Long time existence of smooth solutions for the rapidly rotating shallow-water and Euler equations
aa r X i v : . [ m a t h . A P ] J un LONG TIME EXISTENCE OF SMOOTH SOLUTIONS FORTHE RAPIDLY ROTATING SHALLOW-WATER AND EULER EQUATIONS
BIN CHENG AND EITAN TADMOR
Abstract.
We study the stabilizing effect of rotational forcing in the nonlinear setting of two-dimensional shallow-water and more general models of compressible Euler equations. In [17] we haveshown that the pressureless version of these equations admit global smooth solution for a large set ofsub-critical initial configurations. In the present work we prove that when rotational force dominatesthe pressure, it prolongs the life-span of smooth solutions for t < ∼ ln( δ − ); here δ ≪ δ regime, upon which hinges the long time existence of the exactsmooth solution. These results are in agreement with the close-to periodic dynamics observed in the“near inertial oscillation” (NIO) regime which follows oceanic storms. Indeed, our results indicate theexistence of smooth, “approximate periodic” solution for a time period of days , which is the relevanttime period found in NIO obesrvations. Contents
1. Introduction and statement of main results 12. First approximation– the pressureless system 53. Second approximation – the linearized system 64. Long time existence of approximate periodic solutions 94.1. The shallow-water equations 94.2. The isentropic gasdynamics 114.3. The ideal gasdynamics 135. Appendix. Staying away from vacuum 15References 161.
Introduction and statement of main results
We are concerned here with two-dimensional systems of nonlinear Eulerian equations driven bypressure and rotational forces. It is well-known that in the absence of rotation, these equationsexperience a finite-time breakdown: for generic smooth initial conditions, the corresponding solutionswill lose C -smoothness due to shock formation. The presence of rotational forces, however, hasa stabilizing effect. In particular, the pressureless version of these equations admit global smoothsolutions for a large set of so-called sub-critical initial configurations, [17]. It is therefore a natural Date : November 2, 2018.1991
Mathematics Subject Classification.
Key words and phrases.
Shallow-water equations, rapid rotation, pressureless equations, critical threshold, 2D Eulerequations, long-time existence.
Acknowledgment.
Research was supported in part by NSF grant 04-07704 and ONR grant N00014-91-J-1076. extension to investigate the balance between the regularizing effects of rotation vs. the tendency ofpressure to enforce finite-time breakdown (we mention in passing the recent work [21] on a similarregularizing balance of different competing forces in the 1D Euler-Poisson equations). In this paperwe prove the long-time existence of rapidly rotating flows characterized by “near-by” periodic flows.Thus, rotation prolongs the life-span of smooth solutions over increasingly long time periods, whichgrow longer as the rotation forces become more dominant over pressure.Our model problem is the Rotational Shallow Water (RSW) equations. This system of equationsmodels large scale geophysical motions in a thin layer of fluid under the influence of the Coriolisrotational forcing, (e.g. [18, § § ∂ t h + ∇ · ( h u ) = 0 , (1.1a) ∂ t u + u · ∇ u + g ∇ h − f u ⊥ = 0 . (1.1b)It governs the unknown velocity field u := (cid:0) u (1) ( t, x, y ) , u (2) ( t, x, y ) (cid:1) an height h := h ( t, x, y ), where g and f stand for the gravitational constant and the Coriolis frequency. Recall that equation (1.1a)observes the conservation of mass and equations (1.1b) describe balance of momentum by the pressuregradient, g ∇ h , and rotational forcing, f u ⊥ := f (cid:0) u (2) , − u (1) (cid:1) .For convenience, we rewrite the system (1.1) in terms of rescaled, nondimensional variables. Tothis end, we introduce the characteristic scales, H for total height h , D for height fluctuation h − H , U for velocity u , L for spatial length and correspondingly L/U for time, and we make the change ofvariables u = u ′ (cid:18) t ′ LU , x ′ L, y ′ L (cid:19) U, h = H + h ′ (cid:18) t ′ LU , x ′ L, y ′ L (cid:19) D. Discarding all the primes, we arrive at a nondimensional system, ∂ t h + u · ∇ h + (cid:18) HD + h (cid:19) ∇ · u = 0 ,∂ t u + u · ∇ u + gDU ∇ h − f LU u ⊥ = 0 . We are concerned here with the regime where the pressure gradient and compressibility are of thesame order, gDU ≈ HD . Thus we arrive at the (symmetrizable) RSW system, ∂ t h + u · ∇ h + (cid:18) σ + h (cid:19) ∇ · u = 0 , (1.3a) ∂ t u + u · ∇ u + 1 σ ∇ h − τ J u = 0 , . (1.3b)Here σ and τ , given by(1.3c) σ := U √ gH , τ := Uf L , are respectively, the Froude number measuring the inverse pressure forcing and the Rossby numbermeasuring the inverse rotational forcing. We use J to denote the 2 × J := (cid:18) − (cid:19) .To trace their long-time behavior, we approximate (1.3a), (1.3b) with the successive iterations, ∂ t h j + u j − · ∇ h j + (cid:18) σ + h j (cid:19) ∇ · u j − = 0 , j = 2 , , . . . (1.4a) ∂ t u j + u j · ∇ u j + 1 σ ∇ h j − τ J u j = 0 , j = 1 , , . . . , (1.4b) ONG TIME EXISTENCE OF THE RAPIDLY ROTATING EULER EQUATIONS 3 subject to initial conditions, h j (0 , · ) = h ( · ) and u j (0 , · ) = u ( · ). Observe that, given j , (1.4) areonly weakly coupled through the dependence of u j on h j , so that we only need to specify the initialheight h . Moreover, for σ ≫ τ , the momentum equations (1.3b) are “approximately decoupled”from the mass equation (1.3a) since rotational forcing is substantially dominant over pressure forcing.Therefore, a first approximation of constant height function will enforce this decoupling, serving asthe starting point of the above iterative scheme,(1.4c) h ≡ constant . This, in turn, leads to the first approximate velocity field, u , satisfying the pressureless equations,(1.5) ∂ t u + u · ∇ u − τ J u = 0 , u (0 , · ) = u ( · ) . Liu and Tadmor [17] have shown that there is a “large set” of so-called sub-critical initial configurations u , for which the pressureless equations (1.5) admit global smooth solutions. Moreover, the pressurelessvelocity u ( t, · ) is in fact 2 πτ -periodic in time. The regularity of u is discussed in Section 2.Having the pressureless solution, ( h ≡ constant , u ) as a first approximation for the RSW solution( h, u ), in Section 3 we introduce an improved approximation of the RSW equations, ( h , u ), whichsolves an “adapted” version of the second iteration ( j = 2) of (1.4). This improved approximationsatisfies a specific linearization of the RSW equations around the pressureless velocity u , with onlya one-way coupling between the momentum and the mass equations. Building on the regularity andperiodicity of the pressureless velocity u , we show that the solution of this linearized system subjectto sub-critical initial data ( h , u ), is globally smooth; in fact, both h ( t, · ) and u ( t, · ) retain 2 πτ -periodicity in time.Next, we turn to estimate the deviation between the solution of the linearized RSW system, ( h , u ),and the solution of the full RSW system, ( h, u ). To this end, we introduce a new non-dimensionalparameter δ := τσ = gHf LU , measuring the relative strength of rotation vs. the pressure forcing, and we assume that rotation isthe dominant forcing in the sense that δ ≪
1. Using the standard energy method we show in Theorem4.1 and its corollary that, starting with H m sub-critical initial data, the RSW solution (cid:0) h ( t, · ) , u ( t, · ) (cid:1) remains sufficiently close to (cid:0) h ( t, · ) , u ( t, · ) (cid:1) in the sense that, k h ( t, · ) − h ( t, · ) k H m − + k u ( t, · ) − u ( t, · ) k H m − < ∼ e C t δ (1 − e C t δ ) , where constant C = b C ( m, |∇ u | ∞ , | h | ∞ ) · k u , h k m . In particular, we conclude that for a large setof sub-critical initial data, the RSW equations (1.3) admit smooth, “approximate periodic” solutionsfor long time, t ≤ t δ := ln( δ − ), in the rotationally dominant regime δ ≪ h , u ) nearby the actual flow ( h, u ), with an up-to O ( δ ) ≪ U ’s) and only a thin layer of the oceans is reactive (small aspect ratio H/L ),corresponding to δ = gHfLU ≪
1. Specifically, with Rossby number τ ∼ O (0 .
1) and Froude number
BIN CHENG AND EITAN TADMOR σ ∼ O (1) we find δ ∼ .
1, which yield the existence of smooth, “approximate periodic” solutionfor t ∼ ∂ t ρ + ∇ · ( ρ u ) = 0 , (1.6a) ∂ t u + u · ∇ u + ρ − ∇ e p ( ρ, S ) = f J u , (1.6b) ∂ t S + u · ∇ S = 0 . (1.6c)Here, the physical variables ρ , S are respectively the density and entropy. We use e p ( ρ, S ) for thegas-specific pressure law relating pressure to density and entropy. For the ideal gasdynamics, thepressure law is given as e p := Aρ γ e S where A, γ are two gas-specific physical constants. The isentropicgas equations correspond to constant S , for which the entropy equation (1.6c) becomes redundant.Setting A = g, γ = 2 yields the RSW equations with ρ playing the same role as h .The general Euler system (1.6) can be symmetrized by introducing a “normalized” pressure function, p := √ γγ − e p γ − γ ( ρ, S ) , and by replacing the density equation (1.6a) with a pressure equation,(1.6d) ∂ t p + u · ∇ p + γ − p ∇ · u = 0 . We then nondimensionalize the above system (1.6b), (1.6c) and (1.6d) into ∂ t p + u · ∇ p + γ − (cid:18) σ + p (cid:19) ∇ · u = 0 ,∂ t u + u · ∇ u + γ − (cid:18) σ + p (cid:19) e σS ∇ p = 1 τ J u ,∂ t S + u · ∇ S = 0 . The same methodology introduced for the RSW equations still applies to the more general Eulersystem, independent of the pressure law. In particular, our first approximation, the pressurelesssystem, remains the same as in (1.5) since it ignores any effect of pressure. We then obtain the secondapproximation ( p , u , S ) (or ( p , u ) in the isentropic case) from a specific linearization around thepressureless velocity u . Thanks to the fact that h , p and S share a similar role as passive scalarstransported by u , the same regularity and periodicity argument can be employed for ( p , u , S )in these general cases as for ( h , u ) in the RSW case. The energy estimate, however, needs carefulmodification for the ideal gas equations due to additional nonlinearity. Finally, we conclude in Theorem4.2 and 4.3 that, in the rotationally dominant regime δ ≪
1, the exact solution stays “close” to theglobally smooth, 2 πτ -periodic approximate solution ( p , u , S ) for long time in the sense that, startingwith H m sub-critical data, the following estimate holds true for time t < ∼ ln( δ − ), k p ( t, · ) − p ( t, · ) k m − + k u ( t, · ) − u ( t, · ) k m − + k S ( t, · ) − S ( t, · ) k m − < e C t δ − e C t δ . Our results confirm the stabilization effect of rotation in the nonlinear setting, when it interacts withthe slow components of the system, which otherwise tend to destabilize of the dynamics. The study
ONG TIME EXISTENCE OF THE RAPIDLY ROTATING EULER EQUATIONS 5 of such interaction is essential to the understanding of rotating dynamics, primarily to geophysicalflows. We can mention only few works from the vast literature available on this topic, and we referthe reader to the recent book of Chemin et. al., [6] and the references therein, for a state-of-the art ofthe mathematical theory for rapidly rotating flows. Embid and Majda [7, 8] studied the singular limitof RSW equations under the two regimes τ − ∼ σ − → ∞ and τ − ∼ O (1) , σ − → ∞ . Extensions tomore general skew-symmetric perturbations can be found in the work of Gallagher, e.g. [9]. The seriesof works of Babin, Mahalov and Nicolaenko, consult [1, 2, 3, 4, 5] and references therein, establishlong term stability effects of the rapidly rotating 3D Euler, Navier-Stokes and primitive equations.Finally, we mention the work of Zeitlin, Reznik and Ben Jelloul in [23, 24] which categorizes severalrelevant scaling regimes and correspondingly, derives formal asymptotics in the nonlinear setting.We comment here that the approach pursued in the above literature relies on identifying the limitingsystem as τ →
0, which filters out fast scales. The full system is then approximated to a first order,by this slowly evolving limiting system. A rigorous mathematical foundation along these lines wasdeveloped by Schochet [19], which can be traced back to the earlier works of Klainerman and Majda[13] and Kreiss [14] (see also [20]). The key point was the separation of (linear) fast oscillations fromthe slow scales. The novelty of our approach, inspired by the critical threshold phenomena [16], is toadopt the rapidly oscillating and fully nonlinear pressureless system as a first approximation and thenconsider the full system as a perturbation of this fast scale. This enables us to preserve both slow andfast dynamics, and especially, the rotation-induced time periodicity.2.
First approximation– the pressureless system
We consider the pressureless system(2.1a) ∂ t u + u · ∇ u − τ J u = 0 , subject to initial condition u (0 , · ) = u ( · ). We begin by recalling the main theorem in [17] regardingthe global regularity of the pressureless equations (2.1a). Theorem 2.1.
Consider the pressureless equations (2.1a) subject to C -initial data u (0 , · ) = u ( · ) .Then, the solution u ( t, · ) stays C for all time if and only if the initial data satisfy the criticalthreshold condition, (2.1b) τ ω ( x ) + τ η ( x ) < , for all x ∈ R . Here, ω ( x ) = −∇ × u ( x ) = ∂ y u − ∂ x v is the initial vorticity and η ( x ) := λ − λ is the (possiblycomplex-valued) spectral gap associated with the eigenvalues of gradient matrix ∇ u ( x ) . Moreover,these globally smooth solutions, u ( t, · ) , are πτ -periodic in time. In [17], Liu and Tadmor gave two different proofs of (2.1b). One was based on the spectral dynamicsof λ j ( ∇ u ); another, was based on the flow map associated with (2.1a), and here we note yet anotherversion of the latter, based on the Riccati-type equation satisfied by the gradient matrix M =: ∇ u , M ′ + M = τ − J M.
Here {·} ′ := ∂ t + u · ∇ denotes differentiation along the particle trajectories(2.2) Γ := { ( x, t ) | ˙ x ( t ) = u ( x ( t ) , t ) , x ( t ) = x } . Starting with M = M ( t , x ), the solution of this equation along the corresponding trajectory Γ isgiven by M = e tJ/τ (cid:16) I + τ − J (cid:16) I − e tJ/τ (cid:17) M (cid:17) − M , BIN CHENG AND EITAN TADMOR and a straightforward calculation based on the Cayley-Hamilton Theorem (for computing the inverseof a matrix) shows that(2.3) max t,x |∇ u | = max t,x | M | = max t,x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) polynomial( τ, e tJ/τ , ∇ u )(1 − τ ω − τ η ) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Thus the critical threshold (2.1b) follows. The periodicity of u is proved upon integrating u ′ = τ J u and x ′ = u along particle trajectories Γ . It turns out both x ( t ) and u ( t, x ( t )) are 2 πτ periodic,which clearly implies that u ( t, · ) shares the same periodicity. It follows that there exists a criticalRossby number, τ c := τ c ( ∇ u ) such that the pressureless solution, u ( t, · ), remains smooth for globaltime whenever τ ∈ (0 , τ c ). This emphasizes the stabilization effect of the rotational forcing for a“large” class of sub-critical initial configurations, [17, § τ c need not be small, and in fact, τ c = ∞ for rotational initial data such that η < ω < p − η . Weshall always limit ourselves, however, to a finite value of the critical threshold, τ c .In the next corollary we show that in fact, the pressureless solution retains higher-order smoothnessof the sub-critical initial data. To this end, we introduce the following notations. Notations . Here and below, k · k m denotes the usual H m -Sobolev norm over the 2D torus T and | · | ∞ denotes the L ∞ norm. We abbreviate a < ∼ m b for a ≤ cb whenever the constant c only depends on thedimension m . We let b C denote m -dependent constants that have possible nonlinear dependence onthe initial data | h | ∞ and |∇ u | ∞ . The constant, C := b C · k ( h , u ) k m , will be used for estimatesinvolving Sobolev regularity, emphasizing that C depends linearly on the H m -size of initial data, h and u ) k , and possibly nonlinearly on their L ∞ -size. Corollary 2.1.
Fix an integer m > and consider the pressureless system (2.1a) subject to sub-critical initial data, u ∈ H m . Then, there exists a critical value τ c := τ c ( ∇ u ) < ∞ such that for τ ∈ (0 , τ c ] we have, uniformly in time, |∇ u ( t, · ) | ∞ ≤ b C , (2.4a) k u ( t, · ) k m ≤ C . (2.4b) Proof.
We recall the expression for |∇ u | ∞ in (2.3). By continuity argument, there exists a value τ c such that 1 − τ ω − τ η > / τ ∈ (0 , τ c ) , which in turn implies (2.4a) with a constant b C that depends on |∇ u | ∞ and τ c which also replies on the pointwise value of ∇ u .Having control on the L ∞ norm of ∇ u , we employ the standard energy method to obtain theinequality, ddt k u ( t, · ) k m < ∼ m |∇ u ( t, · ) | L ∞ k u ( t, · ) k m . Since u ( t, · ) is 2 πτ -periodic, it suffices to consider its energy growth over 0 ≤ t < πτ < πτ c .Combining with estimate (2.4a) and solving the above Gronwall inequality, we prove the H m estimate(2.4b). (cid:3) Second approximation – the linearized system
Once we established the global properties of the pressureless velocity u , it can be used as thestarting point for second iteration of (1.4). We begin with the approximate height, h , governed by(1.4a),(3.1) ∂ t h + u · ∇ h + (cid:18) σ + h (cid:19) ∇ · u = 0 , h (0 , · ) = h ( · ) . ONG TIME EXISTENCE OF THE RAPIDLY ROTATING EULER EQUATIONS 7
Recall that u is the solution of the pressureless system (2.1a) subject to sub-critical initial data u ,so that u ( t, · ) is smooth, 2 πτ -periodic in time. The following key lemma shows that the periodicityof u imposes the same periodicity on passive scalars transported by such u ’s. Lemma 3.1.
Let scalar function w be governed by (3.2) ∂ t w + ∇ · ( u w ) = 0 where u ( t, · ) is a globally smooth, πτ -periodic solution of the pressureless equations (2.1a). Then w ( t, · ) is also πτ -periodic.Proof. Let φ := ∇ × u + τ − denote the so-called relative vorticity. By (2.1a) it satisfies the sameequation w does, namely, ∂ t φ + ∇ · ( u φ ) = 0 . Coupled with (3.2), it is easy to verify that the ratio w/φ satisfies a transport equation (cid:0) ∂ t + u · ∇ (cid:1) wφ = 0which in turn implies that w/φ remains constant along the trajectories Γ in (2.2). But (2.1a) tellsus that u ′ = Jτ u , yielding u ( t, x ( t )) = e tτ J u ( x ). We integrate to find, x (2 πτ ) = x (0), namely, thetrajectories come back to their initial positions at t = 2 πτ . Therefore wφ (2 πτ, x ) = wφ (0 , x ) for all x ’s . Since the above argument is time invariant, it implies that w/φ ( t, · ) is 2 πτ -periodic. The conclusionfollows from the fact that u ( t, · ) and thus φ ( t, · ) are 2 πτ -periodic. (cid:3) Equipped with this lemma we conclude the following.
Theorem 3.1.
Consider the mass equation (3.1) on a 2D torus, T , linearized around the pressurelessvelocity field u and subject to sub-critical initial data ( h , u ) ∈ H m ( T ) with m > . It admits aglobally smooth solutions, h ( t, · ) ∈ H m − ( T ) which is πτ -periodic in time, and the following upperbounds hold uniform in time, | h ( t, · ) | ∞ ≤ b C (cid:16) τσ (cid:17) , (3.3a) k h ( t, · ) k m − ≤ C (cid:16) τσ (cid:17) . (3.3b) Proof.
Apply lemma 3.1 with w := σ − + h to (3.1) to conclude that h is also 2 πτ -periodic. Weturn to the examine the regularity of h . First, its L ∞ bound (3.3a) is studied using the L ∞ estimatefor scalar transport equations which yields an inequality for | h | ∞ = | h ( t, · ) | ∞ , ddt | h | ∞ ≤ |∇ · u | ∞ ( σ − + | h | ∞ ) . Combined with the L ∞ estimate of ∇ u in (2.4a), this Gronwall inequality implies | h | ∞ ≤ e b C t | h | ∞ + 1 σ (cid:16) e b C t − (cid:17) . As before, due to the 2 πτ -periodicity of h and the subcritical condition τ ≤ τ c , we can replace thefirst t on the right with τ c , the second t with 2 πτ , and (3.3a) follows. BIN CHENG AND EITAN TADMOR
For the H m − estimate (3.3b), we use the energy method and the Gagliardo-Nirenberg inequalityto obtain a similar inequality for | h | m − = | h ( t, · ) | m − , ddt k h k m − < ∼ m |∇ u | ∞ k h k m − + (cid:18) σ + | h | ∞ (cid:19) k u k m . Applying the estimate on u in (2.4) and the L ∞ estimate on h in (3.3a), we find the above inequalityshares a similar form as the previous one. Thus the estimate (3.3b) follows by the same periodicity andsub-criticality argument as for (3.3a). We note by passing the linear dependence of C on k ( h , u ) k m . (cid:3) To continue with the second approximation, we turn to the approximate momentum equation (1.4b)with j = 2,(3.4) ∂ t u + u · ∇ u + 1 σ ∇ h − τ u = 0 . The following splitting approach will lead to a simplified linearization of (3.4) which is “close” to(3.4) and still maintains the nature of our methodology. The idea is to treat the nonlinear term andthe pressure term in (3.4) separately, resulting in two systems for e v ≈ u and b v ≈ u , ∂ t e v + e v ∇ · e v − τ J e v = 0 , (3.5a) ∂ t b v + 1 σ ∇ h − τ J b v = 0 , (3.5b)subject to the same initial data e v (0 , · ) = b v (0 , · ) = u ( · ) . The first system (3.5a), ignoring the pressure term, is identified as the pressureless system (2.1) andtherefore is solved as e v = u , while the second system (3.5b), ignoring the nonlinear advection term, is solved using the Duhamel’sprinciple, b v ( t, · ) = e tJ/τ u ( t, · ) − Z t e − sJ/σ σ ∇ h ( s, · ) ds ! ≈ e tJ/τ u ( t, · ) − Z t e − sJ/σ σ ∇ h ( t, · ) ds ! = e tJ/τ u ( t, · ) + τσ J ( I − e tJ/τ ) ∇ h ( t, · ) . Here, we make an approximation by replacing h ( s, · ) with h ( t, · ) in the integrand, which introducesan error of order τ , taking into account the 2 πτ period of h ( t, · ).Now, synthesizing the two solutions listed above, we make a correction to b v by replacing e tJ/τ u with u . This gives the very form of our approximate velocity field u (with tolerable abuse of notations)(3.7a) u := u + τσ J ( I − e tJ/τ ) ∇ h ( t, · ) . A straightforward computation shows that this velocity field, u , satisfies the following approximatemomentum equation,(3.7b) ∂ t u + u · ∇ u + 1 σ ∇ h − τ u ⊥ = R ONG TIME EXISTENCE OF THE RAPIDLY ROTATING EULER EQUATIONS 9 where(3.7c) R := τσ J ( I − e tJ/τ )( ∂ t + u · ∇ ) ∇ h ( t, · )(by (3.1)) = − τσ J ( I − e tJ/τ ) (cid:20) ( ∇ u ) ⊤ ∇ h + ∇ (( 1 σ + h ) ∇ · u ) (cid:21) . Combining Theorem 3.1 on h ( t, · ) with Gagliardo-Nirenberg inequality, we arrive at the followingcorollary on periodicity and regularity of u . Corollary 3.1.
Consider the velocity field u in (3.7) subject to sub-critical initial data ( h , u ) ∈ H m ( T ) with m > . Then, u ( t, · ) is a πτ -periodic in time, and the following upper bound, uniformlyin time, holds, k u − u k m − ≤ C τσ (cid:16) τσ (cid:17) . In particular, since k u k m ≤ C for subcritical τ , we conclude that u ( t, · ) has the Sobolev regularity, k u k m − ≤ C (cid:18) τσ + τ σ (cid:19) . We close this section by noting that the second iteration led to an approximate RSW systemlinearized around the pressuerless velocity field, u , (3.1),(3.7), which governs our improved, 2 πτ -periodic approximation, ( h ( t, · ) , u ( t, · )) ∈ H m − ( T ) × H m − ( T ).4. Long time existence of approximate periodic solutions
The shallow-water equations.
How close is ( h ( t, · ) , u ( t, · )) to the exact solution ( h ( t, · ) , u ( t, · ))?Below we shall show that their distance, measured in H m − ( T ), does not exceed e C t δ − e C t δ . Thus,for sufficiently small δ , the RSW solution ( h, u ) is “approximately periodic” which in turn implies itslong time stability. This is the content of our main result. Theorem 4.1.
Consider the rotational shallow water (RSW) equations on a fixed 2D torus, ∂ t h + u · ∇ h + (cid:18) σ + h (cid:19) ∇ · u = 0(4.1a) ∂ t u + u · ∇ u + 1 σ ∇ h − τ J u = 0(4.1b) subject to sub-critical initial data ( h , u ) ∈ H m ( T ) with m > and α := min(1 + σh ( · )) > . Let δ = τσ denote the ratio between the Rossby number τ and the squared Froude number σ , with subcritical τ ≤ τ c ( ∇ u ) so that (2.1b) holds. Assume σ ≤ for substantial amount of pressure forcing in (4.1b).Then, there exists a constant C , depending only on m , τ c , α and in particular depending linearlyon k ( h , u ) k m , such that the RSW equations admit a smooth, “approximate periodic” solution in thesense that there exists a near-by πτ -periodic solution, ( h ( t, · ) , u ( t, · )) , such that (4.2) k p ( t, · ) − p ( t, · ) k m − + k u ( t, · ) − u ( t, · ) k m − ≤ e C t δ − e C t δ . Here p is the “normalized height” such that σp = √ σh , and correspondingly, p satisfies σp = √ σh .It follows that the life span of the RSW solution, t < ∼ t δ := ln( δ − ) is prolonged due to the rapidrotation δ ≪ , and in particular, it tends to infinity when δ → . Proof.
We compare the solution of the RSW system (4.1a),(4.1b) with the solution, ( h , u ), of ap-proximate RSW system (3.1),(3.7). To this end, we rewrite the latter in the equivalent form, ∂ t h + u · ∇ h + (cid:18) σ + h (cid:19) ∇ · u = ( u − u ) · ∇ h + (cid:18) σ + h (cid:19) ∇ · ( u − u )(4.3a) ∂ t u + u · ∇ u + 1 σ ∇ h − τ J u = ( u − u ) · ∇ u + R. (4.3b)The approximate system differs from the exact one, (4.1a),(4.1b), in the residuals on the RHS of(4.3a),(4.3b). We will show that they have an amplitude of order δ . In particular, the comparison in therotationally dominant regime, δ ≪ h , u ). To show that ( h , u ) is indeed an approximatesolution for the RSW equations, we proceed as follows.We first symmetrize the both systems so that we can employ the standard energy method fornonlinear hyperbolic systems. To this end, We set the new variable (“normalized height”) p such that1 + σp = √ σh . Compressing notations with U := ( p, u ) ⊤ , we transform (4.1a),(4.1b) into the symmetric hyperbolic quasilinear system(4.4) ∂ t U + B ( U , ∇ U ) + K [ U ] = 0 . Here B ( F , ∇ G ) := A ( F ) G x + A ( F ) G y where A , A are bounded linear functions with values beingsymmetric matrices, and K [ F ] is a skew-symmetric linear operator so that h K [ F ] , F i = 0. By standardenergy arguments, e.g. [12],[13],[15]), the symmetric form of (4.4) yields an exact RSW solution U ,which stays smooth for finite time t < ∼
1. The essence of our main theorem is that for small δ ’s,rotation prolongs the life span of classical solutions up to t ∼ O (ln δ − ). To this end, we symmetrizethe approximate system (4.3a), (4.3b), using a new variable p such that 1 + σp = √ σh .Compressing notation with U := ( p , u ) ⊤ , we have(4.5) ∂ t U + B ( U , ∇ U ) + K ( U ) = R where the residual R is given by R := (cid:20) ( u − u ) · ∇ p + (cid:0) σ + p (cid:1) ∇ · ( u − u )( u − u ) · ∇ u − R (cid:21) , with R defined in (3.7c). We will show R is small which in turn, using the symmetry of (4.4) and(4.5), will imply that k U − U k m − is equally small. Indeed, thanks to the fact that H m − ( T ) is analgebra for m >
5, every term in the above expression is upper-bounded in H m − , by the quadraticproducts of the terms, k u k m , k p k m − , k u k m − , k u − u k m − , up to a factor of O (1 + σ ). TheSobolev regularity of these terms, u , u and p is guaranteed, respectively, in corollary 2.1, corollary3.1 and theorem 3.1. Moreover, the non-vacuum condition, 1 + σh ≥ α >
0, implies that 1 + σh remains uniformly bounded from below, and by standard arguments (carried out in Appendix A), k p k m − ≤ C (1 + τ /σ ). Summing up, the residual R does not exceed,(4.6) k R k m − ≤ C (cid:18) δ + τσ + ... + τ σ (cid:19) < ∼ C δ, for sub-critical τ ∈ (0 , τ c ) and under scaling assumptions δ < σ < O ( δ )-upperbound holds for the error E := U − U , for a long time, t < ∼ t δ . Indeed, subtracting (4.4) from (4.5), we find the error equation ∂ t E + B ( E , ∇ E ) + K [ E ] = − B ( U , ∇ E ) − B ( E , ∇ U ) + R . ONG TIME EXISTENCE OF THE RAPIDLY ROTATING EULER EQUATIONS 11
By the standard energy method using integration by parts and Sobolev inequalities while utilizing thesymmetric structure of B and the skew-symmetry of K , we arrive at ddt k E k m − < ∼ m k E k m − + k U k m − k E k m − + k R k m − k E k m − . Using the regularity estimates of U = ( p , u ) ⊤ and the upper bounds on R in (4.6), we end upwith an energy inequality for k E ( t, · ) k m − , ddt k E k m − < ∼ m k E k m − + C k E k m − + C δ, k E (0 , · ) k m − = 0 . A straightforward integration of this forced Riccati equation (consult for example, [16, § k E k m − does not exceed(4.7) k U ( t, · ) − U ( t, · ) k m − ≤ e C t δ − e C t δ . In particular, the RSW equations admits an “approximate periodic” H m − ( T )-smooth solutions for t ≤ C ln( dde − ) for δ ≪ (cid:3) Remark 4.1.
The estimate on the actual height function h follows by applying the Gagliardo-Nirenberginequality to h − h = p (1 + σ p ) − p (1 + σ p ) = ( p − p )(1 + σ ( p − p ) + σ p ) , k h ( t, · ) − h ( t, · ) k m − < ∼ e C t δ (1 − e C t δ ) . Our result is closely related to observations of the so called “near-inertial oscillation” (NIO) inoceanography (e.g. [22]). These NIOs are mostly seen after a storm blows over the oceans. Theyexhibit almost periodic dynamics with a period consistent with the Coriolis force and stay stable forabout 20 days which is a long time scale relative to many oceanic processes such as the storm itself.This observation agrees with our theoretical result regarding the stability and periodicity of RSWsolutions. In terms of physical scales, our rotationally dominant condition, δ = gHf LU ≪ , provides aphysical characterization of this phenomenon. Indeed, NIOs are triggered when storms pass by (large U ) and only a thin layer of the oceans is reactive (small aspect ratio H/L ). Upon using the multi-layermodel ([18, § f = 10 − s − , L = 10 m, H = 10 m, U = 1 ms − , g = 0 . ms − (reduced gravity due to density stratification – consult [18, § δ = 0 .
1, and theorem 4.1 implies the existence of smooth, approximate periodic solution over timescale ln( δ − ) L/U ≈ ω = ∂ y u − ∂ x v <
0, which is a preferred scenario ofthe sub-critical condition (2.1b) assumed in theorem 4.1.4.2.
The isentropic gasdynamics.
In this section we extend theorem 4.1 to rotational 2D Eulerequations for isentropic gas, ∂ t ρ + ∇ · ( ρ u ) = 0(4.8a) ∂ t u + u · ∇ u + ρ − ∇ e p ( ρ ) − f u ⊥ = 0 . (4.8b)Here, u := ( u (1) , u (2) ) ⊤ is the velocity field, ρ is the density and e p = e p ( ρ ) is the pressure which forsimplicity, is taken to be that of a polytropic gas, given by the γ -power law,(4.8c) e p ( ρ ) = Aρ γ . The particular case A = g/ , γ = 2, corresponds to the RSW equations (1.1a),(1.1b). The followingargument for long term existence of the 2D rapidly rotating isentropic equations applies, with minormodifications, to the more general pressure laws, e p ( ρ ), which induce the hyperbolicity of (4.8a).We first transform the isentropic Euler equations (4.8a) into their nondimensional form, ∂ t ρ + u · ∇ ρ + (cid:18) σ + ρ (cid:19) ∇ · u = 0 ∂ t u + u · ∇ u + 1 σ ∇ (1 + σρ ) γ − − τ J u = 0where the Mach number σ plays the same role as the Froude number in the RSW equation. In orderto utilize the technique developed in the previous section, we introduce a new variable h by setting1 + σh = (1 + σρ ) γ − , so that the new variables, ( h, u ), satisfy ∂ t h + u · ∇ h + ( γ − (cid:18) σ + h (cid:19) ∇ · u = 0 , (4.10a) ∂ t u + u · ∇ u + 1 σ ∇ h − τ J u = 0 . (4.10b)This is an analog to the RSW equations (4.1a),(4.1b) except for the additional factor ( γ −
1) in themass equation (4.10a). We can therefore duplicate the steps which led to theorem 4.1 to obtain a longtime existence for the rotational Euler equations (4.10a),(4.10b). We proceed as follows.An approximate solution is constructed in two steps. First, we use the 2 πτ -periodic pressurelesssolution, ( h ≡ constant , u ( t, · )) for sub-critical initial data, ( h , u ). Second, we construct a 2 πτ -periodic solution ( h ( t, · ) , u ( t · )) as the solution to an approximate system of the isentropic equations, linearized around the pressureless velocity u , ∂ t h + u · ∇ h + ( γ − (cid:18) σ + h (cid:19) ∇ · u = 0 , u := u + τσ J (cid:16) I − e tJ/τ (cid:17) ∇ h ( t, · ) . In the final step, we compare ( h, u ) with the 2 πτ -periodic approximate solution, ( h , u ). To this end,we symmetrize the corresponding systems using U = ( p, u ) ⊤ with the normalized density function p satisfying 1 + q γ − σp = √ σh . Similarly, the approximate system is symmetrized with thevariables U = ( p , u ) where 1 + q γ − σp = √ σh . We conclude Theorem 4.2.
Consider the rotational isentropic equations on a fixed 2D torus, (4.9a), (4.9a), subjectto sub-critical initial data ( ρ , u ) ∈ H m ( T ) with m > and α := min(1 + σρ ( · )) > .Let δ = τσ denote the ratio between the Rossby and the squared Mach numbers, with sub-critical τ ≤ τ c ( ∇ u ) so that (2.1b) holds. Assume σ < for substantial amount of pressure in (4.9a). Then, there existsa constant C , depending only on m , k ( ρ , u ) k m , τ c and α , such that the RSW equations admit asmooth, “approximate periodic” solution in the sense that there exists a near-by πτ -periodic solution, ( ρ ( t, · ) , u ( t, · )) such that (4.11) k p ( t, · ) − p ( t, · ) k m − + k u ( t, · ) − u ( t, · ) k m − ≤ e C t δ − e C t δ . ONG TIME EXISTENCE OF THE RAPIDLY ROTATING EULER EQUATIONS 13
Here, p is the normalized density function satisfying σp = (1 + σρ ) γ − , and p results from thesame normalization for ρ .It follows that the life span of the isentropic solution, t < ∼ t δ := 1 + ln( δ − ) is prolonged due to therapid rotation δ ≪ , and in particular, it tends to infinity when δ → . Remark 4.2.
For the actual density functions, ρ − ρ = σ [(1 + σp ) γ − − (1 + σp ) γ − ] = R C γ [1 + σ ( θ ( p − p ) + p )] γ − dθ k ρ ( t, · ) − ρ ( t, · ) k m − < ∼ e C t (1 − e C t ) γ − , in the physically relevant regime γ ∈ (1 , . The ideal gasdynamics.
We turn our attention to the full Euler equations in the 2D torus, ∂ t ρ + ∇ · ( ρ u ) = 0 ,∂ t u + u · ∇ u + ρ − ∇ e p ( ρ, S ) = f J u ,∂ t S + u · ∇ S = 0 , where the pressure law is given as a function of the density, ρ and the specific entropy S , e p ( ρ, S ) := ρ γ e S . It can be symmetrized by defining a new variable – the “normalized” pressure function, p := √ γγ − e p γ − γ , and by replacing the density equation (4.12a) by a (normalized) pressure equation, so that the abovesystem is recast into an equivalent and symmetric form, e.g., [12],[11] e S ∂ t p + e S u · ∇ p + C γ e S p ∇ · u = 0 ,∂ t u + u · ∇ u + C γ e S p ∇ p = f J u, C γ := γ − ,∂ t S + u · ∇ S = 0 . It is the exponential function, e S , involved in triple products such as e S p ∇ p , that makes the ideal gassystem a nontrivial generalization of the RSW and isentropic gas equations.We then proceed to the nondimensional form by substitution, u → U u ′ , p → P(1 + σp ′ ) , S = ln( pρ − γ ) → ln(PR − γ ) + σS ′ After discarding all the primes, we arrive at a nondimensional system e σS ∂ t p + e σS u · ∇ p + C γ (cid:18) e σS − σ + e σS p (cid:19) ∇ · u = − C γ σ ∇ · u , (4.13a) ∂ t u + u · ∇ u + C γ (cid:18) e σS − σ + e σS p (cid:19) ∇ p = − C γ σ ∇ p + 1 τ J u, (4.13b) ∂ t S + u · ∇ S = 0 , (4.13c)where σ and τ are respectively, the Mach and the Rossby numbers. With abbreviated notation, U := ( p, u , S ) ⊤ , the equations above amount to a symmetric hyperbolic system written in the compactform,(4.14) A ( S ) ∂ t U + A ( U ) ∂ x U + A ( U ) ∂ y U = K [ U ] . Here, A i ( i = 0 , ,
2) are symmetric-matrix-valued functions, nonlinear in U and in particular A isalways positive definite. The linear operator K is skew-symmetric so that h K [ U ] , U i = 0.Two successive approximations are then constructed based on the iterations (1.4), starting with j = 1, p ≡ constant ,∂ t u + u · ∇ u = 1 τ J u ,S ≡ constant . Identified as the pressureless solution, u is used to linearize the system, resulting in the followingapproximation ∂ t p + u · ∇ p + C γ p ∇ · u = − C γ σ ∇ · u , (4.15a) u − u = τσ J ( I − e tJ/τ ) C γ e σS (1 + σp ) ∇ p , (4.15b) ∂ t S + u · ∇ S = 0(4.15c)The 2 πτ -periodicity and global regularity of U := ( p , u , S ) ⊤ follow along the same lines outlinedfor the RSW equations in section 3 (and therefore omitted), together with the following nonlinearestimate for e σS , k e σS − k m = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j =1 ( σS ) j j ! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m < ∼ m ∞ X j =1 ( C m | σS | ∞ ) j − j ! k σS k m = e C m | σS | ∞ − C m | σS | ∞ k σS k m ;for the latter, we apply recursively the Gagliardo-Nirenberg inequality to typical terms k ( σS ) j k m .Notice the entropy variable (both the exact and approximate ones) always satisfies a transport equationand therefore is conserved along particle trajectories, which implies that the L ∞ norm of the entropyvariable is an invariant. Thus, we arrive at an estimate(4.16) k e σS − k m ≤ σ b C k S k m . Of course, the same type of estimate holds for the approximate entropy, S .Finally, we subtract the approximate system (4.15) from the exact system (4.14), arriving at anerror equation for E := U − U that shares the form as for the RSW system in Section 4.1, exceptthat A i ( U ) − A i ( U ) = A i ( U − U ) due to nonlinearity which is essentially quadratic in the sensethat, k A i ( U ) − A i ( U ) k n < ∼ k U − U k n + k U − U k n , i = 0 , , , k A i ( U ) − A i ( U ) k W , ∞ < ∼ k U − U k W , ∞ + k U − U k W , ∞ , i = 0 , , . where n >
2. This additional nonlinearity manifests itself as three more multiplications in the energyinequality, ddt k E k m − < ∼ k E k m − + ... + k E k m − + δ, k E (0 , · ) k m − = 0 , whose solution (– developed around a simple root of the quintic polynomial on the right), has thesame asymptotic behavior as for the quadratic Riccati equations derived in the previous sections. Consider a typical term of A i , e.g. e σS p . Applying (4.16) together with Gagliardo-Nirenberg inequality to e σS − e σS = e σS ( e σ ( S − S ) − ), we can show k e σS − e σS k n < ∼ k S − S k n . The estimate on k e σS p − e σS p k n then follows byapplying identity ab − a b = ( a − a )( b − b ) + ( a − a ) b + a ( b − b ) together with the triangle inequality and the G-Ninequality. Here regularity of S and p is a priori known. ONG TIME EXISTENCE OF THE RAPIDLY ROTATING EULER EQUATIONS 15
Theorem 4.3.
Consider the (symmetrized) rotational Euler equations on a fixed 2D torus (4.13)subject to sub-critical initial data ( p , u , S ) ∈ H m ( T ) with m > .Let δ = τσ denote the ratio between the Rossby and the squared Mach numbers, with subcritical τ ≤ τ c ( nabla u ) so that (2.1b) holds. Assume σ < for substantial amount of pressure forcing in (4.13b). Then,there exists a constant C , depending only on m , k ( p , u , S ) k m , τ c , such that the ideal gas equationsadmit a smooth, “approximate periodic” solution in the sense that there exists a near-by πτ -periodicsolution, ( p ( t, · ) , u ( t, · ) , S ( t, · )) such that (4.17) k p ( t, · ) − p ( t, · ) k m − + k u ( t, · ) − u ( t, · ) k m − + k S ( t, · ) − S ( t, · ) k m − ≤ e C t δ − e C t δ . It follows that the life span of the ideal gas solution, t < ∼ t δ := ln( δ − ) is prolonged due to the rapidrotation δ ≪ , and in particular, it tends to infinity when δ → . Appendix. Staying away from vacuum
We will show the following proposition on the new variable p defined in section 4.1. Proposition 5.1.
Let p satisfies (5.1) 1 + 12 σp = p σh where h is defined as in (3.1), that is, (5.2) ∂ t h + u · ∇ h + (cid:18) σ + h (cid:19) ∇ · u = 0 subject to initial data h (0 , · ) = h ( · ) that satisfies the non-vacuum condition σh ( · ) ≥ α > .Then, | p | ∞ ≤ b C (cid:16) τσ (cid:17) , k p k n ≤ C (cid:16) τσ (cid:17) . The proof of this proposition follows two steps. First, we show that the L ∞ and H n norms of p (0 , · ) are dominated by h (0 , · ) due to the non-vacuum condition. Second, we derive the equationfor p and obtain regularity estimates using similar techniques from section 4.1.Step 1. For simplicity, we use p := p (0 , · ) and h := h (0 , · ).Solving (5.1) and differentiation yield p = 2 h √ σh + 1 , ∇ p = ∇ h √ σh . Clearly, | p | ∞ ≤ | h | ∞ . The above identities, together with the non-vacuum condition imply k p k ≤ k h k and |∇ p | L ∞ ≤ |∇ h | L ∞ √ α . For higher derivatives of p , we use the following recursive relation. Rewrite (5.1) as p + σp = h and then take the k -th derivative on both sides D k p + 14 σ pD k p + 14 σ (cid:16) D k ( q ) − pD k p (cid:17) = D k h so that taking L norm of this equation yields I − II := (cid:13)(cid:13)(cid:13)(cid:13) (1 + 12 σp ) D k p (cid:13)(cid:13)(cid:13)(cid:13) − σ (cid:13)(cid:13)(cid:13) D k ( q ) − pD k p (cid:13)(cid:13)(cid:13) ≤ k D k h k . Furthermore, we find I ≥ √ α k D k p k by (5.1) and the non-vacuum condition. We also find II < ∼ n |∇ p | ∞ k p k | k |− by Gagliardo-Nirenberg inequalities. Thus we arrive at a recursive relation k p k | k | ≤ b C ( k p k | k |− + k h k | k | )which implies that the H n norm of p (0 , · ) = p is dominated by k h (0 , · ) k n = k h k n .Step 2. We derive an equation for p using relation (5.1) and equation (5.2), ∂ t p + 2 u · ∇ p + (cid:18) σ + p (cid:19) ∇ · u = 0 . This equation resembles the formality of the approximate mass equation (3.1) for h and thus weapply similar technique to arrive at the same regularity estimate for p , | p ( t, · ) | ∞ ≤ b C (cid:16) τσ (cid:17) , k p ( t, · ) k n ≤ C (cid:16) τσ (cid:17) . References [1] A. Babin, A. Mahalov and B. Nicolaenko,
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Department of MathematicsCenter of Scientific Computation And Mathematical Modeling (CSCAMM)University of MarylandCollege Park, MD 20742 USA
E-mail address : [email protected] URL : (Eitan Tadmor) Department of Mathematics, Institute for Physical Science and Technologyand Center of Scientific Computation And Mathematical Modeling (CSCAMM)University of MarylandCollege Park, MD 20742 USA
E-mail address : [email protected] URL ::