Multiple scales and singular limits of perfect fluids

In this article our goal is to study the singular limits for a scaled barotropic Euler system modelling a rotating, compressible and inviscid fluid, where Mach number $=\epsilon^m $, Rossby number $=\epsilon $ and Froude number $=\epsilon^n $ are proportional to a small parameter $\epsilon\rightarrow 0$. The fluid is confined to an infinite slab, the limit behaviour is identified as the incompressible Euler system. For \emph{well--prepared} initial data, the convergence is shown on the life span time interval of the strong solutions of the target system, whereas a class of generalized \emph{dissipative solutions} is considered for the primitive system. The technique can be adapted to the compressible Navier--Stokes system in the subcritical range of the adiabatic exponent $\gamma$ with $1<\gamma\leq\frac{3}{2}$, where the weak solutions are not known to exist.


Introduction
We study models of rotating fluids as described in Chemin et.al. [11]. Let T > 0 and Ω(⊂ R 3 ) = R 2 × (0, 1) be an infinite slab. We consider the scaled compressible Euler equation in time-space cylinder Q T = (0, T) × Ω describing the time evolution of the mass density ̺ = ̺(t, x) and the momentum field m = m(t, x) of a rotating inviscid fluid with axis of rotation b = (0, 0, 1): • Conservation of Mass: (1.1) • Conservation of Momentum: In general, there are infinitely many static solutions for a given potential G.
We assume the far field condition as, |̺ −̺ ǫ | → 0, m → 0 as |x h | → ∞. (1.7) • Initial data: For each ǫ > 0, we supplement the initial data as ̺(0, ·) = ̺ ǫ,0 , m(0, ·) = m ǫ,0 . (1.8) • Choice of G: As a matter of fact, the driving potential G can be seen as a sum of the centrifugal force proportional to the norm of the horizontal component of the spatial variable i.e. (x 2 1 + x 2 2 ) and the gravitational force acting in the vertical direction x 3 . We omit the effect of the centrifugal force in the present paper motivated by certain meteorological models. Instead we consider G(x) = −x 3 in Ω (1.9) corresponding to the gravitational force acting in the vertical direction.
We consider singular limit problem for ǫ → 0 in the multiscale regime: m 2 > n ≥ 1. (1.10) Thus we study the effect of low Mach number limit (also called incompressible limit), low Rossby number limit and low Froude number limit acting simultaneously on the system (1.1)-(1.2). Formally, we observe that low Mach number limit regime indicates the fluid becomes incompressible and low Rossby number limit indicates fast rotation of the fluid and as a consequence of that fluid becomes planner (two-dimensional).
As solutions of the (primitive) compressible Euler systems are expected to develop singularities (shock waves) in a finite time, there are two approaches to deal with the singular limit problem.
I. The first approach consists of considering classical(strong) solutions of the primitive system and expecting it converges to the classical solutions of the target system. Here, the main and highly non-trivial issue is to ensure that the lifespan of the strong solutions is bounded below away from zero uniformly with respect to the singular parameter.
II. The second approach is based on the concept of weak, measure-valued or dissipative solutions of the primitive system. Under proper choice of initial data one can show convergence provided the target system admits smooth solution.
In the case of second approach, most of the results dealing with weak solutions have been studied for compressible Navier-Stokes system with additional consideration of high Reynolds number limit. For rotating fluids there are several results, see Feireisl, Gallagher and Novotný [15], Feireisl et. al. [14], Feireisl and Novotný [21,22] and Li [26].
Since existence of global-in-time weak solution of compressible Euler equation satisfying energy inequality is still open for general initial data. Hence it is important to consider measure-valued solution or newly developed dissipative solution for this system. The concept of measure-valued solutions has been studied in variuos context, like, analysis of numerical schemes etc. In the following articles by Alibert and Bouchitté [1], Gwiazda,Świerczewska-Gwaizda and Wiedemann [23], Březina and Feireisl [8], Březina [7], Basarić [4], Feireisl and Lukáčová-Medvidová [18] we observe the development of theory on measure valued solution for different models describing compressible fluids mainly with the help of Young measures.
The advantages to consider the second approach are, • Weak or measure valued solutions to the primitive system exist globally in time.
Hence the result depends only on the life span of the target problem that may be finite.
• The convergence holds for a large class of generalized solutions which indicates certain stability of the limit solution of the target system.
In particular, the results involving generalized solutions are better in the sense that convergence holds for a larger class of solutions and on the life span of the limit system.
There is a series of works dealing with the low Mach number limit in the framework of measure-valued solutions. In Feireisl, Klingenberg and Markfelder [17], Bruell and Feireisl [9], Březina and Mácha [10], it is shown that measure-valued solution of primitive system which describes some compressible inviscid fluid converges to strong solution of incompressible target system under consideration of suitable initial data. The 'single-scale' limit of our system i.e. m = 1 and G = 0, has been studied by Nečasová and Tong in [28], again with the help of measure-valued solution.
The framework of measure-valued solutions can be applied also in the context of the Navier-Stokes system. Although weak solutions are available here, their existence is constrained by the technical condition for the adiabatic exponent γ > 3 2 . To handle this technical restriction, Feireisl et. al. [16] introduced the concept of dissipative measure-valued solution in terms of the Young measure. Here we use a slightly different approach introducing dissipative solution for Navier-Stokes system without an explicit presence of the Young measure. In such a way, we extend the convergence result to the Navier-Stokes system with high Reynolds number limit in the regime where the existence of weak solutions is not known.
In our approach, it is very important to consider proper initial data mainly termed as well-prepared and ill-prepared initial data. Feireisl and Novotný in [20], explain that for ill-prepared data the presence of Rossby-acoustic waves play an important role in analysis of singular limits. Meanwhile this effect was absent in well-prepared data. Here we deal with the well-prepared initial data.
Our main goal is to prove that under suitable choice of initial data a dissipative solution of compressible rotating Euler system in low Mach and low Rossby regime converges to strong solution of incompressible Euler system in 2D. Hence our plan for the article is, 1. Derivation of limit system.

Definition of dissipative solution.
3. Singular limit for 'well-prepared' data.

Notation:
• To begin, we introduce a function χ = χ(̺) such that • Without loss of generality, we assume the 'normalized' setting for p as Let us define pressure potential as, As a consequence of that we have,

Derivation of Limit systems
Here is an informal justification how we obtain the target system. First we note that (̺ ǫ , 0) is a steady state solution for (1.1)-(1.2).
Let us consider As a consequence of the above we obtain, We have static solution (̺ ǫ , 0) Clearly condition on m and n in (1.10) indicate lim ǫ→0 ∇ x P ′ (̺ ǫ ) = 0. Without loss of generality we assume, Let H be the Helmontz projection, then we have, Assuming m ǫ →̺v in some strong sense, multiplying the above equation by ǫ and using our standard expansion technique, we obtain, Thus we have, Also boundary condition will lead us to conclude The above system is 2D Euler equation.

Choice of Static Solution:
As we have noticed during the informal discussion a static solution (̺ ǫ , 0) satisfies, Finally we choose static solution̺ ǫ with the property, As m > n, asymptotically, the static solution approaches the constant state̺ = 1 as ǫ → 0. Now we will give the definition of dissipative solution for the system of our consideration.

Remark 3.2.
It is important to define the function It follows from the energy inequality (3.9) that for each ǫ > 0, where L 4 is the Lebesgue measure in R 4 . The proof this theorem follows in similar lines of Breit, Feireisl and Hofmanová as in [5,6]. We have to adopt it for unbounded domain as suggested in Basarić [4].

Target System
Taking motivation from [21] we expect the target system as, (4.1) The result stated below by Kato and Lai in [24] ensures the existence and uniqueness of Euler system in 2D: Taking motivation from (2.2) we consider another equation that describes some non-oscillatory part, Also we choose q ǫ (0, ·) = q 0,ǫ such that −∆ x h q 0,ǫ = ǫ m−1̺ curl x h P h (v 0 ).
where, v = (v h , 0) is the unique solution of Euler system with initial data v 0 .
In the following subsections we give the proof.

Relative energy inequality
In our approach, relative energy functional plays an important role. We consider,   Using definition (3.1) we obtain relative energy inequality, (4.8) Following [4] we can extend the above inequality for functions (̺,ũ) having Sobolev regularities, i.e.
We rewrite the above inequality as,

Convergence: Part 1
First withũ = 0,̺ =̺ ǫ as test functions we have the following bounds, ess sup (4.11) Now we want to calculate ̺ ǫ −̺ ǫ (L 2 +L γ )(Ω) . We rewrite, From the above estimates and using fact m >> γ we have, ess sup Now above estimates conclude that, Combining above estimates we obtain, passing to suitable subsequences. Finally we may let ǫ → 0 in the continuity equation to deduce that,

Convergence: Part 2
Here we choose proper test functions and will show that lim ǫ→0 E ǫ (t) = 0.
We consider,ũ , 0), v h as a solution of (4.1), q ǫ solves (4.3) and̺ ǫ satisfies (1.6). Using relation of q ǫ and v we obtain, From the relation ∇ x q ǫ + ǫ m−1 b × (v h , 0) = 0, and (4.2) we can conclude that Also we have q 0,ǫ L ∞ (0,T;L 2 (Ω)) ≤ ǫ m−1 c. Let us calculate each term L i , i = 1(1)8 of (4.14). For term L 1 we have, Consideration of well prepared data yields, From now on we use this generic function ξ(·), such that lim ǫ→0 ξ(ǫ) = 0. Now using our convergence in earlier part and v h solves (4.1) we obtain, It is easy to verify that, We also obtain, |L 2′′ | ≤ ξ(ǫ). Thus we can conclude, Now we also obtain, We want to estimate the term L 4 . First we rewrite it as, We observe that, Using bound of q ǫ as in (4.15) and (4.11), we obtain, We write L 5 as, By using (4.11) and (4.15) we observe, (4.20) Similarly for L 6 we rewrite as, Arguing in the same line as before we have, Now choice of G implies We also have, Thus combining all estimates (4.16)-(4.23) we have, (4.24) Using Grönwall lemma we have, where ξ(ǫ) → 0 as ǫ → 0. Using coercivity of relative energy functional we obtain, where, K ⊂ Ω is a compact set. Hence we can conclude that v = u = v h . Also we have, It ends proof of the theorem.

Extension to the Navier-Stokes System
Here our goal is to give a proper definition of dissipative solution for Navier-Stokes equation. We consider another characteristic number i.e. Reynolds number. In high Reynolds number limit, we will obtain the same target system.

Definition of dissipative Solution for Navier-Stokes system :
Let ̺ be the density and u be the velocity. In time-space cylinder Q T = (0, T) × Ω, we consider: • Conservation of Mass: • Conservation of Momentum: • Constitutive Relation: Here S(∇ x u) is Newtonian stress tensor defined by where µ > 0 and λ > 0 are the shear and bulk viscosity coefficients, respectively.
• The scaled system contains all specified characteristic numbers as in (1.3) along with, Re-Reynolds number. Here we consider, • Far field condition: Let (̺ ǫ , 0) be a static solution. we assume the condition as, • Initial data: For each ǫ > 0, we supplement the initial data as Here we provide the definition of dissipative solution for the Navier-Stokes system.
Remark 5.2. The class of test functions in the momentum equations correspond to the complete slip (Navier slip) boundary conditions. These are necessary to avoid problems with boundary layer.
Let us introduce the conservative variable m δ = ̺ δ u δ . In terms of momentum we rewrite kinetic energy as, As an observation we have the above map is convex l.s.c. From energy inequality, it is worth to notice that it is ∞ only on a measure zero set in (0, T) × Ω. Using convexity of p(·) and [̺, m] → m×m ̺ , and also using fact L 1 (Ω) continuously embedded in M(Ω), we conclude,  We choose, and Clearly the compatibility of two turbulent defect measure is clear from above equations. Arguing similarly as in Breit et. al. [6] we obtain, R m ∈ L ∞ (0, T; M + (Ω; R d×d sym )), R e ∈ L ∞ (0, T; M + (Ω)).
Now we are in a position to conclude that ̺, u, R m and R e is a dissipative solution for the Navier-Stokes equation.
Proof. The proof is similar as before only we have to consider few extra terms, see [21].