# Category Archives: Mathematics Research

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## Haar systems on real lines and adapted to an accretive function

This article is based on the paper of Coifman-Jones-Semmes (JAMS, 1989). One of the fundamental problem is the solvability of the Dirichlet problem for the Poisson equation in a bounded domain(open and connected) in :     Many authors have been studied the solvability of the problem in a various settings. One possible way to… Read More »

## 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 »

## 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 »