Category Archives: Mathematics Research

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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 »

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 »