Full stability of smooth isentropic implosion profiles

Determine whether the smooth self-similar implosion profiles constructed for the isentropic compressible Euler equations are fully stable, rather than merely stable on a finite-codimension manifold, by characterizing the dimension of their unstable manifold.

Background

The paper reviews smooth self-similar isentropic implosion profiles constructed for the compressible Euler equations and notes that existing stability results establish stability only in a finite-codimension weighted Sobolev setting. Those results do not quantify the dimension of the unstable manifold.

Numerical evidence cited in the paper indicates that the profiles may be genuinely unstable even in radial symmetry. Consequently, the broader question of full stability remains explicitly unresolved.

References

We observe that none of these works (both in and outside symmetry) provide any quantitative bound (lower or upper) on the dimension of the unstable manifold, in particular leaving open the question of full stability.

— Nonlinear stability of the Guderley imploding shock wave in radial symmetry  (2610.01690 - Cialdea, 1 Oct 2026) in Section 1, subsection “Isentropic smooth implosions”