Smooth imploding solutions for 3D compressible fluids

TL;DR

This work advances the understanding of singularity formation in 3D compressible flows by constructing exact, smooth self-similar imploding solutions to the 3D isentropic Euler equations for all γ>1, and a sequence for γ=7/5. The authors develop a robust framework combining Frobenius-type expansions, autonomous ODE phase-space analysis with Riemann invariants, and barrier methods to realize a global, smooth self-similar implosion via a connecting orbit through a regular singular point P_s. They establish linear stability and nonlinear stability results, enabling the construction of asymptotically self-similar Navier–Stokes solutions with density-independent viscosity and density bounded away from zero, converging to a constant at infinity. A key innovation is the use of transport structure in Riemann variables to simplify stability and a computer-assisted confirmation of delicate coefficient signs and barrier crossings, including γ=7/5 with large odd indices. Collectively, the results yield the first rigorous examples of singularity formation of this type in a setting with nontrivial vorticity and non-vanishing density at infinity, and they provide a versatile blueprint for related implosion problems and NS-instability analysis.

Abstract

Building upon the pioneering work [Merle, Raphaël, 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 $γ>1$. For the particular case $γ=\frac75$ (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 $γ=\frac75$. 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.

Figures (5)

• Figure 1: Imploding solutions in $(U,S)$ variables. Note that a singular coordinate change has been made in order to compactify the $(U,S)$ coordinates.
• Figure 2: Imploding solutions in $(W,Z)$ variables. Note that a singular coordinate change has been made in order to compactify the $(W,Z)$ coordinates. We have indicated in orange the type of smooth solutions we will find, crossing through $P_s$ with direction $v_-$. On the left of $P_s$ the solution converges to $P_\infty$, while on the right, we show three possibilities for its behaviour (it can start at $D_W = 0$, at $P_0$ or at $D_Z = 0$).
• Figure 3: Field $(N_W D_Z, N_Z D_W)$ in $(W, Z)$ coordinates for $\gamma = \frac{5}{3}$ and $r = \frac{11}{10}$. The shaded area corresponds to the triangle $\mathcal{T}^{(1)}$.
• Figure 4: Region $\mathcal{T}$ for the case $\gamma = 7/5$ and $r$ sufficiently close to $r^\ast$.
• Figure 5: Region $\mathcal{T}$ for the case $\gamma > 1$ and $r \in (r_3, r_4)$.

Theorems & Definitions (207)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Proposition 1.6
• Remark 1.7
• Lemma 2.1
• proof
• Proposition 2.2