Gradient flow of the Chapman-Rubinstein-Schatzman model for signed vortices

Transcription

1 Graient flow of the Chapman-Rubinstein-Schatzman moel for signe vortices Luigi Ambrosio, Eoaro Mainini an Sylvia Serfaty Deicate to the memory of Michelle Schatzman ( ) Abstract We continue the stuy of [AS] on the Chapman-Rubinstein-Schatzman-E evolution moel for superconuctivity, viewe as a graient flow on the space of measures equippe with the quaratic Wasserstein structure. In [AS] we consiere the case of positive (probability) measures, while here we consier general real measures, as in the physical moel. Unerstaning the evolution as a graient flow in this context gives rise to several new questions, in particular how to efine a Wasserstein istance for signe measures. We generalize the minimizing movement scheme of [AGS] in this context, we show the entropy argument of [AS] still carries through, an erive an evolution equation for the measure which contains an error term compare to the Chapman-Rubinstein-Schatzman-E moel. Moreover, we also show the same applies to a very similar issipative moel on the whole plane. 1 Introuction In [CRS], Chapman, Rubinstein an Schatzman (see also E [E]) erive formally the following mean fiel moel for the evolution of the ensity of vortices in a type-ii superconuctor uner the effect of an external magnetic fiel, in the limit where the Ginzburg- Lanau parameter κ tens to + an the number of vortices becomes large: t µ(t) iv( h µ(t) µ(t) ) = 0 in (0, + ) (1.1) µ(0) = µ 0 at t = 0 where h µ is given by { hµ + h µ = µ in h µ = 1 on. (1.2) 1

2 Type-II superconuctors, submitte to an external fiel, have a very particular response: they repel the applie fiel, which only penetrates through vortices. In the above, is the two-imensional omain occupie by the superconucting sample, µ is a signe measure representing the vortex ensity (vortices are punctual objects which carry a quantize topological egree, which can be positive or negative) an h µ is the magnetic fiel inuce in the sample. The bounary conition h µ = 1 correspons to the effect of an external magnetic whose intensity is here normalize to 1. h µ can be viewe as a potential generate by the vortices through the relation (1.2). In [AS] we stuie the problem in the case where µ is a positive measure, which one can normalize to be a probability measure on. Here we examine the signe measure case. More precisely we look for a solution µ(t) to the continuity equation t µ(t) iv(χ h µ(t) µ(t) ) = 0 in D ((0, + ) R 2 ) (1.3) with the initial atum µ(0) = µ 0 in M() H 1 (), where M() enotes the space of boune Raon measures on. Let us recall the efinition of the well-known quaratic Wasserstein istance between two probability measures µ an ν on R n : W 2 (µ, ν) = ( ) 1 inf x y 2 2 γ(x, y) γ Γ(µ,ν) R n R n (1.4) where Γ(µ, ν) enote the set of probability measures on R n R n which have marginals µ an ν. The key point in [AS], was to view (1.1) (1.2) as the graient flow of the energy functional (relate to the stanar Ginzburg-Lanau functional, see [SS1, SS2]) Φ λ (µ) = λ 2 µ () + 1 h µ 2 + h µ 1 2 λ 0, (1.5) 2 for the above quaratic Wasserstein W 2 structure on the space of probability measures on, an then to apply the framework of [AGS] (inspire from the seminal papers [JKO, O]) for constructing graient flows in the Wasserstein spaces, which consists in minimizing recursively ν Φ λ (ν) + W 2 2 (µ k, ν), (1.6) 2τ an then passing to the limit as τ 0. This specific problem pose several ifficulties: the natural energy space was P () H 1 () (where P enotes probability measures) an not the space of absolutely continuous measures; for no α R the energy functional (1.5) is α-isplacement convex in that space; 2

3 the case of measures on a boune omain with the possibility of mass entering or exiting the omain is nonstanar. The results of [AS] can be summarize as follows: if the initial atum is in L then there is existence for a strong formulation of the equation, an uniqueness until some mass reaches the bounary; for general finite energy initial ata, there is existence (but in general no uniqueness) of solutions to the equation in a weak sense, obtaine as the limit of a time-iscrete minimizing movement scheme, an satisfying an energy-issipation relation; there exists a family of entropy functionals which ecrease along the flow, an ensure that if the initial atum is in L p, the solution remains in L p for all time; in aition, in [M1] a global uniqueness result is prove in the case of a convex omain. The ifficulty is the potential presence of mass on, an this result is obtaine through a very precise formulation of bounary conitions, an issue that we are not going to aress in this paper. Here we woul like to pursue the same strategy of viewing (1.1) as a graient flow, but on the space of signe measures. Note that while there have been numerous stuies of PDE s viewe as graient flows on the Wasserstein spaces of probability measures (most of the time for absolutely continuous measures an α-isplacement convex functionals), see [AGS, Vi1, Vi2] an the references therein, there has been absolutely no such stuy in the case of signe measures. This is an open fiel which we believe to be natural since physical moels, such as this one, sometimes also involve signe or charge ensities. As we shall see, our stuy raises as many open questions as it solves. The first question that arises is to efine an analogue of the Wasserstein istance on signe measures (which have equal integrals). While this is obvious in the case of the 1-Wasserstein (or Kantorovich-Rubinstein) istance, it turns out to be really nontrivial for exponents p > 1, as we shall see in Section 2. The first naive attempt one can make is to efine the istance between the signe measures µ an ν such that µ() = ν() by W 2 (µ, ν) := W 2 (µ + + ν, ν + + µ ), where µ +, ν + (resp. µ, ν ) are the positive (resp. negative) parts of the measures µ an ν. (Here, by assumption, µ + + ν an µ + ν + are two positive measures of same mass, one can easily exten the efinition of the stanar W 2 istance to that case.) We will stuy the properties of W 2 in Section 2. It turns out that this efinition has two major flaws: the istance efine this way is not lower semi-continuous, an examples show that the triangle inequality can be violate! One can then think of several ways to fix these problems, obtaine by relaxing the efinition in various ways, which will make the 3

4 istance lower semi-continuous. However, it is still not at all obvious that the triangle inequality hols. So in the en, we postpone the efinition of a canonical 2-Wasserstein istance on signe measures, which we believe to be a problem of inepenent interest, to future work. In the meantime, an expansion of this iscussion incluing several possible concurrent efinitions is escribe in the proceeings paper [M2]. For our purposes of applying the minimizing movement scheme (1.6), it suffices to buil a pseuo-istance which is lower semi-continuous an boune from below by a istance. One of the avantages of the Wasserstein variational approach is the possibility of hanling measure-value solutions an nonsmooth velocity fiels. Here, even thinking of a milly regular µ(t), we o not have regularity of the velocity in (1.3), which is always multiplie by the sign of µ(t). This prevents the application of the DiPerna-Lions theory [DPL] of flows associate to weakly ifferentiable vector fiels. We then procee similarly as in [AS]. The weak formulation of [AS] cannot be use for signe measures, since it uses Delort s convergence theorem [De] which hols only for positive measures. So we have to assume L p (p 4) integrability of the initial ata, an it turns out that the entropy argument of [AS] still carries through (although in a not completely obvious way), an ensures that L p integrability is preserve along the iscrete flow. The Euler-Lagrange equation for iscrete minimizers µ τ can be erive as in [AS]. However, taking the limit as the timestep τ 0, one is confronte with two ifficulties: first, there is no stanar limiting velocity results as in the positive case for writing own a continuity equation. However, we can still pass to the limit by han, without using the general theory of [AGS]. Secon an more importantly, the positive an negative parts µ + τ an µ τ of the iscrete minimizer have weak limits µ + an µ, but these weak limits are not necessarily mutually singular, so it is not clear that µ τ converge weakly to µ + +µ. This kin of strong convergence of the scheme is an open problem of inepenent interest, that we hope to aress in a future paper: at least in principle, one can hope that strong compactness properties hol for the transport-cancellation mechanism, when goo bouns on the velocity (as in our case) are present, so that cancellation is encoe only in the iscrete scheme, an oes not happen in the limit (see also the aitional remarks at the en of the introuction). We also emphasize that the role of the energy term λ µ /2 in (1.5) is not completely clear: while in [AS] it was a null-lagrangian, thought to have an influence only on the rate of mass issipation through the bounary, here it probably has an influence also in cancellations in the iscrete scheme. Because of potential cancellations in the limit, we obtain an evolution equation with a limit term ϱ which is really the limit of µ + + µ above an coul be thought as a kin of efect measure: Theorem 1.1 Let µ 0 L 4 (). The minimizing movement scheme prouces a signe measure µ(t) L 4 () which satisfies µ(0) = µ 0 an t µ(t) iv (χ h µ(t) ϱ(t)) = 0 in D ((0, + ) R 2 ), (1.7) 4

5 where ϱ(t) is a suitable positive measure satisfying ϱ(t) µ(t) in. We can check however that when µ(0) 0 the measure ϱ(t) is equal to µ(t), so we retrieve at least the result of [AS], an we conjecture for the reasons explaine before that the scheme can be improve to obtain ϱ(t) = µ(t) for all t 0. Actually our methos yiel more, namely the system of PDE s t ϱ+ (t) iv (χ h µ(t) ϱ + (t)) = σ(t) t ϱ (t) + iv (χ h µ(t) ϱ (t)) = σ(t). (1.8) This system, where ϱ ± nee not be the positive an negative part of a signe measure ϱ, has an interesting structure in its own right, the coupling being ue to the negative term σ(t) 0 an in the velocity fiel, since µ(t) = ϱ + (t) ϱ (t). At this level, it woul be nice to unerstan uner which assumptions the system preserves orthogonality of ϱ + an ϱ in time. Let us finally turn to the case of the whole plane an comment on relate results in the literature. Moels very similar to (1.1) were previously stuie (see references in [AS], like [DZ, LZ, MZ]). In particular Lin-Zhang [LZ], Masmoui-Zhang [MZ] stuie the equation t µ(t) + iv ( 1 µ(t) µ(t) ) = 0 (1.9) in the whole R 2, which can be viewe as a issipative version of the Euler equation in vorticity formulation. Lin-Zhang focuse on the positive measure case, an Masmoui-Zhang on the signe case. They o not use the graient flow approach, but fin solutions by passing to the limit in some approximating PDEs. In [LZ] results analogous to those we escribe from [AS] were proven. In [MZ] they construct solutions to the equation (1.9) but assuming some W 1,p regularity of the initial measure which is use crucially an ensures goo compactness properties (so, in this case the transport-cancellation mechanism has goo compactness properties). Thus existence of solutions in the general measure case, or in the L p case, is still open. In Section 6, we stuy (1.9) in the whole plane, an exten our graient-flow approach to that case. This poses a slight ifficulty since the obvious energy functional that shoul replace Φ λ, which is 1 h 2 R 2 µ 2, is in general infinite because of the logarithmic behavior of the Green s kernel in imension 2. To remey this, we introuce a renormalize way of computing the energy. We then show that the entropy argument of [AS] can still aapte to that energy, so that all the results of [AS] an those of the present paper are vali for that infinite-plane moel as well. In the case of positive measures, this retrieves solutions of (1.9). 5

6 The paper is organize as follows: in Section 2 we investigate potential istances on signe measures, give counter-examples for W 2 an present several alternative costs that we use later. In Section 3 we present the time-stepping iscrete minimization (or minimizing movement ) scheme an erive the Euler-Lagrange equation satisfie by iscrete minimizers. In Section 4 we prove the same entropy result as in [AS], in the signe case, ensuring that L p regularity is preserve along the iscrete flow, an that the prouct µ h µ makes sense. In Section 5 we pass to the limit in the iscrete minimizers, an obtain the limit evolution equation. In Section 6 we examine the moel (1.9), introuce the renormalize formulation of the energy, an iscuss how to aapt our methos to that case. Acknowlegments: L. Ambrosio an E. Mainini are supporte by a MIUR PRIN2006 grant. S. Serfaty is supporte by an EURYI awar of the European Science Founation. 2 Transport cost for signe measures When trying to generalize the theory of Wasserstein graient flows to signe measures, as we mentione a first ifficulty arises at the theoretical point of view: there is no stanar efinition of p-wasserstein istance on signe measures. Moreover, we o not know how to rephrase the characterization of absolutely continuous curves in the space of measures by means of continuity equations given in [AGS]. On the other han, we can take avantage of the flexibility of the minimizing movements approach. Inee, the minimization problem min φ(ν) + 1 ν X 2τ 2 (ν, µ), µ X (2.1) makes sense in any metric space X, being the corresponing istance, where φ : X R. On top of that, it is not strictly neee for the functional appearing in (2.1) to be a istance. In fact, often the important thing is its behavior on small scales, when ν µ. As in the seminal paper [ATW], one coul also use a non triangular or non symmetric object. Actually, we are going to make use of a functional which, though not a istance, is boune from below by a istance. We begin with the efinition of the ambient space. Let M() enote the set of boune Raon measures over. We enow M() with the stanar weak (or narrow) convergence, given by the uality with continuous an boune functions. Let us efine the following measure subset of M(). M κ, M () := {µ M() : µ() = κ, µ () M}, (2.2) 6

7 where κ R. In the sequel we will often make use of the Hahn ecomposition for a real measure µ, ientifying its positive an negative parts, so that µ = µ + µ, where µ + an µ are two positive measures. This ecomposition is minimal in the sense that, for any other pair of positive measures σ 1, σ 2 such that σ 1 σ 2 = µ, there hol µ + σ 1 an µ σ 2. Moreover, if µ is a positive measure, we will say that µ 0 is a submeasure of µ if µ 0 (A) µ(a) for any µ-measurable set A. A way which seems at first glance natural for efining a 2-Wasserstein istance in M κ, M () is the following. 2.1 First cost Let µ, ν M κ,m (). Define W 2 (µ, ν) := W 2 (µ + + ν, ν + + µ ), (2.3) where W 2 is as in (1.4) (but naturally extene from probability measures to nonnegative measures with a fixe total mass, possibly ifferent from 1). It is immeiate to check that, if µ an ν are nonnegative, W 2 reuces to the Wasserstein istance between positive measures of a given mass κ on. The functional W 2 accounts for the cost of transporting signe measures, an some heuristics on its behavior are worthy. We notice that, when transporting a signe measure µ, its positive an negative masses may change (only µ is fixe, as in (2.2)). So, in orer to connect µ to ν, it may be convenient to transport some part of µ + onto µ, this correspon to auto-annihilation of mass. On the other han, if the total variation of ν is larger than that of µ, one expects that, in the transport given by W 2, a nonzero part will come from moving some part of ν to ν +. From the ynamic point of view, this correspons to some fake zero charge mass which is create an separate into positive an negative mass, while being transporte at a certain cost. Remark 2.1 This framework fits the physical problem we are investigating, since we expect that vortices with opposite egrees can interact like ipoles an cancel each other. Also it is in principle possible that ipoles be create ex-nihilo. Although it is immeiate to verify that W 2 is symmetric an vanishes if an only if µ = ν, W 2 is not a istance. Inee, the following example shows that the triangle inequality fails. On the real line, let µ = δ 0, ν = δ 4 an η = δ 1 δ 2 + δ 3. Clearly W 2 (µ, ν) = W 2 (µ, ν) = 4. But the optimal transport plan between µ + + η an η + + µ is given by δ 0 δ 1 + δ 2 δ 3, so that W 2 (µ, η) = x y 2 (δ 0 δ 1 ) + x y 2 (δ 2 δ 3 ) = 2. R R 7

8 Symmetrically, W 2 (ν, η) = 2, so that W 2 (µ, ν) > W 2 (µ, η) + W 2 (ν, η). On the other han, we notice that if γ Γ 0 (µ + + ν, ν + + µ ) where Γ 0 enotes the set of optimal transport plans, by Höler inequality we have ( 1/2 1 x y γ) 2 x y γ, (2.4) 2M but W 1 (µ, ν) := W 1 (µ + + ν, ν + + µ ) = inf x y γ (2.5) γ Γ(µ + +ν, ν + +µ ) is inee a istance between signe measures. This can be seen by the well-known Kantorovich uality formula, that gives (see for example [Vi1]) W 1 (µ + + ν, ν + + µ ) = sup ϕ Lip(), ϕ Lip 1 ϕ (µ ν). (2.6) Clearly, by looking at the right han sie, we have a istance. Notice in aition that W 1 is not sensitive to the aition of equal masses in the source an in the target (a feature typical of 1-istances), since (2.6) reaily gives W 1 (µ, ν) = W 1 (µ + σ, ν + σ), σ M(R 2 ). (2.7) It is worth analyzing some other features of W 2. In the next proposition we see that W 2 metrizes the weak topology of M κ, M (). Proposition 2.2 Let µ n, µ belong to M κ, M (). Then µ n µ if an only if W 2 (µ n, µ) 0. Proof. Assume that µ n µ. Since µ + n () M an µ n () M, by tightness there exists a subsequence (n k ) such that µ + n k σ + an µ n k σ, with σ + σ = µ. By continuity of the Wasserstein istance, for each limit point we have W 2 (µ + n + µ, µ + + µ n ) W 2 (σ + + µ, µ + + σ ) = 0. Assume that W 2 (µ n, µ) 0, that is W 2 (µ + n + µ, µ + + µ n ) 0. Since W 2 metrizes the weak convergence, there exists a positive measure ϑ such that µ + n + µ ϑ an µ n + µ + ϑ, hence µ + n µ n µ + µ = µ. We have seen that W 2 is not a istance. With a similar simple construction, it is possible to see that the map ν W 2 (ν, µ) is not weakly l.s.c. in M κ, M (). For instance, let µ = δ 1 δ 1 an ν n = δ 1/n δ 1/n, so that ν n ν = 0. Clearly W 2 (ν + +µ, µ + +ν ) = W 2 (µ, µ + ) = 2. But, as n, lim inf W 2 (ν + n + µ, µ + + ν n ) = lim inf 2(n 1)/n = 8

9 2. The point is that if νn ν in M κ, M (), then (ν + n ) an (ν n ) are tight, but the limits are not in general ν + an ν (in the example above, they are not zero). In orer to overcome this problem, we can consier a kin of relaxation of W 2. More etails are given in the proceeings paper [M2]. In orer to eal with optimal transport plans between signe measures, consier partitions of the positive an negative parts of ν an µ of the form µ µ + 1 = µ +, µ 0 + µ 1 = µ, ν ν + 1 = ν +, ν 0 + ν 1 = ν, (2.8) where all the terms are positive measures. Some compatibility conitions have to be taken into account, an precisely ν + 0 () = µ + 0 (), µ 0 () = ν 0 (), µ 1 () = µ + 1 (), ν + 1 () = ν 1 (). (2.9) µ + 0 an µ 0 correspon to the parts that will move to ν + 0, ν 0 respectively an µ + 1, µ 1 (resp. ν + 1, ν 1 ) to the self-cancelling parts. Of course there are many partitions of this kin. Moreover, we have the following Lemma 2.3 (Splitting of the optimal plan) Let γ Γ 0 (ν + + µ, µ + + ν ). Then there exists a partition of the form (2.8)-(2.9) such that γ can be written as the sum of four plans γ + +, γ, γ +, γ + satisfying γ + + Γ 0 (ν + 0, µ + 0 ), γ Γ 0 (µ 0, ν 0 ), γ + Γ 0 (µ 1, µ + 1 ), γ + Γ 0 (ν + 1, ν 1 ). (2.10) Proof. Let ϑ 1 = ν + + µ an ϑ 2 = µ + + ν. It is clear that ν + an µ are both absolutely continuous with respect to ϑ 1. Let f 1, g 1 L 1 (R 2, ϑ 1 ) enote the respective ensities. Similarly, let f 2, g 2 be the ensities of ν an µ + with respect to ϑ 2, so that ν + = f 1 ϑ 1, µ = g 1 ϑ 1, µ + = g 2 ϑ 2, ν = f 2 ϑ 2. Clearly f 1 + g 1 = f 2 + g 2 = 1, so that we can write γ = (f 1 π 1 )(g 2 π 2 )γ + (f 1 π 1 )(f 2 π 2 )γ + (g 1 π 1 )(g 2 π 2 )γ + (g 1 π 1 )(f 2 π 2 )γ. (2.11) Notice that π 1 # ((f 1 π 1 )(g 2 π 2 )γ) = f 1 π 1 # ((g 2 π 2 )γ) f 1 π 1 # γ = f 1ϑ 1 = ν +, Moreover, π 2 # ((f 1 π 1 )(g 2 π 2 )γ) = g 2 π 2 # ((f 1 π 1 )γ) g 2 π 2 # γ = g 2ϑ 2 = µ +. π 1 #((f 1 π 1 )(g 2 π 2 )γ)+π 1 #((f 1 π 1 )(f 2 π 2 )γ) = f 1 π 1 #((g 2 π 2 +f 2 π 2 )γ) = f 1 π 1 #γ = ν +. With the analogous computations for the other terms in the right han sie of (2.11), we see that the marginals of the four plans therein are submeasures of ν +, µ, µ +, ν satisfying (2.8)-(2.9). Hence, in (2.11) γ is written as the sum of four plans on a partition of the esire form. Moreover, each of these plans is optimal, since their sum is. 9

10 2.2 Secon cost In orer to eal with a sequence of measures with ecreasing total mass, we introuce the following simplifie version of W 2. Let µ, ν M κ, M () with ν () µ (). Define W2(ν, 2 µ) = { ( inf lim inf W 2 ν n ν 2 (ν n +, µ + ) + W2 2 (νn, µ ) )}, n ν n + ()=µ+ () (2.12) ν n ()=µ () This way, we see that we may cancel mass only between µ + an µ. This will correspon to the fact that in the evolution we allow for mass cancellation, but not for mass creation. By its very efinition, the map ν W2(ν, 2 µ) is lower semicontinuous. Moreover, since any weak limit point of ν n +, νn is a couple σ +, σ satisfying σ + σ = ν, W2(ν, 2 µ) can also be written as inf σ + σ = ν σ + ()=µ + () σ ()=µ () { W 2 2 (σ +, µ + ) + W 2 2 (σ, µ ) }. (2.13) Tightness an semicontinuity of the stanar Wasserstein istance show that there exists an optimal couple ϑ +, ϑ such that W 2 2(ν, µ) = W 2 2 (ϑ +, µ + ) + W 2 2 (ϑ, µ ), (2.14) where ϑ + ϑ = ν. W 2 is not symmetric. But symmetry is not a key point, since we are going to compute the costs corresponing to subsequent timesteps: an evolution problem has a natural time irection. To connect this efinition with the previous ones, we can easily show the following Proposition 2.4 Let µ, ν M κ, M () an ν () µ (). Then 1 W 2 (ν, µ) 2M W 1(µ, ν). (2.15) Proof. Let ϑ 1, ϑ 2 be the optimal couple for W 2, so that the infimum in (2.13) is attaine. Let γ 1 Γ 0 (µ +, ϑ + ), γ 2 Γ 0 (µ, ϑ ). Then (γ 2 ) 1 Γ 0 (ϑ, µ ) an γ 1 + (γ 2 ) 1 Γ(µ + + ϑ, ϑ + + µ ). Hence W2(µ, 2 ν) = W2 2 (µ +, ϑ + ) + W2 2 (µ, ϑ ) = x y 2 (γ 1 + γ 2 )(x, y) = x y 2 (γ 1 + (γ 2 ) 1 )(x, y) Exploiting (2.4) an (2.7) we get the thesis. 10 W 2 2 (µ + + ϑ, ϑ + + µ ).

11 Notation 2.5 We let ϑ 1 enote the common part of ϑ + an ϑ, so that ϑ + = ν + + ϑ 1 an ϑ = ν + ϑ 1. Moreover, we let γ + Γ 0 (ϑ +, µ + ) an γ Γ 0 (ϑ, µ ) be the two optimal transport plans corresponing to W 2. Thanks to (a simplifie version of) Lemma 2.3, we can write these plans as γ + = γ γ 1 + an γ = γ0 + γ1 where γ 0 + Γ 0 (ν +, µ + 0 ), γ 1 + Γ 0 (ϑ 1, µ + 1 ), γ0 Γ 0 (ν, µ 0 ), γ1 Γ 0 (ϑ 1, µ 1 ), (2.16) an µ µ + 1 = µ + an µ 0 + µ 1 = µ. 3 Fine characterization of iscrete minimizers The functional we are going to analyze is (1.5), efine on signe measures. Notice that Φ 0 is weakly lower semicontinuous, as shown in [AS]. As a consequence, the full Φ λ is still lower semicontinuous, since µ µ () is. Moreover, Proposition 2.1 of [AS] works in the same way also in this case, giving the representation formula Φ λ (µ) = 1 (λ µ () + ) + 2 sup h 1 H 1 0 () { (h 1) µ 1 2 h 2 + h 2 }, (3.1) the supremum being attaine at h = h µ. By means of (3.1), we euce some stanar inequalities, as iscusse in [AS]: for any couple of real measures µ, ν there hols Φ λ (µ) Φ λ (ν) λ 2 µ () λ 2 ν () + (h ν 1) (µ ν). (3.2) On the other han, Φ 0 (µ) Φ 0 (ν) = ν() µ() (h µ + h ν ) (µ ν). (3.3) We are concerne with the iscrete timestepping minimization problem: given µ M κ, M (), solve min Φ λ (ν) + 1 ν M κ, M (), ν () µ () 2τ W2 2(ν, µ). (3.4) In the sequel, given any signe measure µ M κ, M (), we enote by µ its restriction to (i.e. µ = χ µ) an by µ its restriction to, so that we have the orthogonal ecomposition µ = µ + µ. 11

12 In orer to erive an Euler-Lagrange equation, as in [AS], we introuce a perturbe, regularize functional. Let Φ δ λ(ν) = Φ λ ( ν) + δ ν 4 (3.5) if ν L 2 an + otherwise. We have the following result: Lemma 3.1 The perturbe minimization problem min Φ δ λ(ν) + 1 ν M κ, M (), ν () µ () 2τ W2 2(ν, µ) (3.6) has a solution µ δ τ, the family µ δ τ has limit points both for the strong H 1 topology an the M() weak topology, δ ( µδ τ) 4 0 as δ 0, an any limit point µ τ as δ 0 solves (3.4). Proof. The existence of µ δ τ is given by the irect metho, as for the existence of µ τ, using the crucial fact that W 2 (, µ) is lower semi-continuous. Let M δ be the minimum in (3.6) an let M be the minimum of the functional in (3.4). It is clear that M δ M; on the other han, Φ δ λ Φ λ as δ 0 at any amissible point ν such that ν L 4 (). Thus lim sup M δ Φ λ (ν) + 1 δ 0 2τ W2 2(ν, µ) for all ν in M κ, M () such that ν () µ () an ν L 4 (). By ensity we obtain lim sup δ M δ M, therefore M δ M as δ 0. If µ τ is a weak limit point of µ δ τ along some sequence δ i 0, the lower semicontinuity of Φ λ gives, since Φ δ i λ Φ λ for any i, Φ λ (µ τ ) + 1 2τ W2 2(µ τ, µ) lim inf Φ λ(µ δ i τ ) + 1 i 2τ W2 2(µ δ i τ, µ) lim inf M δ i = M, i therefore µ τ is a solution of (3.4). As a consequence lim Φ λ(µ δ i τ ) + δ i µ δ i τ i 2τ W2 2(µ δ i τ, µ) = Φ λ (µ τ ) + 1 2τ W2 2(µ τ, µ). By the lower semicontinuity of Φ λ an ν W2(ν, 2 µ) it follows that Φ λ (µ δ i τ ) Φ λ (µ τ ), W2(µ 2 δ i τ, µ) W2(µ 2 τ, µ) an δ i ( µδ i τ ) 4 0. Now, since Φ λ (ν) is itself the sum of two lower semicontinuous terms, namely Φ 0 (ν) an λ ν ()/2, we obtain lim i λ µδ i τ () = λ µ τ () an lim h δ i µ i τ 2 + (h δ µ i 1)2 = h µτ 2 + (h µτ 1) 2. τ In particular µ δ i τ µ τ strongly in H 1 (). 12

13 Next we erive an Euler equation for problem (3.6), which will give a characterization of the iscrete velocity of the scheme. It is useful to begin with the analysis of the corresponing minimization problem on the whole plane. This way, we can eal with competitors of the form t # ν, which can have some mass outsie. Lemma 3.2 Any minimizer ν of { min Φ δ λ(ν) + 1 } 2τ W2 2(ν, µ) : ν M κ, M (R 2 ), ν (R 2 ) µ (), x 2 ν < + R 2 satisfies (3.7) 3δ (( ν) 4 ) h ν ν = 1 τ π1 #(χ (x)(x y)γ + 0 ) + 1 τ π1 #(χ (x)(x y)γ 0 ) in D (R 2 ), (3.8) where γ + 0 an γ 0 are the optimal transport plans from ν to µ given by splitting, with the notation of (2.16): γ + 0 Γ 0 (ν +, µ + 0 ) an γ 0 Γ 0 (ν, µ 0 ), where µ + 0 an µ 0 are suitable submeasures of µ + an µ respectively. Proof. We perform a variation of the internal part of the optimal measure ν along a smooth vector fiel ξ : R 2 R 2. Let ϑ +, ϑ be the optimal couple for W 2 (ν, µ), such that (2.14) hols. If γ + Γ 0 (ϑ +, µ + ) an γ Γ 0 (ϑ, µ ), we can consier a splitting as (2.16). Accoringly, ϑ 1 enotes the common part of ϑ + an ϑ so ϑ + = ν + + ϑ 1, ϑ = ν + ϑ 1 an W2(ν, 2 µ) = x y 2 (γ γ0 + γ γ1 )(x, y). (3.9) Let an ν ε = ν + (I + εξ) # ν (3.10) ε = {x : x + εξ(x) }. (3.11) For small ε, I + εξ is injective, an it is clear that ν ε (R 2 ) = ν(r 2 ) an ν ε (R 2 ) = ν (R 2 ). Let moreover γ ε + = (I + εξ, I) # (χ γ 0 + ) + χ γ γ 1 + (3.12) γε = (I + εξ, I) # (χ γ0 ) + χ γ0 + γ1. We have W 2 2(ν ε, µ) W 2 2 ( ϑ + + ϑ 1 + (I + εξ) # ν +, µ + ) + W 2 2 ( ϑ + ϑ 1 + (I + εξ) # ν, µ ), but it is clear from (3.12) that γ + ε Γ( ϑ + + ϑ 1 + (I + εξ) # ν +, µ + ) an γ ε Γ( ϑ + ϑ 1 + (I + εξ) # ν, µ ), 13

14 hence W2(ν 2 ε, µ) We write the last integral as x y 2 (γ ε + + γε ) = + + = + + x y 2 (γ + ε + γ ε ). x + εξ(x) y 2 γ x + εξ(x) y 2 γ0 + x y 2 (γ γ1 ) x y 2 γ ε x y 2 γ0 + 2ε x y 2 (γ γ 1 ). Then, recalling also (3.9), we fin W2(ν 2 ε, µ) W2(ν, 2 µ) 2ε ξ(x) (x y) γ ε So we obtain lim sup ε 0 (W 2 2(ν ε, µ) W 2 2(ν, µ)) 2ε x y 2 γ + 0 x y 2 γ 0 ξ(x) (x y) γ + 0 ξ(x) (x y) γ 0 + o(ε) ξ(x) (x y) γ 0 +o(ε). (3.13) ξ(z) [ π 1 # ( χ (x)(x y)(γ γ 0 ) )] (z). (3.14) For the erivative of Φ δ λ (ν ε), we take avantage of the L 4 () convergence of ν ε to ν as ε 0, which gives the W 2,4 () convergence of h νε to h ν an, by smoothness of, the C 1 () convergence as well. We begin by making use of the equality (3.3) about the functional Φ 0 : Φ 0 (ν ε ) Φ 0 (ν) =ν() ν ε () + 1 (h νε + h ν )(ν ε ν) 2 =ν() ν ε () + 1 (h νε (I + εξ) + h ν (I + εξ)) ν 1 (h νε + h ν ) ν 2 ε 2 =ν() ν ε () + 1 (h νε (I + εξ) h νε + h ν (I + εξ) h ν ) ν 2 1 (h νε (I + εξ) + h ν (I + εξ)) ν. 2 \ ε 14

15 But the C 1 () regularity, the fact that h ν = 1 on yiels \ ε (h νε (I+εξ)+h ν (I+εξ)) ν = 2ν(\ ε )+O(ε) ν(\ ε ) = 2 (ν() ν ε ())+o(ε), using ν L 4. As a consequence Φ 0 (ν ε ) Φ 0 (ν) = ε Since ν ε () ν () we also have Φ λ (ν ε ) Φ λ (ν) ε h ν ξ ν + o(ε). h ν ξ ν + o(ε). (3.15) For the regularizing term, we make use of the change of variables formula for the push forwar (see for instance [AGS, Section 5.5]). Since et(j(i + εξ)) = 1 + ε ξ + o(ε), we get [ ] δ ˆν ε 4 ˆν 4 = δ ˆν ε ε [ε 4 ] et 3 (J(I + εξ)) ˆν 4 3δ ˆν 4 ξ + o(1). (3.16) As in the proof of [AS, Proposition 5.1], we combine (3.14), (3.15) an (3.16). By the minimality of ν, an consiering that we can change the sign of the arbitrary vector ξ, we fin the equality 3δ ˆν 4 ξ + h ν ξ ν + 1 ξ [ ( π# 1 χ (x)(x y)(γ γ0 ) )] =0, τ for any ξ C c (R 2 ; R 2 ). The result follows. Corollary 3.3 Let ν M κ, M () be a minimizer of (3.6). Then (3.8) hols. Proof. Let ν P be the minimizer of (3.7). Since any element of M κ, M () is amissible for this problem, there hols Φ δ λ(ν P ) + 1 2τ W2 2(ν P, µ) Φ δ λ(σ) + 1 2τ (σ, µ) σ M κ, M(). (3.17) Let ϑ + P an ϑ P be the optimal couple corresponing to ν P, such that the infimum in the efinition of W 2 is attaine an W 2 2(ν P, µ) = W 2 2 (ϑ + P, µ+ ) + W 2 2 (ϑ P, µ ). Denote by γ + P an γ P the corresponing optimal transport plans. Consier the map Ψ(x, y) = (x, y ), 15

16 where y is equal to y if y, an is equal to the first point on the segment from x to y hitting otherwise; let θ + an θ be the first marginals of Ψ # γ + P an Ψ #γ P respectively, an let ν = θ + θ. It is clear that ν M κ, M (). We claim that ν is the minimum for (3.6). For the proof, notice that ν = ν P (so that Φ δ λ (ν P ) = Φ δ λ (ν)). Moreover W 2 2 (θ +, µ + ) W 2 2 (θ + P, µ+ ) an W 2 2 (θ, µ ) W 2 2 (θ P, µ ), since µ is supporte in an the projection ecreases istances. Since θ + θ = ν, we get W 2 2(ν, µ) W 2 2(ν P, µ). Combining this information with (3.17), the claim is reaily seen to follow. In orer to conclue, it is sufficient to notice that (3.8) epens only on the interior part of the minimizer. 4 The entropy argument One of the key points in this paper consists in showing that the regularity of the initial atum is kept by the iscrete minimizers. This way, we will establish that the analogous result for positive measures (in [AS]) actually extens to the general real measure framework. For this, we nee the regularity of the reference measure µ in (3.6). Hence, in this section we will let µ = µ. From now on, we will say that ϕ : [0, + ) R is an entropy function if it is nonecreasing, C 2 an there hols xϕ (x) = ϕ(x) in [0, 1], 2x 2 ϕ (x) xϕ (x) ϕ(x) (McCann [MC] isplacement convexity). (4.1) Given an entropy ϕ, we let it be extene by oness to (, 0); we will also consier an even convex function ψ on R such that ψ (x) = xϕ (x) ϕ(x) for all x 0. Lemma 4.1 Let ϕ be an entropy an let µ M κ, M () be such that µ = µ L 4 () an ϕ( µ ) <. Then, for any minimizer µδ τ of (3.6), we have ϕ( µ δ τ ) ϕ( µ ). Proof. We know that ν := µ δ τ has L 4 () regularity. But in view of the Euler equation (3.8) we can fin even more regularity. In fact, since ν L 2, we know by Brenier s theorem that χ γ 0 + an χ γ0 are plans inuce by optimal transport maps r 1 an 16

17 r 2, which are boune since is. These maps correspon to the graients of two convex Lipschitz functions (efine on R 2 ). Therefore we have r 1, r 2 BV loc (R 2 ) L (R 2 ) an ( ) π# 1 χ (x)(x y)γ 0 + = (I r1 )ν + ( ) π# 1 χ (x)(x y)γ0 = (I r2 )ν. This way (3.8) becomes 3δ (( ν) 4 ) h ν ν = 1 τ (I r 1)ν τ (I r 2)ν in D (R 2 ). (4.2) Since r 1, r 2 L (), the right han sie is in L 4 (). But since h ν C 0 (), we have h ν ν L 4 (), so that by comparison in (4.2) we fin ν 4 W 1,4 (), an by Sobolev embeing ν C 0 (). Let us now efine r = r 1 χ { ν>0} + r 2 χ { ν<0}. Diviing (4.2) by ν, we obtain that ν-a.e. in 3δsgn( ν) (( ν)4 ) ν which, by efinition of r, correspons to + h ν sgn( ν) = 1 τ (r 1 I)χ { ν>0} + 1 τ (r 2 I)χ { ν<0}, 3δsgn( ν) (( ν)4 ) ν + h ν sgn( ν) = 1 (r I) ν-a.e. (4.3) τ Min that r 1 transports ν + to a submeasure of µ + = µ + L 4 () (an similarly for r 2 ), so r 1# ν + µ + an r 2# ν µ. Since ϕ is nonecreasing on (0, + ), an since the relations (4.1) hol, we have (see [AGS, Lemma ]) ϕ( µ + ) ϕ( ν + ) R 2 ϕ(r 1# ν + ) ϕ( ν + ) R 2 ψ ( ν + ) tr( (r 1 I)) R 2 an ϕ( µ ) ϕ( ν ) R 2 ϕ(r 2# ν ) ϕ( ν ) R 2 ψ ( ν ) tr( (r 2 I)). R 2 We sum the last two inequalities using the fact that ϕ(0) = 0, an euce ϕ( µ ) ϕ( ν ) R 2 ψ ( ν + ) tr( (r 1 I)) R 2 ψ ( ν ) tr( (r 2 I)). R 2 (4.4) But, r 1 an r 2 are graients of convex functions, so that we have tr( (r 1 I)) iv (r 1 I) an tr( (r 2 I)) iv (r 2 I) (these ivergences are to be unerstoo in the 17

18 istributional sense, an are measures since r 1, r 2 are BV ). Now consier the quantity iv ((r 1 I)χ { ν>0} ). Formally by the Volpert formula for BV functions (see [AFP]) we have iv ((r 1 I)χ { ν>0} ) = iv (r 1 I)χ { ν>0} + r 1 I, n { ν=0} H 1 ({ ν = 0}), (4.5) where n enotes the normal. The computation is formal because the level set { ν = 0} nee not be H 1 -rectifiable. But for almost any ε > 0 the bounaries of the sublevels { ν < ε} are, by the BV regularity of ν. Since we are ealing with integrals of the form ψ ( ν) H 1 = 0, { ν <ε} where ψ vanishes in a whole interval containing 0, we can take ε small enough an use the formula above. As a consequence, ψ ( ν + )iv ((r 1 I)χ { ν>0} ) = ψ ( ν + )iv (r 1 I)χ { ν>0}. R 2 R 2 The same hols for ν on { ν < 0}. This way, from (4.4) we euce ϕ( µ ) ϕ( ν ) ψ ( ν + ) tr( (r 1 I)) ψ ( ν ) tr( (r 2 I)) R 2 R 2 R 2 ψ ( ν )iv (r 1 I) ψ ( ν )iv (r 2 I) { ν>0} { ν<0} = ψ ( ν )iv ((r 1 I)χ { ν>0} ) ψ ( ν )iv ((r 2 I)χ { ν<0} ) R 2 R 2 = ψ ( ν )iv (r I) R 2 = ψ ( ν )iv (r I). We make use of (4.3) to estimate the last integral, that is, by means of (4.3) (vali ν-a.e. an since ψ = 0 in a neighborhoo of 0), from the latter inequality we have [ ] ϕ( µ ) ϕ( ν ) τ ψ ( ν )iv 3δsgn( ν) (( ν)4 ) + sgn( ν) h ν. (4.6) R 2 ν Since ψ is o an ν vanishes on R 2 \, arguing as above (using Volpert s formula an ψ = 0 in [ 1, 1]), we fin ψ ( ν )iv (sgn( ν) h ν ) = ψ ( ν) h ν. 18

19 Moreover, by convexity of ψ we obtain ψ ( ν )iv (sgn( ν) h ν ) = ψ ( ν)(h ν ν) ψ(h ν ) ψ( ν). (4.7) Now consier the equation h ν +h ν = ν in. Multiplying it by ψ (h ν ), an integrating by parts yiels ψ (h ν ) h ν 2 + ψ (h ν )(h ν ν) = 0, where we use ψ (h ν ) = ψ (1) = 0 on by continuity of ψ ; so, with the convexity of ψ on R, we obtain ψ (h ν ) h ν 2 ψ( ν) ψ(h ν ). Inserting this inequality into (4.7) we get ψ ( ν )iv (sgn( ν) h ν ) On the other han, the same type of argument also yiels ) ( ) ψ ( ν )iv (sgn( ν) ν4 ν = ψ 4 ( ν )iv ν ν = ψ ( ν ) ν4 ν = g ( ν 4 ) ν4 2, ν ψ (h ν ) h ν 2. (4.8) where g(x) = ψ (x 1/4 ) (hence g 0), an so ) ψ ( ν )iv (sgn( ν) ν4 0. (4.9) ν Inserting (4.8) an (4.9) into (4.6), we fin ϕ( µ ) ϕ( ν ) τ Since ψ 0 (by convexity of ψ) we conclue. ψ (h ν ) h ν 2. (4.10) Corollary 4.2 Let µ = µ L p (), p 4. Then there exists a minimizer µ τ of (3.4) such that µ τ L p (). In particular there hols ϕ( µ τ ) ϕ( µ ) < + for suitable entropies ϕ, enjoying a p-growth at infinity. 19

20 Proof. Let us consier a p-growing entropy: x ( for 0 x 1, ϕ(x) := x p + (p 1) 1 + (p 1)x 1 ) 2 p(1 + x2 ) for x > 1, (4.11) extene by oness to (, 0) (one may check it is inee C 2 ). By Lemma 4.1, we have ϕ( µ δ τ ) ϕ( µ ). On the other han, by Lemma 3.1 we know we have a limit point µ τ of µ δ τ, as δ 0, which minimizes (3.4). But the above inequality gives weak compactness in L p () for the sequence ( µ δ τ). The weak lower semicontinuity in L p () of ν ϕ( ν ) (which hols because ϕ, being an entropy, is convex) allows to conclue. The limiting case as p of the previous corollary gives: Corollary 4.3 Let µ = µ L () an K = max{1, µ }. There exists a minimizer µ τ of (3.4) such that µ τ K, h µτ K. (4.12) Proof. Since K 1 we can construct a sequence (ϕ n ) of p-growing entropies converging monotonically on R + to ϕ(x) := { x for 0 x K + for x > K. (4.13) Let also ψ n be such that ψ n(x) = xϕ n(x) ϕ n (x), with ψ n converging monotonically to + if x > K. By (4.10) we have ϕ n ( µ δ τ ) + τ ψ n(h µ δ τ ) h µ δ τ 2 ϕ n ( µ ). Now we apply Lemma 3.1 to obtain a limit point µ τ of µ δ τ, as δ 0, such that µ τ is a minimizer of (3.4). Then, the weak lower semicontinuity of µ ϕ n( µ ) in L p, the continuity of ψ n an the convergence of h µ δ τ yiel ϕ n ( µ τ ) + τ ψ n(h µτ ) h µτ 2 ϕ n ( µ ) ϕ( µ ). From the convergence properties of ϕ n an ψ n we get µ τ K a.e. in an h µτ K. 20

21 Finally, the L p regularity of a minimizer enables us to use the first variation argument use for the perturbe problem. We are le to: Lemma 4.4 Let p 4 an µ = µ L p (). There exists a minimizer µ τ of (3.4) satisfying h µτ µ τ = 1 τ π1 #(χ (x)(x y)γ + 0 ) + 1 τ π1 #(χ (x)(x y)γ 0 ) in D (R 2 ), (4.14) where γ + 0 Γ 0 (µ + τ, µ + 0 ) an γ 0 Γ 0 (µ τ, µ 0 ), with respect to Notation 2.5. Proof. By the previous corollaries we know that (3.4) possesses a minimizer with L 4 () interior part. Then, we can perform the same variational argument in Lemma 3.2 (with some simplifications, since the term δ µ 4 x is now absent) an Corollary 3.3 to euce (4.14). 5 Back to the continuous moel Let us fix the initial atum µ 0 M κ, M () H 1 () an let µ 0 τ = µ 0. We efine a sequence of iscrete solutions µ k τ. At each step, we minimize starting from the interior part of the previous point, an then we simply a its bounary part. This way, more an more mass is accumulate on the bounary at each step, an never returns to the interior of the omain. This is reminiscent of the analysis of [AS], in the framework of probability measures. Inee, in such context it is proven that no mass enters from the bounary, by means of energy comparison. So, the recursive scheme will be the following. Given a time step τ > 0 an µ k τ M κ, M (), efine ντ k+1 as a minimizer of the iscrete problem min Φ λ (ν) + 1 ν M κ, M (), ν () µ k τ () 2τ W2 2(ν, µ k τ), k N, (5.1) where κ = µ k τ(). Since we minimize starting from the internal part of µ k τ, we can choose ντ k+1 satisfying the regularity properties obtaine by virtue of the entropy argument in the previous section. Then we let M κ, M () µ k+1 τ = ν k+1 τ + µ k τ, k N. (5.2) Also, we efine the piecewise constant interpolation µ τ (t) := µ t/τ τ for any t 0. The following result shows that a minimizing movement oes exist, as the pointwise limit of µ τ (t). Proposition 5.1 (Existence of a limit curve) There exists an infinitesimal sequence (τ n ) such that µ τn (t) converge to some µ(t) M κ, M (), weakly in the sense of measures, for any t 0. Furthermore, µ is uniformly boune in L 4 () if the internal part of µ 0 belongs to L 4 (). 21

22 Proof. Since ντ k+1 is a minimizer of the one-step minimization starting from µ k τ, we have, for any k, W2(ν 2 τ k+1, µ k τ) 2τΦ λ ( µ k τ) 2τΦ λ (ντ k+1 ). Since µ k+1 τ = ν τ k+1, an since Φ λ epens only on the interior part of measures, we fin W 2 2(ν k+1 τ, µ k τ) 2τΦ λ ( µ k τ) 2τΦ λ ( µ k+1 τ ). (5.3) Let us insert (2.15) an take (2.7) into account, so Of course this also implies W 2 1(µ k+1 τ, µ k τ) = W 2 1(ν k+1 τ, µ k τ) 4Mτ ( Φ λ ( µ k τ) Φ λ ( µ k+1 τ ) ). (5.4) Φ λ (µ k τ) Φ λ (µ 0 ), k > 0. (5.5) Let t (k 1 τ, (k 1 + 1)τ] an s (k 2 τ, (k 2 + 1)τ], for some k 1, k 2 > 0, with k 2 > k 1. Using the interpolation of minimizers µ τ (t), summing the relations (5.4) an making use of the triangle inequality (min that W 1 is a istance), along with (5.5) an the positiveness of Φ λ, we have Hence W 2 1( µ τ (t), µ τ (s)) = W 2 1(µ k 1+1 τ, µ k 2+1 τ ) (k 2 k 1 ) 2τ(k 2 k 1 ) k 2 k=k τ(k 2 k 1 )Φ λ (µ 0 ). W 1 ( µ τ (t), µ τ (s)) 2Φ λ (µ 0 ) t s + τ k 2 k=k 1 +1 W 2 1(µ k τ, µ k+1 τ ) ( Φλ (µ k 1+1 τ ) Φ λ (µ k 2+1 τ ) ) s, t [0, + ). The iscrete C 0,1/2 estimate allows to fin (for this see [AS], or also [AGS, Chapter 11] for the precise argument) a subsequence τ n 0 such that in the sense of measures lim n µ τ n (t) = µ(t) t 0. (5.6) This conclues the proof. Notation 5.2 The transportation is escribe by the cost W 2 (ντ k+1, µ k τ), corresponing to an optimal couple of measures (ϑ + ) k+1 τ, (ϑ ) k+1 τ as in (2.14). That is, W 2 2(ν k+1 τ, µ k τ) = W 2 2 ((ϑ + ) k+1 τ, ( µ + ) k τ) + W 2 2 ((ϑ ) k+1 τ, ( µ ) k τ). (5.7) 22

23 With reference to Notation 2.5, we let (ϑ 1 ) k+1 τ be the common part of (ϑ + ) k+1 τ an (ϑ ) k+1 The two terms in the right han sie of (5.7) correspon to optimal plans (γ + ) k+1 τ an (γ ) k+1 τ an can be split as Here (γ + ) k+1 τ = (γ + 0 ) k+1 τ + (γ + 1 ) k+1 τ an (γ ) k+1 τ = (γ 0 ) k+1 τ + (γ 1 ) k+1 τ. (5.8) (γ 0 + ) k+1 τ Γ 0 ((ν + ) k+1 τ, ( µ + 0 ) k τ) an (γ 1 + ) k+1 τ Γ 0 ((ϑ 1 ) k+1 τ, ( µ + 1 ) k τ), where ( µ + 0 ) k τ an ( µ + 1 ) k τ are suitable positive submeasures of ( µ + ) k τ, which is their sum. Similarly for the negative parts. The iscrete velocity of the scheme (3.4) (neglecting the common parts) coul be efine by (x y)/τ with (x, y) ( ) ( ) supp(γ 0 + ) k+1 τ supp(γ 0 ) k+1 τ. The characterization of the iscrete velocity is crucial to interpret our recursive scheme as the iscrete version of a ifferential equation. But we o not have a stanar continuity equation for the signe case, an therefore we can not procee as in [AS] (see Section 6 therein). Instea, we will see how to obtain a partial result by constructing the limiting ifferential equation by han. In [MZ], the authors are able to prouce solutions for a similar moel (see Section 6 below) by means of an explicit, rather than implicit, iscrete scheme. They take avantage of strong regularity hypotheses on the initial atum, which are preserve uring the evolution, guaranteeing goo compactness properties. Here we woul like to aress the case of mere L p initial ata. We start by introucing a basic estimate. With the notation above, we have shown that W 2 2(ν k+1 τ, µ k τ) = τ. x y 2 ( ) (γ 0 + ) k+1 τ + (γ0 ) k+1 τ + (γ 1 + ) k+1 τ + (γ1 ) k+1 τ (x, y). (5.9) From (5.3), summing the telescopic series, we immeiately see that W 2 2(ν k+1 τ, µ k τ) 2τΦ λ (µ 0 ). (5.10) Hence each of the four terms in the right han sie of (5.9) satisfies the same boun. The next proposition shows that there is no contribution from the transport plans γ 1 + an γ1, which can be thought as accounting for self-annihilation of mass, in the subsequent limit process. Proposition 5.3 Let φ Cb 1 (). Then lim τ 0 φ(y) φ(x) (γ + 1 ) k+1 τ (x, y) = 0, 23

24 lim τ 0 φ(y) φ(x) (γ 1 ) k+1 τ (x, y) = 0. Proof. By efinition of W 2, an taking into account the constraint in the iscrete minimization problem ντ k+1 () µ k τ (), we see that for any k there hols (ϑ + ) k+1 τ ()+ (ϑ ) k+1 τ () = µ k τ (). Then, recalling that (ϑ 1 ) k τ is the common part of (ϑ + ) k τ an (ϑ ) k τ, it is clear that (ϑ 1) k τ µ 0 () M, hence (γ 1 + ) k+1 τ ( ) + (γ1 ) k+1 τ ( ) M. (5.11) Now we compute φ(y) φ(x) (γ 1 + ) k+1 τ (x, y) φ Lip = φ Lip x y (γ + 1 ) k+1 τ (x, y) x y (γ + 1 ) k+1 τ (x, y). With Höler s inequality we see that the last term is controlle by ( ) 1/2 ( 1/2 φ Lip x y 2 (γ 1 + ) k+1 τ (x, y) (γ 1 + ) k+1 τ (x, y)). But making use of (5.10) we see that the first factor is boune by 2τΦ λ (µ 0 ), while by (5.11) the secon is less than or equal to M. Hence lim sup φ(y) φ(x) (γ 1 + ) k+1 τ (x, y) lim M φ Lip 2τΦλ (µ 0 ) = 0. τ 0 τ 0 Similarly one shows that obtaining the thesis. lim sup τ 0 We will also nee the following similar result. Proposition 5.4 Let φ Cb 1 (). There hols lim τ 0 φ(y) φ(x) (γ 1 ) k+1 τ (x, y) = 0, φ(x) φ(y) (γ + 0 ) k+1 τ (x, y) = 0, an the same for the analogous sum involving (γ 0 ) k τ. 24

25 Proof. Reasoning as in the previous proposition, one estimates the sum above by ( φ Lip x y 2 (γ + 0 ) k+1 τ (x, y) ) 1/2 ( (γ + 0 ) k+1 τ ( )) 1/2. (5.12) Since no mass on the bounary returns to the interior of the omain uring the iscrete steps, we have (γ 0 + ) k τ( ) = k=1 (ντ k ) + ( ) k=1 ντ k ( ) µ 0 () M. k=1 This shows that the secon factor in (5.12) is uniformly boune. The first one is controlle again by φ Lip 2τΦλ (µ 0 ), as a consequence of (5.9) an (5.10). The same argument gives the thesis if (γ 0 + ) k τ are replace by (γ0 ) k τ. Lemma 5.5 (Convergence of the total variations) Let (τ n ) be given by Proposition 5.1. Then there exist positive measures ϱ + (t), ϱ (t) such that, possibly on a subsequence, there hols µ + τ n (t) ϱ + (t), µ τ n (t) ϱ (t), µ τn (t) ϱ + (t) + ϱ (t). (5.13) Proof. We prove the convergence of the positive parts. By ifference the result follows for the negative parts. Let ϕ be a boune Lipschitz function over. Possibly aing a constant, we can assume that ϕ is nonnegative. Let a k τ := ϕ (ν + ) k+1 τ ϕ ( µ + 0 ) k τ, b k τ := ϕ ( µ + 1 ) k τ. We have, by (5.9), a k τ = ϕ Lip ( (ϕ(x) ϕ(y)) (γ + 0 ) k+1 τ (x, y) ϕ Lip M W2 (ντ k+1, µ k τ), which gives, making use of (5.10), ) 1/2 ( ) 1/2 (γ 0 + ) k+1 τ x y 2 (γ 0 + ) k+1 τ (x, y) (a k τ) 2 τ M τ ϕ 2 Lip W 2 2(µ k+1 τ, µ k τ) 2M ϕ 2 LipΦ λ (µ 0 ). 25

26 As (a k τ) 2 = τ + 0 τ (a t/τ 2 τ ) we infer the L 2 (0, + ) weak compactness of the sequence (a t/τ τ /τ). We can assume, possibly extracting from (τ n ) a subsequence, that (a t/τn τ n /τ n ) weakly converge to some f L 2 (0, + ). In particular we have A τn := a k τδ {kτn} fl 1. Inee, letting ζ C c (0, + ), we have ( ) ζ(t) a k τ n δ {τnk}(t) = a k τ n ζ(τ n k) = 0 Hence, for any t there hols Next, notice that, by (5.2), Since b k τ n 0, we see that lim n t a k τ n δ {kτn}([0, t]) ϕ µ + τ n (t) 0 t, a t/τn τ n τ n t 0 f(y) y. (a k τ n b k τ n )δ {kτn}. ( ) ϕ µ τ + t n (t) A τn ([0, t]) 0. ζ(τ n t/τ n ) t 0 f(t)ζ(t) t. We have a family of monotone functions. We can apply Helly s pointwise compactness theorem (see for instance [AGS, Lemma 3.3.3]) to obtain that, possibly extracting one more subsequence, the pointwise, nonincreasing limit of this family of functions exists. The convergence of A τn now yiels ϕ µ τ + (t) L ϕ (t), t 0. The convergence hols for any positive Lipschitz ϕ, hence for any Lipschitz ϕ. By a iagonal argument we can fin an infinitesimal sequence, that we still enote by τ n, such that ϕ µ τ + n (t) L ϕ (t), t 0 an ϕ D, 26

27 where D is a countable ense subset of Cb 0 () mae of Lipschitz functions. Then, for any t, ϕ L ϕ (t) can be extene uniquely to a weakly continuous linear functional on Cb 0(). By the Riesz representation theorem we conclue that L ϕ(t) = ϕ ϱ+ (t), for some ϱ + M + (), an for any t there hols µ τ + (t) ϱ + (t). Letting ϱ (t) be the pointwise weak limit of µ τ (t) obtaine by the same reasoning, we also infer the convergence of the total variations: µ τ (t) ϱ(t), where ϱ(t) = ϱ + (t) + ϱ (t). Eventually, we are able to prouce a limiting equation. Theorem 5.6 (Equation in the limit) Let µ 0 L 4 (). The minimizing movement µ(t) given by Proposition 5.1 satisfies t µ(t) iv (χ h µ(t) ϱ(t)) = 0 in D ((0, + ) R 2 ), (5.14) where ϱ(t) is a suitable positive measure satisfying ϱ(t) µ(t). Proof. Let (τ n ) be the sequence given by Proposition 5.1 an Lemma 5.5. Just for simplicity of notation, in the sequel we shall write τ instea of τ n. Let φ C 2 (). Let us compute the erivative of the time interpolate measure µ τ (t). We have, in the sense of istributions, φ µ t τ (t) = ( φ µ k+1 τ φ µ k τ ) δ {kτ}. But φ µ k+1 τ φ µ k τ = φ ν k+1 τ φ µ k τ = (φ(y) φ(x)) γ k+1 τ, where, using the notation introuce in (5.8), γ k+1 τ := (γ + 0 ) k+1 τ (γ 0 ) k+1 τ + (γ + 1 ) k+1 τ (γ 1 ) k+1 τ. So we may write φ µ t τ (t) = = δ {kτ} ( δ {kτ} (φ(x) φ(y)) γ k+1 τ (x, y) ) φ(x), x y γτ k+1 (x, y) + R k τ. (5.15) 27

28 Let us estimate the remainer R k τ by writing it in integral form. We have R k τ = φ((1 θ)x + θy)(y x), y x γτ k+1 (x, y)θ φ x y 2 γτ k+1 (x, y) (5.16) By (5.10) we see that φ W 2 2(ν k+1 τ, µ k τ) lim τ 0 R k τ = 0. Together with Proposition 5.3 an Proposition 5.4, this shows that (5.15) can be written, for τ 0, as ( φ µ t τ (t) = δ {kτ} φ(x), x y χ (x) ( ) ) (γ 0 + ) k+1 τ (γ0 ) k+1 τ (x, y) + o(1). (5.17) The Euler equation for iscrete minimizers ντ k of (3.4), since ν τ k = µ k τ, reas (see Lemma 4.4) h µ k τ ( ( µ k τ ) + ( µ k τ) ) = 1 τ ( π 1 # (χ (x)(x y)(γ + 0 ) k τ) + π 1 #(χ (x)(x y)(γ 0 ) k τ) ), but notice that the first term in the right han sie can be ifferent from zero only on supp( µ k τ) +. Similarly for the secon term. Hence we can split the equation in h µ k τ ( µ k τ) + = h µ k τ ( µ k τ) = 1 τ π1 #(χ (x)(x y)(γ + 0 ) k τ), 1 τ π1 #(χ (x)(x y)(γ 0 ) k τ). Substituting in (5.17), we fin φ µ t τ (t) = τδ {kτ} φ(x), hµ k τ (x) µ k τ (x) + o(1). Passing to the limit as τ goes to zero (more precisely, along the sequence τ n ) we get φ µ(t) + φ, h µ(t) ϱ(t) = 0, t where ϱ(t) is given by Lemma 5.5, hence satisfying ϱ(t) µ(t) an ϱ(t)() µ 0 (). 28

29 Remark 5.7 (The positive case) In the case of positive (or negative) measures, it is immeiately seen that both (2.3) an (2.12) reuce to the stanar 2-Wasserstein istance. In particular, if µ is positive an ν is not, by (2.13) we get W 2 (ν, µ) = inf = +. Then, it is clear that, if the initial atum µ 0 is positive, the iscrete minimizers are positive as well. As a consequence, passing to the limit in the iscrete scheme, as shown in [AS] (the schemes coincie in the positive case), prouces a positive solution of t µ(t) iv(χ h µ(t) µ(t)) = 0 in D ((0, + ) R 2 ). If µ 0 L () is compactly supporte such solution is locally unique in time, an also globally unique if suitable bounary conitions hol (see [M1]). Remark 5.8 (System formulation) We coul also erive separately the equations, in the limit as τ 0, for the positive an negative parts of µ τ (t). Let φ C 2 (). Regaring the positive part, we reason as in Theorem 5.6, taking avantage in particular of Proposition 5.4, so that as τ 0 φ µ τ + (t) = t = = ( δ {kτ} ( δ {kτ} ( δ {kτ} φ (µ + ) k+1 τ φ (ν + ) k+1 τ φ (µ + ) k τ ) φ ( µ + 0 ) k τ φ(x), x y (γ 0 ) k+1 τ (x, y) ) φ ( µ + 1 ) k τ + o(1) ) φ ( µ + 1 ) k τ + o(1), (5.18) where ( µ + 0 ) k τ is the part of ( µ + ) k τ that comes from (ν + ) k+1 τ an ( µ + 1 ) k τ is the part of ( µ + ) k τ which gets transporte by (γ 1 + ) k+1 τ, as in Notation 5.2. A term that was not present in (5.17) appears. Let us analyze it. First, notice that, since ( µ + 1 ) k τ accounts for mass cancellation at every step, we have as usual This entails ( µ + 1 ) k τ() µ 0 () M. 0 ( µ + 1 ) t/τ τ τ x t M, so that the sequence ( µ + 1 ) t/τ τ /τ is boune in L 1 ( (0, + )) an hence possesses a subsequence converging weakly in measure. We enote by σ(x, t) a suitable space-time 29