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Formation of Unstable Shocks for 2D Isentropic Compressible Euler

Abstract

In this paper we construct unstable shocks in the context of 2D isentropic compressible Euler in azimuthal symmetry. More specifically, we construct initial data that when viewed in self-similar coordinates, converges asymptotically to the unstable \(C^{\frac{1}{5}}\) self-similar solution to the Burgers’ equation. Moreover, we show the behavior is stable in \(C^8\) modulo a two dimensional linear subspace. Under the azimuthal symmetry assumption, one cannot impose additional symmetry assumptions in order to isolate the corresponding manifold of initial data leading to stability: rather, we rely on modulation variable techniques in conjunction with a Newton scheme.

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Correspondence to Sameer Iyer.

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Communicated by K. Nakanishi

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Tristan Buckmaster partially supported by NSF Grant DMS-1900149 and a Simons Foundation Mathematical and Physical Sciences Collaborative Grant. Sameer Iyer partially supported by NSF Grant DMS-1802940.

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Buckmaster, T., Iyer, S. Formation of Unstable Shocks for 2D Isentropic Compressible Euler. Commun. Math. Phys. 389, 197–271 (2022). https://doi.org/10.1007/s00220-021-04271-z

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  • DOI: https://doi.org/10.1007/s00220-021-04271-z