Arch. Rational Mech. Anal. 167 (2003) 309–336 Digital Object Identifier (DOI) 10.1007/s00205-003-0245-x
The Fully Compressible Semi-Geostrophic System from Meteorology Mike Cullen & Hamed Maroofi Communicated by Y. Brenier
Abstract The semi-geostrophic equations are an approximation to the 3-D Euler equations for an atmosphere where the effects of rotation dominate. They are used by meteorologists to model the formation of fronts. Mathematically rigorous results were obtained by Benamou and Brenier and by Cullen and Gangbo in the incompressible case. In this paper we extend the results of Benamou and Brenier to the fully compressible case and show the existence of weak solutions to a reformulation of these equations in so-called dual variables. The Monge-Kantorovich theory appears as a material tool in our study.
1. Introduction In this paper we study the so-called semi-geostrophic equations, an approximate model in meteorology that has been used to model large-scale behavior of the atmosphere, in particular the formation of atmospheric fronts. First introduced by Eliasen [10] and rediscovered by Hoskins & Bretherton [14], the semi-geostrophic system has been extensively studied by meteorologists concerned with both the theoretical and numerical aspects of the theory [5–9, 15, 16, 18, 20]. Investigating solutions that satisfy a certain energy minimization principle, Cullen & Purser [8] independently established Brenier’s polar-factorization theorem [2]. In a mathematically rigorous way, the semi-geostrophic equations have been first studied in the incompressible case in their Boussinesq form by Benamou & Brenier [1] and Cullen & Gangbo [4]. Benamou and Brenier showed that these equations could be reformulated as a coupled Monge-Ampére/transport problem by an appropriate change of variables. They showed existence of global weak solutions for this formulation using a rigid-wall boundary condition. Cullen and Gangbo studied the so-called semi-geostrophic shallow-water system, a variant of the semigeostrophic system, where potential temperature is assumed to be constant so that
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the 3-D system can be reduced to a 2-D system. Through a change of variables they arrive at a reformulation, for which they show the existence of global weak solutions using a free boundary condition. In this paper we study the fully compressible case (2) and extend the results obtained in [1]. Through a change of variables to geostrophic coordinates (5) and an energy minimization principle, we arrive at a reformulation of the semi-geostrophic system – the semi-geostrophic system in dual variables (16) – for which we show the existence of global weak solutions that are stable in a suitable sense. For the convenience of the reader we list the main physical quantities that appear throughout this paper: v = (v1 , v2 , v3 )T = velocity, vg = (vg1 , vg2 , 0)T = geostrophic velocity, ρ = density, α = 1/ρ = specific volume, p = pressure, = temperature, θ = potential temperature, ν = potential vorticity, σ = θ ρ, φ = geopotential representing gravitation and centrifugal forces, γ = cp /cv = ratio of specific heats ≈ 1.4 for air, cp = specific heat at constant pressure, cv = specific heat at constant volume, p0 = reference pressure fcor = Coriolis parameter R = Boltzmann constant = bounded open set representing the physical domain. The Euler equations for the dynamics and thermodynamics of a compressible adiabatic atmosphere in a rotating reference frame with rotation vector e3 with the hydrostatic approximation (vertical velocity component does not change following the flow) are given by (i)
Dv + fcor e3 × v + ∇φ + α∇p = 0, Dt
(ii)
Dα = α∇ · v, Dt
Dθ (iii) = 0, Dt
(momentum eq.)
(continuity eq.) (1)
(adiabatic assumpt.)
(iv) fcor e3 × vg + ∇φ + α∇p = 0; vg = (vg1 , vg2 , 0)T , where (i)–(iii) are the evolution equations and equation (iv) incorporates the definition of geostrophic velocity and the assumption of hydrostatic balance. The D geopotential φ typically increases with height. The operator Dt = ∂t∂ + v · ∇ is the advection operator. The derivation of the correponding semi-geostrophic equations is standard and can be found for example in [4]. It amounts to replacing the
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advected horizontal velocity component by the advected horizontal geostrophic velocity component without changing the definition of the advection operator. The semi-geostrophic equations are given by (i)
Dvg + fcor e3 × v + ∇φ + α∇p = 0 Dt
(ii)
Dθ = 0, Dt
(iii)
Dα = α∇ · v, Dt
in [0, ∞) × ,
(momentum eq.)
(adiabatic assumpt.) (continuity eq.)
(iv) fcor e3 × vg + ∇φ + α∇p = 0;
vg = (vg1 , vg2 , 0)T ,
(v) v = vg + va ; (2) is a bounded open set in R3 representing the physical domain. The potential φ is given; v, vg , α and θ are the unknowns; va is the ageostrophic wind velocity. This approximation is exact if the ageostrophic velocity does not change following the flow. The energy associated with the flow in (2) is the so-called semi-geostrophic energy and is given by E=
1 2 v + φ + c ρ(x) dx. g v 2
(3)
It consists of three terms representing kinetic, potential and internal energy. The state equations that relate the thermodynamic quantities to each other are given by pα = R;
=θ
p p0
γ −1 γ
.
(4)
Geostrophic coordinates, also called dual coordinates, were first introduced by Yudin [21], and in Hoskins [15] it was shown that a transformation to these coordinates greatly simplified the semi-geostrophic equations (see (12)). In our case, we will use geostrophic coordinates in the horizontal and isentropic (potential temperature) coordinates in the vertical. They are defined as follows: −1 y1 = x1 + fcor vg2 (t, x), −1 y2 = x2 − fcor vg1 (t, x), y3 = θ (t, x).
(5)
In general x = (x1 , x2 , x3 )T will denote physical coordinates and y = (y1 , y2 , y3 )T will denote dual coordinates. Also, most of the time we will leave out time t as an argument. Using (4) and (5) we can write the energy as
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1
E=
=
2 2 fcor {|x1
1 2 2 fcor {|x1
− y1 |2 + |x2 − y2 |2 } + φ(x) + cv ρ(x) dx
− y1 |2 + |x2 − y2 |2 } + φ(x) σ (x)dx + K1 y3
where
(σ (x))γ dx,
(6)
R γ −1 and σ (x) := θ (x)ρ(x). (7) p0 Let us use the symbol T to denote the change of variables from physical to geostrophic coordinates: K1 := cv
T (t, x) = (y1 , y2 , y3 )T = (T1 (t, x), T2 (t, x), T3 (t, x))T . The inverse of T (·), assuming that it exists, is denoted by S(·): S(t, ·) = T −1 (t, ·) = (S1 (t, y), S2 (t, y), S3 (t, y))T . We are looking for stable solutions to system (2). To define the concept of stability, consider the geostrophic energy in (6). It was shown in Shutts & Cullen [20] that states that are critical points of (6) with respect to rearrangements of particles in physical space, that preserve (y1 , y2 , y3 )T and are consistent with equations (2) (ii) and (iii), satisfy (2) (iv), i.e., are states that are in hydrostatic and geostrophic balance. Critical points that are not minima correspond to hydrostatic and geostrophic states, which may be unstable to small perturbations evolving under the 3-D Euler equations (1). The subsequent evolution of these states cannot be described by the semi-geostrophic approximation. Therefore we look for solutions to (2) where at each time t the state of the system minimizes (6), and call such solutions stable solutions. We refer to this requirement as Cullen’s stability condition. To formulate this stability condition mathematically, we recall that the push forward of the measure σ under the map T (·) is the measure ν := T# σ defined by ν(A) := T# σ (A) := σ (T −1 (A)) (8) 3 for all A ⊂ R Borel sets. In the context of semi-geostrophic theory we call ν potential vorticity. The physical significance of this quantity is discussed in [20]. Cullen’s stability condition can then be formulated as follows: at every point in time, the pair (σ, T ) minimizes the energy (6) among all pairs (σ , T ) with σ ∈ Pac ( ) and T # σ = ν. Here Pac ( ) denotes the set of probabilitiy densities in . For a stable solution to (6) the energy can then be written as Eν (σ ) = Wc (σ, ν) + K1 (σ (x))γ dx, (9)
where Wc (σ, ν) is the cheapest cost of transporting the (probability) density σ to the (probability) density ν (assuming that ν is absolutely continuous with respect to Lebesgue measure) with cost c(x, y) and is defined by Wc (σ, ν) = inf c(x, T (x)) σ (x) dx, (10) T # σ =ν
where the cost of transporting one unit of mass from x to y is given by
The Fully Compressible Semi-Geostrophic System from Meteorology
c(x, y) =
1 2 2 fcor {|x1
− y1 |2 + |x2 − y2 |2 } + φ(x) . y3
313
(11)
According to Cullen’s stability condition, the change of variables T (t, ·) from physical coordinates to geostrophic coordinates given by (5) is the optimal transportation map. In Section 4 we will show that given ν, absolutely continuous with respect to Lebesgue measure, there exists a unique σ in Pac ( ) minimizing (9). Furthermore, in Section 3 we will show that, under suitable conditions on the potential φ, given this unique σ there always exists a unique minimizer T in (10). We call T the optimal transportation map in the transport of σ to ν with cost c(x, y). We will also show that its inverse T −1 exists in a suitable sense and that it is the unique (ν a.e.) optimal map in the transport of ν to σ with cost c(y, x) = c(x, y). Assuming that we have a stable solution to (2) we will now derive the semi-geostrophic system in dual variables (16). Using the definition of geostrophic velocity in (2) (iv) the momentum equation (2) (i) can be written as DT = vg . (12) Dt That means that in dual space the flow is completely geostrophic or, in other words, the flow in dual space has vg as its velocity field (see [15, 14]). Using the continuity equation Dα/Dt = α∇ · v and the fact that θ is conserved following the flow, we can see that σ as defined in (7) satisfies the continuity equation ∂σ = −∇ · (σ v). (13) ∂t Equations (8), (12) and (13) imply that the potential vorticity satisfies the continuity equation ∂ν = −∇ · (νw), (14) ∂t where w(t, ·)–the geostrophic velocity in dual variables–is defined by w(t, y) = vg (t, S(t, y)) = fcor (S2 (t, y) − y2 , y1 − S1 (t, y), 0)T .
(15)
In view of (14), (15) and the assumption of stability, we are led to the geostrophic system in dual variables (with an initial condition added): (i)
∂ν = −∇ · (νw) ∂t
in [0, ∞) × R3 ,
(ii) w(t, y) = fcor (S2 (t, y) − y2 , y1 − S1 (t, y), 0)T, (iii) S(t, ·) = T −1 (t, ·), (iv) T (t, ·)# σ (t, ·) = ν(t, ·) unique optimal transportation map with cost c(x, y), (v) σ (t, ·) minimizes Eν(t,·) (·) over Pac ( ). (vi) ν(0, ·) = ν0 (·)
some probability density in Lr ,
r ∈ [1, ∞). (16)
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The unknown is the pair (σ, T ). We obtain ν through ν = T# σ . That (2) formally implies (16) has just been shown and follows from the assumption that the geostrophic flow is stable. It remains to show that (16) formally provides us with a stable solution to (2). Assume we are given a solution (σ, T ) to (16): define θ (t, x) := T3 (t, x), ρ(t, x) := σ (t, x)/θ (t, x) and
vg (t, x) = w(t, T (t, x)). (17)
Let M(t, y) be the flow in dual space: ∂M(t, y) = w(t, M(t, y)); ∂t
M(0, y) = y.
(18)
The flow in physical space is obtained by changing coordinates using S(·, ·): x := N (t, x0 ) := S(t, M(t, y0 )) = S(t, M(t, T (0, x0 ))),
(19)
where y0 = T (0, x0 ). The corresponding velocity field is defined by v(t, x) :=
∂N (t, x0 ) . ∂t
(20)
As a result of minimizing Eν (·) with the unique minimizer σ and corresponding optimal transportation plan T (·) pushing σ forward to ν, we will obtain the following Euler-Lagrange equations (Theorem 4.2): ∇x c(x, T (x)) σ (x) + γ K1 ∇((σ (x))γ −1 ) σ (x) = 0. With (4) and (17) this becomes f e3 × vg + ∇φ + α∇p = 0;
vg3 = 0,
(21)
which is exactly the definition of geostrophic velocity in (2) (iv). Using (17)–(21) and the fact that T and S are inverses of each other, we can then show that (2) is satisfied. The plan to find a weak solution to (16) is as follows: first we discretize equation (16) (i) in time with time step h and find approximate solutions νh to the semi-geostrophic system. This will give the time-discrete scheme as defined at the beginning of Section 5. Given the approximate solution νhk at the k-th time step, we will minimize the geostrophic energy (see Section 4) with the unique minimizer σhk , and given this unique minimizer we will find the optimal transportation map with cost c(x, y) pushing σhk forward to νhk (see Section 3); this will then be used to find the velocity that will advance νh to the (k + 1)-st time-step. Finally, we let the time-step approach 0 and show that the approximate solutions converge to a solution of the semi-geostrophic system in dual variables in some appropriate sense.
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2. Definitions, notation and conventions Important Conventions – Throughout, the symbol C will generically denote a constant that in different situations may depend upon different things, but does not assume any particular value. The objects a, b, . . . that C depends upon will be included as arguments if it is important to indicate that. Notation: C = C(a, b, . . .). – In general we will only deal with measures that are absolutely continuous with respect to Lebesgue measure and have the same mass. Hence, we identify a measure with its density, and by normalization we may assume that we deal with probability densities. – When we say a map is measurable we mean it is Lebesgue measurable, and when we say a set is measurable we mean it is Lebesgue measurable unless otherwise specified. Furthermore, “a.e.” will refer to “Lebesgue a.e..” Definition 1. Given τ > 0 and a bounded domain in Rd and ν0 ∈ L1 ( ) and w(·, ·) such that νw ∈ L1 ((0, τ ) × ), we say that ν(·, ·) satisfies ∂ν = −∇ · (wν) in (0, τ ) × , ∂t ν(0, ·) = ν0 . in the weak sense if, for any test function ψ in C0∞ ([0, τ ) × ), we have τ ∂ψ ν0 (y)ψ(0, y) dy + ν(t, y) (t, y) ∂t 0 + ν(t, y)w(t, y) · ∇ψ(t, y) dy dt = 0. Definition 2. Given a measure µ in Rd , a measurable map F from Rd to Rn induces a measure F# µ on Rn defined by F# µ(B) := µ(F −1 (B))
for all measurable sets B in Rn .
We say that F# µ is the push forward of µ under F . Definition 3. Given a set ⊂ Rd and a function f : → R, we say that f is semi-concave if there is a constant λ > 0 such that x → f (x) − λ2 /2 x 2 is concave in . We say that λ is a constant of semi-concavity. Notations – The open ball of radius R in Rd centered at the origin will be denoted by BR . – The closure of a set A in Rd is denoted by A. – Given an open set O in Rd , Pac (O) will denote the set of absolutely continuous probability measures in Rd whose supports are contained in O. – The support of a measure µ in Rd , denoted by spt(µ), is the smallest closed set containing the total mass of µ. – Given an open set O in Rd , W 1,∞ (O) denotes the space of all functions in L∞ (O) whose weak derivative is also in L∞ (O). – The set of natural numbers will be denoted by N = {1, 2, . . .}. – The characteristic function of a set A in Rd will be denoted by χA . – The Lebesgue measure in Rd will be denoted by | · |; it will be clear from the context what d is.
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3. New results on optimal transport plans In this section we prove some new results on optimal transportation maps, which are used to solve the system of semi-geostrophic equations. Recall that the geostrophic energy in the form (9) contains a term, which is the cheapest cost of transporting one probability density to another. Our contribution in this section is to obtain existence and uniqueness of a minimizer in (10) for c(x, y) of the form (11). The problem is the following: given two bounded open sets and in R3 and two probability densities σ (x) ∈ Pac ( ) and ν(y) ∈ Pac ( ), we consider the optimal transportation problem inf Iσ [T ], (22) T ∈T
where T := {T : → R3 | T # σ = ν} is the set of all transportation maps T pushing σ forward to ν and the transportation cost is given by Iσ [T ] :=
c(x, T (x)) σ (x) dx.
(23)
The cost c(x, y) for transporting one unit of mass from location x to location y is given by (11). We show existence and uniqueness of a minimizer in (22). In view of this result we define the optimal transportation cost Wc (σ, ν) by (10). Existence and uniqueness of an optimal transportation map in (22) is known for the general type of cost c(x, y) = k(x − y) with k(·) satisfying certain convexity or concavity properties (see [3, 11, 12]). In order to avoid technicalities we assume φ is twice continuously differentiable in and satisfies ∂x∂ 3 φ(x) = 0 for all x in ,
(24)
which is sufficiently general for our application and is typically satisfied as φ increases with height. It also implies that the maps y → ∇x c(x, y) and x → ∇y c(x, y) are invertible for all (x, y) ∈ × , which is crucial. We also assume that and are bounded open domains in R3 with (25) ⊂ R2 × [δ, 1/δ] for some δ > 0 and convex. The assumption on is sufficient for our application as y3 represents potential temperature, which can be assumed to be bounded away from 0 and ∞; the assumption on is technical. Our approach is based on a method introduced independently by Caffarelli and Gangbo and later used in Gangbo & McCann [12]. First we recall the Kantorovich dual problem corresponding to (22): sup
J(σ,ν) (f, g),
(f,g)∈Lipc
where
J(σ,ν) (f, g) :=
f (x) σ (x)dx +
g(y) ν(y)dy
and Lipc := {(f, g) |f ∈ W 1,∞ ( ), g ∈ W 1,∞ ( ), f (x) + g(y) c(x, y) ∀(x, y) ∈ × }.
(26)
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The standard Kantorovich relaxation of (22) is given by inf
λ∈(σ,ν)
I [λ],
(27)
where I [λ] :=
c(x, y) dλ(x, y),
and (σ, ν) = {λ | λ probability measure on R3 × R3 with marginals σ and ν}. To see that (27), is a relaxation of (22), note that every T ∈ T generates a measure λT ∈ (σ, ν) defined by λT := (id, T )# σ, with Iσ [T ] = I [λT ]. It is easily seen that sup J(σ,ν) (f, g)
(f,g)∈A
inf
λ∈(σ,ν)
I [λ] inf Iσ [T ]. T ∈T
(28)
Whenever it is clear what σ and ν are, we will simply omit subindices and write I [T ] and J (f, g). We define the following two operators, called infimal convolutions, which are standard in the Monge-Kantorovich theory. Definition 4. Let and be bounded open sets in R3 . Let c(x, y) be given by (11). Assume that f is a function from into R and g is a function from into R. The c-transform of f on is defined as f c (y) := inf {c(x, y) − f (x)} x∈
(y ∈ R3 /{y : y3 = 0}) ;
(29)
the c-transform of g on is defined as gc (x) := inf {c(x, y) − g(y)} y∈
(x ∈ R3 ).
(30)
Remark 1. The use of c as a sub and a super index indicates which domain we are in. Let us summarize the main properties of c-transforms in the following lemma: Lemma 3.1 (Properties of c-transforms). Assume that and satisfy (25). Assume that c(·, ·) is given by (11) and that φ satisfies (24). Then (i) ((f c )c )c = f c and ((gc )c )c = gc ; (ii) f c is Lipschitz in and gc is Lipschitz in and
∇f c L∞ ( ) , ∇gc L∞ ( )
sup {|∇x c(x, y)| + |∇y c(x, y)|} x∈ ,y∈
= C = C( , , c(·, ·));
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(iii) for a.e. y ∈ the infimum in (29) is attained at a unique point x = x(y) and ∇f c (y) = ∇y c(x, y);
(31)
for a.e. x ∈ the infimum in (30) is attained at a unique point y = y(x) and ∇gc (x) = ∇x c(x, y);
(32)
(iv) f c is semi-concave in , gc is semi-concave in and the constant of semiconcavity in both cases depends only on , and c(·, ·). Proof. Parts (i) and (ii) are straightforward. For (iii) observe that since f c is Lipschitz, it is differentiable a.e. in . Given y = (y1 , y2 , y3 )T ∈ , a point of differentiability of f c , the infimum in (29) is attained at some point x = (x 1 , x 2 , x 3 )T ; the function (x, y) → f c (y) + f (x) − c(x, y) attains a maximum at (x, y); this immediately gives (31), in which we can uniquely solve for x because of (24). The other part of (iii) is proved in exactly the same way again utilizing (24) and this time (32). It remains to show (iv). Since c(·, ·) ∈ C 2 ( × ), we find a constant λ > 0 such 2 c(x, y) − λI 0 and D 2 c(x, y) − λI 0 for all (x, y) ∈ × . We have that Dxx yy f c (y) −
λ λ 2 y = inf c(x, y) − y 2 − f (x) . 2 2 x∈
Now the function y → c(x, y) − λ/2 y 2 − f (x) is concave for every x. Hence, y → f c (y) − λ/2y 2 is concave as an infimum of concave functions. Similarly, we can show that x → gc (x) − λ/2 x 2 is concave on . Obviously, λ depends only on c(·, ·), and . Theorem 3.2. Assume and satisfy (25). Let σ ∈ Pac ( ) and ν ∈ Pac ( ). Assume that c(x, y) is given by (11) and that φ satisfies (24). Then the problem (26) of maximizing J(σ,ν) over Lipc has a solution (f0 , g0 ) such that (f0 )c = g0 and (g0 )c = f0 and
f0 W 1,∞ ( ) + g0 W 1,∞ ( ) C = C( , , c(·, ·)). Proof. Given (f, g) ∈ Lipc we always have J (f, g) J (f, f c ) J ((f c )c , f c ),
(33)
Observe also that, if K is a constant, we have (f − K)c = f c + K, (g + K)c = gc − K, J ((f c )c , f c ) = J (((f − K)c )c , (f − K)c ).
(34)
In view of (34) fix a point x ∈ : let (f , g ) be a pair of c-transforms. Now set K = f ( x ), f = f − K and g = g + K. Equations (34) implies that f and g are c-transforms of each other. The fact that f ( x ) = 0 and Lemma 3.1
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(ii) imply that f L∞ ( ) C = C( , , c(·, ·)). Since g = f c we also have
g L∞ ( ) C = C( , , c(·, ·)). Therefore, in view of Lemma 3.1 (ii), a pair of c-transforms (f, g) can always be chosen so that f W 1,∞ +
g W 1,∞ C = C( , , c(·, ·)) without changing the value of J (·, ·).
(35)
Let (fn , gn ) be a maximizing sequence for (26). In view of (33), (35) and Lemma 3.1 (i), (ii) we may assume that (fn )c = gn and (gn )c = fn and
fn W 1,∞ ( ) + gn W 1,∞ ( ) C = C( , , c(·, ·))
∀ n.
By theAscoli-Arzelá Theorem we find functions f0 ∈ W 1,∞ ( ) and g0 ∈ W 1,∞ ( ) such that, up to extraction of a subsequence that we label (fn , gn ) again, fn → f0
uniformly in ,
gn → g0
uniformly in
(n → ∞).
In particular we find that f0 and g0 are c-transforms of each other. By the dominated convergence theorem we deduce that J (fn , gn ) → J (f0 , g0 )
(n → ∞)
so that the pair (f0 , g0 ) is a maximizer for (26).
Theorem 3.3 (Existence and Uniqueness of optimal maps). Assume that and satisfy (25). Let σ ∈ Pac ( ) and ν ∈ Pac ( ). Assume that φ satisfies (24). Then there exist maps T and S, unique σ a.e. and ν a.e., respectively, and a probability measure λ ∈ (σ, ν) such that (i) T is optimal in the transport of σ to ν with cost c(x, y), (ii) S is optimal in the transport of ν to σ with cost c(y, x) = c(x, y), (iii) S and T are inverses, i.e., S(T (x)) = x for σ a.e. x and T(S(y)) = y for ν a.e. y, (iv) λ = (id, T (·))# σ is a minimizer for the relaxed optimal transport problem (27), (v) and there is equality in (28) Proof. (i) Let (f0 , g0 ) be a maximizer (26) such that (f0 )c = g0 and (g0 )c = f0 (Theorem 3.2). Let us derive the Euler-Lagrange equations for this maximizer (we follow here Gangbo [13]). We know from Lemma 3.1 (iii) that there are mappings T (·) and S(·) such that f0 (x) + g0 (T (x)) = c(x, T (x)) (x ∈ ), f0 (S(y)) + g0 (y) = c(S(y), y) (y ∈ ). Let ψ be a smooth function in R3 . For ε > 0 define gε := g0 + εψ,
fε := (gε )c .
Lemma 3.1 (iii) provides us with mappings y ε (x) such that fε (x) = c(x, y ε (x)) − gε (y ε (x))
(x ∈ ).
(36)
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We can show that y ε (x) → T (x)
(ε → 0).
Since (f0 , g0 ) is a minimizer in (26), we have 1 [J (fε , gε ) − J (f0 , g0 )] ε 1 = lim [fε (x) − f0 (x)] σ (x)dx + ψ(y) ν(y)dy ε 0 ε = − ψ(T (x)) σ (x)dx + ψ(y) ν(y)dy,
0 lim
ε 0
(37)
where we have used fε (x) − f0 (x) → −ψ(T (x)) ε
(ε → 0).
As ψ is arbitrary in (37) it also holds for −ψ, and so 0=− ψ(T (x)) σ (x)dx + ψ(y) ν(y)dy,
which says that T# σ = ν. Integrating (36) using (38) we obtain J (f0 , g0 ) = c(x, T (x)) σ (x)dx,
(38)
(39)
which in view of (28) implies that T (·) is a minimizer for (22). Similarly, we can show that S(·) is the optimal map in the push forward of ν to σ with cost c(y, x) := c(x, y). This shows (i) and (ii). To show (iii) we observe that since T# σ = ν and S# ν = σ and the measures live in and , respectively, we may assume that T : → and S : → . Let y be a point of differentiability of g0 and such that there is x ∈ with y = T (x) and x is a point of differentiablity of f0 ; ν a.e. y has this property. Then Lemma 3.1 (iii) implies that g0 (y) = g0 (T (x)) = c(x, T (x)) − f0 (x) = c(x, y) − f0 (x) and g0 (y) = c(S(y), y) − f0 (S(y)) = c(S(T (x)), y) − f0 (S(T (x))). But the x at which the infimum in g0 (y) = (f0 )c (y) is attained is unique, so that S(T (x)) = x and hence, after applying T to this equation, T (S(y)) = y.
(40)
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Similarly, we can show that S(T (x)) = x
(σ a.e. x ∈ ).
(41)
It remains to show uniqueness of an optimal transportation map. Assume there is another optimal transportation map T pushing σ forward to ν. Then we have to have equality in f0 (x) + g0 (T (x)) c(x, T (x)) for σ a.e. x so that f0 (x) = c(x, T (x)) − g0 (T (x)) = c(x, T (x)) − g0 (T (x)). Since the y at which the infimum in f0 (x) = (g0 )c (x) is attained is unique a.e., we obtain T = T σ a.e.. Similarly we show that the optimal transportation map pushing ν forward to σ is unique ν a.e.. This proves uniqueness. For (iv) and (v) observe that (39) immediately gives c(x, T (x)) σ (x)dx = c(x, y) dλT (x, y) = Wc (σ, ν) = J (f0 , g0 ),
where λT := (id, T (·))# σ so that we have equality in the inequalities in (28) and λT is a minimizer in (27). The following theorem contains continuity and convexity results of Wc (σ, ν) that are needed later. Theorem 3.4 (Weak continuity and convexity of Wc (·, ·)). Assume that and satisfy (25). Let ν ∈ Pac ( ). Assume that φ satisfies (24). Then i) The map σ → Wc (σ, ν) is convex on Pac ( ). Furthermore assume σn , σ0 ∈ Pac ( ), νn , ν0 ∈ Pac ( ) are probability densities with σn converging weakly to σ0 and νn converging weakly to ν0 as n tends to ∞. Then (ii) Wc (σn , νn ) → Wc (σ0 , ν0 )
(n → ∞).
Proof. (ii) Let (fn , gn ), λn (n ∈ N) be sequences of solutions to the dual and relaxed problems ((26) and (27)) respectively, corresponding to the problems of optimally transporting σn to νn with cost c(x, y). By the weak compactness of the set of probability densities on × we may extract a subsequence that we label λn again, such that for some λ0 ∈ (σ0 , ν0 ), λn → λ0
weak
(n → ∞).
By Theorem 3.2 we may assume that
fn W 1,∞ ( ) + gn W 1,∞ ( ) C = C( , , c(·, ·)).
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By the Ascoli-Arzelá theorem we can extract uniformly convergent subsequences that we label fn , gn again with fn → f0
and gn → g0
(n → ∞)
with some functions f0 and g0 . It is clear that (f0 , g0 ) ∈ Lipc . Moreover we have J(σn ,νn ) (fn , gn ) → J(σ0 ,ν0 ) (f0 , g0 )
(n → ∞)
and Wc (σn , νn ) = J(σn ,νn ) (fn , gn ) = I [λn ] → I [λ0 ]
(n → ∞),
which with the absolute continuity of σ0 and ν0 and (28) implies that J(σ0 ,ν0 ) (f0 , g0 ) = I [λ0 ] = Wc (σ0 , ν0 ). (i) Let λ be in (0, 1). Let σ0 , σ1 be in Pac ( ) with solutions λ0 , λ1 to the relaxed problems (27) corresponding to optimally transporting σ0 and σ1 to ν with cost c(x, y). Define σt := tσ0 +(1−t)σ1 and λt := tλ0 +(1−t)λ1 . Then λt ∈ (σt , ν), and so, Wc (σt , ν) c(x, y) dλt (x, y) = t c(x, y) dλ0 (x, y) + (1 − t) c(x, y) dλ1 (x, y) = t Wc (σ0 , ν) + (1 − t) Wc (σ1 , ν).
4. The geostrophic energy In this section we take the next step towards finding a weak solution to the semi-geostrophic system in dual variables (16). Recall that we are looking for stable solutions. By Cullen’s stability condition, at each time, given the potential vorticity ν, σ has to minimize the geostrophic energy Eν (·), given by (9). We will show the existence and uniqueness of a minimizer for Eν (·), where the minimization is over Pac ( ). Theorem 4.1 (Existence of a minimizer for Eν (·)). Assume and satisfy (25). Let ν be in Pac ( ). Assume that φ satisfies (24). Then there exists a unique minimizer σ of Eν (·) over Pac ( ). Proof. Observe that Theorem 3.4 and the fact that γ is in (1, 2) imply that the energy is strictly convex, which gives uniqueness. Now let σn ∈ Pac ( ) be a minimizing sequence. For all σ ∈ Pac ( ) and ν ∈ Pac ( ) we have |Wc (σ, ν)|
sup
|c(x, y)| = C = C( , , c(·, ·)).
x∈ ,y∈
We may therefore assume that
σn Lγ ( ) C = C( , , c(·, ·), γ ).
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This allows us to extract a subsequence that we label σn again, which converges γ weakly in Lγ , and hence also weakly toγ a probability measure σ0 in L ( ). It is a standard result that the map σ → σ is weak lower semi-continuous, which with Theorem 3.4 implies that the map σ → Eν (σ ) is weak lower semi-continuous; therefore we have Eν (σ0 ) lim inf Eν (σn ), n→∞
which shows that σ0 is a minimizer for Eν (·).
Definition 5. We denote the unique minimizer of Eν (·) over Pac ( ) by L(ν). Next we write down the Euler-Lagrange equations for a minimizer L(ν) of Eν (·), which will show that L(ν) is actually in W 1,∞ ( ). Theorem 4.2 (Euler-Lagrange equations for a minimizer of Eν (·)). Assume that and satisfy (25). Let ν be in Pac ( ). Assume that φ satisfies (24). Let σ0 = L(ν) be the unique minimizer of Eν (·). Let (f0 , g0 ) be a solution to the dual problem (26) corresponding to the problem of optimally transporting σ0 to ν with cost c(x, y) with optimal transportation map T and inverse S. Then σ0 is Lipschitz continuous and
σ0 W 1,∞ ( ) C = C( , , c(·, ·), γ , K1 ). Furthermore, σ0 satisfies γ
(x ∈ ), ∇f0 (x) σ0 (x) = ∇x c(x, T (x)) σ0 (x) = K1 (1 − γ )∇(σ0 (x)) γ −1 ∇f0 (x) = ∇x c(x, T (x)) = −K1 γ ∇((σ (x)) ) (x ∈ {σ0 > 0}). Proof. Let ψ be a smooth compactly supported vector field in with corresponding flow ∂ M(t, x) = ψ(M(t, x)), ∂t M(0, y) = y. The flow map M(t, ·) is a diffeomorphism of . For t > 0 define σt := M(t, ·)# σ0 so that detDM(t, x) σt (M(t, x)) = σ0 (x).
(42)
We can readily check that detDM(t, x) = 1 + t ∇ · ψ(x) + o(t, x), where sup |o(t, x)| = o(t)
(t → 0).
x∈
The fact that σ0 is a minimizer for Eν (·) implies that 1 1 0 lim inf (Eν (σt ) − Eν (σ0 )) lim sup (Eν (σt ) − Eν (σ0 )). t 0 t t 0 t
(43)
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Using (42) and (43) we can show that lim
(σt (x))
t 0
γ
dx − t
(σ0 (x))
γ
dx
(σ0 (x))γ ∇ · ψ(x) dx. (44)
= (1 − γ )
Since T (M −1 (t, ·))# σt = ν, we have (Wc (σt , ν) − Wc (σ0 , ν) t t 0 −1 c(x, T (M (t, x))) σt (x) dx − c(x, T (x)) σ0 (x) dx lim sup t t 0 (c(M(t, x), T (x)) − c(x, T (x))) σ0 (x) dx = lim sup t t 0 (c(M(t, x), T (x)) − c(x, T (x))) σ0 (x) dx = lim t 0 t ∂M = ∇x c(x, T (x)) · (0, x) σ0 (x) dx ∂t
lim sup
=
∇x c(x, T (x)) · ψ(x) σ0 (x) dx.
(45)
Equations (44) and (45) imply ((1−γ ) K1 (σ0 (x))γ ∇ · ψ(x) dx +∇x c(x, T (x)) · ψ(x) σ0 (x)) dx. (46)
0
Since ψ is arbitrary in (46), it holds for −ψ, as well, and we obtain Eν (σt ) − Eν (σ0 ) 0 = lim t 0 t = ((1 − γ ) K1 (σ0 (x))γ ∇ · ψ(x) dx + ∇x c(x, T (x)) · ψ(x) σ0 (x)) dx
so that γ
∇f0 (x)σ0 (x) = ∇x c(x, T (x))σ0 (x) = K1 (1 − γ ) ∇(σ0 (x))
(47)
in the weak sense (the first equality in (47) follows from Lemma 3.1 (iii) and the construction of T in the proof of Theorem 3.3 (i)). Using the fact that ∇f0 is bounded by Lemma 3.1 (ii), that σ0 ∈ Lγ ⊂ L1 and applying standard Sobolev imbedding theorems, we inductively deduce from (47) that γ
σ0 (·) ∈ W 1,∞ ( ) so that (47) holds pointwise a.e. in .
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By setting σ = 1/| |χ we see that, regardless of what ν is, we have |Eν (L(ν))| c(·, ·) L∞ ( × ) +K1 | |1−γ
(48)
|Wc (σ, ν)| c(·, ·) L∞ ( × ) .
(49)
and In view of (48) and (49) we can find a constant C depending only on , , c(·, ·), γ and K1 such that, for any ν ∈ Pac ( ), inf (L(ν)(x)) C.
(50)
x∈
From (47) we deduce that on the set where σ0 is strictly positive (which is an open set) we have (γ −1) (x)). ∇f0 (x) = −K1 γ ∇(σ0 By Theorem 3.2 we know that ∇f0 L∞ is bounded by a constant that only γ −1 depends upon , and c(·, ·). From Lemma A.1 we deduce that σ0 is Lipschitz on all of with Lipschitz constant no bigger than ∇f0 L∞ ( ) /(K1 γ ), which together with (50) implies that
σ0 L∞ ( ) C = C( , , c(·, ·), γ , K1 ).
(51)
Equations (47) and (51) imply γ
σ0 W 1,∞ ( ) C = C( , , c(·, ·), γ , K1 ). Furthermore, on the set where σ0 is strictly positive we have 2−γ
∇f0 (x) σ0
(x) = K1 (1 − γ ) γ ∇σ0 (x).
(52)
The left-hand side of (52) is bounded by a constant that only depends on , , c(·, ·), γ and K1 . Using Lemma A.1 again, we deduce that σ0 is Lipschitz on all of with
σ0 W 1,∞ ( ) C = C( , , c(·, ·), γ , K1 ). Rewriting (47) gives ∇f0 (x) = ∇c(x, T (x)) = −K1 γ ∇((σ0 (x))γ −1 )
(x ∈ {σ0 > 0}).
(53)
The following lemma is important in the time-discretization as it will allow us to let the time step go to zero and have convergence of approximate solutions of (16) to an exact solution in some appropriate sense. Lemma 4.3 (Stability Lemma). Assume that and satisfy (25). Let νn , ν0 be in Pac ( ) with νn converging weakly to ν0 as n tends to ∞, and that φ satisfies (24). Let σn = L(νn ) (n ∈ (N ∪ {0})). Then the σn converge uniformly to σ0 in for n → ∞.
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Proof. From Theorem 4.2 we know that σn W 1,∞ ( ) C = C( , , c(·, ·), σ ∈ Pac ( ) ∩ γ , K1 ) for all n in N. By the Ascoli-Arzelá theorem we can find a W 1,∞ ( ) such that, up to a subsequence that we label σn again, we have the σn converging uniformly in to σ . We claim that σ is a minimizer for Eν0 (·), so that σ = σ0 by uniqueness of minimizers for Eν0 (·). The continuity properties of Wc (·, ·) (Theorem 3.4) and the uniform convergence of the σn imply that Eνn (σn ) → Eν0 ( σ)
and Eνn (σ0 ) → Eν0 (σ0 )
(n → ∞).
(54)
Assuming that Eν0 ( σ ) > Eν0 (σ0 ) would imply by (54) that Eνn (σ0 ) < Eνn (σn ) for n large enough, which is a contradiction, since σn is a minimizer for Eνn (·). Hence, Eν0 ( σ ) = Eν0 (σ0 ) and σ = σ0 . These arguments show that given any subsequence of {σn } there is a further subsequence converging uniformly to σ0 in . Therefore the whole sequence must converge uniformly to σ0 in .
5. Solution to the semi-geostrophic system in dual variables Recall the geostrophic system in dual variables (16). We show existence of a weak solution to (16) on a fixed time-interval [0, τ ], τ > 0. The physical domain is assumed to be an open bounded convex set in R3 . The approach is to discretize (16) (i) in time with time step h and construct piecewise smooth approximate solutions νh (·, ·) to (16). In view of the proof of existence of an optimal transportation map in Theorem 3.3 and Lemma 3.1 (iii), and the definition of w in (16) (ii) −1 y e × ∇g(y) for some g, which implies (formally) that we see that w = fcor 3 3 the flow in dual space is divergence free. Since the variable y3 represents potential temperature, which can be assumed to be bounded away from 0 and ∞, we assume that δ , 1/ δ ], spt(ν0 ) ⊂ A0 and A0 ⊂ R2 × [ δ < 1 is some constant. For where A0 is some bounded open set in R3 and 0 < 0 < h, let jh : R3 → R be standard mollifiers defined by jh (y) :=
1 y j , h3 h
where j (·) : R3 → R is a nonnegative function of class C ∞ with support in the unit ball of R3 , centered at the origin, which integrates to 1. We define initial potential vorticities for the approximate solutions as follows: νh0 (y) := (jh ν0 )(y) Set
δ := δ /4;
(y ∈ R3 ).
h := δ/4.
Then for 0 < h < h, we have 1 spt(νh0 ) ⊂ A0h ⊂ R2 × δ, , δ
(55)
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where A0h := A0 + B2h is an open bounded set containing the support of νh0 , B2h being the open ball of radius 2h in R3 centered at the origin. It will be important to find bounds on the highest possible rate of spread in time of the supports of νh . We therefore define, for h < h and t ∈ [0, τ ], sh (t) := sup{ |y| : y ∈ spt(νh (t, ·)) }; D1 := sup{ |x| : x ∈ };
D2 := sup { |y| : y ∈ A0h }. 0 0, and that = BR × (δ, 1/δ) ⊂ R3 is an open set with BR being a ball of radius R in R2 centered at the origin and δ > 0. Suppose that {νh }h>0 ⊂ W 1 ((0, τ ) × ), and that {kh }h>0 ⊂ Lr ((0, τ ) × ) satisfy in the weak sense ∂ νh = −∇ · (kh ) in [0, τ ] × , ∂t where νh and kh are set 0 outside . Suppose that there is a constant C such that sup
t∈[0,τ ],h>0
νh (t, ·) Lr ( ) C,
sup
t∈[0,τ ],h>0
kh (t, ·) Lr ( ) C.
Then there exists a sequence {hj } ⊂ (0, 1) converging to 0 as j tends to ∞, a ν ∈ Lr ((0, τ ) × ), and a map k ∈ Lr ((0, τ ) × ) such that ∞ r (i) {νhj }∞ j =1 converges weakly to ν in L ((0, τ ) × ), {khj }j =1 converges weakly r to k in L ((0, τ ) × ) as j tends to ∞; r (ii) {νhj (t, ·)}∞ j =1 converges weakly to ν(t, ·) in L ( ) for every t ∈ [0, τ ], as j tends to ∞.
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We will use this lemma in the following theorem, which is the main result: Theorem 5.5 (Weak solution to the semi-geostrophic equations in dual variables). Let 1 < r < ∞ and ν0 ∈ Lr (A0 ) be an initial potential vorticity with support in A0 , where A0 is a bounded open set in R3 with A0 ⊂ R2 × [ δ , 1/ δ ] for some 0 < δ < 1. Let be defined as in (58). Let be an open bounded convex set in R3 . Assume that c(·, ·) is given by (11) and that φ satisfies (24). Then the system of semi-geostrophic equations in dual variables (16) has a stable weak solution (σ, T ) such that, with ν(t, ·) := T (t, ·)# σ (t, ·) and w as in (16) (ii), (i) ν(·, ·) ∈ Lr ((0, τ ) × ), ν(t, ·) Lr ( ) ν0 (·) Lr ( ) (∀t ∈ [0, τ ]), (ii) σ (t, ·) ∈ W 1,∞ ( ), σ (t, ·) W 1,∞ ( ) C = C( , , c(·, ·), γ , K1 ) (∀t ∈ [0, τ ]), (iii)
w(t, ·) L∞ ( ) C = C( , , fcor ) (∀t ∈ (0, τ )).
Proof. Consider the time-discrete scheme as introduced at the beginning of this section. In view of Lemma 5.3 (i)–(iii) we apply Lemma 5.4 to our setting with ((t, y) ∈ (0, τ ) × ) kh (t, y) := νh (t, y)uh (t, y) to obtain a decreasing sequence {hj } converging to 0 and k(t, y) and ν(t, y) such that νhj → ν
weakly in Lr ((0, τ ) × R3 ),
(70)
νhj uhj → k
weakly in L ((0, τ ) × R ),
(71)
3
r
νhj (t, ·) → ν(t, ·) weakly in Lr (R3 ) ∀t ∈ [0, τ ].
(72)
The convergence (72) and Proposition A.3 imply that W1 (νhj (t, ·), ν(t, ·)) → 0
(∀t ∈ [0, τ ]).
Since by Lemma 5.2 we have, for all t ∈ [0, τ ], W1 (ν hj (t, ·), νhj (t, ·)) 2
fcor (D1 + h + R) ν0 L1 (R3 ) hj , δ2
and since W1 is a metric, we also have, for all t ∈ [0, τ ], W1 (ν hj (t, ·), ν(t, ·)) → 0
(hj → 0)).
This on the other hand implies (see Rachev & Rüschendorf [19]) that ν hj (t, ·) → ν(t, ·)
weakly in Lr .
(73)
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By Lemma 4.3 (stability lemma) we know that, for all t ∈ [0, τ ], σhj (t, ·) → σ (t, ·) and σ hj (t, ·) → σ (t, ·) uniformly in , where σ (t, ·) := L(ν(t, ·)). Denote by T (t, ·) and S(t, ·) the optimal transportation maps pushing σ (t, ·) forward to ν(t, ·) and ν(t, ·) forward to σ (t, ·), respectively. Let (f (t, ·), g(t, ·)) be solutions to the dual problems (26) corresponding to the problems of optimally transporting σ (t, ·) to ν(t, ·) with cost c(x, y). Since spt(ν(t, ·)) ⊂ for all t ∈ [0, τ ], we have by Theorem 3.2
f (t, ·) W 1,∞ ( ) + g(t, ·) W 1,∞ ( ) C = C( , , c(·, ·)). The goal is to show the following claim: Claim. For a.e. t ∈ [0, τ ] we have k(t, ·) = ν(t, ·)w(t, ·)
(a.e. ∈ ),
(74)
where w(t, y) = fcor e3 × (y − S(t, y)).
(75)
Proof of the claim. For t ∈ [kh, (k+1)h) define (f h (t, ·), g h (t, ·)) := (fh (kh, ·), gh (kh, ·)); (f h (t, ·), g h (t, ·)) are solutions to the dual problems (26) corresponding to the problems of optimally transporting σ h (t, ·) to ν h (t, ·) with cost c(x, y). By Theorem 3.2 we have
f hj (t, ·) W 1,∞ ( ) + g hj (t, ·) W 1,∞ ( ) C = C( , , c(·, ·)). Now fix t ∈ [0, τ ]. We may then find a subsequence (f hj (t, ·), g hj (t, ·)) and k k functions (f0 (t, ·), g0 (t, ·)) such that f hj (t, ·) → f0 (t, ·) uniformly in k and g hj (t, ·) → g0 (t, ·) uniformly in . k
From the proof of the weak continuity of Wc (·, ·) in Theorem 3.4, we see that (f0 (t, ·), g0 (t, ·)) is a pair of c-transforms and that it is a solution to the dual problem (26) corresponding to the problem of optimally transporting σ (t, ·) to ν(t, ·) with cost c(x, y). Now let B be the set of points y in at which all the g hj (t, ·) and g0 (t, ·) are differentiable. From Lemma 3.1 (ii) we see that this is a set of full Lebesgue measure in . Fix y ∈ B. Then the sequence {∇g hj (t, y)} is k bounded. Hence a further subsequence {∇g hj (t, y)} converges to some z. Since kl
the functions (g hj (t, ·) − λ/2 (·)2 ) and (g0 (t, ·) − λ/2 (·)2 ) are concave in for kl some λ > 0 (see Lemma 3.1 (iv)), it follows that z is in the superdifferential of (g0 (·) − λ/2 (·)2 ) at y. Since g0 (t, ·) is differentiable at y, we have z = ∇g0 (t, y). This shows that given a subsequence of {∇ghjk (t, y)} there is a further subsequence converging to ∇g0 (t, y). This implies that the whole sequence converges: ∇g hj (t, y) → ∇g0 (t, y) k
(k → ∞).
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Since this is true for every y ∈ B, we deduce that −1 y e × ∇(j −1 fcor 3 3 hjk g hj (t, ·)) → fcor y3 e3 × ∇g0 (t, ·)
in Lp ( )
k
for any p in [1, ∞). Up to extraction of a further subsequence, to which we give the same label, the convergence is also pointwise. This implies that −1 νhjk (t, ·)uhjk (t, ·) = νhjk (t, ·) fcor y3 e3 × ∇(jhjk g hj )(t, ·) −1 → ν(t, ·)fcor y3 e3 × ∇g0 (t, ·)
k
weakly in Lr ( ). But −1 −1 ν(t, ·) fcor y3 e3 × ∇g0 (t, ·) = ν(t, ·) fcor y3 e3 × ∇y c(S(t, ·), ·) = ν(t, ·)w(t, ·).
Hence, we have νhjk (t, ·)uhjk (t, ·) → ν(t, ·)w(t, ·)
weakly in Lr ( ).
The above arguments show that given t and a subsequence of νhj (t, ·)uhj (t, ·) there is a further subsequence converging weakly to ν(t, ·)w(t, ·) in Lr ( ). But this means that we have weak convergence of the whole sequence, and since t ∈ [0, τ ] is arbitrary we obtain νhj (t, ·)uhj (t, ·) → ν(t, ·)w(t, ·)
weakly in Lr ( ) (∀t ∈ [0, τ ]).
(76)
Now let {ψk } be a dense set in Lr ( ), where r denotes the dual exponent to r defined through the equation 1/r + 1/r = 1. Let η be a function in L∞ ((0, τ )). Since k(·, ·) is in Lr ( ) we know that there exists a set E0 ⊂ (0, τ ) of full Lebesgue measure in (0, τ ) such that k(t, ·) ∈ Lr ( )
(for every t ∈ E0 ).
Fix m and observe that because of (71) we have j →∞ η(t) ψm (y) khj (t, y) dy dt −→ (0,τ )
(0,τ )
ψm (y) k(t, y) dy dt.
η(t)
(77) On the other hand, (76) implies j →∞ ψm (y) khj (t, y) dy −→
ψm (y) ν(t, y)w(t, y) dy.
(78)
By Lemma 5.3 (ii) and (iii) we know that khj (t, ·) is bounded in Lr ( ) uniformly in j and t, so we can apply the dominated convergence theorem to deduce that η(t) ψm (y) khj (t, y) dy dt (0,τ ) j →∞ −→ η(t) ψm (y) ν(t, y)w(t, y) dy dt. (79) (0,τ )
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Since η is arbitrary, (77) and (79) imply that there exists a set Em ⊂ (0, τ ) of full Lebesgue measure in (0, τ ) such that ψm (y) ν(t, y)w(t, y) dy = ψm (y) k(t, y) dy (for all t ∈ Em ). (80)
Define
E := ∩∞ m=0 Em .
It is clear that E ⊂ (0, τ ) is a set of full Lebesgue measure in (0, τ ). Using (80) and the density of {ψm } in Lr ( ), we find that, for all ψ in Lr ( ) and t ∈ E, ψ(y) k(t, y) dy = ψ(y) ν(t, y)w(t, y) dy.
This implies that for each t in E there is a set Bt ⊂ of full Lebesgue measure in such that k(t, y) = ν(t, y)w(t, y) (for all y ∈ Bt ). Observe that by redefining k(·, ·) on a set of measure zero in (0, τ ) × we could even assert that (74) holds for every t ∈ [0, τ ]. Now letting hj go to zero in ∂νhj ∂t
= −∇ · (νhj uhj )
and using the fact that νhj → ν0 in Lr , we see that ∂ν = −∇ · (νw) ∂t ν(0, y) = ν0 (y)
in (0, τ ) × R3 ), (y ∈ R3 )
are satisfied in the weak sense. Because of the definition of w in (75) we see that (σ, T ) is a stable solution to (16). Inequality (i) is a consequence of Lemma 5.3 (ii) and (73); (ii) follows immediately from Theorem 4.2, and (iii) is implied by the definition of w in (75) and the fact that S(t, ·) maps into for all t in [0, τ ].
A. Appendix A.1. A technical lemma Lemma A.1. Let be a bounded open and convex set in Rd . Let f ∈ C( ) ∩ 1,∞ (A) be a nonnegative function defined in , where A = {x ∈ : f (x) > 0}. Wloc Assume that ∇f is bounded on A by a constant C. Then f is Lipschitz on all of with Lipschitz constant no bigger than C. Proof. The set A is an open set in . We can write A = ∪∞ k=1 Ak , where the Ak are the open and connected components of A. Let x, y be two points in , and let l be the line segment connectingx and y. Since is convex, l lies entirely inside
The Fully Compressible Semi-Geostrophic System from Meteorology
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k k ∞ . We can write l ∩ A = ∪∞ k=1 ∪j =1 lkj with lj k = (xj , yj ) mutually disjoint and ∪∞ j =1 lj k = l ∩ Ak , where (a, b) denotes the open line segment connecting a and b. Since |∇f (x)| C for all x in A we know that the total variation of f on (xj k , yj k ) is bounded by C|yj k − xj k | for all j, k. On l ∩ ( /A) f is zero, so that there is no change there. Therefore the total variation of f along l is bounded by ∞ j,k=1 C|yj k − xj k | C|y − x|, which implies that |f (y) − f (x)| C|y − x|.
A.2. Background on the Wasserstein-1 distance W1 Definition 6. Given two Borel probability densities µ1 (·) and µ2 (·) in Rd , define inf |x − y| dλ(x, y). W1 (µ1 , µ2 ) := λ∈(µ1 ,µ2 ) Rd ×Rd
Proposition A.2 (Duality). Assume that µ1 (·) and µ2 (·) are two probability densities in Rd that have support in some ball B ⊂ Rd of finite radius. Then (µ1 (x) − µ2 (x)) ψ(x) dx, W1 (µ1 , µ2 ) = sup ψ
Rd
where the supremum is performed over the set of all ψ : B → R that are Lipschitz in B with Lipschitz constant less than or equal to 1. Proof. The proof can be found in [17].
Proposition A.3 (Weak continuity of W1 (·, ·)). Assume that µn and νn in Rd are probability densities with support in some ball B of finite radius in Rd , which converge weakly to some probability densities µ and ν. Then W1 (µn , νn ) → W1 (µ, ν) Proof. See [4].
(n → ∞).
Acknowledgements. We would like to thank W. Gangbo for fruitful discussions and suggestions. We would also like to thank the anonymous referee for pointing out inconsistencies in the notation and for suggesting ways to make the paper shorter and clearer.
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