Stable solutions of the length-preserving Helfrich flow under pressure

Determine whether, for every prescribed pressure P, the one-dimensional length-preserving Helfrich flow with external pressure admits stable steady solutions other than the stable circular or two-fold wrinkling branches, and identify such solutions if they exist.

Background

For the purely mechanical one-dimensional Helfrich model with external osmotic pressure and fixed length, numerical continuation finds the circle and the two-fold wrinkling branch as the only stable branches. Higher-mode branches remain unstable through the computed continuation range, and direct numerical simulations from perturbed higher-mode states converge to the two-fold solution.

The authors consequently leave unresolved whether additional stable solutions can be obtained for a given pressure by methods or initial conditions not explored in their continuation and DNS calculations.

References

Thus, by continuation and bifurcation we can only find $\mS1$ and $\CW_2$ as stable branches, and this raises the question, if for any given $P$ we can find other stable solutions via (length $L=2\pi$ preserving) DNS.

— Breathing and moving vesicles in a geometric mechanochemical model  (2609.09360 - Meiners et al., 8 Sep 2026) in Section 1.2, subsection “Intermezzo, β=0: Destabilization via external pressure”