# Ladyzhenskaya type inequality when d=2

By | February 11, 2016

Problem. If , then we have

To prove this, we need the following lemma.

Lemma. For , we have

Here .

Proof. e proceed by induction on . The case is trivial since and this leads to

We assume that the inequality holds for . Write with , . Set

and

For temporary set , . Then

Since , we have

because .

Thus,

by Hölder inequality. So

By integrating this with respect to and multiple Hölder inequality, we get

Since , we get

Proof of the Ladyzhenskaya’s inequality. For , the result of Lemma is

Put instead of . Then we have

Note that for we have

By Cauchy-Schwarz inequality, we have

for .

Hence,

So

Reference
1. Elias M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1971.

## 2 thoughts on “Ladyzhenskaya type inequality when d=2”

1. Will Kwon Post author

아하 그렇군요ㅋㅋ Ladyzhenskaya inequality에 대해서 교수님께 잠시 여쭤보니 저걸로 2차원 Navier-Stokes의 중요한 결과를 얻어냈다고 하네요.

일반화에 대해서 여쭤보니 조금 더 큰 이론이 필요한거 같고..

저 레포트 잘 읽어보겠습니다 ㅎㅎ