1 Introduction

Long Time Stability for Solutions of a -plane Equation

Abstract.

We prove stability for arbitrarily long times of the zero solution for the so-called -plane equation, which describes the motion of a two-dimensional inviscid, ideal fluid under the influence of the Coriolis effect. The Coriolis force introduces a linear dispersive operator into the 2d incompressible Euler equations, thus making this problem amenable to an analysis from the point of view of nonlinear dispersive equations. The dispersive operator, , exhibits good decay, but has numerous unfavorable properties, chief among which are its anisotropy and its behavior at small frequencies.

2010 Mathematics Subject Classification:
76B03, 76B15, 76U05, 35B34, 35Q31, 35Q35
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1. Introduction

While the study of partial differential equations arising in fluid mechanics has been an active field in the past century, many important and physically relevant questions remain wide open from the point of view of mathematical analysis. Even very fundamental questions such as global-in-time well-posedness remain unresolved. This is particularly true in the study of the long-time behavior of systems without dissipation, such as the Euler equations for incompressible flow,

(1.1)
(1.2)
(1.3)

These model the motion of a so-called ideal, incompressible fluid occupying the whole space , where is the (vector) velocity field of the fluid and is the (scalar) internal pressure.

The most physically relevant cases to study (1.1)-(1.3) are dimensions and . While it isn’t apparent from the system in this form, the mathematical analysis of (1.1)-(1.3) in two dimensions is very different from the analysis in three dimensions. This is easily seen when we pass to the vorticity formulation: Define , the curl of (in two dimensions ). It turns out that in two dimensions satisfies a simple transport equation,

On the other hand, the so-called Biot-Savart law allows us to recover from the (scalar) vorticity using the fact that as .

The fact that is transported by a velocity field which is one degree smoother than (by the Biot-Savart law) allows one to prove that in two dimensions (1.1)-(1.3) is globally well-posed1 in Sobolev spaces, for . However, a priori the norm of the solutions may grow double exponentially in time.

1.1. The -plane model

One mechanism which helps to further stabilize the motion of an ideal fluid is the so-called Coriolis effect, which arises when the fluid is described in a rotating frame of reference, in which it is seen to experience an additional force (called the Coriolis force). On a rotating sphere the magnitude of this effect varies with the latitude and is described by the Coriolis parameter . A common approximation in oceanography and geophysical fluid dynamics (see [19], for example) is to assume that this dependency is linear. In the appropriate coordinate system the Coriolis parameter is then given as , where the variable corresponds to the latitude and is a parameter (from which the model gets its name). In two space dimensions, this is the setting of the current paper (see also [20]).

More explicitly, we are interested in studying the long-time behavior of solutions to the so-called -plane equation

(1.4)

where , denotes the Riesz transform in the first space coordinate, and is a real number determined by the strength of the Coriolis force. For simplicity, we shall take . Henceforth we shall abbreviate the operator on the right hand side by

As described above, this system models the flow of a two-dimensional inviscid fluid under the influence of the Coriolis effect – we have written it here in the vorticity formulation. The asymmetry between the horizontal and vertical motion can be seen clearly in the linear term on the right hand side of the equation. In more technical terms, this breaks the isotropy of the original Euler equation and is a core feature and difficulty of the model.

From a purely hyperbolic systems point of view, i.e. using only based energy estimates, one can show that solutions remain small in energy norms for a time span whose length is inversely proportional to the size of the initial data. There the operator does not play any role since and since commutes with derivatives. However, one can show that such energy norms are controlled by the norm. Since the linear part of the equation exhibits dispersive decay in , this controllability of higher order energy norms by the norm of hints at a first possible improvement through dispersive decay. Indeed, as we already observed in our previous work on the related SQG equation [10, Remark 3.3], the smallness of solutions can be guaranteed for a longer2 period of time.

The Dispersive Approach

In the present article we will improve this result and demonstrate smallness and decay of solutions to the -plane equation for arbitrarily long times. More precisely, we will show (see Theorem 2.1) that for any there exists an such that if the initial data are smaller than (in suitable spaces), then the corresponding solution to the -plane equation stays small and decays until at least time .

We approach the question of well-posedness of the -plane system for small initial data from a perturbative point of view, i.e. we view (1.4) as a nonlinear perturbation of the linear equation

(1.5)

Due to the properties of the operator this puts us in the realm of nonlinear dispersive differential equations: solutions to (1.5) decay in at a rate of . This rate of decay is not integrable, so obtaining good control of solutions to the nonlinear equation (1.4) for long periods of time is a challenge. However, it is the most decay one may hope for in a 2d model, so despite being anisotropic this model is not further degenerate in that sense (unlike the dispersive SQG equation, which has a lower rate of decay of – see [10]).

However, for small frequencies is quite singular: a bound on third order derivatives in an based Besov space is required for the decay given by the linear propagator. For the full, nonlinear -plane equation one is then led to the study of weighted norms with derivatives in order to guarantee the decay.

Further difficulties arise due to the lack of symmetries of this equation. In general, symmetries are a convenient way to obtain weighted estimates by commuting the associated generating vector fields with the equation. As noted above though, (1.4) does not exhibit a rotational symmetry. Only a scaling symmetry holds, which merely gives control3 over a radial weight in . This, however, is not enough to control the decay of solutions. Given the form of , proving weighted estimates on the equation is therefore the key difficulty tackled in this article.

This way we are lead to a detailed analysis of the resonance structure of the equation (in the spirit of the works of Germain, Masmoudi and Shatah – see for example [11]). Here it is crucial to note that the nonlinearity exhibits a null structure that cancels waves of parallel frequencies. However, the anisotropy of makes a precise study of the geometric structure of the resonant sets difficult or even infeasible.

It is also worth noting that, compared to many other nonlinear dispersive equations, this problem is highly quasi-linear in the sense that the nonlinearity has two orders of derivatives more than the linear operator .

Related Works on Rotating Fluids

The -plane approximation has been used in numerous investigations into various types of waves in rotating fluids in two or three spatial dimensions. For the most part the theoretical efforts have been focused on viscous scenarios, such as the rotating Navier-Stokes equations. The dispersion relation there is anisotropic, but less singular4 than the one we study. The viscosity introduces a strong regularizing effect and makes the equations semilinear. In particular, Strichartz estimates can then be used to construct global solutions once the dispersion is sufficiently strong, and to study limiting systems as the dispersion tends to infinity. For the rotating Navier-Stokes equations this was done by Chemin et al. ([6], [7], [5]) and Koh et al. ([18]).

Another important achievement in the study of both the rotating Euler equations and Navier-Stokes equations (in 3d) is the series of works of Babin-Mahalov-Nicolaenco ([1], [2], [4], [3]), which focused on studying the case of fast rotation. They proved that as the speed of rotation increases (depending upon the initial data), the solution of the rotating Navier-Stokes equations becomes globally well-posed and the solution of the Euler equations exists for a longer and longer time. This was shown in many different settings, including various spatial domains such as three dimensional tori ([3]) with non-resonant ([1], [2]) and resonant dimensions ([4]). Apart from the more singular dispersion relation, another key difference between the works of Babin-Mahalov-Nicolaenco and the present article is that we are able to prove almost global existence in the setting of the whole space without viscosity, while their method does not give such a long time of existence and, more importantly, global existence without viscosity seems out of reach using those types of methods. In contrast, our argument does not apply to the periodic case.

Related Works in Dispersive Equations

In recent years there has been a surge of interest in the question of dispersive effects in fluid equations. However, just as is the case for the classical dispersive equations such as the (Fractional) Nonlinear Schrödinger Equation, in many examples (e.g. [12], [17]) the dispersion relation is isotropic. This greatly helps with the resonance analysis and allows for an explicit understanding. In addition, rotational symmetry and scaling symmetries provide conservation laws for the solutions in weighted spaces.

An example of an anisotropic equation that bears closer similarity with the -plane model is the KP-I equation, recently studied by Harrop-Griffiths, Ifrim and Tataru [15]. At the linear level, this 2d model produces the same decay rate as the -plane equation and does not allow for a direct resonance analysis due to the anisotropy. However, in contrast to the -plane equation, the KP-I equation does have enough symmetries to provide control of weighted norms and the main difficulty is to obtain a global decaying solution. Using the method of testing by wave packets, the authors were able to obtain global solutions for small, localized initial data.

1.2. Plan of the Article

We start with the presentation of our main result, Theorem 2.1 in Section 2. This is followed by a short discussion that places it in the relevant context. The rest of the article is then devoted to its proof.

We begin (Section 3) by explaining the general bootstrapping scheme used to prove Theorem 2.1. It proceeds through a combination of bounds on the energy, the dispersion and control of some weighted norms. The individual estimates employed here are the subject of the remaining sections.

Firstly, an energy inequality is established in Section 4. This is a classical blow-up criterion, which shows that the growth of the energy norms is bounded by the exponential of the time integral of the norms of the velocity and vorticity.

We go on to discuss the dispersive decay given by the linear semigroup of the equation in Section 5. We explain the general dispersive result (with a proof in the appendix) and illustrate how we will make use of a more particular version, tailored to our bootstrapping argument in Section 3.

The core of the article then is the proof of the weighted estimates in Sections 6, 7 and 8. We commence by explaining in Section 6 the basics of our approach and give some insight into the difficulties we encounter. This is followed by an explanation of our strategy to prove the main weighted estimates (Propositions 7.1, 7.2 and 7.3).

In Section 7 we embark on the proof with a few observations, which allow us to reduce to one main type of term, which is treated in Section 8. There we illustrate first the various techniques that come in handy and then go on to apply them in the later parts of this section. In particular, the main part of Section 8 is a detailed resonance analysis and relies on the observations and ideas laid out beforehand.

In the appendix we gather the proofs of the full dispersive decay, of a corollary to our main result and of the global well-posedness for large data, as well as some computations that turn out to be useful.

1.3. Notation

For we let be the Littlewood-Paley projectors associated to a smooth, radial bump function with support in the shell and its rescalings , so that one has . Derived from these by summation are the operators , etc. For example, gives the projection onto frequencies larger than in Fourier space (and can alternatively be written using a smooth Fourier multiplier with value 1 on frequencies larger than and supported on frequencies larger than ).

We use standard notation for derivatives, but make use of the slight abbreviation to denote by a scalar derivative in in any direction. Moreover, we let .

For a set we denote by its characteristic function.

Whenever a parameter is carried through inequalities explicitly, we assume that constants implicit in the corresponding are independent of it.

2. Main Result

The main result of this article is the following

Theorem 2.1.

For any there exist and such that for all and , if satisfies

then there exist and a unique solution to the initial value problem

(2.1)

Moreover, for the solution decays in the sense that

(2.2)

and the energy as well as a weight with two or three derivatives on the profile remain bounded:

(2.3)
Remark 2.2 (Decay of Riesz transforms of and ).

Note that and only differ by a Riesz transform - since we obtain decay of and through based bounds on a weight on the profile, the proof of the theorem actually shows that any Riesz transforms of and decay at the same rate, respectively – see also Remark 5.3. The only reason we include both and here is that they come up explicitly in the estimates.

The question of stability of the zero solution was touched upon briefly in our previous work [10], where we showed that smallness of in for is propagated for at least a time . However, this result does not make use of any finer structure of the equation than the blow-up criterion for the energy and does not give decay of the norm.

We point out that even for large data global solutions to the -plane equation (1.4) can be constructed for in , – see Theorem B.1. However, a priori, the norms of the solution could grow faster than double exponentially in time, which is completely different from the situation we have in the small data case. The proof of this result proceeds in analogy with that for the 2d Euler equation. We include it in Appendix B for ease of reference.

About the Proof of Theorem 2.1.

The local well-posedness of (2.1) for is well-known. It can be seen by invoking the Sobolev embedding for the exponential in the energy estimate (4.1), for example. Thus the crucial part of the theorem is to obtain decay and existence on a “long” time interval .

For this we employ a bootstrapping argument, in which a decay assumption (similar to, but weaker than (2.2)) for some time interval gives the bounds (2.3) on that time interval. This in turn will yield the stronger decay (2.2), which allows us to prolong the solution and then repeat the argument by the continuity of , as long as .

We make use of Fourier space methods to understand the detailed structure of the nonlinearity through a resonance analysis, the null structure being a crucial element. The present rate of dispersive decay, , is not time integrable, which prevents us from proving estimates on a global time scale. More precisely, for the present quadratic nonlinearity, the bilinear estimates of type have a logarithmic growth when integrated in time. Together with a loss of derivatives in in our weighted estimates, this forces us to use a lot of derivatives for the energy estimates. However, the constants involved therein grow exponentially with the amount of derivatives, so that we are not able to close our estimates on an exponentially long time interval.

As outlined above in 1.2, the proof is spread out over the rest of this article: We begin by demonstrating how a bootstrapping scheme combines our estimates for the energy, dispersion and weights to yield Theorem 2.1. Subsequently, these ingredients are proved: Section 4 establishes energy estimates, after which we demonstrate in Section 5 how to obtain and use the dispersive estimate for the linear semigroup . Finally, Sections 6, 7 and 8 furnish the proof of the weighted estimates. ∎

Remark 2.3 (Symmetries).

As mentioned in the introduction, apart from time translation, the only continuous symmetry the -plane equation in (2.1) exhibits is a scaling symmetry. The former gives the conservation of and in ,

As for the latter, one can see that if solves (1.4), then so does for any . Differentiating with respect to at gives the generating vector field of this symmetry as , and we have

From this we obtain the conservation of : . Moreover, through the commutation of with the linear semigroup and the equation one can so obtain a bound on (under the appropriate decay assumptions).

As an aside we note that from the proof of Theorem 2.1 it also follows that the Fourier transform of the profile also stays bounded in (with two derivatives):

Corollary 2.4.

In the setting of Theorem 2.1, if one also has , then

The proof of this corollary builds on the techniques developed for Theorem 2.1 and is given in detail in Appendix C.

3. Bootstrap

We now demonstrate the bootstrap argument used to prove Theorem 2.1, deferring the details of the relevant estimates for the dispersion, energy and for weights on the profile function to the later sections of this article. The general approach here is a typical continuity argument for dispersive equations and has been used successfully in a plethora of other cases.

We will prove Theorem 2.1 by showing that for there exist and (depending on ) such that for its claim holds for , which can be made arbitrarily large.

Bootstrapping Scheme.

Given and satisfying the assumptions in Theorem 2.1 we make the bootstrapping assumption that for

(3.1)

From this we will deduce that the energy remains bounded, i.e.

(3.2)

where is chosen sufficiently large.

Then we shall go on to conclude the bound for two derivatives on a weight on the profile

(3.3)

from which it will follow that also three derivatives on a weight on the profile remain under control:

(3.4)

From this it then follows using the dispersive estimate that the following stronger decay estimate holds:

By a continuity argument this allows us to close our decay, energy and weighted estimates on the time interval .

The purpose of the present section is to show how these steps (3.2), (3.3) and (3.4) can be derived by combining results established in later sections.

Energy Estimate

The blow-up criterion (4.1) in our energy estimate (Lemma 4.1) gives

(3.5)

where . This we aim to bound by .

Weighted Estimate

Combining Propositions 7.2 and 7.3 yields the estimate

with to be chosen later. Upon choosing and carrying out a time integration this reduces to the bound

An application of Grönwall’s inequality then gives

(3.6)

using that with the exponent for small enough. This we aim to bound by .

Assuming this bound on two derivatives and a weight we may use Proposition 7.1 in direct analogy to conclude that

where the extra term comes from the estimate on two weights. We thus obtain

(3.7)

Also this will be bounded by .

Dispersive Estimate

To close the bootstrap argument we need to show that from the bounds on the energy and the weights ((3.2), (3.3) and (3.4)) we can actually deduce the stronger decay estimate

To this end we invoke Lemma 5.2 and deduce that for the choice as above we have

with .

To close we thus need to satisfy the condition

(3.8)

Closing the Bootstrap

We are now in the position to show how the bootstrap can be closed. This is merely a matter of collecting the conditions established above and showing that they can indeed be satisfied.

Given we will establish that for small enough the bootstrap (3.1)-(3.4) closes on a time interval once we choose large enough, thus concluding our proof of Theorem 2.1.

Firstly, to control the energy and two or three derivatives with a weight on the profile for we require by (3.5)-(3.7) that

(3.9)
(3.10)
(3.11)

Secondly, to close the dispersive estimate (3.1) we require by (3.8) that

(3.12)

Claim: There exists a such that for the above conditions are met.

Proof.

With this choice we have:

  1. Condition (3.9) is equivalent to , which in turn is simply the requirement that . Clearly for any choice of and this can be met by choosing small enough (depending on ).

  2. Conditions (3.10) and (3.11) will be satisfied if we have

    i.e. if . For this will be satisfied once , which can be seen to hold for small enough.

  3. To satisfy condition (3.12) we need that . Hence up to the constant we require . One checks directly that this can be satisfied by choosing (and thus also ) small enough.

This completes the proof of Theorem 2.1. ∎

4. Energy Estimates

To obtain energy estimates we differentiate the equation and get the following standard blow-up criterion:

Lemma 4.1.

Let be a solution of the -plane equation (1.4) on a time interval containing . Then for any we have the bound

(4.1)

for some universal constant independent of .

Since our arguments will require a large number of derivatives, here it is important to keep track of the size of the constants with respect to .

Proof.

In this proof, all implicit constants are assumed to be uniform in .

For any , , we let be a multiindex of order and proceed by differentiating the equation by , multiply by and integrate over to obtain

We note that , so that in estimating we only have to bound terms of the form , where , and . However, we point out that there are up to such terms. To bound one of them we use Hölder’s inequality and Gagliardo-Nirenberg interpolation to conclude then that

since the conditions and imply that .

Thus

from which summation over and Grönwall’s Lemma give (4.1). ∎

5. Dispersive Estimate

To understand the decay properties of the semigroup we adopt the point of view of oscillatory integrals. We write

Asymptotics for such integrals have been studied extensively (e.g. in [16]). Since the Hessian of the phase of this integral is non-degenerate, the proof of the following proposition is a standard argument using stationary phase and Littlewood-Paley techniques and has already been hinted at in our previous work [10] – we include it in Appendix A for completeness’ sake. The need to control three derivatives of can be seen by a scaling argument.

Proposition 5.1.

There is a constant such that for any we have5

(5.1)

More importantly, the goal of this section is to show how we can employ the weighted estimates to obtain decay of in . The main issue that arises is that the control of one weight in does not give integrability, i.e.  does not control . Thus we cannot use (5.1) directly. However, as will be clear from the details in Sections 6ff., it is not an easy matter to obtain other weighted estimates that do not grow fast, so we will work with only one weight and accept the loss this entails. The idea here is to avoid estimates either using energy estimates for high frequencies, or spaces with large for low frequencies.

Lemma 5.2.

For any and we have

(5.2)

where and can be chosen later.

Remark 5.3.

The fact that these bounds employ based spaces shows that the above arguments can equally be applied to Riesz transforms of to obtain their decay under the action of the semigroup . In particular, this lemma shows that we can control in using one weight with three derivatives in (and an energy term). The velocity on the other hand is one degree smoother and contains a Riesz transform: Recall that , so this can be controlled using one weight with only two derivatives.

As we will see in Sections 6-8, our proof of Theorem 2.1 first establishes bounds for two derivatives on a weight of the profile , which then gives bounds for three derivatives on a weight of . In terms of the variables of the equation, we first obtain control of the decay of , which then allows us to control that of .

Proof of Lemma 5.2.

Firstly we note that for high frequencies we may invoke the energy estimates via (7.4) and obtain

so that we may restrict to frequencies less than or equal to some .

Next we notice that by Bernstein’s inequality in two space dimensions we have for any

allowing us to work with lower integrabilities than infinity. As for the dispersive estimate we write

as long as . A direct computation shows that

Furthermore we may interpolate between and (see Proposition 5.1) to conclude that

Combining this with the above estimates when and yields

(5.3)

with .

Now we use that by Hölder’s inequality we can control by near the origin and by one weight in near infinity. More precisely, we have the estimates

and

Optimizing with respect to suggests the choice , which leaves us with

(5.4)

In particular, notice that as .

Returning to the above dispersive estimate we thus see that

(5.5)

Here the last inequality holds since one can combine the inequalities of Hölder and Bernstein to show that in one has for . ∎

6. Towards the Weighted Estimate

Our goal for this and the following two sections is to provide bounds for a weight with two or three derivatives on the profile function, i.e. bounds for and in . This is done first for two derivatives (Propositions 7.3 and 7.3), from which we can deduce estimates for the case of three derivatives (Proposition 7.1). We thus obtain first a bound for , which then yields a bound for (see also Remark 5.3).

For this we will essentially use estimates of the type . Informally speaking, in terms of our bootstrapping scheme in Section 3, every bilinear term then contains a decay factor of through dispersion (see (5.1)) and gives back the weight in or an energy term, both of which are assumed to remain small.

We start with a general discussion of our approach and give a brief overview of the structure of the remaining chapters and the difficulties we encounter.

6.1. The Duhamel Term and Remarks on the Present Approach

The Duhamel formula for the -plane equation on the profile in Fourier space reads

(6.1)

where the phase is given as

and the multiplier is

or (by changing variables )

We note that these have a null structure which annihilates waves with parallel frequencies.

As is clear from the Duhamel formula (6.1), all estimates on just amount to assumptions on the initial data, so we may restrict ourselves to bounding the time integral term

for .

Up to powers of this leaves us with the following three main types of terms6 to be dealt with:

(6.2)
(6.3)
(6.4)

Of these, (6.2) is the most difficult one due to an extra factor of from the differentiation . For (6.3) we may just invoke the energy estimates (see Section 7.2), while (6.4) gives back the weight (once we notice that we may assume and to be of comparable size, as remarked in Section 7.1). In particular we note that for these two terms the cases of two or three derivatives are completely analogous. It is the difficulties encountered in (6.2) that guide the structure of the following chapters.

More precisely, for (6.2) a detailed analysis of the resonance structure of the equation is needed. This in turn gives rise to Fourier multipliers in bilinear terms. However, due to the anisotropy and singularity of the linear propagator, the asymptotics of these do not fall into the classical categories of well-studied multipliers. In particular, the authors are not aware of any techniques that would allow to directly treat the bilinear terms that so arise in a satisfactory way.

We conclude this section with some remarks regarding the particulary difficult terms of the type of (6.2) and useful techniques for the bilinear estimates.

Difficulties in (6.2)

As we will see through the reductions in Section 7, this term is the heart of the matter and will be discussed in detail in Section 8. The basic issue is that we need to control the additional factor in the time integration:

As is well known, the typical ways of dealing with this are integrations by parts in either space or time (i.e. exploiting the lack of space or time resonances in various domains of Fourier space). In our case the anisotropy of the equation and the degeneracy/singularity of the linear propagator make this a difficult task. In particular, we cannot even calculate the time resonances explicitly.

Up to singularities when , or we have:

  • Time resonances : They contain the sets and , for example, but their explicit computation is practically infeasible.

  • Space resonances : These are given by .

  • Spacetime resonances