Abstract
This study deals with higher-order asymptotic equations for the water-waves problem. We considered the higher-order/extended Boussinesq equations over a flat bottom topography in the well-known long wave regime. Providing an existence and uniqueness of solution on a relevant time scale of order 1/√ε and showing that the solution’s behavior is close to the solution of the water waves equations with a better precision corresponding to initial data, the asymptotic model is well-posed in the sense of Hadamard. Then we compared several water waves solitary solutions with respect to the numerical solution of our model. At last, we solve explicitly this model and validate the results numerically.
Highlights
The water-wave equations In this paper, we investigate the one-dimensional flow of the free surface of a homogeneous, immiscible fluid moving above a flat topography z = −h0
Under the extra assumption ε ∼ μ, we can neglect the terms which are of order O(μ2) in the Green-Naghdi equations; the standard Boussinesq equations reads:
An inconvenient feature appears in this left-most term due to the positive sign in front of the elliptic forth-order linear operator T which ravel the way towards well-posedness using energy estimate method
Summary
The free surface is parametrized by the graph of the function ζ(t, x) denoting the variation with respect to its rest state z = 0 (see Fig. 1). The fluid occupies the strictly connected (ζ(t, x) + h0 > 0) domain Ωt at time t ≥ 0 denoted by: Ωt = {(x, z) ∈ R2; −h0 ≤ z ≤ ζ(t, x)}.
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