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EXISTENCE OF EULERIAN SOLUTIONS

TO THE SEMIGEOSTROPHIC EQUATIONS IN PHYSICAL SPACE: THE 2-DIMENSIONAL PERIODIC CASE

with ut and pt periodic.

Energetic considerations (see [11, Section 3.2]) 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 (nor-malized) Lebesgue measure on the torus, then formally2ρt := (∇Pt)♯LT2 satisfies the following dual problem (see the Appendix):

Hence ∇Pt can be viewed as a map from T2to T2and ρt is a well defined measure on T2. One can also verify easily that the inverse map ∇P∗(1.3) as a PDE on T2, i.e., using test functions which are Z2-periodic in space. tsatisfies (1.4) as well. Accordingly, we shall understand

The dual problem (1.3) is nowadays pretty well understood. In particular, Benamou and Brenier

velocity ut in (1.5) is a well defined L1function (see the proof of Theorem 1.2). Moreover, following some ideas developed in [17] we can show that the first term is also L1, thus giving a meaning to

ut (see Proposition 3.3). At this point we can prove that the pair (pt, ut) is actually a distributional

solution of system (1.2). Let us recall, following [12], the proper definition of weak Eulerian solution of (1.2).

Definition 1.1. Let p : T2× (0, ∞) → R and u : T2× (0, ∞) → R2. We say that (p, u) is a weak Eulerian solution of (1.2) if:
- |u| ∈ L∞((0, ∞), L1(T2)), p ∈ L∞((0, ∞), W1,∞(T2)), and pt(x) + |x|2/2 is convex for any t ≥ 0;
- For every φ ∈ C∞c(T2 × [0, ∞)), it holds

Theorem 1.2. Let p0 : R2→ R be a Z2-periodic function such that p0(x) + |x|2/2 is convex, and assume that the measure (Id + ∇p0)♯L2is absolutely continuous with respect to L2with density ρ0, namely
(Id + ∇p0)♯L2= ρ0L2.

Moreover, let us assume that both ρ0 and 1/ρ0 belong to L∞(R2).

2. Optimal transport maps on the torus and their regularity

The following theorem can be found in [10] (see for instance [14, Section 2] for the notion of Alexandrov solution of the Monge-Amp`ere equation).

y∈R2 x · y − P ∗(y),
P(x) = sup

Combining the previous theorem and the known regularity results for strictly convex Alexandrov

∥P∥C1,β ≤ C.

(ii) P ∈ W2,1(T2), and for any k ∈ N there exists a constant C = C(λ, Λ, k) such that �T2 |∇2P| logk +|∇2P| dx ≤ C.

3. The dual problem and the regularity of the velocity field

In this section we recall some properties of solutions of (1.3), and we show the L1integrability of

(3.1) � �T2�∂tϕt(x) + ∇ϕt(x) · Ut(x)�ρt(x) dx dt +�T2 ϕ0(x)ρ0(x) dx = 0

for every ϕ ∈ C∞Finally, the following regularity properties hold: c(R2 × [0, ∞)) Z2-periodic in the space variable.

and 3.6. Further regularity properties of ∇Pt and ∇P∗twith respect to time will be proved in Propositions 3.3

In the proof of Theorem 1.2 we will need to test with functions which are merely W1,1. This is

4Here Pw(T2) is the space of probability measures on the torus endowed with the weak topology induced by the duality with C(T2)

in [15] allows to show the validity of the natural a priori estimate on the left hand side in (3.2).

Proposition 3.3 (Time regularity of optimal maps). Let ρt and Pt be as in Theorem 3.1. Then

To prove Proposition 3.3, we need some preliminary results.

Lemma 3.4. For every k ∈ N we have

ab logk +(ab) ≤ ab�log+� b a�+ 2 log+(a)�k

≤ 2k−1ab�logk +� b a�+ 2klogk +(a)�

U ∈ C∞(T2× [0, ∞); R2) satisfy

0 < λ ≤ ρt(x) ≤ Λ < ∞ ∀ (x, t) ∈ T2× [0, ∞),

(i) P∗t−�−T2P∗t∈ Liploc([0, ∞); Ck(T2)) for any k ∈ N.

Hence, from (3.6), (3.7), (3.5), and (3.9), it follows that

i,j=1�Mij(∇2P∗t(x)) ∂t∂ijP ∗t(x).

Taking into account the continuity equation and the well-known divergence-free property of the co-

(3.14) ∂tρt + ∇ · (Utρt) = 0 ,

(3.15) (∇Pt)♯LT2 = ρtLT2,

Multiplying (3.4) by ∂tP∗�T2 P ∗t= 0 for all t ≥ 0, so that by Lemma 3.5 we have ∂tP ∗

tand integrating by parts, we get
t∈ C∞(T2). Fix t ≥ 0.

(3.17) −�T2 ρt∂t∇P ∗t· (∇2P ∗t)−1/2(∇2P ∗t)1/2Utdx

≤��T2 ρt|(∇2P ∗t)−1/2∂t∇P ∗t|2 dx�1/2 ��T2 ρt|(∇2P ∗t)1/2Ut|2 dx�1/2 .

(3.19) �T2 ρt|(∇2P ∗t)−1/2∂t∇P ∗t|2 dx ≤ max�ρt|Ut|2� �T2 |∇2P ∗t| dx.

We now apply Lemma 3.4 with a = |(∇2P∗existence of a constant C(k) such that t)1/2| and b = |(∇2P ∗t)−1/2∂t∇Pt∗(x)| to deduce the

(3.20) ≤ C(k)��T2 ρt|∇2P ∗t| log2k +(|∇2P ∗t|) dx +�T2 ρt|(∇2P ∗t)−1/2∂t∇P ∗t|2 dx�

which proves (3.2).≤ C(k)��T2 ρt|∇2P ∗t| log2k +(|∇2P ∗t|) dx + max�ρt|Ut|2� �T2 |∇2P ∗t| dx�,

ρn:= ρ ∗ σn, Un(x) := (ρU) ∗ σn

ρ ∗ σn
.

are identifying periodic functions with functions defined on the torus). Let Pn tbe the only convex

function such that (∇Pn

for any t > 0.

C := C(λ, Λ, k) such that Moreover, by Theorems 2.1 and 2.2(ii), for every k ∈ N there exists a constant

(3.23) �T2 |∇2P n∗t| dx →�T2 |∇2P ∗t| dx.

Finally, since the function (w, t) �→ F(w, t) = |w|2/t is convex on R2× (0, ∞), by Jensen inequality we get

since the function w �→ |w| logk semicontinuity theorem [1, Theorem 5.8] to the functions φ(t)ρn +(|w|/r) is convex for every r ∈ (0, ∞) we can apply Ioffe lower

t∂t∇P n∗and φ(t)ρn tto infer

� T φ(t)�T2 ρt|∂t∇P ∗t| logk +(|∂t∇P ∗t|) dx dt

Since this holds for every φ ∈ C∞
≤ C(k)� T φ(t)��

and arguing again as in the proof of [17, Theorem 5.1], the following more general statement holds

(compare with [17, Theorem 5.1, Equations (27) and (29)]):

(∇Pt)♯LT2 = ρtLT2,

and denote by P∗tits convex conjugate.

(3.27) �T2 |∂t∇Pt| logk +(|∂t∇Pt|) dx

≤ C(k)��T2 |∇2Pt| log2k +(|∇2Pt|) dx + ess sup T2 �ρt|vt|2� �T2 |∇2P ∗t| dx�.

smooth and hence, arguing as in Lemma 3.5, we have that Pt, P∗

changing variables in the the left hand side of (3.19) we get t∈ Liploc([0, ∞), C∞(T2)). Now,

and [∂t∇P∗t](∇Pt) + [∇2P ∗

t◦ ∇Pt= Id, Equation (3.28) becomes
t](∇Pt)∂t∇Pt= 0

Proof of Theorem 1.2. First of all notice that, thanks to Theorem 2.2(i) and Proposition 3.3, it holds

|∇2P∗t|, |∂t∇P ∗t| ∈ L∞loc([0, ∞), L1(T2)). Moreover, since (∇Pt)♯LT2 = ρtLT2, it is immediate to check

By Theorem 2.1 and the periodicity of φ, ϕt(y) is Z2-periodic in the space variable. Moreover ϕt

is compactly supported in time, and Proposition 3.3 implies that ϕ ∈ W1,1(R2× [0, ∞)). So, by

12 L. AMBROSIO, M. COLOMBO, G. DE PHILIPPIS, AND A. FIGALLI

which, taking into account the formula (1.5) for u, after rearranging the terms turns out to be equal + J(∇Pt(x) − x)

+

Hence, combining (4.3), (4.4), (4.5), and (3.1), we obtain the validity of (1.6).� ∞�T2�J∇pt(x)�∂tφt(x) + ut(x) · ∇φt(x)�+

�−∇pt(x) − Jut(x)�φt(x)�dx dt.

0 =� ∞�T2 {∂tϕt(y) + ∇ϕt(y) · Ut(y)} ρt(y) dy dt

=� ∞φ′(t)�T2 ψ(x) dx dt

times independent of the test function ψ, thus proving (1.7). □

5. Existence of a Regular Lagrangian Flow for the semigeostrophic velocity field

(5.1) Ft(x) = x +� t bs(Fs(x))dx ∀t ∈ [0, ∞).

EXISTENCE OF EULERIAN SOLUTIONS OF THE SEMIGEOSTROPHIC EQUATIONS 13

ut(x) = [∂t∇P∗t](∇Pt(x)) + [∇2P ∗t](∇Pt(x))J(∇Pt(x) − x), (5.2)
where Pt and P∗ Existence for weaker notion of Lagrangian flow of the semigeostrophic equations was proved by tare as in Theorem 1.2. Recall also that, under these assumptions, |u| ∈ L∞loc([0, ∞), L1(T2)).

Cullen and Feldman, see [12, Definition 2.4], but since at that time the results of [14] were not available the velocity could not be defined, not even as a function. Hence, they had to adopt a more indirect definition. We shall prove indeed that their flow is a flow according to Definition 5.1. We discuss the uniqueness issue in the last section.

is invertible in the sense that for all t ≥ 0 there exist Borel maps F∗Ft(F∗t) = Id a.e. in T2. tsuch that F ∗t(Ft) = Id and

Proof. Let us consider the velocity field in the dual variables Ut(x) = J(x − ∇P∗t(x)). Since P∗t

The validity of property (b) in Definition 5.1 and the invertibility of F follow from the same arguments

of [12, Propositions 2.14 and 2.17]. Hence we only have to show that property (a) in Definition 5.1

n→∞�T2� T�|Qn t− ∇P ∗t| + |∂tQn t− ∂t∇P ∗t| + |∇Qn t− ∇2P ∗t|�dy dt.

= lim

Hence, since G is a RLF and by assumption + |[∇Qn t](Gt(x)) − [∇2P ∗t](Gt(x))|�dt → 0.

(∇P0)LT2 ≪ LT2,

EXISTENCE OF EULERIAN SOLUTIONS OF THE SEMIGEOSTROPHIC EQUATIONS 15

6. Open problems

Conjecture. Let f ∈ W1,1((0, T) × T2; R2) ∩ C([0, T] × T2; R2), and let Ht be a measure-preserving Lagrangian flow relative to b. Assume that

(6.1) [∂tft](Ht(x)) + [∇ft](Ht(x))bt(Ht(x)) ∈ L1(0, T) for a.e. x ∈ T2.

Notice that this formal derivation holds independently of u (only the divergence-free condition of u is needed), and that u does not appear explicitly in (1.3).

References

[5] L.Ambrosio, N.Gigli, G.Savar´e: Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. Preprint, 2011.

[6] J.-D.Benamou, Y.Brenier: Weak existence for the semigeostrophic equation formulated as a coupled Monge- Amp`ere/transport problem. SIAM J. Appl. Math., 58 (1998), 1450–1461.

[11] M.Cullen: A mathematical theory of large-scale atmosphere/ocean flow. Imperial College Press (2006). [12] M.Cullen, M.Feldman: Lagrangian solutions of semigeostrophic equations in physical space. SIAM J. Math. Anal., 37 (2006), 1371–1395.

[13] G.Crippa, C.De Lellis: Estimates and regularity results for the DiPerna–Lions flow. J. Reine Angew. Math., 616 (2008), 15–46.

[19] R. J. Mc Cann Polar factorization of maps on manifolds . Geom. Funct. Anal. 11 (2001), 589608
[20] C.Villani: Optimal Transport. Old and new. Grundlehren der Mathematischen Wissenschaften, 338. Springer- Verlag, Berlin, (2009)

Scuola Normale Superiore, p.za dei Cavalieri 7, I-56126 Pisa, Italy E-mail address: l.ambrosio@sns.it