On Lagrangian solutions for the semi-geostrophic system with singular initial data

Transcription

1 On Lagrangian solutions for the semi-geostrophic system with singular initial data Mikhail Feldman Department of Mathematics University of Wisconsin-Madison Madison, WI 5376, USA Adrian Tudorascu Department of Mathematics West Virginia University Morgantown, WV 2656, USA March 14, 212 Abstract We show that weak (Eulerian) solutions for the Semi-Geostrophic system in physical space exhibiting some mild regularity in time cannot yield point masses in dual space. However, such solutions are physically relevant to the model. Thus, we discuss a natural generalization of weak Lagrangian solutions in the physical space to include the possibility of singular measures in dual space. We prove existence of such solutions in the case of discrete measures in dual space. We also prove that weak Lagrangian solutions in physical space determine solutions in the dual space. This implies conservation of geostrophic energy along the Lagrangian trajectories in the physical space. 1 Introduction The Semi-Geostrophic (abbreviated SG in this work) equations have been proposed as simplifications of the primitive equations (Boussinesq) when the rate of change of momentum is much smaller than the Coriolis force (small Rossby number) [15]. The advected quantity momentum is approximated by its geostrophic value, but the trajectories are not. Throughout the entire paper, IR 3 is a given open, bounded set, and T (, ) is fixed. A Key words: SG system, flows of maps, optimal mass transport, Wasserstein metric, optimal maps, absolutely continuous curves. 1

2 2 version of the 3D Semi-Geostrophic system [5, 8] is D t X = J [ X x ], X = P, u =, u ν =, on [, T ), P (, ) = P in, (1) where P is convex defined on. One looks for solutions (P, u) satisfying the Cullen-Purser stability condition (see, e.g., [9]), which amounts to imposing that P t ( ) := P (t, ) be convex for all t [, T ). Here, 1 J = 1. Henceforth, we shall assume without loss of generality that L 3 () = 1 (otherwise all the measures considered will have total mass equal to the volume of ) and denote χ := L 3. If P t# χ =: α t are all absolutely continuous with respect to L 3, then one can use the Legendre-Fenchel transforms Pt of P t to formally rewrite (1) as the so-called SG in dual variables t α + (Uα) = in [, T ) IR 3, (2) P (t, )#χ = α(t, ) for any t [, T ); (3) U(t, X) = J[X P (t, X)], (4) α(, X) = α (X) for a.e. X IR 3. (5) Existence of solutions of the problem recast in dual variables was obtained by J. D. Benamou, Y. Brenier [5], and for some related models by M. Cullen and W. Gangbo [9], and M. Cullen and H. Maroofi [11]. They considered case when α = P # χ L q (; IR 3 ) for some q > 1, and the solution satisfies α(t, ) L q for all t. These results were extended to the case q = 1 in [17]. These solutions are not known to be regular enough to be translated into (Eulerian) solutions of the problem in physical space. Existence of Eulerian solutions in the two-dimensional periodic case when the density in the dual space is bounded away from zero and infinity was obtained recently by L. Ambrosio, M. Colombo, G. De Philippis, A. Figalli [2] based on the results of G. De Philippis and M. Figalli [12] on regularity of solutions for the Monge-Ampere equation. Existence of Eulerian solutions in the more general case when α (L 1 L q )(IR 3 ) for some q [1, ] and α is presently not known. Another approach is to consider Lagrangian solutions in the physical space. Such solutions were introduced by M. Cullen and M. Feldman [8], and existence of Lagrangian solutions of (1) was shown in [8] for the case α = P # χ L q (; IR 3 ) for q > 1, on the basis of Ambrosio s theory on transport equations and flows of BV vector fields [1]. These results were extended to the case q = 1 in [13]. The definition of weak Lagrangian solutions in the physical space is following (where we slightly modify the definition given in [8] by relaxing assumptions on P ). Definition 1.1. Let P H 1 () be convex, F : [, T ) be a Borel map such that F C([, T ); L 2 (; IR 3 )), and let P C([, T ); H 1 ()). Assume P (t, ) is convex in for each t [, T ). Then the pair (P, F ) is called a weak Lagrangian solution of (1) in [, T ) if i. F (, x) = x, P (, x) = P (x) for a.e. x,

3 3 ii. for any t > the mapping F t = F (t, ) : is Lebesgue measure preserving, in the sense that F t# χ = χ; iii. There exists a Borel map F : [, T ) such that for every t (, T ) the map F t = F (t, ) : is Lebesgue measure preserving: F t# χ = χ, and satisfies F t F t (x) = x and F t F t (x) = x for a.e. x ; iv. The function is a distributional solution of Z(t, x) = P (t, F t (x)) (6) t Z(t, x) = J [ Z(t, x) F (t, x) ] in [, T ), Z(, x) = P (x) in. (7) Note that the sense in which (7) must be satisfied is [Z(t, x) t ϕ(t, x) + J [ Z(t, x) F (t, x) ] ϕ(t, x)]dxdt + T for any ϕ C 1 c ([, T ) ; IR 3 ). P (x) ϕ(, x)dx = (8) In this paper we consider the case of singular initial data, i.e. when α = P # χ is a singular measure. The dual problem in this case was studied by G. Loeper [16], and L. Ambrosio and W. Gangbo [3]. Note that equation (2) represents the fact that the dual flow t α t is weakly (in the sense of distributions) transported by the dual velocity U defined by (4). The change of variable X = P t (x) is reversible (although there may not be enough smoothness to transport the dual space solutions back to physical space even in this case) if and only if Pt pushes α t forward to χ, one such situation being provided if α t L 3. In general, if Pt is not necessarily the a.e. inverse of P t, then Pt may not necessarily push α t forward to χ, α t may not necessarily be absolutely continuous with respect to L 3 and the equation (4) in the dual-variable system must be generalized to (see [3]) U(t, X) = J [ X γ t (X) ], (9) where γ t is the barycentric projection onto α t of the (unique) optimal Kantorovich plan [19] γ t := ( P t Id ) # χ having α t and χ as first and second marginals, respectively. It is defined by (see [3], [4]) ξ(x) γ t (X)dα t (X) = ξ(x) ydγ t (X, y) (1) IR 3 IR 3 for all continuous ξ : IR 3 IR 3 of at most quadratic growth. Since χ is absolutely continuous with respect to L 3, we deduce ξ(x) γ t (X)dα t (X) = ξ( P t (x)) xdx. (11) IR 3 To justify generalizing (4) to (9), let us assume (1) has a smooth, stable solution (P, u). Set Y := P and compute, for any ζ Cc (IR 3 ), d ζ(x)dα t (X) = t Y t (x) ζ(y t (x))dx dt IR 3 [ = (u(t, x) )Yt (x) ] ζ(y t (x))dx + J [ Y t (x) x ] ζ(y t (x))dx.

4 4 Since u is divergence-free and Y t = 2 P t is symmetric, we deduce that the first integral in the right hand side of the second equation of the above display is [( Yt (x) ) t ] [( u(t, x) ζ(yt (x))dx = Yt (x) ) t ζ(yt (x)) ] u(t, x)dx = [ ζ(y t (x)) ] u(t, x)dx =. As for the second integral, we have, due to J t = J, Y t# χ = α t and (11), J [ Y t (x) x ] ζ(y t (x))dx = Y t (x) (J ζ)(y t (x))dx + x (J ζ)(y t (x))dx = (JX) ζ(x)dα t (X) [J γ t (X)] ζ(x)dα t (X). IR 3 IR 3 Thus, which implies d ζ(x)dα t (X) = J [ X γ t (X) ] ζ(x)dα t (X), (12) dt IR 3 IR 3 is satisfied in the sense of distributions. t α(t, X) + X (J [ X γ(t, X) ] α(t, X) ) = (13) In this paper we define weak Lagrangian solutions in physical space in case of singular initial data, and prove existence of such solutions in the case of discrete measures. Moreover, we show conservation of geostrophic energy along the Lagrangian trajectories in the physical space. The rest of the paper is organized as following. In the next section we collect some results on the existence and properties of solutions in dual space. In Section 3 we prove that, under a mild time-regularity assumption, Eulerian solutions do not exist if the dual-space solutions have point masses at non-negligibly many times. This will advocate finding a suitable notion of weak Lagrangian solutions in physical case that makes sense in the general case of measurevalued solutions α in the dual space. This will be achieved in Section 4. There we will also prove that such solutions exist in the case where α t are convex combinations of point masses, or equivalently, when P is the maximum of finitely many affine functions. In Section 5 we will show that weak Lagrangian solutions in physical space can be translated into weak solutions of the problem in the dual space. This will lead to conservation of energy for weak Lagrangian solutions for the Semi-Geostrophic system. 2 Solutions in dual space In this section we collect a number of results on the existence and some properties of solutions in dual space. Our main source is [3]. Before that, let us recall the definitions of some important objects. In the spirit of [4], one defines AC p (, T ; P 2 (IR 3 )) (for 1 p ) as the set of all paths µ : [, T ] t µ t P 2 (IR 3 ) for which there exists β L p (, T ) such that W 2 (µ s, µ t ) t s β(τ)dτ for all s t T, where W 2 is the quadratic Wasserstein distance [19]. The smallest of the functions β satisfying the inequality above is called the metric derivative of the curve µ, it is denoted by µ and it

5 5 satisfies [4] that µ (t) = lim s t W 2 (µ s, µ t ) s t for a.e. t (, T ). There exists a Borel velocity v : (, T ) IR 3 IR 3 transporting µ in the sense of distributions, i.e. t µ + x (vµ) = in D ((, T ) IR 3 ), (14) such that v(t, ) L 2 (µ t ; IR 3 ) and v t L 2 (µ t;ir 3 ) = µ (t) for a.e. t (, T ). It turns out that this velocity (called of minimal norm ) minimizes v t L 2 (µ t;ir 3 ) among all possible Borel velocities (i.e. satisfying (14)), and it can be uniquely selected (µ t a.e.) for a.e. t (, T ) by requiring that it belong to the closure of Cc (IR 3 ) in L 2 (µ t ; IR 3 ), which is denoted [4] by T µt P 2 (IR 3 ) and called the tangent space to P 2 (IR 3 ) at µ t. Finally, if F : P 2 (IR 3 ) IR is lower semicontinuos with respect to the topology induced by the Wasserstein distance W 2, then we can define the subdifferential of F at some µ P 2 (IR 3 ) as the set of all ξ L 2 (µ; IR 3 ) such that F (ν) F (µ) + ξ(x) (y x)γ(dx, dy) + o ( W 2 (µ, ν) ) IR 3 IR 3 for all ν P 2 (IR 3 ) and all the optimal transport plans [19] γ between µ and ν. This set is denoted by F (µ) and its element of minimal L 2 (µ; IR 3 ) norm is denoted by w F (µ) and called [3] the Wasserstein gradient of F at µ. Adapting to our context (by, for example, replacing J by J t = J which still satisfies the required orthogonality property in [3]), a Hamiltonian ODE solution/trajectory for the lower semicontinuous Hamiltonian H : P 2 (IR 3 ) IR is defined as follows. Definition 2.1. A curve µ AC 1 (, T ; P 2 (IR 3 )) with the property that there exists a vector field v : (, T ) IR 3 IR 3 such that and (, T ) t v t L 2 (µ t;ir 3 ) belongs to L1 (, T ) (15) t µ (Jvµ) =, µ = µ as distributions, (16) v t T µt P 2 (IR 3 ) H(µ t ) for a.e. t (, T ), (17) is called a solution of the Hamiltonian ODE associated to the Hamiltonian H, starting at µ. Fix ν P 2 (IR 3 ) and define H ν : P 2 (IR 3 ) IR, H ν (µ) = 1 2 W 2 2 (µ, ν), (18) which is, obviously, continuous. Then H ν (µ) consists of all functions of the type γ Id, where γ is any optimal plan between µ and ν [19], and γ is its barycentric projection onto µ, i.e. given by (see, e.g. [4]) y ξ(x)γ(dx, dy) = IR 3 IR 3 γ(x) ξ(x)µ(dx) IR 3 for all ξ C(IR 3 ; IR 3 ) of at most quadratic growth. In light of the fact that the y marginal of γ is ν, the above definition implies γ L 2 (µ;ir 3 ) Id L 2 (ν;ir 3 ). Thus, W2 2 (µ, ν) = x 2 µ(dx) + IR 3 y 2 ν(dy) 2 IR 3 x γ(x)µ(dx) γ Id 2 IR 3 L 2 (µ;ir 3 )

6 6 for all such optimal plans, which implies w H ν (µ) L 2 (µ;ir 3 ) 1 + 2H ν ( µ) =: a( µ) whenever W 2 (µ, µ) 1. (19) Then one combines Lemma 7.6 [3] and Theorem 7.4 [3] to obtain on [, 1/a( µ) =: T ] a solution µ AC (, T ; P 2 (IR 3 )) (Lipschitz curve) of the Hamiltonian ODE starting at µ and satisfying that it is a( µ) Lipschitz and conservative, i.e. [, T ] t H ν (µ t ) is constant. Thus, a(µ T ) = a( µ), which means that (19) is also satisfied with the same bounds if we replace µ by µ T. Thus, in light of the same results from [3], we infer that we can extend the solution to [T, 2T ] (whereas it preserves the Lipschitz bound and conserves the Hamiltonian). By induction, we obtain: Theorem 2.2. For any µ P 2 (IR 3 ) there exists a solution µ AC (, ; P 2 (IR 3 )) of (16) and (17), which starts at µ, conserves [, ) t H ν (µ t ), and is globally a( µ) Lipschitz. Let us now specialize to the case ν = χ (defined in the introduction) and denote H : P 2 (IR 3 ) IR, H(µ) = 1 2 W 2 2 (µ, χ). (2) The absolute continuity of χ with respect to L 3 adds the benefit of the fact that there is a unique optimal plan now between µ and χ for every µ P 2 (IR 3 ), namely γ µ = ( Φ Id) # χ, where Φ is the optimal map pushing χ forward to µ. Thus, we have H(µ) T µ P 2 (IR 3 ) = { γ µ Id} = { w H(µ)} for all µ P 2 (IR 3 ). This means that Theorem 2.2 has the following: Corollary 2.3. Let α P 2 (IR 3 ) be given. Then there exists a distributional solution α AC (, ; P 2 (IR 3 )) for (13) with α() = α, where γ t is the barycentric projection of the optimal plan between α t and χ. In other words, the curve α is globally Lipschitz and satisfies the system (2), (3), (9), (5) in the sense of distributions in (, ) IR 3. Also, the Hamiltonian energy t H(α t ) is conserved. Remark 2.4. Since [, ) t α t P 2 (IR 3 ) is continuous, we use Proposition 3.2 [18] to conclude that there is a family P C([, ); H 1 ()) of convex functions P (t, ) such that x P (t, ) =: P t pushes χ forward to α t optimally for all t. 3 Lagrangian vs Eulerian As of this date, the only weak Lagrangian solutions in the physical space have been shown to exist in the case P # χ =: α L 1 (). This is achieved in [13] by improving the L q (q > 1) result in [8]. The solution (P, F ) constructed in these references satisfies P t# χ =: α t L q () for almost all t (, T ). However, note that we do not impose that our solution satisfy any of these conditions, i.e. the measures α t (for t [, T )) may be singular. In [8] the authors start their construction from an appropriate solution α t to the problem in the dual space, then they set Z(t, x) := P t F t (x), where F t := Pt Φ t P. Crucial to their argument of why (P, F ) is a weak Lagrangian solution in the physical space are the facts that Pt# α t = χ and that the flow Φ of J[Id Pt ] satisfies Φ t# χ = χ. As for the definition of weak solutions in the dual space, we have already showed in the Introduction that in the general case (in which α t is not necessarily a function) Pt is replaced by the barycentric projection γ t. It is easy to see that when γ t and Pt are distinct α t a.e., we have α t γ t# χ and the construction in [8] fails. This does not necessarily happen when α t is not a function, but rather when there is no optimal map pushing the measure α t forward to χ.

7 7 Remark 3.1. It is not known whether weak Lagrangian solutions in the physical space exist in the case P t# χ is a singular measure (not a function). In [13] the authors show that a stability result in L 1 is available, i.e., if α n converges to α in L 1 (IR 3 ) and all their supports lie within a given ball, then the weak Lagrangian solutions in the physical space corresponding to α n converge to that corresponding to α. The strong L q (; IR 3 ) (q 1) convergence of the Lagrangian flows Ft n to F t is proved. Thus, as an immediate consequence, the existence result in the case α L q (IR 3 ), q > 1 obtained in [8] is generalized to q 1. Then they argue by means of counterexample that a similar stability result in the space of measures is unavailable while keeping the strong convergence of sequences of Lagrangian flows as a conclusion. Considering the nature of their counterexample (where α = δ z ), the conclusion is to be expected, as the approximating αt n converge as measures to Dirac measures δ z(t) satisfying the Semi-Geostrophic system in the dual space. Note that the corresponding optimal maps pushing χ forward to these measures are P t z(t) and, thus, the first equation in (7) would imply that the first two components F t are independent of x, which would violate (ii) in Definition 1.1. Our point in the previous paragraph was that since the limiting solution in the dual space cannot give rise to a weak Lagrangian solution in the physical space, it is hardly surprising that the approximating Lagrangian flows in the physical space do not converge strongly to anything. That does not mean that given, say, α = δ z a weak Lagrangian solution in the physical space with P z does not exist. Nevertheless, in what follows we argue that good Lagrangian solutions in the physical space cannot give rise to solutions in the dual space which are too singular. By good Lagrangian solution we understand one for which t F L ((, T ) ; IR 3 ), so that by setting u(t, x) := t F (t, F (t, x)) we obtain [8] that the pair (P, u) is a weak (Eulerian) solution of the Semi-Geostrophic system (1) in the physical space. Let us begin by recalling the definition of a weak solution of (1). Definition 3.2. Let u : [, T ) IR 3 and P : [, T ) IR satisfy u L 1 ([, T ) ; IR 3 ), P L ([, T ) ) C([, T ); L 1 ()), and P (t, ) is convex in for every t [, ). The pair (P, u) is a weak Eulerian solution of (1) if T for any φ C 1 c ([, T ) ; IR 3 ), and for any ψ C 1 c ([, T ) ). { P (t, x) [ t φ(t, x) + φ(t, x)u(t, x)] (21) +J[ P (t, x) x] φ(t, x)}dxdt + P (x) φ(, x)dx = T u(t, x) ψ(t, x)dxdt = (22) We would now like to make a stronger case for the consideration of weak Lagrangian solutions, by showing that one cannot have weak Eulerian solutions exhibiting mild time regularity and spatial flat parts except, possibly, at negligibly many times. We begin with Proposition 3.4 below, whose proof is a relatively straight-forward adaptation of Corollary 2.3 and Proposition 2.4 in [18]. The only difference is that in said reference the set O is a subset of (, T ) of full measure.

8 8 Before stating the result, we need some preliminary observations. If X H 1 (, T ; L 2 (; IR 3 )), we denote by Ẋ L2 (, T ; L 2 (; IR 3 )) its functional derivative, defined by lim X t+h X t h h Ẋt = for L 1 a.e. t (, T ). L 2 (;IR 3 ) In the next lemma, we extend X to a map in AC 2 (IR; L 2 (; IR 3 )) by setting X t = X + for t and X t = X T for t T. Lemma 3.3. Let X H 1 (, T ; L 2 (; IR 3 )) and T lim h Ẋ be its functional derivative. Then X t+h x X t x h Ẋtx 2 dxdt =. (23) As a consequence, there exist sequences h + k +, h k and a measurable subset A IR such that L 4 ((IR ) \ A) = and for all (t, x) A. lim k X t+h + x X t x k h + = lim k k X t+h x X t x k h = Ẋtx (24) k The proof in [14] needs no modification. The philosophy behind this result is that, in some specified sense, Ẋ can be viewed as almost a classical pointwise time-derivative of X. Also, since X H 1 (, T ; L 2 (; IR 3 )), we have that it admits a Borel representative. Equation (23) shows that Ẋ itself has that property. Throughout the paper we identify both X and Ẋ with their Borel representatives. Proposition 3.4. Let X H 1 (, T ; L 2 (; IR 3 )) be such that X t = P t, where P t is convex for all t (, T ). Let A (, T ) as in Lemma 3.3. Furthermore, let O A be a Borel set with L 4 (O) > and such that L 3 ([X t x]) > for all (t, x) O, where [X t x] := {y : X t y = X t x}. Then, there exists a Borel map w : (, T ) IR 3 IR 3 such that Ẋ(t, x) = w(t, X(t, x)) for L 4 a.e. (t, x) O. (25) Proof: ( Let ) λ denote the L 4 measure restricted to O, Ψ : O (, T ) IR 3 given by Ψ(t, x) = t, X(t, x), and set ϑ := Ψ# λ. Denote by η the vector-measure whose density with respect to λ is Ẋ, then set σ := Ψ # η. Clearly, σ ϑ, which means there exists a Borel vector field w : (, T ) IR 3 IR 3 such that dσ = wdϑ. The disintegration theorem (see, for example, Theorem [4]) applies to the Borel vector field Ψ and the measure λ. Thus, for ϑ a.e. (t, y) (, T ) IR 3, there exists a unique Borel probability measure λ t,y on O such ( that the map (t, y) λ t,y (B) is Borel measurable for each Borel set B O. Furthermore, λ t,y Ψ 1 (t, y) ) = 1 for ϑ a.e. (t, y) (, T ) IR 3 and O f(t, x) dλ(t, x) = T IR 3 ( ) f(t, x) λ t,y (dt, dx) ϑ(dt, dy) Ψ 1 (t,y) for every Borel measurable f : (, T ) [, ]. We showed in [18], Theorem 2.2, that w(t, X(t, x)) = Ẋ(t, z) dλ t,x(t,x) (t, z) for λ a.e. (t, x) O. (26) Ψ 1 (t,s(t,x))

9 9 Note that (t, z) Ψ 1 (t, X(t, x)) is equivalent to (t, z) O and X(t, z) = X(t, x), so we have Φ 1 (t, X(t, x)) = O [X t x]. Then we apply Proposition 2.4 in [18] to get that Ẋ(t, x) = Ẋ(t, z) for all (t, z) O [X t x]. According to (26), we get(25). QED. Before proving the main theorem of this section, we need a measurability lemma. Lemma 3.5. Let α AC 2 (, T ; P 2 (IR 3 )) for some T >. Then the set D(α) := { (t, X) (, T ) IR 3 : X is an atom of α t } (27) is Borel. Proof: Denote by C c + the nonnegative cone of C c (IR 3 ). Consider, for every positive integer m and every ξ C c +, the set { Dm ξ := (t, X) (, T ) IR 3 : ξ(y )dα t (Y ) 1 } IR 3 m ξ(x). Note that the absolute continuity in time of the left hand side of the above inequality [4] and the continuity in X of the right hand side imply that the difference is a continuous function of (t, X). Therefore, Dm ξ is the nonnegative set of a continuous function, which makes it a closed subset of (, T ) IR 3. Thus, { D m := (t, X) (, T ) IR 3 : ξ(y )dα t (Y ) 1 } IR 3 m ξ(x) for all ξ C+ c is closed as well, by being an arbitrary intersection of closed sets. Since D(α) = m 1 D m, the proof is concluded. QED. Now we prove the main result of this section: that weak (Eulerian) solutions in the physical space exhibiting some mild regularity in time cannot give rise to very irregular solutions in dual space. Theorem 3.6. Let (P, u) be a weak solution for the Semi-Geostrophic system in the physical space such that P H 1 (, T ; L 2 (; IR 3 )). Then α t := P t# χ is atom-free for L 1 a.e. t (, T ). Proof: Set Ψ(t, x) := (t, X(t, x)). Since Ψ is a Borel map on (, T ), due to Lemma 3.5 we infer that O := Ψ 1( D(α) ) is a Borel subset of (, T ). One can see that O = {(t, x) (, T ) : L 3( [X t x] ) > }. By Fubini s Theorem and by the convexity of the potentials whose gradients push χ forward to α t, we infer that there exists a Borel subset T (, T ) such that O = t T ( {t} Ot ), (28)

10 1 where O t is the union of all (at most countably many) convex subsets of of positive L 3 measure on each of which X t is constant. Assume by contradiction that L 1 (T ) >. We further throw out of T the L 1 negligible set of times at which ζ(x)u(t, x)dx IR 3 (29) for all ζ Cc (; IR 3 ) (via the separability of this space with respect to the sup norm), but we keep the notation T for the remaining subset. Consider A as the Borel subset of (, T ) of full measure defined in Lemma 3.3, i.e. where the time pseudo-derivative of X in the sense of (24) exists. The set A O is a Borel set (which we still denote by O) with L 4 (O) >. According to Proposition 3.4, we infer there exists a Borel map w : (, T ) IR 3 IR 3 such that (25) is satisfied. By taking φ(t, x) = ξ(t)ζ(x) with ξ Cc (, T ) and ζ Cc (; IR 3 ) in (21) we discover that t X(t, x) ζ(x)dx is absolutely continuous and d X(t, x) ζ(x)dx = {X(t, x) [ ζ(x)u(t, x)] + J[X(t, x) x] ζ(x)}dx dt for a.e. t (, T ) and every ζ Cc (; IR 3 ) (via the usual argument involving the separability of this space endowed with the sup norm). Throwing out, if necessary, a negligible set of times, we conclude Ẋ(t, x) ζ(x)dx = {X(t, x) [ ζ(x)u(t, x)] + J[X(t, x) x] ζ(x)}dx for a.e. t (, T ) and every ζ Cc (; IR 3 ). Choose such a t that also lies in T and consider now only test functions ζ Cc (ω ; IR 3 ), where ω is a connected component of O t (which is a convex subset of of positive L 3 measure on which X t is constant). Since X t c IR 3 in ω, we infer w(t, c) ζ(x)dx = c ω ζ(x)u(t, x)dx + ω J[c x] ζ(x)}dx ω for all ζ Cc (ω ; IR 3 ). Due to (29) the first term in the right hand side vanishes and since the equality holds for all ζ Cc (ω ; IR 3 ), we deduce w(t, c) = J[c x] for a.e. x ω, which contradicts the fact that ω has nonempty interior. QED. Remark: Thus, in order to accommodate singular solutions in dual space, we see the need for defining Lagrangian solutions instead of Eulerian ones in the physical space. Whereas solutions in the dual space may come in any form or shape (from pure Dirac deltas to functions), only the absolutely continuous ones with respect to the Lebesgue measures have been so far known to give rise to Lagrangian solutions in the physical space [8], [13]. In the next section we discuss an extension to this notion and prove some existence results.

11 11 4 Weak Lagrangian solutions in physical space for the case of singular initial data In [8], Lagrangian solutions in the physical space with initial data P # χ = α L p (IR 3 ) (p > 1), were constructed by the following procedure. First, time-stepping approximation from [9] combined with results of [1] yield existence of a solution (α, P ) of the dual space system (2) (4) with initial data (5), and a locally bounded map Φ : (, T ) IR 3 IR 3 satisfying such that Φ = U(, Φ), L 4 a.e. in (, T ) P (), Φ(, X) = X for α a.e. X IR 3, (3) α t = Φ t# α for all t >. There also exists a Borel map Φ such that Φ t preserve L 3 and Φ t Φ t = Id = Φ t Φ t a.e. in IR 3. The physical space flow is defined as F t := P t Φ t P. (31) Then it is shown that (P, F ) satisfies all the requirements of Definition 1.1. It is not known whether weak Lagrangian solutions in the physical space exist in the case P t# χ is a singular measure (not a function). In [13] the authors show that a stability result in L 1 is available, i.e., if α n converges to α in L 1 (IR 3 ) and all their supports lie within a given ball, then the weak Lagrangian solutions in the physical space corresponding to α n converge to that corresponding to α. The strong L p (; IR 3 ) (p 1) convergence of the Lagrangian flows Ft n to F t is proved. Thus, as an immediate consequence, the existence result in the case α L p (IR 3 ), p > 1 obtained in [8] is generalized to p 1. Then, they argue by means of counterexample that a similar stability result in the space of measures is unavailable while keeping the strong convergence of sequences of Lagrangian flows as a conclusion. Considering the nature of their counterexample (where α = δ z ), the conclusion is to be expected, as the approximating αt n converge as measures to Dirac measures δ z(t) satisfying the Semi-Geostrophic system in the dual space. Note that the corresponding optimal maps pushing χ forward to these measures are P t z(t) and, thus, the first equation in (7) would imply that the first two components F t (x) are constant, and thus, independent of x, which would violate ii in Definition 1.1. Thus, it appears natural that Definition 1.1 is unusable if one endeavors to discuss a general notion (i.e. to include the case of general measures α t ) of weak Lagrangian solutions in physical space. In order to see what is needed in general, let us discuss what is reasonable to expect in the case α = δ z (the generalization of the counterexample in [13] cited above) for some z IR 3. One can readily check that if ż(t) = Jz(t), z() = z, then α t := δ z(t) solves the SG in dual space. Whereas it is clear that there are no L 3 measure preserving maps F t satisfying Definition 1.1 in this case, let us go back to the definition of F t = Pt Φ t P from [8]. Since in the general case (of measure-valued solutions) the place of Pt is taken by the barycentric projection γ t, it is natural to ask if one can keep (7) valid by putting F t := γ t Φ t P. (32) Of course, for this function to be well-defined, one needs the Lagrangian flow in dual space Φ t to be defined at z for all t [, T ]. Note that P t (X) = X z(t) implies γ t (z(t)) =, γ t (X) = X z(t) X z(t) if X z(t).

12 12 If we fix IR 3 X z and solve Φ(t, X) = J[Φ(t, X) γ t (Φ(t, X))] with Φ(, X) = X, (33) then we find the unique solution (which stays away from the singularity z(t) of Pt ) cos ct sin ct Φ(t, X) = z(t) + M c (t)(x z ), where M c = sin ct cos ct 1, c := 1 X z. Thus, lim Φ(t, X) = z(t) X z and it becomes natural to define Φ(t, z ) = z(t). Consequently, we have defined a unique Lagrangian flow Φ continuous in the spatial variable, and solving (33) in the classical sense. Indeed, note that γ t (z(t)) = and ż(t) = Jz(t) imply that (33) holds even for X = z. With this Φ in hand, F t is well-defined by (32) as F t (x) = γ t (Φ(t, z )) = γ t (z(t)) = for all x. Instead of preserving χ, F t now is a map pushing forward χ to the reduced domain measure µ t := γ t# α t = δ. Clearly, this construction is compatible with the one in [8]. Indeed, if all measures α t are absolutely continuous with respect to the Lebesgue measure, then γ t# α t = Pt# α t = χ, i.e. µ t = χ for all t [, T ]. Remark 4.1. The computations above pinpoint the solution predicted by the theory of regular Lagrangian flows [1]. One can easily check that U(t, X) := J[X γ t (X)] satisfies the conditions in [1], therefore a unique solution to the ODE exists and is unique for Lebesgue-a.e. X IR 3 such that Φ t preserves L 3. Therefore, when α is a singular measure (not absolutely continuous with respect to the Lebesgue measure), the set {X IR 3 : Φ(t, X) exists and is unique} may not contain all (if any) of the support of α. However, we see that in this very special case (α is just a point mass), the regular Lagrangian flow can be uniquely extended by continuity to the whole IR 3 (thus including the support of α ). We expect this to be true for the case where α is an arbitrary convex combination of Dirac masses. In order to see the motivation for Definition 4.8 of Lagrangian solutions for the case of singular initial data given below, we first study properties of flows in physical space given by (32) in the case when α = P # χ is possibly a singular measure, under the following assumptions: (i) P H 1 () is convex, α = P # χ P 2 (IR 3 ); (ii) There exists a Borel map Φ : [, T ] IR 3 IR 3 such that (33) holds in the integral sense for α -a.e. X IR 3 ; (iii) Family of measures α t := Φ t# α for t [, T ) is a solution in D (, T ) IR 3 ) of equation (2) with U defined by (9), (1), where γ t is the unique optimal transport plan between α t and χ; this also defines convex P t in ) for each t such that γ t := ( P t Id ) # χ; (iv) α AC (, ; P 2 (IR 3 )). (34) Note that assumptions (34) say that there exists a Lagrangian solution of the dual problem with initial data P. In the case when P W 1, () and α = P # χ L p (IR 3 ) for p (1, ] existence of Φ t such that (34) is satisfied is shown in [8]. Below it is convenient to work with a Borel representative of the barycentric projection γ. That is why we prove the following lemma:

13 13 Lemma 4.2. There exists a Borel measurable function defined on (, T ) IR 3 which for L 1 -a.e. t (, T ) coincides with γ(t, X) for α(t, )-a.e. X IR 3. Proof: Let ϑ P ( (, T ) IR 3 ) be the Borel probability given by T ϕ(t, X, y)ϑ(dt, dx, dy) = 1 T ϕ(t, X, y)γ(t, dx, dy)dt IR 3 T IR 3 for any continuous and bounded ϕ. Since α(t, ) is the X-marginal of γ(t,, ), the (t, X) marginal of ϑ is ϑ P ( (, T ) IR 3) given by T ζ(t, X) ϑ(dt, dx) = 1 T ζ(t, X)α(t, dx)dt. IR 3 T IR 3 Thus, by disintegrating ϑ we get T IR 3 ϕ(t, X, y)ϑ(dt, dx, dy) = 1 T T IR 3 ( ) ϕ(t, X, y)ϑ(t, X; dy) α(t, dx)dt, where ϑ(t, X; ) are Borel probabilities on such that the map (t, X) ϑ(t, X; B) is Borel for any Borel set B. In particular, the maps (t, X) f(y)ϑ(t, X; dy) ( are Borel for all f C b ; IR 3 ). By taking ϕ(t, X, y) = u(t)ξ(x) y and using (1), we conclude that γ(t, X) = yϑ(t, X; dy) for L 1 a.e. t (, T ) and α(t, ) a.e. X IR 3. This finishes the proof. QED. Remark 4.3. In light of Lemma 4.2 and equation (9), we see immediately that there exists a Borel measurable function defined on (, T ) IR 3 which for L 1 -a.e. t (, T ) coincides with U(t, X) for α(t, )-a.e. X IR 3. Remark 4.4. In view of the above lemma, from now on we shall use the notation γ to denote the Borel representative of the barycentric projection defined by (1). Likewise, by following the same proof, it is easy to see that the barycentric projection of ( Id P t χ onto its first )# marginal (namely, χ) can also be extended to a Borel map from (, T ) into IR 3. Since this coincides with P t (x) for L 4 -a.e. (t, x) (, T ), we shall assume in the remainder of the paper that P is Borel measurable in both variables. We also note the following: Lemma 4.5. For (t, X) [, T ) IR 3 denote t,x := {x : P t (x) exists and P t (x) = X}. Then In particular, γ t (X) t,x for α(t, )-a.e. X IR 3. (35) ( P t γ t )(X) = X for every t (, T ) and α(t, )-a.e. X IR 3. (36)

14 14 Proof: Fix t (, T ). Since γ t = ( P t Id) # χ and π 1# γ t = α t, by disintegrating γ t as in Theorem [4], we get γ t = γ t,x dα t (X), IR 3 where γ t,x is a family of probability measures on such that the map X γ t,x (B) is Borel for any Borel set B, and γ t,x (IR 3 \ t,x ) = for α(t, )-a.e. X IR 3. From (1) and the disintegration, we get γ t (X) = y dγ t,x (y) = y dγ t,x (y) for α(t, )-a.e. X IR 3. t,x The convexity of P t ( ) implies that t,x is a convex set (of dimension k(t, X) {, 1, 2, 3}) for every X such that t,x. Thus, γ t (X) t,x for α(t, )-a.e. X IR 3, and for such X we get ( P t γ t )(X) = X. QED. The continuity of F in time was proved in [8], [13] in the case of absolutely continuous α t. In the general we cannot expect that, which is clear from the structure (32) of F t, especially looking at the example when P is linear, i.e. α = δ z discussed above: streamlines in the physical space are concentrated on the barycenters γ t (X) of the sets P t (X) where X supp (α t ). Thus it is natural to expect continuity of F t relative to P t. Indeed, we have the following: Proposition 4.6. Assume that (34) hold. Then the map F defined in (32) satisfies lim ξ( P t F t (x)) F t (x)dx = ξ( P t F t (x)) F t (x)dx (37) t t for all t (, ) and all ξ C c (IR 3 ; IR 3 ). and Furthermore, lim ξ( P t F t (x)) F t (x)dx = t t for all t (, ) and all ξ C c (IR 3 ; IR 3 ). ξ( P t F t (x)) F t (x)dx, (38) lim ξ( P (x)) F t (x)dx = ξ( P (x)) x dx (39) t + Proof: The (unique) optimal transport plan between its X-marginal α t and it y-marginal χ is γ t = ( P t Id) # χ. Let t [, ). Since W 2 (α t, α t ) as t t (which we define as meaning t + if t = ), we infer that the second moments of γ t are uniformly bounded as t t. In fact, they converge to the second moment of γ t. Then, by Remark [4], we conclude {γ t } t t is tight in P(IR 3 ), so out of any sequence t n t we extract a subsequence (not relabeled) which converges narrowly to some Π P(IR 3 ). Clearly, its marginals with respect to X and y are α t and χ, respectively. By lower semicontinuity, we deduce that Π must be optimal between α t and χ. By uniqueness of optimal maps, we see that Π = γ t. Therefore,

15 15 due to the convergence of the second moments, we infer that γ t converges to γ t in P 2 (IR 3 ). We use the definition (32) of F t, the fact that P # χ = α and Φ t# α = α t, and (1) to get ξ(φ t P (x)) F t (x)dx = ξ(x) yγ t (dx, dy). IR 3 IR 3 By the continuity of t γ t proved above, we deduce lim ξ(φ t P (x)) F t (x)dx = ξ(x) yγ t (dx, dy) = t t IR 3 ξ(φ t P (x)) F t (x)dx. (4) Next we note that (Φ t P ) # χ = α t. Combining this with (36), we obtain for every t (, T ): P t γ t Φ t P (x) = Φ t P (x) for a.e. x. Then using (32) we obtain for every t (, T ) Now (4) implies (37). Furthermore, (3) implies This yields P t F t (x) = Φ t P (x) for a.e. x. (41) t Φ t (X) = Φ t (X) + U(s, Φ s (X))ds for α a.e. X IR 3 and all t t. t ξ(φ t (X)) ξ(φ t (X)) 2 ξ 2 ( t (t t ) ξ 2 t U(s, Φ s (X)) ds t ) 2 t U(s, Φ s (X)) 2 ds for α -a.e. X IR 3. Note that (, T ) IR 3 (t, X) U(t, Φ t (X)) is Borel (as composition of Borel maps). Next we use P # χ = α and Φ s# α = α s to get ξ Φ t P ξ Φ t P L 2 (;IR 3 ) ( t ) 1/2 t t ξ U(s, Y ) 2 dα s (Y )ds. IR 3 Then, using (41) we obtain ξ P t F t ξ P t F t L 2 (;IR 3 ) ( t ) 1/2 t t ξ U(s, Y ) 2 dα s (Y )ds. IR 3 Since U t L 2 (α t;ir 3 ) L (, ) and F is bounded uniformly in time-space, we use (37) and the inequality displayed above to deduce (38). For t = we use (11) and the fact that Φ (X) = X for α a.e. X. QED. t t Definition 4.7. Let P : [, ) IR be such that P C([, ); H 1 ()) and P (t, ) is convex in for each t [, ). A map F : [, ) is called weakly P continuous if (38), (39) hold for all t (, ) and all ξ C c (IR 3 ; IR 3 ).

16 16 Furthermore, we note that considering the example of initial data in the dual space being a linear combination of Dirac masses (see also Proposition 4.1 below), the solution α t at each time t is also a linear combination of Dirac masses concentrated in the time-dependent location, it is not clear whether distinct initial location of Dirac masses should imply that their locations are distinct at all times. Thus we cannot expect existence of the map F as in Definition 1.1(iii). In light of the above, we are ready to generalize Definition 1.1 as follows: Definition 4.8. Let P : [, ) IR be such that P C([, ); H 1 ()) and P (t, ) is convex in for each t [, ). Let F : [, ) be a weakly P -continuous Borel map. Denote by γ t the barycentric projection of the measure γ t := ( P t Id ) χ, defined in (1), and # set α t := P t# χ, µ t := γ t# α t. (42) Then the pair (P, F ) is called a weak Lagrangian solution of (1) in [, T ) if i. F (, x) = x for µ -a.e. x, P (, x) = P (x) for a.e. x, ii. for any t the mapping F t = F (t, ) : satisfies F t# χ = µ t and F t# µ = µ t ; iii. The function Z : (, T ) IR 3 defined by Z t = P t F t (43) lies, along with F, in L (, T ; L 2 (; IR 3 )) and is a distributional solution of (7) in the sense of (8). The following proposition gives sufficient conditions for the existence of weak Lagrangian solutions in physical space. Note that these conditions are exactly what was proved in [8] in the case α t L q (IR 3 ). Proposition 4.9. Assume that (34) hold. Define F by (32). Then the pair (P, F ) is a weak Lagrangian solution in physical space in the sense of Definition 4.8, where F is given by (32). Furthermore, if there exists a Borel map Φ : [, T ] IR 3 IR 3 such that Φ # α t = α for all t [, T ), then there exists a Borel mapping F : [, T ] satisfying F t# µ t = µ and F t F t = Id µ -a.e., and F t F t = Id µ t -a.e. (44) Proof: Proposition 4.6 implies that F is weakly P -continuous. From (36) and since P t# χ = α t, we have: P t γ t P t (x) = P t (x) for every t (, T ) and χ a.e. x. (45) Now, in order to prove (i) of Definition 4.8 for F, we find that γ (Φ P γ ) P (x) = γ P (x) for χ-a.e. x, due to (45) for t = and the hypothesis on Φ. Since µ = γ P # χ, then (i) follows. To prove (ii) we check that F t# χ = ( γ t Φ t P ) # χ = ( γ t Φ t ) # α = γ t# α t = µ t,

17 17 where we used the definitions (42), and that Φ t# α = α t. Similarly, F t# µ = ( γ t Φ t P ) # µ = ( γ t Φ t P γ ) # α = ( γ t Φ t ) # α = γ t# α t = µ t, where we used in the second line the property (36) for t =. Next we prove (iii) of Definition 4.8. We first note that Z and F are Borel (therefore, Lebesgue) measurable as compositions of Borel maps (see Lemma 4.2). By the definition of γ, we have ( ) 1/2 ( ) 1/2 ξ(x) γ t (X)dα t (X) ξ(x) IR IR 2 dγ t (X, y) y 2 dγ t (X, y) 3 3 ( 1/2 = ξ L 2 (α t;ir 3 ) y dy) 2. It follows that for all t (, T ) we have γ t L 2 (α t ; IR 3 ) with γ t L 2 (α t;ir 3 ) R (L 3 ()) 1/2, (46) where < R < is large enough such that B(, R ). Thus, there exists C IR independent of t such that F t (x) 2 dx = γ t Φ t 2 dα = IR 3 γ t 2 dα t = γ t 2 IR 3 L 2 (α t;ir 3 ) C. Also, using that F t# χ = µ t as proved above, and also using (36), we have for Z t = P t F t : Z t (x) 2 dx = P t (y) 2 dµ t (y) = P t ( γ t (X)) 2 dα t (X) = X 2 dα t (X) = P t (x) 2 dx C, IR 3 where C < is a constant coming from the fact that P C([, T ]; H 1 ()). Thus, both Z and F belong to L (, T ; L 2 (; IR 3 )). Next we note that assumption (ii) of (34) implies that for for χ-a.e. x, the function Φ(, P (x)) is a weak solution of the problem Φ(t, P (x)) = J[Φ(t, P (x)) γ t (Φ(t, P (x)))] with Φ(, P (x)) = P (x). (47) From (41), for each t [, T ) the equality Z t (x) = Φ t P holds for a.e. x. Thus using the integrability of Z and F proved above, using a function ϕ C 1 ([, T ) IR3 ) in the weak form of (47) and integrating with respect to x, we get (8). Finally, in order to prove (44) under the additional assumption of the existence of Φ, we set and compute F t (x) := γ Φ t P t F t# µ t = ( γ Φ t P t γ t ) # α t = ( γ Φ t ) # α t = γ # α = µ. (48)

18 18 To finish proving (44), note that F t F t (y) = y for µ -a.e. y amounts to for χ-a.e. x. Since F t F t γ P (x) = γ P (x) (49) (F t F t γ )( P (x)) = ( γ Φ t P t γ t Φ t P γ )( P (x)) and P t γ t, P γ, Φ t Φ t are all equal to the identity on the corresponding domains, we deduce (49). QED. In the case of discrete measures we can prove that the construction works. Even though not explicitly present in [3], a simple argument inserted in the proof of Theorem 7.4 from said reference yields that the solutions to the Hamiltonian ODE constructed there are convex combinations of point masses provided that the initial measures are of the same form (coefficients of the convex combinations are time-invariant). Our Hamiltonian in (2) satisfies all the requirements for Theorem 7.4 [3] to apply (see Lemma 7.6 [3]). Proposition 4.1. Let x = (x 1,..., xn ) IR3n be arbitrary for some integer n 1, and let µ = n c i δ x i (5) i=1 be a convex combination of the Diracs at these points, i.e. nonnegative constants c i satisfy n i=1 c i = 1. Then the solution constructed in Theorem 7.4 [3] for the initial-value problem associated to the Hamiltonian ODE as in Definition 2.1 for the Hamiltonian in (2) is of the form n α t = c i δ x i (t), (51) where [, T ] t x(t) is in W 1, (, T ; IR 3n ) and x() = x. i=1 Proof: Let m be a positive integer and set h = T/m. Then take w m := J H(ᾱ), where H(α) denotes the element of H(α) with least L 2 (α; IR 3 )-norm, and set, for all t [, h], α m t = (Id + tw m ) # ᾱ, ν m t = (Id + tw m ) # (w m ᾱ) and w m t := dνm t dα m t, where we used the fact (see Lemma 7.1 [3]) that νt m αt m to get the Radon-Nykodim derivative wt m. On the next time subinterval [h, 2h] one defines αt m and wt m similarly by using αh m and wh m instead of ᾱ and wm, and t h instead of t. In general, the construction can be extended to [kh, (k + 1)h] for k =,..., m 1 by using αkh m and wm kh, t kh instead of t, and repeating the steps above. It is proved in [3] that the paths of measures t αt m are uniformly bounded in P 2 (IR 3 ) and uniformly Lipschitz continuous. For a subsequence m j we have a limiting t α t, which is shown to satisfy the Hamiltonian ODE with α = ᾱ. Since α m j t = [Id + (t kh)w m j kh ] #α m j kh for t [kh, (k + 1)h], we deduce that all probabilities α m j t are convex combinations of n Dirac masses if α m j kh is (with same coefficients). This is true for all k =,..., m j 1, so we deduce that

19 19 it holds for α m j t all t [, T ]. The uniform bounds on αt m mentioned above translate into the uniform L (, T ; IR 3n ) bounds on t x m j (t) = (x mj,1 (t),..., x mj,n (t)) (where x mj,i (t) are the points in the support of α m j t ). Furthermore, the uniform Lipschitz continuity of the paths t αt m in the Wasserstein space P 2 (IR 3 ) gives a finite constant C > for which W 2 (α m j t, α m j s ) C t s for all t, s [, T ]. Now fix t (, T ). Note that x m are piecewise linear and continuous in time, thus it is clear that C 2 t t 2 W2 2 (α m j t, α m j t ) = n c i x mj,i (t) x mj,i (t ) 2 for all t close enough to t. i=1 Thus, the vector functions t x m j (t) are uniformly Lipschitz. By Ascoli-Arzela s theorem, a subsequence converges in the sup norm to a function x W 1, (, T ; IR 3n ), which implies the limiting measures α t found above must coincide with the averages of the Dirac masses at x 1 (t),..., x n (t) for all t [, T ]. This follows from W2 2 (α m j t, β t ) n c i x mj,i (t) x i (t) 2 as j, i=1 where β t is the convex combination of the Dirac masses at x i (t) with coefficients c 1,..., c n. QED. So far we are not aware of any reason why distinct initial x k should give rise to distinct xk (t) at all later times. As a consequence, existence of the map Φ is uncertain (since transport maps from an average of n points masses to one of m point masses exist if and only if n m). Proposition Let n be a positive integer and α t := n c i δ x i (t), for t [, T ) i=1 be the solution of SG in dual space constructed in Proposition 4.1 with initial data α := n c i δ x i, i=1 where x i are arbitrary in IR3, i = 1,..., n, and c i are nonnegative and n i=1 c i = 1. Then the map Φ n : [, T ] IR 3 IR 3 given by Φ n (t, X) = X if X x i, and Φn (t, x i ) = xi (t) for t [, T ], i = 1,..., n satisfies (34), (ii) (iii). Proof: It is obvious that Φ n is Borel and Φ n satisfies Φ n t# α = α t. To show that Φ n solves (33) we start from the fact that α t solves the system (16), (17). In fact, the Hamiltonian (2) has the property that T α P 2 (IR 3 ) H(α) = { γ Id } (see, e.g. [3]), so α t solves (13) in the sense of distributions. This is equivalent to t ξ(x)dα t (X) is absolutely continuous IR 3 and for a.e. t (, T ) we have d ξ(x)dα t (X) = ξ(x) U(t, X)dα t (X) for all ξ Cc 1 (IR 3 ), dt IR 3 IR 3

20 2 for U given in (9). According to Proposition 4.1, we have that t x i (t) is in W 1, (, T ; IR 3 ) for all i = 1,..., n, which implies n c i ξ(x i (t)) ẋ i (t) = i=1 This leads to the desired conclusion. n c i ξ(x i (t)) U(t, x i (t)) for all ξ Cc 1 (IR 3 ). k=1 We are now in a position to formulate: QED. Corollary Let P be the maximum of finitely many affine functions: for some integer n 1 and a i IR 3, b i IR 1 for i = 1,..., n P (x) = max (a i x + b i ), for x. i=1,...,n Then there exists a weak Lagrangian solution for (1) in the sense of Definition 4.8 with P (, ) = P a.e. in. 5 Return to dual space and conservation of energy We show that weak Lagrangian solutions give rise to solutions in dual space. Theorem 5.1. Let (P, F ) be a weak Lagrangian solution of (1) in the sense of Definition 4.8 and set α t := P t# χ. Then α is a distributional solution of (13). Proof: We need to show that for every ϕ C ((, T ) IR3 ) T IR 3 ( t ϕ + J[X γ t (X)] ϕ)dα t (X)dt =. (52) By the density argument, it is sufficient to show that for Fix such ξ, ζ. ϕ(t, X) = ζ(t)ξ(x), for all ξ C (IR 3 ), ζ C (, T ). From (42), ( P t γ t ) # α t = P t# µ t. Then (36) yields P t# µ t = α t. Then F t# χ = µ t implies Z t# χ = (P t F t ) # χ = P t# µ t = α t, i.e. Now we calculate using (53): T ζ (t)ξ(z t (x))dxdt = Z t# χ = α t. (53) T IR 3 ζ (t)ξ(x)dα t dt. (54) On the other hand, we can show that integrating by parts in t and using the regularity of Z, F and equation (8), we get T T ζ (t)ξ(z t (x))dxdt = ζ(t) ξ(z t (x)) J[Z t (x) F t (x)]dxdt. (55)

21 21 Indeed, let η ε (t, x) be the family of standard mollifiers in time-space. We extend Z(t, x) to IR 1 IR 3 by defining it to be zero outside of (, T ), and define Z ε = η ε Z(t, x) on (, T ) IR 3, where the convolution is with respect to (t, x). Then Z ε C ((, T ) IR 3 ). Also, (8) implies that the distributional derivative t Z(t, x) in (, T ) IR 3 is J(Z F ) (extended by zero outside of [, T ] ), and the functions Z, Z F are in L 2 ((, T ) ; IR 3 ) by (iii) of Definition 4.8. Let [a, b] (, T ) be such that supp (ζ) [a, b]. Then for sufficiently small ε, t Z ε = J(Z F ) η ε (t, x) in (a, b) IR 3, and (Z ε, t Z ε ) (Z, J(Z F )) in L 2 ((a, b) ; IR 3 ) as ε. Then, integrating by parts to get T T ζ (t)ξ(zt ε (x))dxdt = ζ(t) ξ(zt ε (x)) t Zt ε (x)dxdt, (56) and using that ξ, ξ, D 2 ξ are bounded, we get that the left and right hand sides of the above equality converge to the left and right hand sides of (55), respectively. Indeed, denoting by R ε and R the right-hand sides of (56) and (55) respectively, we have b ( R R ε D 2 ξ L (IR 3 ) Zε t (x) Z t (x) Z t (x) F t (x) a ) + Dξ L (IR 3 ) tzt ε (x) J[Z t (x) F t (x)] dxdt ( C Z t (x) F t (x) L 2 ((,T ) ) Zt ε Z t L 2 ((a,b) ) ) + t Zt ε (x) J[Z t (x) F t (x)] L 2 ((a,b) ). Convergence of the left-hand sides is proved similarly. This shows (55). Next, (36) implies ( γ t P t γ t )(X) = γ t (X) for every t (, T ) and α(t, )-a.e. X IR 3. Using γ t# α t = µ t, we get In view of F t #χ = µ t, we have γ t P t (x) = x for every t (, T ) and µ t -a.e. x. γ t Z t (x) = γ t P t F t (x) = F t (x) for every t (, T ) and χ-a.e. x. Then we can rewrite (55) as T T ζ (t)ξ(z t (x))dxdt = ζ(t) ξ(z t (x)) J[Z t (x) γ t Z t (x)]dxdt, and using (53) to change variables in the right-hand side, we get T T ζ (t)ξ(z t (x))dxdt = ζ(t) ξ(x) J[X γ t (X)]dα t (X)dt, IR 3 Combining with (54), we get T IR 3 ζ (t)ξ(x)dα t dt = T IR 3 ζ(t) ξ(x) J[X γ t (X)]dα t (X)dt

22 22 which is (52) in the case ϕ(t, X) = ζ(t)ξ(x). QED. We finish with an observation concerning energy conservation along weak Lagrangian solutions of (1). Corollary 5.2. Let (P, F ) be a weak Lagrangian solution of (1) in the sense of Definition 4.8. Then the function [, T ) t y Pt (y) 2 dy is constant. (57) Proof: Due to (46), we infer that the dual-space velocity U(t, X) = X γ t (X) satisfies But ( ) 1/2 U(t, ) L 2 (α t;ir 3 ) R [L 3 ()] 1/2 + X 2 dα t (X). IR 3 X 2 dα t (X) = P t 2 for all t [, T ), IR 3 L 2 (;IR 3 ) which, since P C([, ); H 1 ()), implies the local boundedness in time of the L 2 (α t ; IR 3 ) norm of the velocity U(t, ), boundedness required by Theorem 5.2 in [3]. Furthermore, it follows that the path t α t lies in AC 2 (, T ; P 2 (IR 3 )), see e.g. [3, page 24]. Thus, in light of Theorem 5.1, we can apply Theorem 5.2 in [3] to conclude. QED. Acknowledgements The authors would like to thank M. Cullen for his valuable suggestions and comments. The work of Mikhail Feldman was supported in part by the National Science Foundation under Grants DMS-8245, DMS-11126, and the Vilas Award by the University of Wisconsin- Madison. Adrian Tudorascu gratefully acknowledges the support provided by the Department of Mathematics of West Virginia University. References [1] L. Ambrosio. Transport equation and Cauchy problem for BV vector fields. Invent. Math., 158 (24), [2] L. Ambrosio, M. Colombo, G. De Philippis, A. Figalli. Existence of Eulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic case. Preprint arxiv: [3] L. Ambrosio, W. Gangbo. Hamiltonian ODE in the Wasserstein spaces of probability measures. Comm. Pure Appl. Math. 61 (28), [4] L. Ambrosio, N. Gigli and G. Savaré. Gradient flows in metric spaces and the Wasserstein spaces of probability measures. Lectures in Mathematics, ETH Zurich, Birkhäuser, 25. [5] J.-D. Benamou, Y. Brenier. Weak existence for the Semi-Geostrophic equations formulated as a coupled Monge-Ampere/transport problem. SIAM J. Appl. Math., 58 (1998), no. 5,