A global existence result for the semigeostrophic equations in three dimensional convex domains

Exploiting recent regularity estimates for the Monge-Amp\`ere equation, under some suitable assumptions on the initial data we prove global-in-time existence of Eulerian distributional solutions to the semigeostrophic equations in 3-dimensional convex domains.


Introduction
A simplified model for the motion of large scale atmospheric/oceanic flows inside a domain Ω ⊂ R 3 is given by the semigeostrophic equations.
Let Ω ⊆ R 3 be bounded open set with Lipschitz boundary. Then the semigeostrophic equations inside Ω are: Here p 0 is the initial condition for p, 1 ν Ω is the unit outward normal to ∂Ω, e 3 = (0, 0, 1) T is the third vector of the canonical basis in R 3 , J is the matrix given by and the functions u t , p t , and m t represent respectively the velocity, the pressure and the density of the atmosphere, while u g t is the so-called semi-geostrophic wind. 2 Clearly the pressure is defined up to a (time-dependent) additive constant.
Substituting the relation u g t = J∇p t and introducing the function (1.2) P t (x) := p t (x) + 1 2 (x 2 1 + x 2 2 ), the system (1.1) can be rewritten in Ω × [0, ∞) as . Notice that, given a solution (P, u) of (1.3), one easily recovers a solution of (1.1): indeed p t can be obtained from P t through (1.2) and the density m t is given by m t = ∂ 3 P t (in particular, the third component of the first equation in (1.3) tells us that ∂ t m t + u t · ∇ m t = 0 is satisfied).
Energetic considerations (see [11,Section 3.2]) show that it is natural to assume that the function P t is convex on Ω. This condition, first introduced by Cullen and Purser, is related in [14,24] to a physical stability required for the semigeostrophic approximation to be appropriate. If we denote with L Ω the (normalized) Lebesgue measure on Ω, then formally ρ t := (∇P t ) ♯ L Ω (see, for example, [1, Appendix A]) satisfies the following dual problem and (∇P t ) ♯ L Ω is the push-forward of the measure L Ω through the map ∇P t : Ω → R 3 defined as The dual problem is pretty well understood, and admits a solution obtained via time discretization (see [5,13]). Moreover, at least formally, given a solution P t of the dual problem (1.4) and setting , the couple (P t , u t ) solves the semi-geostrophic problem (1.1). However, because of the low regularity of the function P t the previous velocity field may a priori not be well defined, and this creates serious difficulties for recovering a "real solution" from a "dual solution".
Still, a recent regularity result [15] can be applied to show that the map P t is W 2,1 in space, so that we can give a meaning to the second term in the definition of u t . More precisely, in [15] it is shown that |D 2 u| log k + |D 2 u| ∈ L 1 loc for any k, and following ideas developed in [1,20], we will be able to show that the function P t is regular enough also in time, so that the couple (P t , u t ) is a true distributional solution of (1.1).
Let us point out that the regularity result in [15] has been recently extended, independently in [17] and [23], to |D 2 u| ∈ L γ , where γ > 1 depends on the local L ∞ norm of log ρ t . However, as we will better explain in Remark 3.6, in our situation there is no advantage in using this improvement, since the fact that γ depends on log ρ t ∞,loc makes the estimates less readable. For this reason we will rely only on the L log L integrability given by [15], as we previously did in [1].
The first existence result about distributional solutions to the semigeostrophic equation is presented in [1], where the analysis is carried out on the 2-dimensional torus (see also [21] where a short time existence result of smooth solutions is proved in dual variables, and because of smoothness the existence can be easily transferred to the initial variables).
The 3-dimensional case on the whole space R 3 , which is more physically relevant, presents additional difficulties. First, the equation (1.1) is much less symmetric compared to its 2-dimensional counterpart, because the action of Coriolis force Ju t regards only the first and the second space components. Moreover, even considering regular initial data and velocities, regularity results require a finer regularization scheme, due to the non-compactness of the ambient space.
Our proofs are also based on some additional hypotheses on the decay of the probability measure ρ 0 = (∇P 0 ) ♯ L Ω . This decay condition happens to be stable in time on solutions of the dual equation (1.4), and allows us to perform a regularization scheme.
It would be extremely interesting to consider compactly supported initial data ρ 0 = (∇P 0 ) ♯ L Ω . However the nontrivial evolution of the support of the solution ρ t under (1.4) prevents us to apply the results in [15] (which actually would be false in this situation), so at the moment this case seems to require completely new ideas and ingredients.
Remark 1.2. This definition is the classical notion of distributional solution for (1.3) except for the fact that the boundary condition u t ·ν Ω = 0 is not taken into account. In this sense it may look natural to consider ψ ∈ C ∞ (Ω) in (1.7), but since we are only able to prove that the velocity u t is locally in L 1 , Equation (1.7) makes sense only with compactly supported ψ. On the other hand, as we shall explain in Remark 1.4, we will be able to prove that there exists a measure preserving Lagrangian flow F t : Ω → Ω associated to u t , and such existence result can be interpreted as a very weak formulation of the constraint u t · ν Ω = 0. As pointed out to us by Cullen, this weak boundary condition is actually very natural: indeed, the classical boundary condition would prevent the formation of "frontal singularities" (which are physically expected to occur), i.e. the fluid initially at the boundary would not be able to move into the interior of the fluid, while this is allowed by our weak version of the boundary condition.
We can now state our main result.
Let Ω ⊆ R 3 be a convex bounded open set, and let L Ω be the normalized Lebesgue measure restricted to Ω, that is L Ω (Ω) = 1. Let ρ 0 be a probability density on Then the vector field u t in (1.5) is well defined, and the couple (P t , u t ) is a weak Eulerian solution of (1.3) in the sense of Definition 1.1. Remark 1.4. Following Cullen and Feldman one can give also a notion of Lagrangian solution of the semigeostrophic equation. More precisely they show the existence of a measure preserving flow F t : Ω → Ω which solves a sort of Lagrangian version of (1.1) (see [12] and [1, Section 5] for a more precise discussion). Actually the flow they constructed has the explicit expression , in the sense of Ambrosio, Di Perna and Lions (see [3,4,18]). In [1, Section 5] we showed, in the two dimensional periodic setting, that for almost every x the map t → F t (x) is absolutely continuous with derivative given by u t (F t (x)). The proof of this fact can be almost verbatim extended to our contest, showing that, for almost every x ∈ Ω, t → F t (x) is locally absolutely continuous in [0, ∞) with derivative given by u t (F t (x)). We leave the proof of this fact to the interested reader. Finally we remark that the uniqueness of such a flow (both according to the definition given in [12] or in [1]) is unknown.
Acknowledgement. L.A., G.D.P., and A.F. acknowledge the support of the ERC ADG GeMeThNES. A.F. was also supported by the NSF Grant DMS-0969962. M.C. and G.D.P. also want to acknowledge the hospitality of the University of Texas at Austin, where part of this work has been done.
2. Regularity of optimal transport maps between convex sets of R 3 Throughout this paper, Ω ⊆ R 3 is a bounded convex open set, d Ω > 0 is fixed in such a way that Ω ⊂ B(0, d Ω ), and L Ω denotes the normalized Lebesgue measure restricted to Ω.
In this section we recall some regularity results for optimal transport maps in R 3 needed in the paper.
(i) There exists a unique optimal transport map between µ and ν, namely a unique (up to an additive constant) convex function P * : .
Let us also assume that Ω 0 , Ω 1 are bounded and uniformly convex, and there exists a constant C which depends only on α, Moreover, there exist positive constants c 1 and c 2 and κ, depending only on λ 0 , λ 1 , ρ C 0,α , and σ C 0,α , such that The first statement is standard optimal transport theory, see [10,22], except for the fact that we are not assuming that the second moment of µ is finite, thus the classical Wasserstein distance from µ and ν can be infinite. Nevertheless the existence of an "optimal" map is provided by [22]. The W 2,1 part of the second statement follows from a recent regularity result about solutions of the Monge-Ampère equation [15], while the C 1,β regularity was proven by Caffarelli in [6,9,10]. The regularity up to the boundary and the oblique derivative condition of the third statement have been proven by Caffarelli [7] and Urbas [25].
Remark 2.2. By compactness and a standard contradiction argument, the constants C 1 and C 2 in the statement (ii) of the previous theorem remain uniformly bounded if Ω 1 varies in a compact class (with respect, for instance, to the Hausdorff distance) of convex sets. In particular, let Ω n 1 be a sequence of open convex sets which converges to Ω 1 with respect to the Hausdorff distance and σ n a sequence of densities supported on Ω n 1 with λ 1 ≤ σ n ≤ Λ 1 on Ω n 1 which converge to σ in L 1 (R 3 ). Then the estimates in Theorem 2.1(ii) hold true with constants independent of n. Remark 2.3. As already mentioned in the introduction, in statement (ii) the optimal regularity is that for every Ω ⋐ Ω 0 and 0 < λ ≤ ρ(x) ≤ Λ < ∞ in Ω, there exist γ(λ, Λ, λ 1 , Λ 1 ) > 1 and C(Ω, Ω 1 , λ, Λ, λ 1 , Λ 1 ), such that However, as explained in Remark 3.6, this improvement does not give any advantage.

The dual problem and the regularity of the velocity field
In this section we recall some properties of solutions of (1.4), and we show the L 1 integrability of the velocity field u t defined in (1.5).
We have the following result whose proof follows adapting the argument of [5,13], where compactly supported initial data are considered. Since the velocity U t has at most linear growth, the speed of propagation is locally finite and the proof readily extends to general probability densities. Theorem 3.1 (Existence of solutions of (1.4)). Let P 0 : R 3 → R be a convex function such that (∇P 0 ) ♯ L Ω ≪ L 3 . Then there exist convex functions P t , P * t : , and ρ t is a distributional solution to (1.4), namely . Moreover, the following regularity properties hold: is the space of probability measures endowed with the weak topology induced by the duality with C 0 (R 3 ); Observe that, by Theorem 3.1(ii), t → ρ t L 3 is weakly continuous, so ρ t is a well-defined function for every t ≥ 0. Further regularity properties of P t and P * t with respect to time will be proven in Proposition 3.5.
In the proof of Theorem 1.3 we will need to test with functions which are merely W 1,1 with compact support. This is made possible by a simple approximation argument which we leave to the reader, see [1, Lemma 3.2].
Lemma 3.2. Let ρ t and P t be as in Theorem 3.1. Then (3.1) holds for every ϕ ∈ W 1,1 (R 3 × [0, ∞)) which is compactly supported in time and space, where now ϕ 0 (x) has to be understood in the sense of traces.
Lemma 3.4 (Decay estimates on ρ t ). Let v t : R 3 × [0, ∞) → R 3 be a C ∞ velocity field and suppose that sup for suitable constants N, A, D. Let ρ 0 be a probability density, and let ρ t be the solution of the continuity equation starting from ρ 0 . Then: (i) For every r > 0 and t ∈ [0, ∞) it holds (ii) Let us assume that there exist d 0 ∈ [0, ∞) and M ∈ [0, ∞) such that Then for every t ∈ [0, ∞) we have that (iii) Let us assume that there exists R > 0 such that ρ 0 is smooth in B(0, R), vanishes outside B(0, R), and that v t is compactly supported inside B(0, R) for all t ≥ 0. Then ρ t is smooth inside B(0, R) and vanishes outside B(0, R) for all t ≥ 0.
) be the flow associated to the velocity field v t , namely the solution to For every t ≥ 0 the map t → X t (x) is invertible in R 3 , with inverse denoted by X −1 t . The solution to the continuity equation (3.13) is given by ρ t = X t♯ ρ 0 , and from the well-known theory of characteristics it can be written explicitly using the flow: Since the divergence is bounded, we therefore obtain )e N t Now we deduce the statements of the lemma from the properties of the flow X t .
(i) From (3.21) we have that which proves (3.14). From the equation (3.19) we obtain which can be rewritten as From the first inequality we get Hence from (3.21) and (3.23) we obtain that, for every x ∈ B(0, r), which proves (3.15).
(iii) If v t = 0 in a neighborhood of ∂B(0, R) it can be easily verified that the flow maps X t : R 3 → R 3 leave both B(0, R) and its complement invariant. Moreover the smoothness of v t implies that also X t is smooth. Therefore all the properties of ρ t follow directly from (3.20).
We are now ready to prove the regularity of ∇P * t .
Step 1: The smooth case. In the first part of the proof we assume that Ω is a convex smooth domain, and, besides (3.25), that for some R > 0 the following additional properties hold: for some constants N, λ, Λ, and we prove that (3.26) holds for every t ∈ [0, T ]. Notice that in this step we do not assume any coupling between the velocity U t and the transport map ∇P * t . In the second step we prove the general case through an approximation argument.
By a standard density argument it follows that the above equation holds outside a negligible set of times independent of the test function ψ, thus proving (1.7).