Well-posedness And Ill-posedness For A Fifth-order Shallow Water Wave Equation

As is known to all, with the development of nonlinear sciences, there are many nonlinear evolution equations which play a great role in different physical backgrounds to emerge. One important example is shallow water wave equations. Though look like simple, they have a lot of valuable properties. Researchers from different fields have studyed it all along. Among those properties, the one called well-posedness is essentially important in the study of equations. It was defined by Jacques Solomon Hadamard, he thought that the solution of mathematical models derived from physical phenomena should have three properties:a. Ex-istence; b. Uniqueness; c. The dependence on the initial data is Lipschitz. If the well-posedness of the equation is obtained, we can apply a stable algorithm on a computer to get the solution, otherwise we need express the solution by a formula for convenience of dealing with numerical values. In this dissertation, we consider a fifth-order shallow water wave equation which was introduced by Tian et al. in [28] for the purpose of understanding the role of nonlinear disper-sive and nonlinear convection effects in K(2,2,1). They established the local well-posedness result in Hs with any s≥-11/16 by the Fourier restriction norm method. Our goal is to improve existing low regularity well-posedness results. In the first chapter, we introduce two harmonic analysis tools which are Bourgain space technique and Tao's [k;Ζ] multiplier norm method for studying dispersive equation, their definitions and elementary properties will be given. In the second chapter, we establish a new bilinear estimate in suitable Bourgain spaces by using a fundamental estimate on dyadic blocks for the Kawahara equation which was obtained by the [k;Ζ] multiplier norm method of Tao [25], then the local well-posedness of the Cauchy problem for a fifth-order shallow water wave equation in Hs(R) with s>-5/4 is achieved by the Fourier restriction norm method. And some ill-posedness in Hs(R) with s<-5/4 is derived from a general principle of Bejenaru and Tao.For the purpose of extending the local well-posedness result to s=-5/4, in the third chapter, we can achieve an ideal bilinear estimate by using a new space—the l1-analogue of Xs,b space Fs which is based on the Littlewood-Paley decom-position, then the sharp local well-posedness result will be obtained through the standard fixed point argument.