Classical Euler flows generate the strong Guderley imploding shock wave

TL;DR

The paper rigorously constructs a basin of attraction for the Guderley imploding shock by evolving shock-free radial data backward in time to a preshock cusp, then forward to recover the classical Guderley profile at a finite time. It introduces a symmetric, DRV-focused framework and an auxiliary shock-trajectory control via a function $g(t)$ that governs shock strength decay, enabling a controlled transition from strong to preshock states. A detailed bootstrap analysis in both exterior and interior regions, RH inversion, and a Goursat-type exterior problem establish well-posed backward evolution and precise regularity of the preshock—$z$ exhibits a $C^{1/3}$ cusp, while $w$ and $b$ achieve $C^{1,1/3}$ regularity. Finally, a backward regularization step using fast acoustic coordinates converts the preshock cusp into $C^{1,1/3}$ initial data, delivering a global-in-time Euler solution that evolves from regular initial data into the Guderley implosion and its reflected blast, thereby linking smooth data to the canonical strong shock regime with rigorous PDE control.

Abstract

We prove that Guderley's self-similar imploding shock solution for the compressible Euler equations with ideal--gas law ($γ>1$) arises from classical, radially symmetric, shock--free data. For such data prescribed at initial time $\mathrm{T_{in}} < 0$, we prove that the flow remains smooth up to a first singular time $t=\mathrm{T}_* \in (\mathrm{T_{in}}, 0)$, where a preshock forms with a $C^{1/3}$ cusp in the fast acoustic variable. From this preshock a unique, initially weak, regular shock is born, whose strength can be made arbitrarily large on a controlled time interval; the front then deforms onto the Guderley shock and implodes at the origin at the collapse time $t=0$. There exists a matching time $t=\mathrm{T_{fin}} \in (\mathrm{T}_*,0)$ such that on $[\mathrm{T_{fin}},0)$ the solution coincides exactly with the classical Guderley self--similar profile, and at $t=\mathrm{T_{fin}}$ the shock trajectory matches the self--similar front to all orders. As $t \to 0^-$, the Euler solution implodes at the center, and continues for $t>0$ as a reflected blast wave, providing a global-in-time unique Euler solution which evolves from regular initial conditions.

Figures (7)

• Figure 1: A schematic illustrating the dual nature of the Guderley solution. At early stages (right), various initial conditions are attracted towards the universal self-similar solution. At late stages (left), as the shock approaches collapse, the inherent instability of this solution amplifies any small residual perturbation, causing the flow to diverge from the ideal path.
• Figure 2: A schematic of the converging shock front. The shock moves left into the quiescent Region 1 (the core), leaving behind the compressed, hot gas of Region 2 (the outer layer it has passed through).
• Figure 3: The global picture showing the main result. The initial data is specified at time $t=\mathrm{T_{in}}$; this data is $C^2$-smooth except for the points $\{r_{\mathrm{in}}^{(i)} \}_{i=2}^{3}$, which are the backwards-in-time images of the preshock along the wave speeds $\{\lambda_i\}_{i=2}^3$; at these locations, the initial data has $C^{1,{\frac{1}{3}}}$ regularity. Shock formation occurs on the time interval $[\mathrm{T_{in}},\mathrm{T}_*)$. At time $t=\mathrm{T}_*$ and position $r=r_*$ a $C^{\frac{1}{3}}$ preshock develops for the Riemann-variable $z$; the Riemann-variables $w$ and $b$ remain $C^{1,{\frac{1}{3}}}$-smooth. Shock development occurs on the time interval $(\mathrm{T}_*,0)$; this interval is sub-divided intro three pieces. On $(\mathrm{T}_*,\mathrm{T}_{\! \circ})$ we transition from the preshock into a weak-shock, for all three Riemann variables; on either side of the shock surface $\{r = \mathsf{s}(t)\}$ the solution is $C^\infty$ smooth in space. On $(\mathrm{T}_{\! \circ},\mathrm{T_{fin}})$ we transition from a weak-shock into the exact Guderley shock state (the self-similar profile evaluated at $\mathrm{T_{fin}}$). On $(\mathrm{T_{fin}},0)$ the dynamics is precisely described by the exact Guderley imploding shock. We emphasize that in the light blue region of spacetime, the solution is at "rest", meaning that velocity and sound speed equal vanish identically, while the density is a constant. In the yellow region of spacetime, the solution exactly coincides with the Guderley solution, when restricted to that spacetime. We also emphasize that the shock surface $\{r=\mathsf{s}(t) \colon t\in (\mathrm{T}_{\! \circ},\mathrm{T_{fin}})\} \cup \{ r = {\mathsf{c}} (t) \colon t \in (\mathrm{T}_*, \mathrm{T}_{\! \circ}]\}$ cannot be determined solely by knowledge of the Guderley data along the time-slice$\{t = \mathrm{T_{fin}}\}$. Hence, we choose the shock surface $\{ r=\mathsf{s}(t) \colon t \in (\mathrm{T}_{\! \circ}, \mathrm{T_{fin}})\}\cup \{ r = {\mathsf{c}} (t) \colon t \in (\mathrm{T}_*, \mathrm{T}_{\! \circ}]\}$ and the trace $\lambda_1( {\mathsf{c}} (t)^+, t)$ for $t \in (\mathrm{T}_*, \mathrm{T}_{\! \circ})$. This gives us all the data required in order to uniquely solve the Euler equations in the white region.
• Figure 4: In thick red, we have displayed the classical Guderley shock curve $\mathsf{g}$ for $t > \mathrm{T_{fin}}$. In thin red, we have continued the Guderley shock curve for $t \in [\mathrm{T}_*,\mathrm{T_{fin}}]$, in order to emphasize the relative placement of the new, chosen, shock curve $\mathsf{s}$, which we have drawn in orange. The green $\psi$ characteristic is the backwards-in-time fast-acoustic characteristic emanating from $(\mathsf{g}(\mathrm{T_{fin}})^+,\mathrm{T_{fin}}) = (\mathsf{s}(\mathrm{T_{fin}})^+,\mathrm{T_{fin}})$; the intersection of this characteristic curve with $\{t = \mathrm{T}_*\}$ imposes an upper bound for the choice of $\mathsf{s}(\mathrm{T}_*)$. In this step, we solve the Euler equations in the white shaded region $\Omega^-$, by tracing back characteristics backwards-in-time to $\{t = \mathrm{T}_*\}$.
• Figure 5: We have drawn in black the new shock curve ${\mathsf{c}}$, while in orange we have displayed the old shock curve $\mathsf{s}$. In this step we solve the equations in the light-pink shadowed region $\mathcal{D}$. The boundary value for $\lambda_1$ is given along the black ${\mathsf{c}}$ curve, while the data for $w, b$ is given along the green $\psi$ characteristic in green.

Theorems & Definitions (79)

• Definition 1.1: Regular shock solution BuDrShVi2022
• Definition 1.2: Regular shock solution emanating from a preshock
• Theorem 1.3
• Corollary 1.4: Global-in-time Euler evolution from classical data
• proof : Proof of \ref{['cor:global']}
• Remark 1.5: Sharp Regularity of the Initial Data: Finitely Differentiable Structure
• Remark 1.6: Cancellation of Weak Characteristic Singularities
• Remark 1.7: Non-uniqueness of the initial data
• Remark 1.8: Forward-in-time uniqueness
• Remark 1.9: Qualitative nature of the initial data