Well-posedness for the fifth-order shallow water equations

作者：Jia Y; Huo Z^*

来源：Journal of Differential Equations, 2009, 246(6): 2448-2467.

DOI：10.1016/j.jde.2008.10.027

• Bilinear estimates
• Fifth-order shallow water equations
• Fourier restriction norm
• Local well-posedness
• Trilinear estimates

The Cauchy problems for some kind of fifth-order shallow water equations∂t u + α ∂x5 u + β ∂x3 u + γ ∂x u + F (u, ∂x u, ∂x2 u) = 0, x, t ∈ R × R, are considered by the Fourier restriction norm method, where nonlinear terms F (u, ∂x u, ∂x2 u) are μ ∂x (uk), k = 2, 3, μ u ∂x2 u or μ ∂x u ∂x2 u respectively. The local well-posedness is established for data in Hs (R) with s - frac(7, 4) for the Kawahara equation (F = μ ∂x (u2)) and is established for data in Hs (R) with s ≥ - frac(1, 4) for the modified Kawahara equation (F = μ ∂x (u3)), respectively. Moreover, the local result is established for data in Hs (R) with s 0 if F = μ u ∂x2 u and is established for data in Hs (R) with s - frac(1, 4) if F = μ ∂x u ∂x2 u, respectively.