Notes on INS (4): Useful inequalities & spaces

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 1 \leq p < n. Then there exists a positive constant C, depending only on p and n, such that \|v\|_{L^{p*}(\Omega)} \leq C \| Dv\|_{L^p(\Omega)} for all v \in C_0^1(\Omega) (space of continuously differentiable functions with compact support), where \Omega =  \mathbb{R}^n and p* = np/(n-p).

Calderon-Zygmund (CZ). For any p > 1 there is a constant C (which depends on p and on the dimension of the space) such that if \Delta w = u  (in some domain, with some reasonable boundary conditions …) then \|D^2 w\|_{L^p} \leq C \|u\|_{L^p}  . In other words, all second partial derivatives of w have Lp norm bounded by the Lp norm of u. When p=2 the above inequality is actually an equality: \|D^2  w\|_{L^2} = \|u\|_{L^2}

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 B_r of radius r (here in 3-dim space):

\int_{B_r} |v|^q dx \leq c (\int_{B_r} |\nabla v|^2 dx)^{a}.(\int_{B_r} |v|^2 dx)^{q/2-a} + (c/r^{2a})(\int_{B_r} |v|^2 dx)^{q/2} for all 2 \leq q \leq 6, a = 3 (q-2)/4.

Remark: the above inequality appears in Lin’s paper (1998), but the inequality in the paper contains a typo. In the limit case (q=6, a=3) we get GNS inequality.

Helly’s compactness theorem: The natural embedding of H^1(\Omega) in L^2_{loc}(\Omega) is compact.

Proof of inequalities

Proof of Gagliardo-Nirenberg-Sobolev Inequality:

Step 1: The case p=1

For each i = 1,\hdots, n we can write

v(x) = \int_{-\infty}^{x_i} v_{x_i} (x_1, \hdots , y, \hdots ,   x_n) d y

so

|v(x)| \leq \int_{-\infty}^{\infty} |v_{x_i} (x_1, \hdots , y,   \hdots , x_n)| dy

Taking the product of the above inequalities, we get

|v(x)|^{n/(n-1)} \leq \prod_{i=1}^n (\int_{-\infty}^{\infty}   |v_{x_i} (x_1, \hdots , y, \hdots , x_n)| dy)^{1/(n-1)}

Integrate the above inequality over \Omega = \mathbb{R}^n, and use Holder’s inequality repetitively –> desired result.

Holder’s inequality: \int |f_2 \hdots f_n| dx_1 \leq   \prod_{i=2}^n (\int |f_i|^{n-1} dx_1)^{1/(n-1)}

Step 2. The case 1 < p < n

Put w = |v|^\gamma where \gamma > 1 is to be determined. Then

|Dw| = \gamma |v|^{\gamma-1}|Dv| and

(\int_\Omega |w(x)|^{n/(n-1)} dx)^{(n-1)/n} \leq \int_\Omega   |Dw|dx (per the case p=1).

Thus we have

(\int_\Omega |v(x)|^{\gamma n/(n-1)} dx)^{(n-1)/n} \leq \gamma   \int_\Omega |v|^{\gamma-1}|Dv| dx \leq \gamma (\int_\Omega |v(x)|^{(\gamma -1) p/(p-1)}   dx)^{(p-1)/p}) (\int_\Omega |Dv(x)|^{p} dx)^{1/p}

Now choose \gamma so that \gamma n/(n-1) = (\gamma -1)   p/(p-1), i.e.

\gamma = p(n-1)/(n-p) > 1 we will get:

\|v\|_{L^{p*}(\Omega)} \leq \gamma \|Dv\|_{L^p}(\Omega)

Proof of Calderon-Zygmund Inequality:

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

w (x) = \int_{\Omega} \Gamma(x-y) u(y) dy

where \Gamma(x-y) proportional to 1/|x-y|^{2-n} is a fundamental solution to Laplace’s equation.

For the proof (see Gilbarg & Trudinger):

– The case p=2: If u is smooth and compactly-supported, then the equality follows from Stokes formula. If u 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 X be a metric space such that for each \epsilon > 0 there exist \delta  > 0,  a metric space W, and a mapping \phi: X \to W such that \phi(X) is totally bounded, and for any x,y \in  X with d(x,y) < \delta we have d(\phi(x),\phi(y))  < \epsilon. Then X is totally bounded.

The proof is almost obvious.

Theorem (Arzela-Ascoli, 1884-1894). Let M be a compact topological space. Then a subset of C(M) 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 \mathcal{F} \subset C(M) be such a set of functions. For each \epsilon, find a finite set of points x_1,\hdots,x_k in M and neighborhoods U_i around them which cover M, such that for any x,y \in U_i and any f \in \mathcal{F} we have |f(x) - f(y)| <  \epsilon. Then define the map \phi: \mathcal{F} \to  \mathbb{R}^k by \phi(f) = (f(x_1),\hdots, f(x_k)). These maps will satisfy the conditions of the lemma.

Theorem (Fréchet 1908). A subset of \ell^p is totally bounded if and only if:

i) It’s pointwise bounded, and

ii) For every \epsilon > 0 there is some $n$ such that, for every x in the given subset,

\sum_{i > n} |x|_i^p < \epsilon


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