Subject Classification: Primary: 35L67; Secondary: 74J40.

    Citation:

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