薛丁格方程的 Strichartz 估計與長波短波交互作用方程式的半古典極限

Abstract

此篇文章分為兩個部分。第一部分主要討論薛丁格方程上的Strichartz估計，我們先從量綱分析的角度觀察不等式中指數對 (p,q) 所需滿足的關係式，再給予嚴格的証明。從而結論在推導中可允許的 (p,q) 符合量綱分析的結果。 第二部分討論長波短波交互作用方程式的半古典極限。首先利用Madelung轉換，討論方程式的流體結構與守恆律。再透過修正的Madelung轉換與能量估計，證明局部古典解的存在性與唯一性。最後證明半古典極限解的存在性。

There are two parts in this paper. In part I, we discuss the Strichartz estimates on Schrödinger equation. First, we observe the restrictions on exponent pair (p,q) from the viewpoint of dimension. Then we also provide a rigid proof, and conclude that the so-called admissible pair coincides with the arguments of dimensional analysis. In part II, we study the semiclassical limit of the three coupled long wave-short wave interaction equations. First, we employ the Madelung transformation to discuss the hydrodynamical structures and the conservation laws. Then, we apply the modified Madelung transformation and energy estimates to justify the existence and uniqueness of the local classical solution. Finally, we prove the existence of the semiclassical limit of the solution.