Life Span and Consistency of Solutions for Whitham–Boussinesq Type Equations

Abstract

In this dissertation we consider a Whitham-Boussinesq type system modeling surface water waves of an inviscid incompressible fluid layer. The system describes the evolution with time of surface waves of a liquid layer in d +1 dimensional physical space. We are interested in well-posedness at a low level regularity. We diagonalized the system and reduce it to integral form. Then we derived solution propagators. We established a lower bound for derivatives of all order for the Fourier symbol of the pseudo-differential operator that appears in linearized water wave type equations for both purely gravity waves and capillary-gravity waves. We then derived dispersive and Strichartz estimate for linearized water wave type equations in R d . We also derived bilinear estimates which are important to bound nonlinear terms. Using these estimates together with contraction mapping principle we prove low regularity well posedness for Whitham-Boussinesq type system in R d , d ≥ 3. This dissertation also contains a study of the long time existence of solutions of Whitham–Boussinesq type system in one dimensional space. Using fixed point argument we prove that the system is well-posed on the time scale of order O(1/ √ ε), where ε > 0 is the shallowness parameter measuring the ratio of amplitude of the wave to mean depth of fluid. Thus if the amplitude of the wave is very small compared to the mean depth of the fluid that is 0 < ε ≤ 1 then the solution exists for arbitrarily large time. We also show that the solution to the Whitham–Boussinesq type system approximates the solution of a Boussinesq system on the time scale of order O(1/ √ ε) for one dimensional space.