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Local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation in Besov spaces

Gang WuJia Yuan — 2007

Applicationes Mathematicae

We study local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation t u - ³ t x x u + 2 κ x u + x [ g ( u ) / 2 ] = γ ( 2 x u ² x x u + u ³ x x x u ) for the initial data u₀(x) in the Besov space B p , r s ( ) with max(3/2,1 + 1/p) < s ≤ m and (p,r) ∈ [1,∞]², where g:ℝ → ℝ is a given C m -function (m ≥ 4) with g(0)=g’(0)=0, and κ ≥ 0 and γ ∈ ℝ are fixed constants. Using estimates for the transport equation in the framework of Besov spaces, compactness arguments and Littlewood-Paley theory, we get a local well-posedness result.

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