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Smooth imploding solutions for 3D compressible fluids

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arxiv 2208.09445 v2 pith:CW5C7HPG submitted 2022-08-19 math.AP math-phmath.MP

Tristan Buckmaster , Gonzalo Cao-Labora , Javier Gómez-Serrano This is my paper · ORCID
classification math.APmath-phmath.MP
keywords solutionsimplodingcompressiblegammamathself-similarsmoothcase
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abstract

Building upon the pioneering work [Merle, Rapha\"el, Rodnianski, and Szeftel, Ann. of Math., 196(2):567-778, 2022, Ann. of Math., 196(2):779-889, 2022, Invent. Math., 227(1):247-413, 2022] we construct exact, smooth self-similar imploding solutions to the 3D isentropic compressible Euler equations for ideal gases for all adiabatic exponents $\gamma>1$. For the particular case $\gamma=\frac75$ (corresponding to a diatomic gas, e.g. oxygen, hydrogen, nitrogen), akin to the previous result, we show the existence of a sequence of smooth, self-similar imploding solutions. In addition, we provide simplified proofs of linear stability and non-linear stability, which allow us to construct asymptotically self-similar imploding solutions to the compressible Navier-Stokes equations with density independent viscosity for the case $\gamma=\frac75$. Moreover, the solutions constructed have density bounded away from zero and converge to a constant at infinity, representing the first example of singularity formation in such a setting.

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Cited by 2 Pith papers

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  1. Blow-up of the 3-D compressible Navier-Stokes equations for monatomic gases

    math.AP 2025-01 conditional novelty 8.0 of 10

    For gamma = 5/3, corresponding to a monatomic gas, there exist smooth initial data for which the 3-D compressible Navier-Stokes equations blow up in finite time in a self-similar implosion.

  2. Gradient catastrophes and an infinite hierarchy of H\"older cusp-singularities for 1D Euler

    math.AP 2024-12 conditional novelty 8.0 of 10

    For every integer n>=1, smooth 1D Euler data can form a pre-shock cusp with Holder exponent 1/(2n+1), and the set of such data is a codimension-(2n-2) Banach manifold in W^{2n+2,infinity}.

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