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.

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,

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