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This paper proves the existence and uniqueness of a time-invariant measure for the 2D Navier-Stokes equations on the sphere under a random kick-force and a time-periodic deterministic force. Several examples of deterministic force satisfying the necessary conditions for a unique invariant measure to exist are given. The support of the measure is examined and given explicitly for several cases.

The existence and uniqueness of a time-invariant measure for the Navier-Stokes equations has been the subject of much recent research. A major advance was achieved in [

This paper extends the work in [

The first section uses a combination of the approaches in [

The second section presents the main theorem, which establishes the existence and uniqueness of an invariant measure for the kicked equations with a time-periodic deterministic external force. The proof of the main theorem is done by proving that necessary conditions hold for the applicability of Theorem 3.2.5 in [

The third section recalls work done by the author in [

Let ^{3}. Let ^{3}. Let

form a basis for the tangent space of M, denoted TM, and induce on M the Reimannian metric

The Navier-Stokes equations on the rotating sphere are

where n is the normal vector to the sphere,

The operators

where

To define the covariant derivative

For a vector

For a vector field normal to M, the curl is well-defined and is tangent to M. However, for a vector field in TM the curl is not well-defined but the third component of the curl, denoted curl_{n}, is well-defined. Due to this, define the following operators ([

Definition 1. Let u be a smooth vector field on M with values in TM and let

where on the right side Curl denotes the standard curl operator in

The covariant derivative and vector Laplacian are now defined in terms of the curl and curl_{n} operators ([

Definition 2. The covariant derivative on the sphere is given by

Remark 3. As with the curl and curl_{n} operators, it is possible to define the gradient, divergence, and covariant derivative in terms of extensions (see [

Thus both curl and curl_{n}, and thus the gradient, divergence, and covariant derivative, can be defined without resorting to extensions.

Definition 4. The vector Laplacian on the sphere is given by ([

Thus, the Navier-Stokes equations on the two-dimensional sphere, i.e., for vector fields on M, are:

Let

with the induced norm on

Let ^{s} have norm

and

By the Hodge Decomposition Theorem, the space of smooth vector fields on M can be decomposed as ([

Define the following closed subspaces of

Definition 5.

with norm

Note, H is the L^{2} closure of V_{0} and thus

Definition 6.

with norm

Note, V is the H^{1} closure of V_{0} and thus ^{1} norm for divergence-free vector fields.

Definition 7. For a vector field u, define the Laplacian on divergence-free vector fields as

Furthermore, if

The following theorem implies that the analysis used for the stochastic Navier-Stokes system on flat domains can be used for the system on the sphere. Its proof is identical to the case of flat domains with smooth boundary conditions, see [

Theorem 8. The operator

Let

where

where

where

We now state the existence and uniqueness of solutions to the deterministic Navier-Stokes equations in terms of the projected equations, as is standard.

Theorem 9. Suppose

and

The proof is the same as the case of bounded domains with smooth boundaries and periodic boundary conditions (see [

Let the deterministic force

Definition 10. Let

Theorem 11. Suppose there exists a globally defined solution to the Navier-Stokes equation with initial condition in H, has

If the stability radius δ is large enough or the period is small enough then

While the theorem in [^{1}, this is for

It is well-known that if the force is small enough (see Remark 33) then the stability radius is infinite and thus there is a unique exponentially stable solution with the same period as the force that all other solutions converge to. We conclude this section by examining two more cases where the stability radius is infinite. The proofs of the following lemmas are found in the Appendix. The main idea behind both of the lemmas is that the (spherical) scalar Laplacian commutes with longitudinal derivatives, allowing for terms in the calculations only dependent on latitude to vanish.

Definition 12. A solution to the Navier-Stokes equations, u, is called zonal if for each fixed t,

Lemma 1. Suppose that the time-periodic force

Remark 13. For a stationary force, it is sufficient that the force is zonal to have a stationary zonal solution ([

Analogously, the Stoke’s equation

Thus, to have a zonal solution it is sufficient that force is zonal. (The proof that the equations form an isomorphism is analogous to the result in [

Lemma 2. Suppose that

Definition 14. We define an almost zonal solution to be a solution guaranteed by Lemma 2.

It is worth noting that while Lemma 2 allows for nonzonal solutions, they are only a “small” perturbation from being zonal.

This section presents the main theorem on the existence and uniqueness of a (time-)invariant measure for the Navier-Stokes system with random kicks and a time-periodic deterministic force, where time-invariance is understood to mean that the random variables generated by restricting the solutions to instants of time proportional to the period of the deterministic forcing term have a unique stationary probability distribution which all other distributions converge to exponentially (i.e. it is exponentially mixing). A similar result in [

Consider the Navier-Stokes system with forcing

The notation from now on will be:

・

・ For simplicity of notation take the period as

・

Then

In other words, the solution between kicks is given by the flow of the deterministic system with time-periodic forcing. Notice that due to the periodicity of the force, if all the kicks were zero then for any positive integer n,

Following [

Condition 15. Let

for

support in the interval

For a given positive integer k and

where

The Markov semigroup

where

Definition 16. A measure

The next two definitions deal with behavior of the deterministic flow and are necessary for the statement of the main theorem.

Definition 17. We say that there is an asymptotically stable solution if for some

where

Notice that an asymptotically stable solution is exponentially stable for any radius

At minimum the following satisfy condition (16):

・

・ Time-periodic forces that give zonal flow of the form g(t)curl sin(ϕ), see Lemma 1.

・ Time-periodic forces that give almost zonal flow, see Lemma 2.

While both zonal and almost zonal flow are of potential meteorological interest, Condition (16) is actually restrictive since it guarantees that the exponentially stable solution is unique and that all other solutions converge to it exponentially. Thus it is of interest to consider more general deterministic forces than just the ones that satisfy Condition (16).

Note that since the Navier-Stokes equations have an absorbing set ([

Definition 18. Let

In other words, if the finite-dimensional projections are “close enough”, then the solutions converge.

A finitely stable point captures the same concept as determining modes ([

While the assumption of a finitely stable point allows for the possibility of multiple solutions, the assumption also has the disadvantage that we will need additional assumptions on the structure of the kicks.

Definition 19. The following is called the big kick assumption. Let M be as in Definition 18. For some

where

By Equations (54) and (41) the

Main Theorem 20. Let the kicks satisfy Condition (15) and let

・ there exists at least one finitely-stable point and the big kick assumption holds or

・ there is an asymptotically stable solution.

Then there is N such that if

1) The system (13) has invariant measure

2) The invariant measure is unique.

3) For any

The constant

The main theorem will follow from applying a modified version of Theorem 3.2.5 in [

Condition 21. For any R and r with

where

Condition 22. For any

where

Assume the kicked flow also satisfies:

Condition 23. For K, the support of the distribution of

and for any B bounded in H let

Then there exists

In addition, assume that the kicked flow satisfies the following type of controllability.

Condition 24. For any d > 0 and R > 0 there exists integer

In other words, the kicked flow from two different initial conditions has a positive probability of becoming arbitrarily close together in finite time.

We now formulate a modified version of Theorem 3.2.5 from [

Theorem 25. If the forced-kicked system (13) satisfies Conditions 21, 22, 23, and 24 and the kicks satisfy Condition 15 then there is

The constant

While Theorem 25 can be proved using the same approach as in [

Recall that a pair of random variables

Lemma 3. Under the conditions of Theorem 25, there exists a constant

where C > 0 does not depend on

Since the conditions on the deterministic solution operator are imposed on each fixed time interval and the operator is the same for each interval, the kicked-equations have the same form as the time-independent and zero-force cases. Thus, with the exception that constants now depend on the norm of the deterministic force, the proof of Lemma 3, which depends on conditions 20 and 21, is identical to the proof in [

It should be noted that the choice of N in Theorem 25 comes from the construction of the coupling in Lemma 3 and the construction only needs that N is sufficiently large.

Remark 26. In [

Given Lemma 3, the remainder of the proof proceeds under the following two cases:

1) If

2) If

The above argument gives the main idea behind the following lemma ([

Lemma 4. Let

25 for any

1) The maps

2) There exists a constant

3) If

Due to Lemma 3 which establishes the existence of a coupling, the proof of this lemma is very similar to the one in [

Having established Theorem 25, it only remains to check that the conditions hold for the kicked Navier- Stokes equations. It is straightforward that Condition 21 implies Condition 23. Furthermore, since Conditions 21 and 22 are well known and analogous to results for the torus these are included in the Appendix for completion. Instead only Condition 24 is proved here.

In order to establish Condition 24 the following is needed ([

Lemma 5. For any

uniformly in

The proof of Condition (24) uses the main idea behind Lemma 3.1 in [

Lemma 6. Suppose that there exists an asymptotically stable solution, then for any d > 0 and R > 0 there exists integer

Proof. First fix all realization of the kicks as the zero realization. Then by assumption there exists a time l such that

By continuity of the flow there is a

By Lemma 5 the probability of

as desired.

Now recall that the N in Theorem 25 is from the construction of the coupling in Lemma 3. Let N' be the maximum of the N from the big kick assumption (and thus ³ M) and the N generated by Lemma 3.

Lemma 7. Let

Proof. Let δ be the radius for the finitely stable point, u, and fix all realizations of the kicks as the zero realization. By (53) there exists a time l such that

By the big kick assumption, there exists a kick

Again fix all realizations as the zero realization. By the assumption of a finitely stable point, there exists a time k such that

Thus there exists a time

By continuity,

By Lemma 5 there is a positive probability of the kicks satisfying the inequalities.

This completes the proof of Condition 24 and thus there is uniqueness of invariant measure in H.

Before stating the main result of this section, we recall some definitions and straightforward results about the support of a measure.

Definition 27. The support of a measure

To continue we need Lemma 5.5 from [

Definition 28. For

The set of attainability for a set of points is the union of the set of attainability for all points in the set.

Lemma 8. For any

It is worth noting that the definition of the set of attainability is similar to Condition 23 except that the ball is centered at y instead of 0.

Remark 29. The support of the measure for the Navier-Stokes equations is concentrated on V ([

where

When there is an asymptotically stable solution the support is contained in a ball of radius

centered at the limiting solution ([

We next extend the standard definitions of wandering and nonwandering points ([

Definition 30. Let

Definition 31. A point

A point is defined as wandering or nonwandering based on the behavior of nearby points. One consequence of this is that for a stationary force an unstable stationary solution is now a wandering point unlike for the deterministic setting.

The following result was proved in [

Theorem 32. Let A be the set of attainability from the the set of nonwandering points. Then any

We outline the proof below (which is similar to the steps in [

・ By time-invariance, for any

Thus, by integrating over a subset of H instead

・ Thus, it is sufficient to show that for any _{1}, t_{2} such that

・ By the definition of the set of attainability, there is a nonwandering point y such that a is accessible from y. Furthermore, since for any

it is enough that

Lemma 9. Let y be a nonwandering point. For any

The proof is very similar to that of Condition 24 and thus only a sketch is given. By the definition of a nonwandering point, the intersection of any open ball (for example of radius

has a non-empty intersection with the deterministic flow of the set at some time

Lemma 10. Let A be the set of attainability from the set of nonwandering points. For any _{2} and all

The proof is nearly a repeat of the argument made in [

Due to the existence of an asymptotically stable solution when the force is small enough, gives a zonal solution of the form g(t)curl sin(ϕ), or gives an almost zonal solution, the following holds.

Corollary 1. If the force

1) is small enough―see Remark 33;

2) yields a zonal solution of the form g(t)curl sin(ϕ);

3) yields an almost zonal solution;

then the support of the measure is the set of attainability from the unique exponentially stable periodic solution.

While there is invariant measure for the kicked Navier-Stokes equations with a bounded time-periodic deterministic force, it is only possible to give a clear description of the support of the measure in a few limited situations. Furthermore, if there is an asymptotically stable solution then the support can be considered to nearly be the unique stable periodic solution since the kicks can be taken arbitrarily small (with the first N dimensions nonzero). Unfortunately, for more general forces the support of the measure is not as clear. For example, it is not as clear what nonwandering points may exist. In addition, while the assumption of a finitely stable point is more general than the assumption of a globally attracting solution and gives that there is a (at least one) periodic solution (possibly with the same period as the force), the size requirement on the kick is problematic both for understanding the support of the measure and for meteorological considerations.

It is possible, however, that the kicks may be allowed to be smaller. The big kick assumption is introduced to ensure that a kick can, with positive probability, send the flow into the neighborhood of any point in the deterministic absorbing ball. The necessity of the big kick assumption comes from the deterministic setting where a Dirac measure at any stationary solution is a time-invariant measure, giving non uniqueness if there are multiple stationary solutions. Thus, for example, if there are two stable stationary solutions the kicks must be (at minimum) large enough to send the flow from inside the radius of stability of one into the radius of stability of the other. The big kick assumption is sufficient to do this, but a smaller kick may suffice.

Of course, the results presented in this paper also apply to time-independent and zero forcing deterministic forces since they are trivially time-periodic. Furthermore, the majority of the results presented in this paper apply to the Navier-Stokes equations on the torus. For example, while a zonal and almost zonal solution no longer makes sense on the torus, if the force still yields an unique asymptotically stable solution then the support of the measure is again straightforward to describe.

We thank the Editor and the referee for their comments.

GregoryVarner, (2015) Unique Measure for the Time-Periodic Navier-Stokes on the Sphere Navier-Stokes on the Sphere. Applied Mathematics,06,1809-1830. doi: 10.4236/am.2015.611160

We now present estimates that will be needed to establish Conditions 21 and 22 and Lemmas 1 and 2. With the exception of Equations (48) and (49) of Lemma 12 and Lemma 13 these estimates are analogous to standard estimates on flat-domains with periodic boundary conditions.

Lemma 11. For

where ^{1}-norm on V.

The proof is identical to the case of flat domains due to the existence of an orthonormal basis. Furthermore,

Lemma 12. For

If

Furthermore, let

Proof. Since the proof of (42) and (45) are identical to the ones in [

It suffices to show that (48) holds for u= curl sin(ϕ). Since the sphere is simply connected, for a divergence-free vector field u, there is a flow function

where ∆ is the spherical Laplacian for functions.

For the following calculation, we will need the following information about the spherical Jacobian ([

The proof of Equation (48) uses an argument similar to [

The following lemma will allow for the Coriolis term

Lemma 13. For smooth vector fields u, the following holds for

We now turn to the proofs of Conditions 21 and 22 and Lemmas 1 and 2. Since many of the calculations are standard, only the main steps are given. Recall that

Let

Lemma 14. The following inequalities hold for the deterministic 2D Navier-Stokes equation on the sphere for all

where

Moreover, for any

Proof. The proof follows the estimates in [^{2} inner product of the Navier-Stokes equation with u. By (52) and (42)

This gives

establishing (53).

For (54), take the L^{2} inner product with Au. By (52) and (45)

Therefore

establishing (54).

For (55), note that integrating (56) from

(54) implies that for any

Integrating (61) with respect to

establishing (55).

Now consider the difference between two solutions

Lemma 15. For any

whenever

Proof. Taking the L^{2} inner product with w

By (42) and (44),

By the Cauchy inequality

and thus by (41)

By (60)

Thus the exponential is less than or equal to some constant (depending on R and the norms of f) for any fixed

Remark 33. By (69) in order to ensure (16) it is sufficient that

If Equation (16) is satisfied, there is a unique globally exponentially stable solution that is periodic with the same period as the force.

A.3. Proof of Condition 22Lemma 16. Let

Proof. Integrating (58) from s to t gives

Using (55) gives for

Integrating (67) from 1/2 to 1 and by the Mean Value Theorem there is

Taking the L^{2} inner product of (63) with Aw gives

By (47) and (46) respectively, the right side of (74) is bounded above by

Therefore

and

By (53) and (72) this is bounded above by

By (60) and (73) this is bounded above by

which establishes (70).

Let

Then

where the last step is by (70). For any t ≥ 1 a N can be found (depending on t, R, and f) such that the

The proof is analogous to a calculation in [

Let

where

Dropping the primes for ease of notation and taking the inner product with Au gives

Since

Thus the solution is asymptotically attracting in H.

A.5. Proof of Lemma 2The proof uses a different approach than the analogous result in [

Let u be the unique zonal solution for the Navier-Stokes equations with force f from Lemma 1. Suppose g is such that there exists

Let

Let

Take the inner product with Aq. By (45)

Thus

For any

Since

Thus if the norms of ^{1} (and thus in H).

It remains to express the norms of

Since

Integrating from

Similarly using (41) and integrating (92) from

Thus by Cauchy’s inequality

Thus the term in the exponential in (90) is bounded above by

Therefore there is