The Low Regularity For The Fifth-order MKdV Equation

In this dissertation we mainly study the low regularity for the fifth-order modi-fied Korteweg-de Vries (mKdV for short) equation. Consider the following Cauchy problem For different cases of the initial data, we investigate the global well-posedness of (0.0.2)in Hs(R) and Hs(T), respectively.The low regularity, belonging to the well-posedness theory, is one of the basic problems in nonlinear dispersive equations. There are two strategies for this issue:Bourgain's Fourier truncation method and the I-method together with the multilin-ear multiplier theory. The first chapter is preliminaries, including some notations, as well as some related definitions and lemmas. Meanwhile, we introduce the research progress on the low regularity of some classical dispersive equations.Chapter two considers the global well-posedness of the Cauchy problem (0.0.2)in Sobolev space HS(R) with negative indices. Using the first generation I-method with the multilinear multiplier analysis, we prove that the Cauchy problem (0.0.2) is globally well-posed in Hs(R) for s>-3/22 .Firstly, based on a trilinear estimate and the contraction principle, we establish a variant of local well-posedness theory. Secondly,we define the first-generation modified energy and rewrite its increment as a function-al of multilinear multipliers. Thirdly, we perform some multilinear estimates in the localized Bourgain space and obtain an upper bound, N-1/2, on the increment of the modified energy. Lastly, we apply the rescaling and iteration argument to prove the global results.Chapter three devotes to study the global well-posedness of the periodic Cauchy problem (0.0.2) below the energy space. We prove (0.0.2) is globally well-posed in Hs(T) for s>1 by making use of the second generation I-method. However, it is more difficult to estimate the multiplier M4, so we give a delicate discussion on different frequencies. On the other hand, when we control the growth of the modified energy in the modified Bourgain space Ys, we establish some refined trilinear estimates. With these crucial improvements, we derive the almost conserved quantity for the second generation modified energy since its increment has an upper bound N-2.Chapter four establishes the exotic Strichartz estimates for the Beam and fourth-order Schr(?)dinger equations with null initial conditions. Based on Foschi's arguments about the exotic Strichartz estimates for inhomogeneous Schr(?)dinger equations, we utilize the stationary phase arguments, which has been developed by Kenig-Ponce-Vega, to get the dispersive inequalities for high and low frequencies. By the Littlewood-Paley decomposition, we deduce our final results.The last Chapter summarizes the thesis and gives some suggestions on further work, such as the low regularity of the Cauchy problem for the coupled system of the Schr(?)dinger-KdV equations, the well-posedness for the nonlinear Schr(?)dinger equa-tions on various manifolds and so on.