## Differential and Integral Equations

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

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

#### 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

**Permanent link to this document**

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

**Mathematical Reviews number (MathSciNet)**

MR2906543

**Zentralblatt MATH identifier**

1249.35287

**Subjects**

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]

#### Citation

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