Simply take as , is understood as a vector function.

is the closure of under the norm of

is defined for arbitrary domain or , unless

.

Consider the steady Stokes problem

Reasonable solution of this problem is from physical reason. So it is also “natural” to require . However, if the is too bad(say not lipschitz), we can not talk about the in a right sense. So we have to require is in a reasonable space and is has some regularities. In order to overcome this difficulty, we force for any domain, bounded or unbounded, except .

**Formulation I:** A generalized solution is and , for all .

In this formulation, we force . If , which means

then is uniquely solved. Additionally, we do not need to make any assumption on near the boundary of .

**Formulation II: **A generalized solution is and , for all .

In this formulation, in order to get from , we need , . But this require . We can not isolate finding of from the of . Moreover, it is not natural to require for unbounded domain from the physical point of view.

**Formulation III**: A generalized solution is and , for all .(see[2])

In this formulation, we are able to isolate and . But this may results in has more than one solutions. According to the paper of Heywood, the number of solution is the number of cosets . He actually gives an example such that is one-dimensional.

So the real question is whether coincides with . [1] says that if is “star shaped” bounded domains, or has a compact lipschitz boundary.

[2] proves that if is the exterior domain with boundaries, then they are the same space.

**Application:**

When one try to solve , first one takes some solenoidal extension of inside of . If one forces , one can find a solution of in the form of , where . However, there is question here, if we find another extension of , are we necessarily find the same solution?

In another way, we can find solution of by hydrodynamic-potentials method, is this solution the same with what we have obtained before?

All of these question is related to .

[1]O.A. Ladyzhenskaya, V.A. Solonnikov. Some problems for vector analysis and generalized formulations of boundary-value problems for the navier-stokes equations.

[2]J.G. Heywood. On uniqueness questions in the theory of viscous flow.