Several Basic Inequalities for PDEs

Edited by Leun Kim

In mathematics, especially in analysis and PDEs, we evaluate or compare some mathematical quantities in order to reach a desired conclusion. Of course, equality is the best one, however, in many situations we cannot reach our goal only with equalities. In these cases, we may apply inequalities which are weaker, but still powerful.

In this post, I introduce several basic inequalities which are frequently used in many papers on PDEs in which I am majoring now. I listed the inequalities, in the order of frequency of use, appeared in PDE papers which I have read. Since all the listed inequalities are very elementary, I do not mention the proofs, but I write some descriptions for them.

For the simplicity, we assume that the functions $f,g$ and $h$ are in a suitable function space and denote the standard Sobolev space $H^s = H^{s,2}$.



1. Sobolev Inequality

The first prize is the Sobolev inequality which is also called Sobolev embedding theorem. There are a lot of versions of the Sobolev inequality, but the most frequently used form is $H^{s,p} \hookrightarrow L^q$ version:

For $n \in \mathbb N$, if $1\leq p<\infty$ and $0<s<n/p$, then $$ \|f\|_{L^{\frac{np}{n-sp}}(\mathbb R^n)}\lesssim\|f\|_{H^{s,p}(\mathbb R^n)}. $$In fact, I think the most frequently used one is the $H^{s,p} \hookrightarrow L^\infty$ version which is very simple but sharp:

For $n\in\mathbb N$, $k\in\mathbb N_0$, if $p>1$ and $s>k+n/p$, then $f\in C_b^k(\mathbb R^n)$, vanishes at the infinity, and
$$\|\nabla^\alpha f\|_{L^\infty(\mathbb R^n)}\lesssim\|f\|_{H^{s,p}(\mathbb R^n)} $$
holds for $|\alpha|\le k$.

Most mathematicians in this field naturally apply various Sobolev-type inequalities, even without any mentions in the calculation on their papers.
For example, one may adopt a suitable function space, say $X_T$ with a suitable norm something like
$$ \|f\|_{X_T} = \sup_{t\in[0,T]} \left(a(t)\|f(t,\cdot)\|_{L^\infty(\mathbb R^n)} + b(t)\|f(t,\cdot)\|_{H^s(\mathbb R^n)} + \cdots \right)$$
for some $a(t),b(t)$ to obtain appropriate a priori estimate for evolution equations. In general, we have to estimate $L^\infty$ term under a certain assumption. In this case, the Sobolev inequality can be used to estimate the $L^\infty$ terms by $H^s$ terms for small $t$ like
$$
\|f(t,\cdot)\|_{L^\infty(\mathbb R^n)}
\lesssim
\cdots
\lesssim
a(t)^{-1} \|f(t,\cdot)\|_{H^s(\mathbb R^n)}
\lesssim
\cdots
\lesssim
\text{functions of } \|f(t,\cdot)\|_{X_T}
$$
In fact, the case of large $t$ is difficult to estimate in general. Like this, in most cases, a PDE paper includes at least several Sobolev inequalities in the lines of calculation.



2. Energy Inequality

The second one is the energy inequality, sometimes called by energy method or energy estimates. In general, the energy method can be applied to almost all PDEs since the method does not require any strong conditions. Of course, we cannot obtain powerful results only with the standard energy method, however, in most cases the energy method plays a role to perform various methods in PDE. For the simplest example, we consider the following nonlinear wave equation
$$
\square f = N(f)
$$
where $\square=\partial_t^2 – \Delta$ with the initial condition $(f,\partial_t f)(0,\cdot) = (f_0(\cdot), g_0(\cdot))$ and a nonlinear term $N$. Then as in the Sobolev inequality, we may have to estimate $H^s$ terms in $X_T$ above to get a suitable priori estimate. To do this, we may need a standard energy inequality:
$$
\|\partial f (t) \|_{L^2(\mathbb R^n)} \lesssim \|\partial f(0)\|_{L^2(\mathbb R^n)} + \int_0^t \|N(f(\tau))\|_{L^2(\mathbb R^n)}d\tau
$$
for $t< \infty$. This is the straight forward result by multiplying $\partial_t f$ both sides to the equation and integrating by parts in the space. Of course, we can do the same steps for the $H^s$ norm. The energy inequality also can be obtained when we apply the vector field method. For example we consider the following nonlinear Klein-Gordon equation: $$ (\square+1)f = N(f) $$ with the initial condition as before. Then by the substitution $u = \frac{1}{2} (f + i\langle i\nabla \rangle^{-1}\partial_t f)$, we may write the equation as $$ (\partial_t + i\langle i\nabla \rangle)u = \widetilde{N}(u) $$ for some $\widetilde{N}$ where $\langle \cdot \rangle = \sqrt{1+|\cdot|^2}$. Multiplying both sides of this equation by $e^{i\langle i\nabla \rangle t}$, using the identity $e^{i\langle i\nabla \rangle t} (\partial_t + i\langle i\nabla \rangle) = \partial_t e^{i\langle i\nabla \rangle t}$, and integrating both sides with respect to time, we obtain $$ u(t) = e^{-i\langle i\nabla \rangle t} u(0) + \int_0^t e^{i\langle i\nabla \rangle (\tau-t)} \widetilde{N}(u(\tau))d\tau. $$ Finally, taking $H^s$ norm to the both sides, we get the standard energy inequality as follows: $$ \| u(t) \|_{H^s(\mathbb R^n)} \le \| u(0) \|_{H^s(\mathbb R^n)} + \int_0^t \|\widetilde{N}(u(\tau))\|_{H^s(\mathbb R^n)}d\tau. $$ In general, the next step is to estimate the second integral term under a suitable assumption.


3. Cauchy-Schwarz Inequality

In PDE field, the Cauchy-Schwarz Inequality with $L^2$ inner product is also useful:
$$
\int_{\mathbb R^n} f \overline{g} \le \| f \|_{L^2(\mathbb R^n)} \| g\|_{L^2(\mathbb R^n)}.
$$
This inequality is really elementary, but applied in many lines of calculations.



4. Hölder Inequality

The Hölder inequality is the generalization of the Cauchy-Schwarz inequality, that is:
$$
\| fg \|_{L^1(\mathbb R^n)} \le \| f \|_{L^p(\mathbb R^n)} \| g\|_{L^q(\mathbb R^n)}
$$
where $1 \le p,q \le \infty$ with $1/p+1/q=1$. And the generalized version of $m \ge 2$ functions is also frequently used. Note that the Hölder inequality or Cauchy-Schwarz inequality makes us possible to deal with $L^{\text{(space that I want)}}$ norms with some restrictions instead of $L^1$.



5. Young Inequality

The Young inequality is also useful in many lines of calculations:
$$
\forall \varepsilon>0, \quad |fg| \le \frac{|f|^2}{2\varepsilon} + \frac{\varepsilon |g|^2}{2}.
$$
We usually use the Young inequality to separate two functions if we know some information of the separated functions respectively. Also the Young inequality may play a role in applying the Gronwall inequality in some situations.



6. Gronwall Inequality

The next one is Gronwall’s inequality which is also called Gronwall’s lemma:

Assume $h\geqslant 0$ and that
$$
f(t) \leqslant g (t) + \int_{0}^{t} h(s) f(s) ds.
$$
Then the following holds:
(1) $f(t) \leqslant g (t) + \int_{0}^t g(s) h(s) e^{\int_{s}^t h(r) dr} ds$
(2) If $g$ is non-decreasing, $f(t) \leqslant g(t) e^{\int_{0}^t h(r) dr}$.

In many cases, the condition in (2) is satisfied. Usually the Gronwall inequality is used to deriving an energy inequality in this field. Also we note that there is differential version of the Gronwall inequality.




7. Gagliardo-Nirenberg Inequality

The last one is Gagliardo-Nirenberg’s inequality:

For $1\le r,p \le \infty$, $m\in\mathbb N$,
$$
\| \nabla^j f \|_{L^q(\mathbb R^n)} \lesssim \| \nabla^m f \|_{L^p(\mathbb R^n)}^{j/m} \| f \|_{L^r(\mathbb R^n)}^{1-j/m}
$$
holds, where $j\in \{0,1,\cdots,m\}$ and
$$
\frac{1}{q} = \frac{j}{m} \frac{1}{p} + \left(1-\frac{j}{m} \right) \frac{1}{r}.
$$

Almost all mathematicians in this field use this inequality naturally on their works. So this inequality is also essential to research PDEs. And there are plenty of interpolation inequalities of Gagliardo-Nirenberg type.

If you familiarize yourself with only these several inequalities, I think you may read and understand lots of recent papers on PDE. In the following article, I will introduce more specific inequalities and discuss about them.

 

 
I was born and raised in Daegu, S. Korea. I majored in electronics and math in Seoul from 2007 to 2012. I've had a great interest in math since freshman year, and I studied PDE in Osaka, Japan from 2012-2014. I worked at a science museum and HUFS from 2014 in Seoul. Now I'm studying PDE in Tokyo, Japan. I also developed an interest in music, as I met a great piano teacher Oh in 2001, and joined an indie metal band in 2008. In my spare time, I enjoy various things, such as listening music, blogging, traveling, taking photos, and playing Go and Holdem. Please do not hesitate to contact me with comments, email, guestbook, and social medias.



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