On the well-posedness of the hyperelastic rod equation

It is shown that the data-to-solution map for the hyperelastic rod equation is not uniformly continuous on bounded sets of Sobolev spaces with exponent greater than 3/2 in the periodic case and non-periodic cases. The proof is based on the method of approximate solutions and well-posedness estimates for the solution and its lifespan. Building upon this work, we also prove 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 exponent 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.