On weighted estimates for the stream function of axially symmetric solutions to the Navier-Stokes equations in a bounded cylinder

Higher-order estimates in weighted Sobolev spaces for solutions to a singular elliptic equation for the stream function in an axially symmetric cylinder are provided. These estimates are essential for investigating the existence of axially symmetric solutions to incompressible Navier-Stokes equations in axially symmetric cylinders. To derive the estimates the technique of Kondratiev is incorporated. The weight has a form of a power function of the distance to the axis of symmetry.


Introduction
In this note we derive estimates for solutions to the following problem where Ω ⊂ R 3 is a bounded cylinder with boundary S. Before we go into any geometrical details (see (1.6)) we briefly justify why this problem is highly important in mathematical fluid mechanics.
Let w = w(x, t) be any vector-valued function of x and t.Then in cylindrical coordinates w is expressed in standard basis as follows (1.2) w = w r (r, ϕ, z, t)ē r + w ϕ (r, ϕ, z, t)ē ϕ + w z (r, ϕ, z, t)ē z .
In the mathematical theory of fluid mechanics we call function rw ϕ the swirl.Let v and p denote the velocity field of an incompressible fluid and the pressure, respectively.Let rot v be the vorticity vector.Then, the Navier-Stokes equations read where f is the external force field and n is the unit outward vector normal to S and Ω is the same domain as in (1.1).
The problem of regularity of axially-symmetric solutions to (1.3) in general is open.Since 1968 (see [1] and [2]) it is known that the Navier-Stokes equations have regular axially-symmetric solutions in R 3 provided that v ϕ | t=0 = 0 and f ϕ = 0 (hence the swirl is zero).In case of non-vanishing swirl there are some partial results, e.g.[3], [4], [5], [6], though this list is far from complete.
One way to investigate the existence of solutions to (1.3) is to start with the following observation: if v is axially symmetric solution to (1.3), then in light of (1.2) we have v = v r (r, z, t)ē r + v ϕ (r, z, t)ē ϕ + v z (r, z, t)ē z and rot v = −v ϕ,z (r, z, t)ē r + ω(r, z, t)ē ϕ + 1 r (rv ϕ ) ,r (r, z, t)ē z where (1.4) Expressing (1.3) 2 in the cylindrical coordinates yields (rv r ) ,r + (rv z ) ,z = 0 and combining this equation with (1.4) suggests introducing a stream function ψ such that we see that this stream function satisfies (1.1).Note that (1.1) 2 implies from (1.3) 3 .This explains why (1.1) is of primary interest.Solutions to this problem are essential for establishing global, regular and axially-symmetric solutions to the Navier-Stokes equations with non-vanishing swirl.We will demonstrate this idea for the case of small swirl in [7].Having proper estimates for solutions to (1.1) the proof in [7] is elementary.
There is a challenge in investigating (1.1), which we shall now discuss.Let a > 0 and R > 0. Then a bounded cylinder Ω in cylindrical coordinates is given by (1.6) where S = ∂Ω = S 1 ∪ S 2 and From the above description of Ω it follows that the terms 1 r 2 ψ and 1 r ψ r might be undefined for r = 0.This is a key challenge.There are a few possibilities for overcoming this issue: • one could remove the ǫ-neighborhood of r = 0, derive necessary estimates and pass with ǫ → 0 + (see e.g.[1]), • consider 1 r 1−ǫ ψ, derive necessary estimates and pass with ǫ → 0 + at the end (see e.g.[8]), • use weighted Sobolev spaces.We adopt the third approach.The classical results for the Poisson equation tell that if ω ∈ H 1 , then ψ ∈ H 3 .We would expect a similar outcome but we need to handle 1 r and similar terms carefully.If we were interested in basic energy estimates we could proceed the standard way: multiply (1.1) by ψ, integrate by parts, use the Hölder and Cauchy inequalities.This would be justified because in light of [9] and [10,Remark 2.4] we have provided that ψ is introduced through (1.5) and v is an axially symmetric vector field of class where a 1 and a 3 are smooth functions.Since basic energy estimates are not enough in our case, more sophisticated tools and techniques are needed.Weighted Sobolev spaces seem to be the right choice.
To conduct our analysis we introduce the quantity ψ 1 = ψ/r.We see that it satisfies Since ψ = ψ r r = ψ 1 r and r is bounded by R we see that any estimates for ψ 1 are immediately applicable to ψ.In fact, in [7] we need estimates for ψ 1 because this function appears naturally in some auxiliary problems.
As we can see both above problems are similar.What differs them is the domain.In case of Ω (2) we can safely use the classical theory for the Poisson equation.
Since r 0 > 0 we instantly deduce that problem (1.15) can be solved classically.
For studying the existence and properties of solutions to (1.14) we need the weighted Sobolev spaces.They are defined at the beginning of Section 2. In addition we will be utilizing the Kondratiev technique (see [11]).It offers a way to deal with expression of the form u r α when α > 0. We saw in (1.7) that ψ 1 is well defined at r = 0 but in case of the weighted Sobolev space H 3 0 we would need to handle ψ1 r 3 in L 2 .Function ψ 1 does not have such an order of vanishing when r → 0 + , thus it has to be modified in a certain way.These kinds of modifications form the essence of this note.
The very first theorem we prove is the following: Theorem 1. Suppose that ψ 1 is a solution to (1.9).Assume that ω 1 ∈ L 2,µ (Ω), µ ∈ (0, 1).Then the estimate holds where In light of (1.8) we cannot expect ψ 1 ∈ H 2 µ (0, R) for almost all z.However, this should be the case for the difference In a similar manner we obtain a higher order regularity Theorem 2. Let ψ 1 be a solution to (1.9).Let ω 1 ∈ H 1 µ (Ω), µ ∈ (0, 1).Then . The above theorems are useful but we need the estimates when µ = 0. We cannot simply pass with µ → 0 because ψ 1 − ψ 1 (0) / ∈ H 2 0 nor H 3 0 .Instead we construct two auxiliary functions χ and η that we subtract from ψ 1 (this construction is presented in Lemmas 3.6 and 3.7).This allows us to derive necessary estimates in H 3 0 .We emphasize that H 3 0 denotes a weighted Sobolev space (with the weight µ = 0; see Section 2) as opposed to a Sobolev space with zero traces.
In the below theorems we assume that ψ 1 is a weak solution to (1.9).Basic energy estimates and the existence of weak solutions are discussed in Section 2.
Theorem 3. Suppose that ψ 1 is a weak solution to (1.9).Let ω 1 ∈ L 2 (Ω) and introduce where K(τ ) is a smooth function with a compact support such that Then In case of H 3 0 we have Theorem 4. Let ψ 1 be a weak solution to (1.9).Let and K is the same as in Theorem 3.
At this point the estimates from Theorems 3 and 4 may look surprising.In [7] we show how to eliminate ψ 1 (0), χ and η by the data.
At the end of the Introduction it is worth mentioning that we could continue the process of deriving higher-order estimates for ψ 1 .In light of (1.8) it would require more subtractions from ψ 1 when r = 0.However, we do not see any potential gain nor immediate applications for such estimates.

Notation and auxiliary results
Notation.By c we mean a generic constant which may vary from line to line.
We also use N = {1, 2, . ..} and In is convenient to write: r.h.s.-the right-hand side and l.h.s.-the left-hand side.
and inversely The above considerations imply equivalence (2.1) for k ≤ 2. Similarly we prove equivalence (2.1) for k ≥ 3. Fourier transform.Let f ∈ S(R), where S(R) is the Schwartz space of all complexvalued rapidly decreasing infinitely differentiable functions on R. Then the Fourier transform and its inverse are defined by Using the Fourier transform we introduce equivalent norms to (2.2) convenient for examining solutions of differential equations.Hence, by the Parseval identity we have where the r.h.s.norm is equivalent to norm (2.2) under the equivalence (2.1).This ends Remark 2.2.
Energy estimates and weak solutions.
Then there exists a solution to problem (1.9) such that ψ 1 ∈ H 1 (Ω) and the estimate holds where Moreover, we have also where Ω (2) was introduced in (1.12).
Proof.Multiplying (1.9) by ψ 1 , integrating over Ω and using the boundary condition and the Poincaré inequality we derive (2.4).Then the existence follows from Fredholm alternative.
In light of (1.11) and (1.13) we obtain Using (2.4) we conclude the proof.
Proof.Multiply (1.1) by −ψ ,zz and integrate over Ω yields (2.9) Integrating by parts in the first term and using the boundary conditions, we get Similarly the second term in (2.9) vanishes.Hence, (2.9) implies (2.8) and concludes the proof.
Remark 2.8.If we set r = 1 − α and f (x) = x 0 g(y) dy in Lemma 2.7 we obtain 3. L 2 -weighted estimates with respect to r for solutions to (1.14)In this section we derive various estimates with respect to r for solutions to (1.14) in the weighted Sobolev spaces using the technique of Kondratiev (see [11]).These estimates lay foundations for the proofs of Theorems 1, 2, 3 and 4. The key idea is to treat variable z as a parameter.
First, we rewrite (1.14) in the form For a fixed z ∈ (−a, a) we treat (3.1) as a Multiplying (3.1) 1 by r 2 we obtain −r 2 u ,rr − 3ru ,r = r 2 (f + u ,zz ) ≡ g(r, z) or equivalently Introduce the new variable Since r∂ r = −∂ τ we see that (3.3) takes the form Utilizing the Fourier transform (see (2.2)) to (3.4) we get For λ / ∈ {0, −2i} we have Assume that R(λ) does not have poles on the line ℑλ = 1 + k − µ.Then, there exists a unique solution to Passing to variable r yields Continuing, we get where the relation g = r 2 (f + u ,zz ) was used.
We will be using the notation from Remark 3.2.Lemma 3.3.Let k = 0. Then there exists a constant c 0 such that If k = 1, then we also have Proof.Consider the case k = 0. Function ĝ′ is analytic for any h ∈ (h 1 , h 2 ) and We also have (see Fig. 1) Passing with N → ∞ yields where Hence (3.8) holds.
The restriction follows from Remark 2.4.
Functions u 1 and ū1 are valid candidates for weak solutions to (3.2) because they do not vanish on r = 0.
Repeating the proof of Lemma 2.5 we can show existence of weak solutions to (1.14) and the estimates Applying (3.6) for u = u 1 and µ = µ 1 and using (3.11) yields where µ 1 ∈ (1, 2).The above inequality reflects increasing regularity of weak solutions to (1.14).
Our aim is to find estimates in weighted Sobolev spaces for weak solutions to problem (1.14).Let u be such a weak solutions.We already know that u satisfies (3.10) and (3.11).Recalling properties of u 1 and u 2 and assuming that f ∈ L 2 (−a, a; L 2,µ (R + )), µ ∈ (0, 1) we can conclude that u satisfies Recalling properties of ū1 and ū2 and assuming that , where µ ∈ (0, 1).
Recall that u is a solution to and f ∈ L 2 (−a, a; H 1 (R + )).Then there exists a function where K(r) is a smooth function with a compact support near r = 0 such that and the function Proof.Since u ∈ L 2 (−a, a; H 3 (R + )) we can work with C(−a, a; C ∞ 0 (R + )) and then use the density argument.
We construct function η as a solution to the equation Integrating this equation we obtain (3.15).
To prove (3.16) and (3.17) we use Lemma 3.5 for k = 3.To ensure its assumptions are met we check that where Remark 2.4 implies that u ,r r=0 and Applying Lemma 3.5 and integrating (3.18) with respect to z we derive (3.16) and (3.17).This ends the proof.
Then, there exists a function where K is defined in Lemma 3.6 and the function we prove this lemma for functions from C(−a, a; C ∞ 0 (R + )) and use the density argument.
We construct function χ as a solution to Integrating (3.22) with respect to r yields (3.19).
To prove (3.20) and (3.21) we use Lemma 3.5 for k = 2.We need to check its assumptions.We have . Integrating (3.23) with respect to z and applying Lemma 3.5 for k = 2 we conclude the proof.
Recall that ψ 1 is a solution to Lemma 3.8.For solutions to (3.24) the following estimates hold.
Proof.First we show (3.25).Multiplying (3.24) by ψ 1,zz and integrating over Ω yields (3.27) The first term in (3.27) equals where the first term is equal to Applying the Hölder and Young inequalities to the r.h.s of (3.29) we obtain (3.30) Multiplying (3.24) by 1 r ψ 1,r and integrating over Ω yields The first term on the r.Using the above results in (3.31) yields Applying the Hölder and Young inequalities to the r..s of (3.32) we obtain From (3.24) we infer that Combining the above inequalit with (3.30) yields (3.25).
Next we show (3.26).Differentiating (3.24) with respect to z, multiplying by −ψ 1,zzz and integrating over Ω we obtain Integrating by parts in the first term yields In view of the above calculations equality (3.33) takes the form Applying the Hölder and Young inequalities to the r.h.s of (3.34) gives The above inequality implies (3.26) and concludes the proof.

4.
Estimates with respect to z for solutions to (1.9) Consider problem (1.9) in the form (4.1) Ω). Then the following estimate holds Proof.Multiply (4.1) 1 by −ψ 1,zz r 2µ and integrate over Ω.Then we have Integrating by parts in the second term on the l.h.s.we obtain We easily see that where the first integral vanishes under the same arguments used for I 1 .
The last term on the l.h.s. of (4.In virtue of boundary condition u| r=R = 0 and Remark 2.4 the first integral on the r.h.s. of (4.7) vanishes.
Integrating by parts in the last term on the l.h.s. of (4.5) and using (4.7), we obtain