H\"older Continuity of the Data to Solution Map for HR in the Weak Topology

TL;DR

This paper proves that the data-to-solution map for the hyperelastic rod equation exhibits Hölder continuity when measured in weaker Sobolev norms, using energy and commutator estimates.

Contribution

It establishes Hölder continuity of the data-to-solution map for the hyperelastic rod equation in weaker Sobolev topologies, extending understanding of solution stability.

Findings

Hölder continuity holds for Sobolev exponents s > 3/2

Continuity is valid in both periodic and non-periodic cases

Proof relies on energy estimates and commutator estimates

Abstract

It is shown that the data to solution map for the hyperelastic rod equation is H\"older continuous from bounded sets of Sobolev spaces with exponent $s > 3/2$ measured in a weaker Sobolev norm with index $r < s$ in both the periodic and non-periodic cases. The proof is based on energy estimates coupled with a delicate commutator estimate and multiplier estimate.