Well-posedness for the fifth-order shallow water equations

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 (u^k), k = 2, 3, μ u ∂_x² u or μ ∂_x u ∂_x² u respectively. The local well-posedness is established for data in H^s (R) with s > - frac(7, 4) for the Kawahara equation (F = μ ∂_x (u²)) and is established for data in H^s (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 H^s (R) with s > 0 if F = μ u ∂_x² u and is established for data in H^s (R) with s > - frac(1, 4) if F = μ ∂_x u ∂_x² u, respectively. © 2008 Elsevier Inc. All rights reserved.