| Summary: | 碩士 === 國立清華大學 === 數學系 === 102 === This dissertation is focusing on introducing and proving Strichartz estimates for linear Schrodinger equation and wave equation and their application to the Cauchy problem of Schrodinger equation.
We will use some techniques of real analysis to prove them, so at the rst, we will introduce some tools of real analysis such as some operations, theorems of real analysis, and any tools we will use in order to avoid confusion.
Second, we focus on proving Strichartz estimate for linear Schrodinger equation and wave equation. For Schrodinger eqaution ,the most important tool we apply is Hardy-Littlewood-Sobolev Theorem which helps us transform a property of solutions of Schrodinger equation into a estimate under the so-called mixed norm.
For Wave equation, we focus on R3, and the main idea is we consider homogeneous case with given initial datum and inhomogeneous case but with zero initial datum, respectively, and derive estimates we want respectively, nally combine them and apply Triangle inequality to get the Strichartz Estimate.
And in the last section, we show a application of linear Strichartz estimate for a Schrodinger equation which indicates under some certain conditions and specic spaces, the local well-posedness of a nonlinear Schrodinger equation will be assured.
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