Low Froude Number Limit of the Rotating Shallow Water and Euler Equations

Transcription

1 Low Froude Number Limit of the Rotating Shallow Water and Euler Equations Kung-Chien Wu Department of Pure Mathematics and Mathematical Statistics University of Cambridge, Wilberforce Road Cambridge, CB3 OWA, UK Abstract We perform the mathematical derivation of the rotating lake equations anelastic system) from the classical solution of the rotating shallow water and Euler equations when the Froude number tends to zero. Keywords and phrases. Shallow water equations, Euler equations, low Froude number limit, incompressible limit, lake equations, anelastic system. 200 Mathematical Subject Classification. 35Q3, 76B5. Introduction The isentropic Euler equations with additional rotating forcing read, in a two dimensional bounded domain : t h + hu) = 0, t hu) + hu u ) ) + hu h γ Ro + h h γ 0 = 0, F r 2 hu) n = 0, h t=0 = hx, 0), u t=0 = ux, 0), kcw28@dpmms.cam.ac.uk.)

2 where γ 2. In particular, when γ = 2,.) is the inviscid rotating shallow water equations which are frequently used for modeling both oceanographic and atmospheric fluid flow in the midlatitudes with relatively large length and time scales [8, 6, 20]. In this case the unknowns are h = ht, x), the height of water and u = ut, x) = u t, x), u 2 t, x)), the horizontal component of the fluid velocity. The orthogonal velocity is denoted by u = u 2, u ) and the strictly positive function h 0 = h 0 x) describes the bottom topography. Note that the two parameters Ro and F r are respectively, the Rossby number measuring the inverse rotational forcing and the Froude number measuring the inverse pressure forcing, they penalize Coriolis force hu and the pressure forcing h ) h γ h γ 0 respectively. In real geophysical interest, e.g., the large-scale motions in the atmosphere, at least one of the parameters Ro or F r is very small which will lead asymptotically to reduced models. As noted by Majda in [6], although the rotating shallow water equations are mathematical similar to the compressible flow equations, whether or not compressibility effects are important depends on the scales associated with the fluid motion. In gas dynamics, the measure of the importance of the compressibility effects is given by the Mach number. For rotating shallow water equations, the Froude number plays the analogous role as the Mach number. In this paper, we will consider the low Froude number limit, i.e. F r 0 of.). For simplicity of notations, we may assume Ro =, F r = ε, and rewrite.) as t h ε + h ε u ε ) = 0, t h ε u ε ) + h ε u ε u ε) + h ε u ε ) + h ε ) h ε ) γ h γ 0 = 0, ε 2 h ε u ε ) n = 0, h ε t=0 = h ε 0x), u ε t=0 = u ε 0x)..2) When ε is a fixed number, under the assumption of initial conditions h ε 0 c > 0, h ε 0, u ε 0) H 3 ) H 3 )) 2 and some appropriate compatibility conditions on, Beirao da Veiga proved the local existence and uniqueness of classical solution of.2) in [2]. Moreover, the solution satisfies the energy equality d e ε, t)dx = 0, e ε = dt 2 hε u ε 2 + ε 2 Θhε ),.3) 2

3 where the potential energy Θh ε ) = ) h ε ) γ + γ )h γ 0 γh ε h γ 0 γ is a convex function with minimum occurring at h ε = h 0 and satisfies Θh ε ) 0. Formally, letting ε 0, h ε will converge to h 0 from the energy equality.3) and the limiting velocity u will solve the rotating lake equations or anelastic system) h 0 u) = 0, t h 0 u) + h 0 u u ) + h 0 u + h 0 π = 0,.4) h 0 u) n = 0, u t=0 = u 0 x), h 0 u 0 ) = 0. Thus, the rotating lake equations may be seen as the low Froude number limit of the usual inviscid rotating shallow water and Euler equations when the initial height converges to a nonconstant function h 0 x) depending on the space variable see [5] for the case without rotating forcing). Note that for non-varying bottom h 0 =,.4) reverts to the rotating incompressible Euler equations. Before the presentation of the main result of this paper, let us make the following assumptions on the initial conditions: A) h ε 0 c > 0, h ε 0, u ε 0) H 3 ) H 3 )) 2 and some appropriate compatibility conditions on, this guarantees the local existence and uniqueness of classical solution of the inviscid rotating shallow water equations.2). A2) ε 2 Θhε 0)dx 0 as ε 0, this means the initial potential energy converges to 0 as ε goes to zero. A3) h ε 0 u ε 0 h 0 u 0 in L 2 ) as ε 0, this means the initial kinetic energy is well prepared. A4) h 0 c > 0, u 0 H 3 )) 2, h 0 u 0 ) = 0 and some appropriate compatibility conditions on, this guarantees the existence and uniqueness of classical solution of the rotating lake equations.4) and the well prepared initial condition. Under the assumption A4), Levermore, etc. [0] proved the existence and uniqueness of classical solution of the lake equations.4). The main result of this paper is stated as follows: 3

4 Theorem. Let h ε, u ε ) H 3 ) H 3 )) 2 be the solution of.2) and the initial conditions h ε 0, u ε 0) satisfy the assumptions A) A4), then there exists T > 0 such that h ε h 0 ), t) L γ ) 0,.5) h ε u ε h 0 u), t) L 2γ γ+ ) 0.6) as ε 0, where u H 3 )) 2 is a classical solution of the rotating lake equations.4). The question of the singular limits, e.g. incompressible, low Froude number limits, in fluid mechanics has received considerable attention. For low Mach number or incompressible limit, some fundamental facts on this problem have been established by Klainerman and Majda in [9] see also [5]). The basic result, which has been proven in various contexts, is that slightly compressible fluid flows are close to incompressible flows even though the equations for the latter are related to those for the former via a singular limit. This justifies the use of the incompressible flow equations for certain real fluids that are actually slightly compressible. For weak solutions, this problem was done by P.L. Lions and Masmoudi in [4] where Leray global weak solutions of the incompressible Navier-Stokes equation are recovered from the global weak solutions of the compressible Navier-Stokes equation see also [6] for the quasi-neutral limit of the Navier-Stokes-Poisson system). The modulated energy method is a popular way to study the hydrodynamic limits, it was introduced by Brenier [3] to prove the convergence of the Vlasov-Poisson system to the incompressible Euler equation. It is also applied to study various singular limits of the other equations, for example the Schrödinger-Poisson equation [2], the Gross-Pitaevskii equation [3], the Klein-Gordon equation [2] and the quantum hydrodynamic model []. In fact, we will employ this method to study the low Froude number limit of the rotating shallow water and Euler equations. We limit ourselves in this paper to the case when the initial data is well prepared see assumption A4)). For general not well-prepared initial condition as mentioned in [, 7], we must consider the oscillation part generated by the nondivergence free part of the initial momentum. Indeed, this is a challenge problem and will be our next research project. The density variations in real fluids are related to both pressure and entropy variations, even in the low Mach or Froude number limit and the 4

5 limiting density may not necessary be a constant is the main issue of the recent research about the singular limit problems. The only known results concerning the non-constant limiting density we will refer to [4, 5] where the viscous shallow water equation is discussed for the periodic domain. We also refer to [9] for non-isentropic Euler equation and [] for full Navier- Stokes equation. Moreover, we mention the works about anelastic system by Feireisl, etc. [7] and Masmoudi [8], they extended the classical Leray s global weak solutions of the incompressible Navier-Stokes equation to the anelastic system. In this paper, we use the modulated energy functional to control the propagation of the height h ε and velocity u ε, we have to check the evolution of the modulated energy and calculate the kinetic part R, potential part R 2 and rotating part R 3 carefully as showed in 2.9). Fortunately, we can treat the kinetic part R similar to [3,, 2, 2], and control R 2 and R 3 successfully. Note that it is easy to control R 2 if the bottom topography h 0 =. Besides the introduction, section 2 is devoted to the rigorous proof of the main theorem. 2 Proof of the theorem The assumptions of initial conditions A2) A3) give the uniform bound of initial energy, by the energy estimate.3), we have uniform bound of total energy e ε, t)dx = e ε, 0)dx C. 2.) Especially, we will have Θh ε )dx = Oε 2 ), h ε h 0 ), t) L γ ) = O ε 2 γ ), t [0, T ] 2.2) by the following elementary convexity inequality h ε h 0 γ Θh ε ) for γ 2. γ Now, we define the modulated energy as follows: H ε t) = h ε u ε u 2 dx + 2 ε 2 5 Θh ε )dx. 2.3)

6 The modulated energy H ε t) can be further rewritten as H ε t) = e ε dx h ε u u ε dx + h ε u 2 dx. 2.4) 2 Differentiating the modulated energy 2.4) with respect to t and using energy equation.3), we obtain d dt Hε t) = d h ε u u ε dx + d dt dt 2 hε u 2 dx I + I ) By momentum equation.2) 2, integration by parts and the boundary condition of h ε u ε, we obtain I = h ε t u u ε dx h ε u ε u ε) : udx + h ε u u ε ) dx + ε 2 ) h ε u h ε ) γ h γ 0 dx. 2.6) Next employing the continuity equation.2), integration by parts and using the boundary condition of h ε u ε, we have I 2 = h ε u t udx + 2 u 2 h ε u ε )dx. 2.7) Consequently, by 2.5) 2.7) we have where d dt Hε t) = 2 u 2 h ε u ε )dx + t u h ε u h ε u ε )dx + R = h ε u ε u ε) : udx, R 2 = ) h ε u h ε ) γ h γ ε 2 0 dx, R 3 = h ε u u ε ) dx. 3 R i, i= 2.8) 2.9) 6

7 To deal with the kinetic part R, we rewrite R as R = h ε u ε u) u ε u) ) : udx + h ε u u ) : udx h ε u ε u ) : udx. h ε u u ε) : udx 2.0) One can calculate, using an integration by parts and the boundary conditions of h ε u ε and u, and h ε u u ε) : udx = h ε u u ) : udx 2 u 2 h ε u ε )dx, 2.) h ε u ε u ) : udx = [ u ) ] u h ε u h ε u ε )dx, this means that R = h ε u ε u) u ε u) ) : udx + 2 u 2 h ε u ε )dx + To deal with the potential part R 2, we need [ u ) ] u h ε u h ε u ε )dx. 2.2) 2.3) h ε u h ε ) γ = γ γ u hε ) γ, 2.4) and using the divergence free of h 0 u to obtain h ε u h γ 0 = γ )h ε h γ 0 u. 2.5) Combing 2.4) and 2.5) together we have ) h ε u h ε ) γ h γ 0 = γ γ [ u h ε ) γ + γh ε h γ 0 u ]. 2.6) 7

8 Moreover, using integration by parts, the boundary condition of u, and divergence free of h 0 u, we have h 0 ) γ udx = u h 0 ) γ dx = γ γ h 0 u h 0 ) γ dx = ) Consequently, by 2.6) and 2.7) R 2 = γ [ ] ε 2 γ u h ε ) γ γh ε h γ 0 + γ )h γ 0 dx. 2.8) For the rotating part R 3, one can prove that R 3 = h ε u u ε ) dx = u h ε u ε )dx = u h ε u h ε u ε )dx, 2.9) where the anti-symmetric property u u ε ) = u u ε and the orthogonal property u u = 0 have been used. Combing above equalities, we have d dt Hε t) = h ε u ε u) u ε u) ) : udx γ [ ] h ε ) γ γh ε h γ ε γ )h γ 0 udx γ + [ t u + u ) ] u + u h ε u h ε u ε )dx. 2.20) We can estimate the first two integral of right side of 2.20), it can be bounded by u L )H ε t), Moreover, 2.20) can be transformed into d dt Hε t) C H ε t) π h ε u h ε u ε )dx, 2.2) 8

9 where π is the pressure of the lake equations anelastic system).4). Now we will estimate the second term of right side of 2.2). By 2.2), divergence free of h 0 u and Hölder inequality, we arrive at the inequality h ε u πdx = h ε h 0 )u πdx ε 2 γ u L ) π L γ γ ). 2.22) To go further, we need the relation which follows from the continuity equation.2), integration by parts and the boundary condition of h ε u ε ) h ε u ε ) πdx = π t h ε h 0 )dx = d dt πh ε h 0 )dx t πh ε h 0 )dx. 2.23) The last integral of 2.23) can be estimated by Hölder inequality t πh ε h 0 )dx ε 2 γ t π γ, 2.24) L γ ) We have to introduce one more correction term of the modulated energy defined by W ε t) = h ε h 0 )πdx. 2.25) The correction term W ε t) can be served as the acoustic part density fluctuation) of the modulated energy H ε t). This term describes the propagation of the density fluctuation in order to obtain the incompressible limitsee [2] for Klein-Gordon equation). This is similar to the low Mach number limit of the compressible fluid [4, 4, 5]. Hence for t [0, T ) we have Integrating this inequality yields d ) H ε t) + W ε t) C H ε t) + Oε γ 2 ). 2.26) dt t H ε t) H ε 0) W ε t) + C H ε τ)dτ + Oε γ 2 ). 2.27) 9 0

10 One can show that W ε t) = Oε γ 2 ), and hence t H ε t) C H ε τ)dτ + H ε 0) + Oε γ 2 ). 2.28) 0 In order to obtain the convergence result, we need to estimate the initial modulated energy functional H ε 0). It is easy to see that h ε 0u ε 0 h ε 0u 0 L 2 ) h ε 0u ε 0 h 0 u 0 L 2 ) + h 0 h ε 0)u 0 L 2 ), 2.29) and the first term of right hand side of 2.29) converges to 0 by assumption A3). For the second term, using the finite measure of, assumption A) and an elementary inequality we have x a 2 a x a 2, x 0, a c > 0, h 0 h ε 0)u 0 L 2 ) u 0 L ) h 0 h ε 0 L 2 ) c u 0 L ) h 0 h ε 0 L 2 ) 2.30) C) u 0 L ) h 0 h ε 0 L γ ), which converges to 0 by assumption A2), and hence H ε 0) 0 as ε 0. Applying the Gronwall inequality, we can show that H ε t) 0 for t [0, T ). It is easy to rewrite the modulated energy 2.3) as H ε t) = 2 h ε hε u ε h ε u) 2 dx + Θh ε )dx, 2.3) ε 2 then we have h ε hε u ε h ε u) 2 dx ) as ε 0. Therefore we can deduce from 2.32) and Hölder inequality that h ε u ε h 0 u 2γ h ε L γ+ ) L 2γ ) h ε hε u ε h ε u) L 2 ) 2.33) + h ε h 0 L γ ) u L 2 ), 0

11 which converges to zero as ε 0. This completes the proof of the theorem. ACKNOWLEDGMENTS It is a pleasure to thank Professor Chi-Kun Lin for stimulating discussion concerning this paper. This work is supported by the Tsz-Tza Foundation in institute of mathematics, Academia Sinica, Taipei, Taiwan. The author would like to thank Dr. Clément Mouhot for his kind invitation to visit Cambridge during academic year. References [] T. Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Rational Mech. Anal., ), 73. [2] H. Beirao da Veiga, Data dependence in the mathematical theory of compressible inviscid fluids, Arch. Rational Mech. Anal., 9992), [3] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Commun. in Partial Differential Equations, ), [4] D. Bresch, B. Desjardins, E. Grenier and C. K. Lin, Low Mach number limit of viscous polytropic flows: Formal Asymptotics in the periodic case, Studies in Appl. Math., ), [5] D. Bresch, M. Gisclon and C. K. Lin, An example of low Mach number effects for compressible flows with nonconstant density height) limit, Mathematical Modelling and Numerical Analysis, ), [6] D. Donatelli, P. Marcati, A quasi-neutral type limit of the Navier-Stokes- Poisson system with large data, Nonlinearity, 22008), [7] E. Feireisl, A. Novotny, H. Petzeltova, Low Mach number limit for the Navier-Stokes system on unbounded domains under strong stratification, Commun. in Partial Differential Equations, 35200), [8] H. P. Greenspan, The Theory of Rotating Fluids, Cambridge University Press, London 968).

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