## Introduction

In the abstract study of partial differential equations, we often times throw the concept of weak derivatives around as if they acted just like classical derivatives. One such example is the product rule. For example, when studying evolution equations (time dependent partial differential equations such as the Navier-Stokes equations), books blow past the following equality

where is some sort of inner product or integration and is some sort of associated norm. What really happened is a coupling of a weak product rule and differentiating under the integral sign. But, to make these assumptions, we are often thinking of a “smoothed” version of the equation, where we have replaced all the nasty functions (often lying in some Hilbert space or Banach space) with appropriate “smoothified” cousins.

What does it mean for the weak derivative respect the product rule? That is, for weakly differentiable functions and with weak derivatives and , respectively, is it the case that the weak derivative of , their product exists? If so, is it given by

?

In fact, if only one of them is smooth, we can easily show that, yes, this is the case. What do we mean by smooth?

*Definition:* Let be open. We say that is smooth if it is infinitely differentiable (in a classical sense). In this case, we will write that .

**Theorem 1: **Let be open. Let be a weakly differentiable, and let . Then, their product is weakly differentiable. Moreover, the weak derivative of is given by

.

*Proof:* Let be a test function. Then, we have that . Moreover,

.

Thus, rearranging this, we have that

.

Using the weak differentiability of on the first term on the right-hand side, we see that

and the theorem is proved.

*QED.*

However, we don’t need to assume that is smooth. We just have to replace it with something smooth and take limits. That is the magic of mollification.

## Mollification

Mollification is the process of using convolution to replace a function with a smooth version of it which has nice limiting properties. The function we are convolutioning against is known as a *mollifier*.

*Definition: *A function is called a mollifier if

- .
- .
- For each integrable ,

as .

We will use a function called the standard mollifier. That is, given by

where is chosen so that . We have not established property 3 in the definition of a mollifier. That, we will leave until later. The graph for in one dimension is as follows:

Now, we see that the rescaled version of given by looks as follows in one dimension ():

Using this rescaling, we have that has support in , and using a simple change of variables (),

.

*Definition: *Let $\Omega\subseteq\mathbb{R}^n$ be open. Let be integrable. The mollification of , written as is given by

for , .

Note that using differentiation under the integral sign, is smooth. We do this by passing the derivatives under the integral sign and onto the mollifier (which is smooth). That is, for any multi-index ,

.

## Convergence Results

Next, we investigate the convergence of to as . As usual, we start with the continuous case.

**Theorem 2: **Let . Then, as uniformly on compact subsets of .

*Proof: *Let be compact. Let (to make sense of the following integrals). Note that using the fact that , we have that

using the change of variables . Since , we have that

.

Thus, taking absolute values, we have that

.

Since is compact, is uniformly continuous on . Thus, for any , there is a sufficiently small so that

.

*QED.*

Now, we extend to more general functions. See this article for a refresher on spaces.

**Theorem 3: **Let for some open set and . Then, , and in .

*Proof: *If we extend to zero outside of , we may assume without loss of generality that . We will start by showing that

.

First, using the change of variables and the with the conjugate exponent of (),

.

Using an application of Hölder’s inequality, this becomes

.

Therefore, by Fubini’s theorem,

by the shift-invariance of the integral. Thus, we use that to finish the proof that .

Let . Using the density of in , we can find a so that . This result can be found in any measure theory book. For instance, Donald Cohn’s *Measure Theory*. So, by the results above, we get that

since as the reader can easily check. Letting gives us that using Theorem 2. So, for small enough , we can make the above terms less than .

*QED.*

## The Product Rule

We are almost ready to prove the product rule for weak derivatives. We just need to know how mollification and weak derivatives interact.

**Theorem 4: **Let have weak derivative for some multi-index . Then,

.

Note that the left-hand side of the preceeding equation is the derivative of the smooth function in the classical sense, and the right-hand side is the mollification of the weak derivative . In the below proof, we will specify that is the (weak) derivative in the -variable, and is the (weak) derivative in the -variable when such ambiguities must be dealt with.

*Proof: *Using differentiation under the integral sign, we have that

by the chain rule in the classical sense. Note that . So, using the weak derivative property of , we get that

.

*QED.*

Now, we have all the machinery necessary to prove our initial theorem for the product rule. Note that we have to make the assumption that the integral of the product makes sense.

**Theorem 5: **Let be weakly differentiable. Then, if the product is integrable, it is weakly differentiable with

.

*Proof: *Replacing with , the mollified version of , we can use Theorem 1 to get that

.

By Theorem 4, . We can then say that the right-hand side converges to using Theorem 3. Also, we can show that in using the fact that in implies that there is some subsequence almost everywhere (see any introductory book on measure theory). This, and the dominated convergence theorem give the result.

*QED.*