Sobolev regularity for the Monge-Ampère equation, with application to the semigeostrophic equations

In this note we review some recent results on the Sobolev regularity of solutions to the MongeAmpère equation, and show how these estimates can be used to prove some global existence results for the semigeostrophic equations. 1 The Monge-Ampère equation The Monge-Ampère equation arises in connections with several problems from geometry and analysis (regularity for optimal transport maps, the Minkowski problem, the affine sphere problem, etc.) The regularity theory for this equation has been widely studied. In particular, Caffarelli developed in [4, 6, 5] a regularity theory for Alexandrov/viscosity solutions, showing that convex solutions of { det(D2u) = f in Ω, u = 0 on ∂Ω (1.1) are locally C1,α provided 0 < λ ≤ f ≤ Λ for some λ,Λ ∈ R. Moreover, for any p > 1 there exists δ = δ(p) > 0 such that u ∈W 2,p loc (Ω) provided |f − 1| ≤ δ. Then, few years later, Wang [17] showed that for any p > 1 there exists a function f satisfying 0 < λ ≤ f ≤ Λ such that u 6∈W 2,p loc (Ω). This counterexample shows that the results of Caffarelli were more or less optimal. However, an important question which remained open was whether solutions of (1.1) with 0 < λ ≤ f ≤ Λ could be at least W 2,1 loc , or even W 2,1+ loc for some = (n, λ,Λ) > 0. In the next section we motivate this W 2,1 loc question, showing how a positive answer to this question can be used to obtain some global existence results for the semigeostrophic equations [1, 2]. Then, following [11, 12], in Section 3 we prove that solutions to (1.1) are actually W 2,1+ε loc , and we show how the very same proof can be used to obtain Caffarelli’s W 2,p loc estimates. 2 The semigeostrophic equations A motivation for being interested in the W 2,1 loc regularity of solutions to (1.1) comes from the semigeostrophic equations: The semigeostrophic equations are a simple model used in meteorology ∗The University of Texas at Austin, Mathematics Dept. RLM 8.100, 2515 Speedway Stop C1200, Austin, Texas 78712-1202 USA. ([email protected]) †The author is supported by NSF Grant DMS-0969962. 1 to describe large scale atmospheric flows. As explained for instance in [3, Section 2.2] (see also [9] for a more complete exposition), these equations can be derived from the 3-d incompressible Euler equations, with Boussinesq and hydrostatic approximations, subject to a strong Coriolis force. Since for large scale atmospheric flows the Coriolis force dominates the advection term, the flow is mostly bi-dimensional. For this reason, the study of the semigeostrophic equations in 2-d or 3-d is pretty similar, and in order to simplify our presentation we focus here on the 2-dimentional periodic case. The semigeostrophic system can be written as  ∂t∇pt + (ut · ∇)∇pt +∇pt + ut = 0 ∇ · ut = 0 p0 = p̄ (2.1) where ut : R2 → R2 and pt : R2 → R are periodic functions corresponding respectively the velocity and the pressure. As shown in [9], energetic considerations show that it is natural to assume that pt is (−1)-convex, i.e., the function Pt(x) := pt(x) + |x|2/2 is convex on R2. If we denote with LT2 the Lebesgue measure on the 2-dimensional torus, then formally ρt := (∇Pt)]LT2 satisfies the following dual problem:  ∂tρt +∇ · (U tρt) = 0 U t(x) = ( x−∇P ∗ t (x) )⊥ ρt = (∇Pt)]LT2 P0(x) = p̄(x) + |x|2/2, (2.2) where P ∗ t is the convex conjugate of Pt, namely P ∗ t (y) := sup x∈R2 { y · x− Pt(x) } . The dual problem (2.2) is nowadays pretty well understood. In particular, Benamou and Brenier proved in [3] existence of weak solutions to (2.2). On the contrary, much less is known about the original system (2.1). Formally, given a solution ρt of (2.2) and defining Pt through the relation ρt = (∇Pt)]LT2 (namely the optimal transport map from ρt to LT2 for the quadratic cost on the torus), the pair (pt,ut) given by { pt(x) := Pt(x)− |x|2/2 ut(x) := ∂t∇P ∗ t (∇Pt(x)) +D2P ∗ t (∇Pt(x)) ( ∇Pt(x)− x )⊥ (2.3) solves (2.1). Being P ∗ t just a convex function, a priori D 2P ∗ t is a matrix-valued measure, thus it is not clear the meaning to give to the previous formula. However, since ρt solves a continuity equation with a divergence free vector field (notice that U t is the rotated gradient of the function |x|2/2 − P ∗ t (x), see (2.2)), we know that 0 < λ ≤ ρt ≤ Λ ∀ t > 0 (2.4) 2 provided this bound holds at t = 0. In addition, the relation ρt = (∇Pt)]LT2 implies that (∇P ∗ t )]ρt = LT2 (since ∇P ∗ t is the inverse of ∇Pt), from which it follows [8] that P ∗ t solves in the Alexandrov sense the Monge-Ampère equation det(DP ∗ t ) = ρt (see Section 3.1 for the definition of Alexandrov solution). Hence, it becomes clear now our initial question on the W 2,1 regularity of solutions to the Monge-Ampère equation: if we can prove that under (2.4) we have D2P ∗ t ∈ L1, then we have hopes to give a meaning to the velocity field ut defined in (2.3), and then prove that (pt,ut) solve (2.1). 2.1 Space-time Sobolev regularity of ∇P ∗ t In [11] we proved not only that solutions to (1.1) with 0 < λ ≤ f ≤ Λ are W 2,1 loc , but that actually, for any k > 0, ∫ Ω′ |Du| log(2 + |Du|) <∞ ∀Ω′ ⊂⊂ Ω. (2.5) The proof of this estimate strongly exploits the affine invariance of Monge-Ampère, and can actually be pushed forward to show that solutions are W 2,1+ε loc for some ε = ε(n, λ,Λ) > 0 [12, 16]. As shown in [1, Theorem 2.2], this estimate immediately extends to solutions on the torus, so in particular it applies to P ∗ t . Thanks to this fact, it is easy to see that the second term in the definition of ut (see (2.3)) is well-defined and belongs to L 1. To deal with the term ∂t∇P ∗ t , we need a second argument. We use log+ to denote the positive part of the logarithm, i.e., log+(t) = max{log(t), 0}. The following estimate is proved in [1, Proposition 3.3], following an idea introduced in [15, Theorem 5.1]: Proposition 2.1. For every k ∈ N there exists a constant Ck such that, for almost every t ≥ 0, ∫ T2 ρt|∂t∇P ∗ t | log+(|∂t∇P ∗ t |) dx ≤ Ck (∫ T2 ρt|DP ∗ t | log + (|DP ∗ t |) dx+ ∥∥ρt|U t|2∥∥L∞(T2) ∫ T2 |DP ∗ t | dx ) . (2.6) Remark 2.2. Let us mention that, by the W 2,1+ε loc regularity of P ∗ t , one could actually prove that ∫ T2 ρt|∂t∇P ∗ t | dx ≤ C, κ := 2 + 2ε 2 + ε > 1. Although this estimate is stronger, it is less suited when one investigates the problem in the whole space [2]: indeed, in that case one would obtain that, for any R > 0, there exist κR > 1 and CR > 0 such that ∫ B(0,R) ρt|∂t∇P ∗ t |R dx ≤ CR (i.e., the integrability exponent depends on R), while the estimates with the logarithm reads ∫ B(0,R) ρt|∂t∇P ∗ t | log+(|∂t∇P ∗ t |) dx ≤ Ck,R, and this makes it simpler to use. 3 Sketch of the proof of (2.6). In order to justify the following computations one needs to perform a careful regularization argument. Here we show just the formal argument, referring to [1] for a detailed proof. First of all, by differentiating in time the relation det(D2P ∗ t ) = ρt we get 2 ∑ i,j=1 Mij(D P ∗ t (x)) ∂t∂ijP ∗ t (x) = ∂tρt, where Mij(A) := ∂ det(A) ∂Aij is the cofactor matrix of A. Taking into account (2.2) and the well-known divergence-free property of the cofactor matrix ∑ i ∂iMij(D Pt ∗(x)) = 0, j = 1, 2, we can rewrite the above equation as