Nonlinear stability of the Guderley imploding shock wave in radial symmetry

We prove nonlinear stability of the Guderley imploding shock under radial perturbations for the three-dimensional compressible Euler equations in the adiabatic range $1<\gamma \le \frac{5}{3}$. The perturbed solutions develop an asymptotically self-similar implosion in finite time. Our result allows perturbations with small, nonzero constant pressure in the quiescent region ahead of the shock. The proof combines weighted $L^{\mathrm{\infty }}$ estimates along characteristics with a computer-assisted proof of stability inequalities for the Guderley profile.