# Category Archives: 2018 Math

## Crawling ink spots lemma and its application

The crawling ink spots lemma is useful to exploit a level set argument. The argument is orginally due to Safonov and Krylov (1980). Lemma 1. Let and be two open sets satisfying for all , there is a ball such that and     for all such that     we have . Then for some… Read More »

## Characterization of \$\BMO^{-1}\$

We fix some notations. For with , we write     We denote by which is a collection of all open balls in . Let be an open subset of . By , we denote the homogeneous Sobolev space defined as the completion of the complex-valued functions in the seminorm . The dual space is… Read More »

## Aubin-Lions Lemma

In this article, we prove the celebrated compactness lemma which will be used to show the global existence of weak solution of Navier-Stokes equation. Lemma 1. Let and be three Banach spaces with     Suppose is reflexive. Then for each , there is a constant such that     Proof. If the statement were… Read More »

## Strichartz estimates for Schrodinger equation

Theorem 1 (Strichartz esitmates for Schrödinger). Fix and and call a pair of exponents admissible if , and . Then for any admissible exponents and , we have the homogeneous Strichartz estimate (1)   the dual homogeneous Strichartz estimate (2)   and the inhomogeneous (or retarded) Strichartz estimate (3)   We illustrate some application of… Read More »

## Grading policy

한 학기동안 저의 절대적인 채점기준입니다. 이를 지키지 않을 경우 불이익은 학생에게 있습니다. 채점의 기준은 교과서입니다. / 증명이 안 좋은 것도 사실이고, 책의 증명이 틀린 부분도 있어서, 그 점은 제가 감안해서 반영합니다. 다만 제가 교수님이 뭘 세부적으로 수업을 했는지 알 수도 없는 상황이고, 그렇다고 제가 수업에 들어가서 그 수업을 듣는다는 것은 제 입장에서도 많은 시간낭비입니다. 문장으로 작성하십시오. 각… Read More »