We consider the initial-value problem for the bidirectional Whitham equation, a system which combines the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow-water nonlinearity. We prove local well-posedness in classical Sobolev spaces in the localised as well as the periodic case, using a square-root type transformation to symmetrise the system. The existence theory requires a non-vanishing surface elevation, indicating that the problem is ill-posed for more general initial data.2010 Mathematics Subject Classification. 76B15; 76B03, 35S30, 35A20. Key words and phrases. Whitham-type equations, dispersive equations, well-posedness. M.E. and Y.W. acknowledge the support by grants nos. 231668 and 250070 from the Research Council of Norway. shock waves developing from initial depressions in shallow water, Phys. D, 333 (2016), pp. 276-284.
This paper is concerned with decay and symmetry properties of solitary wave
solutions to a nonlocal shallow water wave model. It is shown that all
supercritical solitary wave solutions are symmetric and monotone on either side
of the crest. The proof is based on a priori decay estimates and the method of
moving planes. Furthermore, a close relation between symmetric and traveling
wave solutions is established
Abstract. This paper studies the Cauchy problem for systems of semi-linear wave equations on R 3+1 with nonlinear terms satisfying the null conditions. We construct future global-in-time classical solutions with arbitrarily large initial energy. The choice of the large Cauchy initial data is inspired by Christodoulou's characteristic initial data in his work [2] on formation of black-holes. The main innovation of the current work is that we discovered a relaxed energy ansatz which allows us to prove decay-in-time-estimate. Therefore, the new estimates can also be applied in studying the Cauchy problem for Einstein equations.
We consider the initial-value problem for the bidirectional Whitham equation, a system which combines the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow-water nonlinearity. We prove local well-posedness in classical Sobolev spaces, using a square-root type transformation to symmetrise the system.
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