Abstract
We regard the Cauchy problem for a particular Whitham–Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. The system can be seen as a weak nonlocal dispersive perturbation of the shallow water system. The proof of well-posedness relies on energy estimates. However, due to the symmetry lack of the nonlinear part, in order to close the a priori estimates one has to modify the traditional energy norm in use. Hamiltonian conservation provides with global well-posedness at least for small initial data in the one dimensional settings.
Highlights
Consideration is given to the following one-dimensional Whitham-type system
In case of the trivial surface tension κ = 0, System (1.1) was proposed in [6] as an approximate model for the study of water waves to provide a two-directional alternative to the well-known Whitham equation [23]
For any f ∈ H a(R), g ∈ H b(R) and h ∈ H c(R) the following inequality holds provided that f gh L1 fHagHbhHca
Summary
Where D = −i∂x and tanh D are Fourier multiplier operators in the space of tempered distributions S (R). The positive parameter κ stands for the surface tension here. The space variable is x ∈ R and the time variable is t ∈ R. The unknowns η, v are real valued functions of these variables. We pick the initial values η(0), v(0). Corresponding to the time moment t = 0 in Sobolev spaces as follows η(0) = η0 ∈ H s+1/2(R), v(0) = v0 ∈ H s (R),
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