Smooth self-similar imploding profiles to 3D compressible Euler

Abstract

The aim of this note is to present the recent results by Buckmaster, Cao-Labora, and Gómez-Serrano [Smooth imploding solutions for 3D compressible fluids, Arxiv preprint arXiv:2208.09445, 2022] concerning the existence of “imploding singularities” for the 3D isentropic compressible Euler and Navier-Stokes equations. Our work builds upon the pioneering work of Merle, Raphaël, Rodnianski and Szeftel [Invent. Math. 227 (2022), pp. 247–413; Ann. of Math. (2) 196 (2022), pp. 567–778; Ann. of Math. (2) 196 (2022), pp. 779–889] and proves the existence of self-similar profiles for all adiabatic exponents γ > 1 \gamma >1 in the case of Euler; as well as proving asymptotic self-similar blow-up for γ = 7 5 \gamma =\frac 75 in the case of Navier-Stokes. Importantly, for the Navier-Stokes equation, the solution is constructed to have density bounded away from zero and constant at infinity, the first example of blow-up in such a setting. For simplicity, we will focus our exposition on the compressible Euler equations.

The relativistic Euler equations: ESI notes on their geo-analytic structures and implications for shocks in 1D and multi-dimensions

“…It follows that h( X, X) = µ 2 . With the help of (19), one can show that XU = 1. That is, X is a geometric replacement 14 for ∂ ∂U , and among the elements of Z , it is the one vectorfield that is transversal to the characteristics.…”

“…In (198), A 1 is the universal positive constant mentioned above and • • • denotes similar or easier 19 error terms. From (198) and Grönwall's inequality, we deduce that:…”

“…In the breakthrough works [73,74], the authors proved implosion singularity formation for some initially C ∞ spherically symmetric solutions to the compressible Euler equations and Navier-Stokes equations under adiabatic equations of state p = ρ γ for γ > 1, outside of a countable set of γ-values. In [19], the results were extended to allow for all γ > 1. Implosion singularities are much more severe than shocks in the sense that the density itself blows up in finite time at the center of symmetry (in shock singularities, the density's gradient blows up but the density itself remains bounded).…”