Study on the low regularity Cauchy problem for fifth-order dispersive equations

Abstract

In this thesis, we are going to mainly discuss about the low regularity Cauchy problem for fifth order dispersive equations, in particular, the (integrable) fifth-order modified KdV equation $\partial_tu$ - $\partial_x^{5}u$ + $40u\partial_xu\partial_x^{2}u$ + 10u^2\partial_x^{3}u$ + $10(\partial_xu)^3$ - 30u^4\partial_xu$ = 0 and the (integrable) fifth-order KdV equation $\partial_tu$ - $\partial_x^{5}u$ + $30u^2\partial_xu$ + $20u\partial_xu\partial_x^{2}u$ + $10u\partial_x^{3}u$ = 0 under the periodic boundary condition. Both equations are locally well-popsed via the standard energy method in the Sobolev space $H^s(T)$ for s > 2 and $s \ge 2$, respectively, and the fifth-oreder KdV equation is, in particular, globally well-posed in the energy space $H^2(T)$ thanks to the conservation law of the Hamiltonian.

In Chapter 1, we provide general theory concerning the low regularity Cauchy problem for dispersive equations, in particular, the Fourier restriction norm method. Moreover, we introduce KdV equation as the complete integrable system and provide a short proof of the local well-posedness, which is based on the Picard iteration method in addition to the $X^{s,b}$ space. The fifth-order KdV and modified KdV equations are also introduced in this chapter. Reviews and main idea to show the local well-posedness of certain equations will be provided.

In Chapter 2, we are going to show that nonlinear estimates for the fifth-order KdV and modified KdV equations fail in the standard $X^{s,b}$ space for any regularity. Also, we construct the short time $X^{s,b}$-type function space to overcome the failure of nonlinear estimates and introduce good properties of this new space. At the end of this chapter, we provide a short proof of the local well-posedness for the non-periodic fifth-order KdV equation by using the short-times $X^{s,b}$ space, which is contained in the author's first work [12]. In Chapters 3 and 4, we provide main ingredients in this dissertation: Nonlinear estimates and energy estimates concerning both the fifth-order modified KdV and the fifth-order KdV equations, and proof of the local well-posedness for the fifth- order modified KdV equation on T. In particular, we show the unconditional local well-posedness for the fifth-order modified KdV equation for s > 7/2 in Chapter 3. Chapters 3 and 4 are based on the author's works [27] and [28], respectively. In Appendix, for the completeness of the proof of the local well-posedness for the KdV equation on R in Section 1.2, we introduce Tao's [k

Z]-multiplier, and using this, we prove the bilinear estimate.