Existence and uniqueness of stationary Navier-Stokes equation

By | January 8, 2018

Let be a bounded domain in or . We consider the Dirichlet boundary value problem for the stationary Navier-Stokes equation:

where is a viscosity constant. We call this problem as (NS).

Assume . Now we consider a weak formulation of (NS).

A function is called a weak solution of (NS) if in and satisfies

for any .

The definition is well established. Take inner product to the first equation in (NS) with and integrate it by parts. For the first part,

(1)

For the second part,

For the thrid part, it vanishes since . Also, for , for , by the Sobolev embedding theorem and the boundedness of , . Hence \eqref{eq:weak-sol-st-NS} makes sense.

From now, we assume . In 1933, J. Leray proved the existence of weak solution of (NS). To present the theorem of Leray, we define some terminology and the fundamental fixed point theorem proved by Leray and Schauder.

Theorem (Leray-Schauder’s fixed point theorem). Let be a Banach space. Suppose that the compact operator satisfies the following: there exists a constant such that if is a solution of , and

then there exists in satisfying .

If is reflexive and is completely continuous, then is compact. So in this case, we can apply the Leray-Schauder principle for such operator. See this article for proof.

Theorem. Let be a bounded Lipschitz domain. Then there exists at least one weak solution of (NS).

Proof. Since is a bounded Lipschitz domain, due to Poincar\’e’s inequality, one can check that is an inner product on .

Also, for any , . Indeed, for any ,

So .

Now for each by the existence result on Stokes equation, there exists a unique weak solution satisfying

(2)

for all . By Lemma ??, the above identity holds for any .

Since , by the Riesz representation theorem, there exists a unique in such that

for all . By the uniqueness, the operator is well-defined. Similarly, there exists a unique such that

for all with . So \eqref{eq:perturb-Stokes} can be written as

i.e., in operator form. Note that the existence of weak solution of (NS) is equivalent to the existence of a fixed point of the above operator equation. If there exists such that , then . So becomes a weak solution of (NS).

To use the Leray-Schauder fixed point theorem, we need to show that the operator defined by is a compact operator and a priori estimate.

We show that is completely continuous on . Let in . Then there exists a constant such that for all . For simplicity, let . Since is bounded Lipschitz domain, the Rellich-Kondrashov theorem shows that is compactly embedded in . Thus, strongly in . Now

So as , in . Note

Put . Then we get

So

as . This shows that is completely continuous.

Now we left to show a priori estimate.

First, note that for any ,

The last identity holds because of divergence free condition. By density, for any .

Now suppose satisfies , . So

Here we used

So

(3)

Here the constant does not depend on . Therefore, by the Leray-Schauder fixed point theorem, has a fixed point . This is the desired weak solution of (NS).

Now we prove the uniqueness.

Theorem. Let be a bounded Lipschitz domain and suppose that

where is a constant in the inequality

(4)

for any . Then (NS) has a unique weak solution.

Remark. The inequality (4) holds since is bounded and the Sobolev embedding theorem.

Proof. Suppose and are two different solutions to (NS). Then

Apply (3) and (4)}. Then we get

Now by assumption, we get

which is a contradiction. Therefore, the solution of (NS) is unique.