Last updated: 25/Oct/2010

There are two many inequalities used in the theory of Navier-Stokes equations. I’ll have to keep track of them. So this is the place where I put the inequalities.

**Gagliardo-Nirenberg-Sobolev (GNS)**. *Assume . Then there exists a positive constant , depending only on and , such that for all (space of continuously differentiable functions with compact support), where and .*

**Calderon-Zygmund (CZ)**. For any there is a constant (which depends on and on the dimension of the space) such that if (in some domain, with some reasonable boundary conditions …) then . In other words, all second partial derivatives of have Lp norm bounded by the Lp norm of . When the above inequality is actually an equality:

Reference for CZ: Gilbarg & Trudinger, Elliptic PDEs of second order, Section 9.4.

**LqL2dL2-Br**. This inequality interpolates between Lp norm of a function, ands L2 norms of it and its differential, on a ball of radius (here in 3-dim space):

for all .

Remark: the above inequality appears in Lin’s paper (1998), but the inequality in the paper contains a typo. In the limit case () we get GNS inequality.

**Helly’s compactness theorem:** The natural embedding of in is compact.

**Proof of inequalities**

**Proof of Gagliardo-Nirenberg-Sobolev Inequality:
**

*Step 1: The case *

For each we can write

so

Taking the product of the above inequalities, we get

Integrate the above inequality over , and use Holder’s inequality repetitively –> desired result.

Holder’s inequality:

*Step 2. The case *

Put where is to be determined. Then

and

(per the case ).

Thus we have

Now choose so that , i.e.

we will get:

**Proof of Calderon-Zygmund Inequality:**

can be obtained from by the formula (called the **Newtonian potential** of )

where proportional to is a fundamental solution to Laplace’s equation.

For the proof (see Gilbarg & Trudinger):

– The case : If is smooth and compactly-supported, then the equality follows from Stokes formula. If is more general, then use approximations by compacted-supported smooth functions.

– General case: use Marcinkiewicz interpolation theorem.

**About Helly’s compactness theorem**

Key ingredient in Glimm’s proof of existence of solutions of hyperbolic PDEs ? Helley’s theorem can be deduced from Kolmogorov-Fréchet theorem (goes back to Kolmogorov’s paper 1931: necessary and sufficient conditions for a subset in Lp to be compact) as an easy corollary. Ref: THE KOLMOGOROV–RIESZ COMPACTNESS THEOREM (http://arxiv.org/pdf/0906.4883v2)

**Totally bounded**: A metric space is called *totally bounded*, if its completion is compact. (If it is a subspace of a complete metric space, then totally-bounded means precompact)

**Lemma (bounding lemma)**. *Let be a metric space such that for each there exist , a metric space , and a mapping such that is totally bounded, and for any with we have Then is totally bounded.*

The proof is almost obvious.

**Theorem (Arzela-Ascoli, 1884-1894)**. *Let be a compact topological space. Then a subset of is totally bounded in the supremum norm if, and only if:
(i) it is pointwise bounded, and
(ii) it is equicontinuous.*

**Proof**. The “only if” part is direct. For the “if” part, use the bounding lemma: Let be such a set of functions. For each , find a finite set of points in and neighborhoods around them which cover , such that for any and any we have . Then define the map by . These maps will satisfy the conditions of the lemma.

**Theorem (Fréchet 1908)**. *A subset of is totally bounded if and only if:*

*i) It’s pointwise bounded, and*

*ii) For every there is some $n$ such that, for every in the given subset,*

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