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The Cauchy problem on large time for a Boussinesq-Peregrine equation with large topography variations. (English) Zbl 1372.35231

Summary: We prove, in this paper, a long time existence result for a modified Boussinesq-Peregrine equation in dimension \(1\), describing the motion of water waves in shallow water, in the case of a non flat bottom. More precisely, the dimensionless equations depend strongly on three parameters \(\epsilon,\mu,\beta\) measuring the amplitude of the waves, the shallowness and the amplitude of the bathymetric variations, respectively. For the Boussinesq-Peregrine model, one has small amplitude variations (\(\epsilon = O(\mu)\)). We first give a local existence result for the original Boussinesq Peregrine equation as derived by J. Boussinesq [Liouville J. (2) 17, 167–176 (1872; JFM 04.0512.02); C. R. Acad. Sci., Paris 73, 256–260 (1871; JFM 03.0486.02)] and D. H. Peregrine [J. Fluid Mech. 27, 815–827 (1967; Zbl 0163.21105)] in all dimensions. We then introduce a new model which has formally the same precision as the Boussinesq-Peregrine equation, and give a local existence result in all dimensions. We finally prove a local existence result on a time interval of size \(\frac{1}{\epsilon}\) in dimension \(1\) for this new equation, without any assumption on the smallness of the bathymetry \(\beta\), which is an improvement of the long time existence result for the Boussinesq systems in the case of flat bottom (\(\beta=0\)) by J.-C. Saut and L. Xu [J. Math. Pures Appl. (9) 97, No. 6, 635–662 (2012; Zbl 1245.35090)].

MSC:

35Q35 PDEs in connection with fluid mechanics
35B25 Singular perturbations in context of PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
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