Self-similar solutions to the compressible Euler equations and their instabilities

Biasi, Anxo

doi:10.1016/j.cnsns.2021.106014

Cited by 11 publications

(3 citation statements)

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“…The smoothness of the self-similar solutions across the sonic point is in fact a crucial ingredient in the proof of their (finite codimension) nonlinear stability [24]. For some related numerical results see the subsequent work [1].…”

Section: 21)mentioning

confidence: 98%

“…We will assume ε > 0 is small enough so that 2−γ −C 0 ε > 3 4 (2−γ ) > 4−3γ 3 . Note that if y ∈ F, then as y * ∈ Y and ω(y) > 4−3γ 3 , we must have ρ(y) > ρ(y 0 ) by Lemma 4.9, and so also 1 G(y) < α. Our goal is to prove that F = (s(y * ), y 0 ] (by showing that F is relatively open in (s(y * ), y 0 ]).…”

Section: Proof (Proof Of Lemma 418)mentioning

confidence: 99%

“…) by the following interval arithmetic: quad9(w,g) = (−6 * g^2+8 * g+54) * w^2+(−6 * g^3+16 * g^2+2 * g−16) * w +3 * (g−1) * (2−g) * (g^2−g+2) q9(v)=quad9(v[1]−v[2] ,v[2]) minimise(q9,V, tol=1e−3) shows a minimum in the range [1.5754, 1.58222], concluding the proof.C.3. Proof ofLemma 3.We prove(3.16) in the case m = 1.…”

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confidence: 99%

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Gravitational Collapse for Polytropic Gaseous Stars: Self-Similar Solutions

Guo

Hadžić

Jang

et al. 2022

Arch Rational Mech Anal

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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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Section: 21)mentioning

confidence: 98%

Section: Proof (Proof Of Lemma 418)mentioning

confidence: 99%