Differential and Integral Equations

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

Katelyn Grayshan

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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

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

First available in Project Euclid: 20 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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