# Category Archives: Navier-Stokes equation

## 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 »

## Existence and uniqueness of stationary Navier-Stokes equation

Let be a bounded domain in or . We consider the Dirichlet boundary value problem for the stationary Navier-Stokes equation:     where is a viscosity constant. We call this problem as (NS). Assume . Now we consider a weak formulation of (NS). A function is called a weak solution of (NS) if in and… Read More »

## Leray-Schauder fixed point theorem

First, we derive the Schauder fixed point theorem. Theorem 1 (Schauder fixed point theorem). Let be a compact convex set in a Banach space and let be a continuous mapping of into itself. Then has a fixed point, that is, for some . Proof. Fix . Since is compact, has a finite subcover which covers… Read More »

## Notes 2. generalized de Rham theorem

다음의 정리는 미적분학 시간에 배운 정리다. Proposition. For a smooth vector field , the following hold: If , there exists  such that . If , then there exists a scalar field such that . 위의 정리는 어떤 벡터장이 conservative field임을 알려주는 조건이 된다. 이와 관련된 기하학적인 방향으로의 일반화는 다음과 같은 정리가 있다. Theorem(de Rham). For… Read More »