Derivation of numerical regularity bounds from smooth initial data

Determine whether the discrete regularity assumptions imposed on the numerical membrane and protein solutions can be established from sufficiently smooth initial data for the stabilized alternating ETD1 and Strang-ETDRK2 schemes applied to the coupled multicomponent-vesicle phase-field system.

Background

The discrete energy-dissipation proofs assume uniform discrete Sobolev bounds for every membrane and protein stage generated by the alternating ETD1 and Strang-ETDRK2 schemes. These assumptions imply the uniform LinftyL^infty bounds needed to control the nonlinear terms in the energy estimates.

The paper states that the relevant uniform bounds are not proved analytically; numerical experiments suggest that they may hold independently of the mesh size and time step. Establishing the assumed regularity from sufficiently smooth initial data would remove a key conditional hypothesis in the discrete energy-stability analysis.

References

Although these uniform bounds are not proved in the current work, systematic numerical experiments for various spatial mesh sizes $h$ and temporal step sizes $\tau$ suggest that such bounds exist. In the future work, we will further investigate whether the regularity assumptions in Assumption~\ref{ass:actual-stage-bounds} can be established from sufficiently smooth initial data.

— A Thermodynamically Consistent Model for Multicomponent Vesicles  (2609.10168 - Luo et al., 9 Sep 2026) in Remark 4.1, Section 4, subsection “Energy Dissipation of the Stabilized Alternating ETD Schemes”