The Construction And Well-posedness Of Solutions For Some Types Of Higher-order Nonlinear Shallow Water Wave Models

The study of shallow water wave equations is an important topic in the fields of nonlinear science and partial differential equations.This dissertation mainly studies three types of higherorder nonlinear shallow water wave equations,and systematically research the derivation of models,the well-posedness of the initial value problems and the properties of solutions,including the Camassa-Holm-type shallow water wave equation with non-zero vorticity,the highly nonlinear shallow water wave equation for large-amplitude waves with the Coriolis effect,and the higher-order μ-Camassa-Holm equation.The main contributions of this dissertation are as follows:In Chapter 2,we investigate a Camassa-Holm-type shallow water wave equation with nonzero vorticity.First,based on the theory of shallow water waves,the model is derived from the two-dimensional,incompressible,rotating flow in moderately nonlinear regime,via the method of double asymptotic expansions.As a higher-order generalized CH-type equation,the existence of this model is based on the choice of depth,and describes the elevation of horizontal component of the velocity field at a specific depth below the surface,moving over a shear flow.Secondly,by comparing the depth of the model with the depth of the CH equation with non-zero vorticity and the rotation-Camassa-Holm(r-CH)equation,we investigate the effect of non-zero vorticity and nonlocal higher nonlinearities on the variation of depth intuitively.Furthermore,we simply explain the difference and interaction between these two physical quantities,by considering the effect of shear flow and Coriolis force in the background of equatorial water waves.Finally,for the case of θ=1,the traveling wave solutions of this model are classified.In Chapter 3,we study a new highly nonlinear shallow water wave equation for largeamplitude waves with the Coriolis effect.First,based on the theory of shallow water waves,a higher-order Camassa-Holm-type shallow water wave equation is obtained in the shallow-water regime for waves of large amplitude,using the method of double asymptotic expansions and the f-plane approximation.As an equatorial water wave model,its fluid motion is controlled not only by the effect of gravity,but also by the Coriolis effect caused by the Earth rotation.Such a model describes the unidirectional and irrotational propagation of the large amplitude waves near the equator.The influence of the Earth’s rotation and the increase of amplitude in dynamic water wave problems on the structure of the model is studied.Secondly,we investigate the influence of nonlinearities and Coriolis effect on the local well-posedness and higher-order regularity of the equation.Finally,the blow-up mechanism and blow-up phenomenon for solutions are also discussed.In Chapter 4,we study a higher-order μ-Camassa-Holm equation.As a higher-order extension of the μ-CH equation,this equation preserves the relevant typical properties of the μ-CH equation and the modified μ-CH equation.We first show that the equation admits the peaked traveling wave solution in the sense of distribution.Secondly,the local well-posedness of the Cauchy problem in the Sobolev space Hs(s>3/2)is obtained by utilizing the Kato’s method and establishing the estimates of higher-order nonlinearities.Finally,we provide the blow-up criterion for strong solutions with certain initial profiles.