## Differential and Integral Equations

### Continuity properties of the data-to-solution map for the Periodic $b$-family equation

Katelyn Grayshan

#### Abstract

For Sobolev exponents greater than $3/2$, it is proved that the data-to-solution map for the $b$-family equation is continuous from $H^s({\mathbb{T}})$ into $C([0,T]; H^s({\mathbb{T}}))$ but not uniformly continuous. The proof of non-uniform dependence on initial data is based on approximate solutions and delicate commutator and multiplier estimates. The novelty of the proof lies in the fact that it makes no use of conserved quantities. Furthermore, it is shown that the solution map is Hölder continuous in a weaker topology.

#### Article information

Source
Differential Integral Equations, Volume 25, Number 1/2 (2012), 1-20.

Dates
First available in Project Euclid: 20 December 2012

https://projecteuclid.org/euclid.die/1356012822

Mathematical Reviews number (MathSciNet)
MR2906543

Zentralblatt MATH identifier
1249.35287

Subjects
Grayshan, Katelyn. Continuity properties of the data-to-solution map for the Periodic $b$-family equation. Differential Integral Equations 25 (2012), no. 1/2, 1--20. https://projecteuclid.org/euclid.die/1356012822