On nonlinear cross-diffusion systems: an optimal transport approach
aa r X i v : . [ m a t h . A P ] M a r ON NONLINEAR CROSS-DIFFUSION SYSTEMS: AN OPTIMAL TRANSPORTAPPROACH
INWON KIM AND ALP ´AR RICH ´ARD M´ESZ ´AROS
Abstract.
We study a nonlinear, degenerate cross-diffusion model which involves two densities withtwo different drift velocities. A general framework is introduced based on its gradient flow structure inWasserstein space to derive a notion of discrete-time solutions. Its continuum limit, due to the possiblemixing of the densities, only solves a weaker version of the original system. In one space dimension, wefind a stable initial configuration which allows the densities to be segregated. This leads to the evolutionof a stable interface between the two densities, and to a stronger convergence result to the continuumlimit. In particular derivation of a standard weak solution to the system is available. We also studythe incompressible limit of the system, which addresses transport under a height constraint on the totaldensity. In one space dimension we show that the problem leads to a two-phase Hele-Shaw type flow. Introduction
Let Ω be a bounded domain in R d with C boundary, and let T > m > , T ] × Ω:(PME m ) ∂ t ρ − ∇ · (cid:0) ( ∇ p + ∇ Φ ) ρ (cid:1) = 0; ∂ t ρ − ∇ · (cid:0) ( ∇ p + ∇ Φ ) ρ (cid:1) = 0 , where Φ , Φ : Ω → R are given and the common diffusion term is generated by the pressure variable(1.1) p := mm − ρ + ρ ) m − . In this article the system is subject to no flux condition on [0 , T ] × ∂ Ω and is equipped with initialnonnegative densities ρ , ρ ∈ L (Ω).Formally (PME m ) can be seen as the gradient flow in Wasserstein (product) space of the free energy(1.2) ( ρ , ρ ) ˆ Ω m − ρ + ρ ) m d x + ˆ Ω Φ ρ d x + ˆ Ω Φ ρ d x. Staying at the formal level, in the incompressible limit as m → + ∞ where the first term in free energyturns into the constraint ρ + ρ ≤
1, the corresponding system for the limiting density pair ( ρ , ∞ , ρ , ∞ )is(PME ∞ ) ( ∂ t ρ , ∞ − ∇ · (cid:0) ( ∇ p ∞ + ∇ Φ ) ρ , ∞ (cid:1) = 0; ∂ t ρ , ∞ − ∇ · (cid:0) ( ∇ p ∞ + ∇ Φ ) ρ , ∞ (cid:1) = 0 , where the pressure p ∞ is supported in the region { ρ , ∞ + ρ , ∞ = 1 } . When the densities ρ i, ∞ ’s arecharacteristic functions with separate supports, the problem corresponds to a two-phase Hele-Shaw typeflow with drifts.Our goal in this paper is to study the problems (PME m ) and (PME ∞ ) in the context of the aforemen-tioned gradient flow, and verify the above heuristics. More precisely we will formulate the problem interms of the discrete-time gradient flow (i.e. JKO or minimizing movement scheme) of the aforementionedfree energy (1.2), posed in the product space equipped with the 2-Wasserstein metric. Then we will studythe solutions of this discrete scheme as the time step goes to zero. We will show that the limiting pairof densities ( ρ , ρ ) satisfies a set of transport equation that will reduce to (PME m ) under a strongerconvergence assumption (for the precise statement we refer to Theorem 3.8). To strengthen this result, it Date : March 20, 2018.Inwon Kim is supported by the NSF grant DMS-1566578. seems necessary to consider “stable” initial configurations which avoids mixing: see the discussion below.It turns out that in one dimension, in the setting of stable initial configurations which avoids mixing, astronger convergence result holds and as a consequence the continuum limit densities satisfy (PME m ) inthe standard weak sense. Below is a summary of our main results: precise statements are contained inthe quoted theorems.The main results of the paper are obtained in one space dimension. Here we assume that the densitywith stronger drift in x direction sits on the right side on the x -axis, i.e.,(1.3) − ∂ x Φ ≥ − ∂ x Φ and x ≥ x for x i ∈ { ρ i > } , i = 1 , . Under the assumption (1.3) the following theorems hold:
Theorem 1.1 (Segregation of solutions: Proposition 4.1, Theorem 4.2) . For given m ∈ (1 , ∞ ] and τ > ,let ( ρ ,τ , ρ ,τ ) be the time-discrete solutions given by the minimizing movement scheme with time stepsize τ > , as given in (MM m ) . Then, the pair stays ordered for all times t > , i.e. ρ ,τt is supportedto the right of the support of ρ ,τt for all t > . Moreover, as τ ↓ each density ρ i,τ converges weaklyin L m − ([0 , T ] × Ω) , along a subsequence, to ρ i,m for i = 1 , . Also, along a subsequence ρ i,τ convergespointwise a.e. to ρ i,m for i = 1 , . The pair of limiting densities ( ρ ,m , ρ ,m ) solves (PME m ) in the weaksense and it stays ordered for all times. Theorem 1.2 (Convergence of weak solutions as m → ∞ : Theorem 4.7) . Let ( ρ ,m , ρ ,m ) be as givenabove for given m > . Then as m → ∞ and along a subsequence, the density pairs converge weakly in L p ([0 , T ] × Ω) for any < p < ∞ to ( ρ , ∞ , ρ , ∞ ) , which is a weak solution of (PME ∞ ) . Theorem 1.3 (Patch solutions for (PME ∞ ): Proposition 4.9) . Let m = ∞ and suppose, in additionto (1.3) , that ∂ xx Φ i ≥ and ρ i = χ ( a i (0) ,b i (0)) . Then ρ i, ∞ remains a patch for all t > , i.e. ρ i, ∞ t = χ ( a i ( t ) ,b i ( t )) , i = 1 , . The density pair in this case is a solution to a two-phase Hele-Shaw type flow withdrifts. Remark 1.1.
For the linear diffusion m = 1 the logarithmic entropy ´ ρ ln ρ d x replaces m − ´ ρ m d x inthe free energy, resulting in slightly different, however mostly parallel, analysis. Let us point out that inthis case the sum of the densities is always positive, and thus they will always form an interface betweeneach other in the event of segregation. Let us discuss now the existing results from the literature that are relevant to our work. The singledensity version of the system (PME ∞ ) has been introduced in [30] in the gradient flow setting, the freeboundary characterization and its links to (PME m ) has been studied in [1]. These and similar systemsreceived a lot of attention in the past a few years (see for instance [31, 32, 33] and the references therein).These models are strongly related to the so-called Hele-Shaw models, as we can see in [1] (for otherreferences we direct the reader to [23, 24, 35, 39] and to the references therein).Cross diffusion systems arise naturally from mathematical biology. These appear either as systemsof reaction-diffusion equations (as in [28, 22] for instance) or systems of advection-diffusion equations(as in [7, 6, 16, 45] for instance). These systems appear also in fluid mechanics, such as the thin filmapproximation of the Muskat problem studied in [20, 27]. Beside the PDE approach for degenerateparabolic systems (used in most of the above references), more recently the optimal transport and gradientflow theories have been adopted to study these systems. For a non-exhaustive list of the fast-growingreferences in this direction we refer to [5, 8, 9, 10, 12, 13, 14, 15, 16, 18, 25, 26, 27].Most of the aforementioned papers concern systems including separate diffusion terms for the evolutionof each densities. Such feature enables tracking of separate densities in the evolution. This is in contrastto our case where recovering separate densities out of the dynamic system appears to be out of reachunless the densities are guaranteed to be segregated. There are very few results available for systemswithout separate diffusions, we mention here a few particular papers in this direction. In [6] the authorsstudy the well-posedness of the system (PME m ) in one space dimension when m = 2, Φ = Φ ≡ m = ∞ with gravity potentials Φ i ( x ) = C i x and with full saturation, i.e. with the condition ρ + ρ = 1. There the mixing profile of one density correponds to an entropy solution of aBurger’s type equation. This interesting description of mixing phenomena remains open to be extendedbeyond the specific setting given in the paper. Lastly in the recent paper [9] the authors study existenceand segregation properties of one dimensional stationary solutions for systems of similar form to ours,when m = 2 and the drift is generated by interaction energies. Main difficulties and ingredients
As mentioned above, our main challenge lies in the fact that the densities may mix into each otherduring the evolution, which indeed happens with the “unstable” initial configurations where the densitiesare initially positioned in the opposite order to the equilibrium solution (see the discussion in Section 2.5).Such situation indicates low regularity of each densities, hindering the system from being well-posed.Indeed, in general we are only able to obtain strong convergence on the sum of the two density variablesin the continuum limit, as we will see in Theorem 3.8. Naturally this reasoning leads to the questionof whether one can formulate a “stable” initial configuration to obtain a stronger result. This question,while under investigation by the authors, stands open beyond the one dimensional result for segregatedsolutions, Theorems 1.1, 1.2, 1.3.In terms of gradient flows, the challenge lies in the lack of available estimates or convexity properties.Though not surprising in the context of the above discussion, this is an interesting contrast to the singlespecies case, where for instance stability of the discrete gradient flow solutions based on λ -convexityproperties played an important role in the analysis. For us the higher order space regularity estimatesin the JKO scheme are available only for functions of the sum of the two densities. For similar modelsconsidered in [25] and [27] this difficulty was overruled by presence of separate diffusions, or “separateentropies” of the form ε ( ´ f ( ρ ) + g ( ρ )d x ) in the free energy (1.2), however estimates obtained here donot carry through as ε ↓
0. Let us also mention that the flow interchange technique introduced in [29],which has been quite successful to analyze some non-convex gradient flow systems such as in [19] or [27],does not appear to be applicable to our system. Thus here we derive all our estimates relying only onthe first order optimality conditions satisfied by the discrete in time minimizers in the JKO scheme (seefor instance the proof of Theorem 2.4). This procedure is rather natural yet appears to be unexploitedin the literature for similar models.
Structure of the paper
In Section 2 the discrete-time scheme for the gradient flow is introduced, set in the W -product space.In Section 2.4 the properties of discrete-time minimizers are studied. Here we observe that while thetotal density ρ + ρ is relatively regular (Lipschitz continuous), separate densities may be segregatedand discontinuous. The segregation of densities with respect to the ordering properties of their potentialsare more obvious in Section 2.5, where one discusses the equilibrium solutions. Such segregation andordering property suggests that fingering and mixing is inevitable for densities starting from “unstable”initial configurations, to position themselves into the stationary profile.In Section 3 we analyze the continuum limit of discrete-time solutions by studying their convergencemodes as the time step size is sent to zero. We show that the limit solution satisfies a system of transportequations which can be interpreted as a generalized solutions for the system (PME m ). We also introducethe standard notion of weak solution for our systems and show that the continuum limit satisfies thisnotion when pointwise convergence holds for separate densities. It remains an open question whether thedensities indeed converge pointwise, i.e. whether we can track down the position of each density in theevolution of the problem in general framework or in general dimension.Section 4 is devoted to the analysis in one space dimension, where we consider stable initial configura-tions that line up with the strength order of the drift potentials. In this setting we are able to guaranteethat solutions stay segregated with an evolving interface between them. As a consequence it follows thatpointwise convergence holds for each densities, which in turn yields the existence of weak solutions forthe system (PME m ). The continuum solutions of (PME m ) are then shown to converge as m tends toinfinity to a weak solution of (PME ∞ ) along a subsequence. Furthermore when the drift is compressive(or incompressible), we show that patch solutions appear, yielding a solution to the two-phase Hele-Shawflow. I. KIM AND A.R. M´ESZ´AROS
Finally, in the Appendices A and B we recall some results from the theory of optimal transport and arefined version of Aubin-Lions lemma respectively.
Acknowledgements
The authors are thankful to G. Carlier, D. Matthes, F. Santambrogio and Y. Yao for many valuablediscussions at different stages of the preparation of this paper. The authors warmly thank the referee forhis/her constructive comments and remarks.2.
Minimizing movement schemes and properties of the minimizers
Setting and notations.
Let us introduce the setting of the problem and some notations. LetΩ ⊆ R d be a bounded domain with smooth boundary. We denote by P (Ω) the space of probabilitymeasures on Ω. For M > P M (Ω) the space of finite nonnegative Radon measures on Ω( M + (Ω)) with mass M .For a Borel measurable map T : Ω → Ω and µ, ν ∈ P M (Ω), we say that T pushes forward µ onto ν ,and write ν = T µ if ν ( B ) = µ ( T − ( B )) for every B ⊆ Ω Borel measurable set. Using test functions,the definition of pushforward translates to ˆ Ω φ ( y )d ν ( y ) = ˆ Ω φ ( T ( x ))d µ ( x ) , ∀ φ : Ω → R , bounded and measurable . We equip the space P M (Ω) with the well-known 2-Wasserstein distance W ,M , i.e. For µ, ν ∈ P M (Ω) ,W ,M ( µ, ν ) := min (cid:26) ˆ Ω × Ω | x − y | d γ : γ ∈ Π M ( µ, ν ) (cid:27) , where Π M ( µ, ν ) is the set of the so-called transport plans , i.e. Π M ( µ, ν ) := { γ ∈ P M (Ω × Ω) : ( π x ) γ = µ, ( π y ) γ = ν } . In particular if µ ≪ L d Ω then the previous problem has a unique solution, which isof the form γ T := (id , T ) µ . Here, in particular we adjusted the usual distance defined on probabilitymeasures to measures having mass M >
0. Since it shall be clear from the context, from now on we write W instead of W ,M . On the forthcoming pages we shall use classical results from the optimal transporttheory. All of these can be found for instance in [43, 3, 44].We denote by M d (Ω) the space of finite vector-valued Radon measures on Ω . If E ∈ M d (Ω) , wedenote by | E | its variation. We denote the subspaces of absolutely continuous measures (w.r.t. L d Ω)by P ac (Ω) , P ac ,M (Ω) , etc.; we always identify these absolutely continuous measures with their densitiesand write ρ instead of ρ · L d or ρ d x. If ρ ∈ M ac+ (Ω) and c ≥ { ρ > c } we mean the set (up to L d -negligible sets) where ρ ( x ) > c a.e. In particular a property holds a.e. in { ρ > } if and only if itholds ρ − a.e. Notice also that { ρ > } ⊆ a.e. spt( ρ ) . For a measurable set B ⊂ R d , we denote the set of itsLebesgue point by Leb( B ) . Minimizing movements.
The heart of our analysis is the well-known minimizing movement or JKO scheme (see for instance [2, 3, 42, 21]) on a product Wasserstein space.Let us introduce the functionals. We consider F m , F ∞ : P M (Ω) × P M (Ω) → R ∪ { + ∞} and G : P M (Ω) × P M (Ω) → R to be defined as(2.1) F m ( ρ ) = ˆ Ω m − ρ ( x ) + ρ ( x )) m d x, if ( ρ + ρ ) m ∈ L (Ω) , + ∞ , otherwise , (2.2) F ∞ ( ρ ) = , if k ρ + ρ k L ∞ ≤ , + ∞ , otherwise , and G : P M (Ω) × P M (Ω) → R (2.3) G ( ρ ) = ˆ Ω Φ ( x )d ρ ( x ) + ˆ Ω Φ ( x )d ρ ( x ) , where ρ := ( ρ , ρ ), m > , Φ : Ω → R are given continuous potentials. Notice that F ∞ is the indicator function (in the sense of convex analysis) of the set K := (cid:8) ( ρ , ρ ) ∈ P M , ac (Ω) × P M , ac (Ω) : ρ + ρ ≤ . e . (cid:9) . It is classical that F m , F ∞ and G are l.s.c. w.r.t. the weak convergence of measures on P M (Ω) × P M (Ω). It is immediate to see that they are convex, moreover F m is also strictly convex (in the usualsense) on P M (Ω) × P M (Ω). We remark also that in general F m is not displacement convex (in thesense of [34]) on the product space. To see this, let us consider for simplicity m = 2. In this case,for ρ , ρ ∈ L (Ω) we can write F ( ρ , ρ ) = ´ Ω ( ρ ) d x + ´ Ω ( ρ ) d x + 2 ´ Ω ρ ρ d x. If F would be λ -displacement convex (for some λ ∈ R ), then the map ρ
7→ F ( ρ , ρ ) would share at least the samemodulus of convexity for any ρ ∈ P M (Ω) ∩ L (Ω) fixed. While the first term in the developmentof F is 0-displacement convex and the second term is a constant for fixed ρ , the last term would be λ -displacement convex if and only if ρ would be λ -convex, i.e. D ρ ≥ λI d in the sense of distributions.However, ρ can be chosen in a way that the lower bound on its Hessian is arbitrarily negative. Therefore,this term fails to be λ -displacement convex for any λ ∈ R and so does the functional F .We proceed as in the classical setting (see for instance [3, 21]): we define a recursive sequence of densitiesassociated to a fixed time step τ , then we introduce suitable interpolations between these densities andtake the limit as τ ↓ τ > N ∈ N such that N τ = T. Let ( ρ , ρ ) be two given initial densities. For all k ∈ { , . . . , N } we define ρ τk := ( ρ ,τk , ρ ,τk ) as ρ τ = ( ρ ,τ , ρ ,τ ) := ( ρ , ρ )and for k ≥ m ) ρ τk +1 = ( ρ ,τk +1 , ρ ,τk +1 ) = argmin ρ ∈ P M (Ω) × P M (Ω) (cid:26) F m ( ρ ) + G ( ρ ) + 12 τ W ( ρ , ρ τk ) (cid:27) . In this scheme either m > m = ∞ . Here W denotes the Wasserstein distance on theproduct space P M (Ω) × P M (Ω), i.e. W ( µ , ν ) := W ( µ , ν ) + W ( µ , ν ) , where µ := ( µ , µ ), and ν := ( ν , ν ) . We state the following well-known lemma.
Lemma 2.1.
The objective functional in the minimization problem (MM m ) is l.s.c. and bounded frombelow and P M (Ω) × P M (Ω) is compact, thus the optimizer exists. Moreover, F m ( m ∈ [1 , + ∞ ] ) and G are convex functionals and the functional ρ W ( ρ , µ ) is strictly convex whenever µ = ( µ , µ ) hasabsolutely continuous density coordinates (see for instance [43] ). Therefore, if the densities ( ρ , ρ ) areabsolutely continuous w.r.t. L d Ω , the optimizer ρ τk is also unique at each step. Different diffusion coefficients for the two densities.
In many cross-diffusion models (comingmainly from mathematical biology or fluid mechanics, see for instance in [6, 28]) considered in theliterature, it is important to have different diffusion coefficients for the two densities. In our setting, thiscould be formulated as follows. Given κ , κ positive constants, consider a system similar to (PME m ) or(PME ∞ ), i.e.(2.4) ( ∂ t ρ − ∇ · (cid:0) κ ∇ pρ + ∇ Φ ρ (cid:1) = 0 ∂ t ρ − ∇ · (cid:0) κ ∇ pρ + ∇ Φ ρ (cid:1) = 0on [0 , T ] × Ω , where p := mm − ρ + ρ ) m − , with m > , Φ : Ω → R are givenpotentials. Observe that (PME m ) corresponds to κ = κ = 1. Actually, even for κ = κ , this systementers naturally into the framework of gradient flows considered in this paper. Indeed, we can define theminimizing movement scheme as( ρ ,τk +1 , ρ ,τk +1 ) = argmin ( ρ ,ρ ) (cid:26) F ( ρ , ρ ) + ˆ Ω Φ κ ρ d x + ˆ Ω Φ κ ρ d x + 12 τ κ W ( ρ , ρ ,τk ) + 12 τ κ W ( ρ , ρ ,τk ) (cid:27) . Actually a part of the analysis that we perform in the forthcoming sections will be valid in this case aswell. In particular the results from Section 3 can be easily adapted to the system (2.4).
I. KIM AND A.R. M´ESZ´AROS
Properties of the minimizers.
We discuss now some properties of the minimizers in (MM m ).For this, let us consider the following hypotheses(H mρ ) ( ρ , ρ ∈ L m (Ω) , if m ∈ (1 , + ∞ ) , k ρ + ρ k L ∞ (Ω) ≤ L d (Ω) > M + M , if m = + ∞ ;Notice that the structural condition L d (Ω) > M + M in the case of m = + ∞ is needed in order tohave nontrivial competitors that satisfy the upper bound constraint.(H Φ ) Φ , Φ ∈ W , ∞ (Ω) . First, let us derive the first order necessary optimality conditions for the minimizers in (MM m ). Lemma 2.2 (Optimality conditions: m finite) . Let m ∈ (1 , + ∞ ) and let Φ and Φ satisfy (H Φ ) and ( ρ , ρ ) satisfy (H mρ ) . Let ( ρ , ρ ) be the unique minimizer in (MM m ) with k = 0 . Then (1) there exist Kantorovich potentials ϕ i , i = 1 , , in the transport of ρ i onto ρ i and C i ∈ R ( i = 1 , )such that (2.5) mm − ρ + ρ ) m − = max (cid:0) C − Φ − ϕ /τ ; C − Φ − ϕ /τ ; 0 (cid:1) , In particular ( ρ + ρ ) m − is Lipschitz continuous, ρ + ρ ∈ C , / ( m − (Ω) , and these regularitiesdegenerate as τ ↓ . (2) One can differentiate the above equality a.e. and the optimal transport maps T i ( i = 1 , ) in thetransport of ρ i onto ρ i have the form T i = id + τ (cid:18) mm − ∇ ( ρ + ρ ) m − + ∇ Φ i (cid:19) Proof.
The proof of these results are just easy adaptations of the ones from Lemma A.1, thus we omit it. (cid:3)
Lemma 2.3 (Optimality conditions: m = ∞ ) . Let m = ∞ and let Φ and Φ satisfy (H Φ ) and ( ρ , ρ ) satisfy (H mρ ) . Let ( ρ , ρ ) be the unique minimizer in (MM m ) with k = 0 . Then (1) there exist Kantorovich potentials ϕ i in the transport of ρ i onto ρ i ( i = 1 , ) such that (2.6) ˆ Ω (Φ + ϕ /τ )( µ − ρ )d x + ˆ Ω (Φ + ϕ /τ )( µ − ρ )d x ≥ , for any ( µ , µ ) ∈ P M (Ω) × P M (Ω) such that µ + µ ≤ a.e. in Ω . (2) There exists a Lipschitz continuous pressure function p that can be defined via the Kantorovichpotentials ϕ , ϕ from (1) as (2.7) ∇ p = −∇ ϕ i /τ − ∇ Φ i , ρ i − a . e ., i = 1 , , and p ≥ and p (1 − ( ρ + ρ )) = 0 a.e. in Ω . In particular, the optimal transport map T i in thetransportation of ρ i onto ρ i ( i = 1 , ) has the form T i = id + τ ( ∇ p + ∇ Φ i ) . Proof.
The proof of the above results are adaptations of the ones from [30, Lemma 3.1-3.2] and [25,Lemma 6.11-Proposition 6.12], so we omit it. (cid:3)
Equilibrium solutions when m < + ∞ . Let us study the equilibrium solutions ( ρ , ρ ) of thescheme (MM m ), meaning that ( ρ , ρ ) is a minimizer of the free energy F + G . This exists by the l.s.c.and boundedness from below of the functional and the compactness of P M (Ω) × P M (Ω). Then writingdown the first order optimality conditions as in Lemma 2.2, one obtains that(2.8) ( mm − ( ρ + ρ ) m − = C i − Φ i in { ρ i > } , mm − ( ρ + ρ ) m − ≥ C i − Φ i , in { ρ i = 0 } , or in short mm − ρ + ρ ) m − = max[ C − Φ ; C − Φ ; 0] , for i = 1 , C , C ∈ R . For simplicity in this informal discussion one may supposethat both Φ and Φ are strictly convex with a unique minimizer in Ω. Otherwise the constants C i mayvary on each connected component of { ρ i > } . Observe that the above conditions imply in particularthat whenever the potentials Φ and Φ are different and their difference is not only a constant, then thephases ρ and ρ are separated, i.e. L d (cid:0) { ρ > } ∩ { ρ > } (cid:1) = 0 . Moreover, in general the interface { ρ > }∩{ ρ > } is present and on the interface the densities ρ i ( i = 1 ,
2) are positive. For instance thisis the case when we take potentials Φ ( x ) = | x | and Φ = 2 | x | and C , C are such that 0 < C < C and both densities are present.In fact, with the above choice of potentials Φ i , i = 1 ,
2, suppose that we start our minimizing move-ments with initial configuration of densities ρ = χ {| x |≤ } and ρ = χ { < | x | < } . In the equilibrium limitwe have { ρ > } = {| x | ≤ r } and { ρ > } = { r ≤ | x | ≤ r } for some 0 < r < r . Thus, ifsolutions ( ρ , ρ ) of the system (PME m ) exist with these initial data and potentials, heuristically it isinevitable that the supports of ρ t and ρ t get mixed for some finite time t >
0, while ρ “filtrates” through ρ to change the ordering of their supports from the initial configuration. Such situation indicates lowregularity for each density, and illustrates the difficulty in obtaining a strong notion of limit solutionsfor (PME m ) in the continuum limit. Indeed in general we are only able to obtain a very weak notion ofsolutions in the continuum limit, as we will see in Theorem 3.8. Deriving this weak notion of solutionsin general settings is our first main result in the paper. To the best of the authors’ knowledge, theredoes not seem to be a PDE approach to yield well-posedness on the continuum PDE (PME m ), especiallywhen ∇ Φ = ∇ Φ .On the other hand, if the initial configuration of above example is in line with the potentials, i.e. if weswitch the roles of ρ and ρ , we expect the solutions to be well-behaved and to stay separated throughoutthe evolution, with stable interface in between them. It turns out that we can indeed show such separationin one spacial dimension. In this case stronger results are available, and one can derive stronger notion ofsolutions as well as the properties of the solutions and their interfaces in the incompressible limit m → ∞ ,which in some cases leads to a type of two-phase Hele-Shaw flow with drifts (see Section 4.4).2.6. Regularity of the minimizers in the (MM m ) scheme.Theorem 2.4. Let m ∈ (1 , + ∞ ) . Let ( ρ , ρ ) ∈ P M (Ω) × P M (Ω) satisfying (H mρ ) and let (H Φ ) befulfilled. Let ( ρ , ρ ) be the minimizer in (MM m ) constructed with the help of ( ρ , ρ ) . Then (2.9) ρ , ρ ∈ L m (Ω) and (2.10) ( ρ + ρ ) m − / ∈ H (Ω) . If m = + ∞ , ρ + ρ ≤ a.e. in Ω .Proof. First, setting ρ = ( ρ , ρ ) and ρ = ( ρ , ρ ) , by the optimality of ρ in (MM m ) w.r.t. ρ , oneobtains F m ( ρ ) = 1 m − ˆ Ω ( ρ + ρ ) m d x ≤ τ W ( ρ , ρ ) + F m ( ρ ) + G ( ρ ) − G ( ρ ) ≤ τ W ( ρ , ρ ) + F m ( ρ ) + 2( M k Φ k L ∞ + M k Φ k L ∞ ) , which by the assumptions (H mρ ) and (H Φ ) implies (2.9) for m finite. If m = ∞ , then clearly ρ + ρ ≤ mm − ρ + ρ ) m − + Φ i + ϕ i τ = C i , in { ρ i > } , i = 1 , , where ϕ i is a Kantorovich potential in the optimal transport of ρ i onto ρ i . This potential is linked tothe optimal transport map between these densities as T i ( x ) = x − ∇ ϕ i ( x ) . So, by Lemma 2.2(2) one canwrite(2.11) − mm − ∇ ( ρ + ρ ) m − − ∇ Φ i = ∇ ϕ i τ , ρ i − a . e ., i = 1 , . I. KIM AND A.R. M´ESZ´AROS
Since the r.h.s. of (2.11) is in L ρ i (Ω) with ´ Ω 1 τ |∇ ϕ i | ρ i d x = τ W ( ρ i , ρ i ) and ∇ Φ i ∈ L ρ i (Ω; R d ) wehave the estimation ˆ Ω (cid:12)(cid:12) ∇ ( ρ + ρ ) m − (cid:12)(cid:12) ρ i d x ≤ m − m (cid:18) τ W ( ρ i , ρ i ) + M i k∇ Φ i k L ∞ (cid:19) . Adding up the two inequalities for i = 1 , , one obtains after rearranging(2.12) ˆ Ω (cid:12)(cid:12)(cid:12) ∇ ( ρ + ρ ) m − / (cid:12)(cid:12)(cid:12) d x ≤ m − / m (cid:18) τ W ( ρ , ρ ) + M k∇ Φ k L ∞ + M k∇ Φ k L ∞ (cid:19) . By the estimation (2.9) ρ + ρ is bounded in L m (Ω), so by the fact that Ω is compact, ρ + ρ is summablein L q (Ω) for any 1 ≤ q ≤ m. This means in particular that the average can be bounded as Ω ( ρ + ρ ) m − / d x ≤ k ρ + ρ k m − / L m L d (Ω) / (2 m ) − hence Poincar´e’s inequality yields that k ( ρ + ρ ) m − / k L (Ω) is bounded, more precisely k ( ρ + ρ ) m − / k L (Ω) ≤ C Ω k∇ ( ρ + ρ ) m − / k L (Ω) + L d (Ω) Ω ( ρ + ρ ) m − / d x = C Ω k∇ ( ρ + ρ ) m − / k L (Ω) + k ρ + ρ k m − / L m L d (Ω) / (2 m ) − / where C Ω > . Thus, (2.10) follows. (cid:3) The continuum limit solutions in general dimension
In this section we study the convergence of the time-discrete solutions in the continuum limit. Thelimit solutions can be interpreted as a very weak solution for both systems (PME m ) and (PME ∞ ) in thefollowing sense: Definition 3.1 (Notion of weak solution) . By a weak solution of system (PME m ) we mean a pair ( ρ , ρ ) such that ρ i ∈ AC ([0 , T ]; P M i (Ω)) ∩ L m − ([0 , T ] × Ω) , and setting p := mm − ( ρ + ρ ) m − , ∇ pρ i ∈ L r ([0 , T ] × Ω; R d ) , for some < r < ( i = 1 , ). Moreover ρ i | t =0 = ρ i ( i = 1 , ) and the equation (Weak) − ˆ ts ˆ Ω ρ i ∂ t φ d x d τ + ˆ ts ˆ Ω v i · ∇ φρ i d x d τ = ˆ Ω ρ is ( x ) φ ( s, x )d x − ˆ Ω ρ it ( x ) φ ( t, x )d x, holds true for all φ ∈ C ([0 , T ] × Ω) and for all ≤ s < t ≤ T, where v i := ∇ p + ∇ Φ i . Similarly, by a weak solution of (PME ∞ ) we mean a triple ( ρ , ∞ , ρ , ∞ , p ∞ ) such that ρ i, ∞ ∈ AC ([0 , T ]; P M i (Ω)) ∩ L ∞ ([0 , T ] × Ω) , i = 1 , with k ρ , ∞ + ρ , ∞ k L ∞ ≤ , p ∞ ∈ L ([0 , T ]; H (Ω)) , p ∞ ≥ and p ∞ (1 − ρ , ∞ − ρ , ∞ ) = 0 a.e. in [0 , T ] × Ω . Moreover ρ i | t =0 = ρ i ( i = 1 , ) and the equation (Weak) holds true with p replaced by p ∞ for all φ ∈ C ([0 , T ] × Ω) and for all ≤ s < t ≤ T. We underline that the above weak formulations encode in particular no-flux boundary conditions on [0 , T ] × ∂ Ω . Remark 3.2. (a)
Notice that by density arguments, in the definition of the weak solution of (PME m ) one can consider φ ∈ W , ([0 , T ]; L q (Ω)) ∩ L q ([0 , T ]; W ,q (Ω)) where q = max { r ′ , (2 m − ′ } andin the case of (PME ∞ ) one can consider test functions in W , ([0 , T ]; L (Ω)) ∩ L ([0 , T ]; H (Ω)) . (b) Also, by the fact that we impose that the densities are absolutely continuous curves in the Wasser-stein space , imposing the initial conditions is meaningful. (c) The uniqueness question of weak solutions seems to be very delicate and challenging. See the Appendix on optimal transportation
Interpolations between the densities.
Let m ∈ (1 , + ∞ ] and let us consider ( ρ , ρ ) and Φ andΦ satisfying the hypotheses (H mρ ) and (H Φ ) respectively. We consider also T > τ > N ∈ N such that N τ = T and the densities ( ρ ,τk , ρ ,τk ) Nk =0 obtained via the (MM m )scheme starting from ( ρ , ρ ) . We denote the optimal transport maps and the corresponding Kantorovichpotentials between two consecutive densities ρ i,τk +1 and ρ i,τk by T ik and ϕ ik respectively ( k ∈ { , . . . , N − } , i = 1 , m = ∞ , we consider also the pressure variables p τk ( k ∈ { , . . . , N } ) constructed as in Lemma2.3.Since ∇ ϕ ik τ = id − T ik τ can be seen as a discrete velocity (displacement divided by time), it is reasonableto define the discrete velocity of the particles of the i th fluid located at x ∈ Ω (for a.e. x ∈ Ω) as(3.1) v i,τk ( x ) := ( − mm − ∇ ( ρ ,τk +1 ( x ) + ρ ,τk +1 ( x )) m − − ∇ Φ i ( x ) , if m ∈ (1 , + ∞ ) , −∇ p τk +1 ( x ) − ∇ Φ i ( x ) , if m = ∞ . As technical tools, we shall consider continuous and piecewise constant interpolations between thediscrete densities. We will also work with the associated velocities and momenta. These constructionsand the estimates on them are standard for experts and are very similar to the ones from [43, Chapter8.3] and from [30]. We refer to [42] as well, as an overview of these techniques.
Continuous interpolations . Using McCann’s interpolation – as it is done for instance in [43, Chapter8.3] – we can consider families of continuous interpolations [0 , T ] ∋ t ( ρ ,τt , ρ ,τt ) ∈ P M (Ω) × P M (Ω)between the discrete in time densities parametrized with τ > . We denote the corresponding timedependent families of velocities and momenta by v i,τ , E i,τ .It is worth to notice that the above construction implies in particular that ( ρ i,τ , E i,τ ) ( i = 1 ,
2) solvesthe continuity equation(3.2) ∂ t ρ i,τ + ∇ · E i,τ = 0 . on [0 , T ] × Ω in the weak sense, i.e.(3.3) ˆ ts ˆ Ω ρ i ∂ t φ d x d τ + ˆ ts ˆ Ω E i · ∇ φ d x d τ = − ˆ Ω ρ is ( x ) φ ( s, x )d x + ˆ Ω ρ it ( x ) φ ( t, x )d x for all φ ∈ C ([0 , T ] × Ω) and 0 ≤ s < t ≤ T . Piecewise constant interpolations . We consider a second family of interpolations, simply taking(3.4) ˜ ρ i,τt := ρ i,τk +1 , ˜ v i,τt := v i,τk , and e E i,τt := ˜ ρ i,τt ˜ v i,τt for t ∈ [ kτ, ( k + 1) τ ) . We consider the piecewise constant interpolation for the pressure variable (see Lemma 2.3) as well, i.e.(3.5)˜ p τt := ( C − Φ − ϕ k /τ ) + , in { ρ ,τk +1 > } , ( C − Φ − ϕ k /τ ) + , in { ρ ,τk +1 > } , , in Ω \ (cid:16) { ρ ,τk +1 > } ∪ { ρ ,τk +1 > } (cid:17) , for t ∈ [ kτ, ( k +1) τ ) , k ∈ { , . . . , N − } . In addition we set ˜ ρ τt := (˜ ρ ,τt , ˜ ρ ,τt ) for all t ∈ [0 , T ] . We remark that by construction one has ˜ ρ i,τt = ρ i,τt for t = kτ, k ∈ { , . . . N } . A priori estimates for the interpolations.
We discuss now some estimates on the interpolationsthat will be useful to pass to the limit as τ ↓ . In general, all the constants in the estimates depend on thedata ρ , ρ , Φ , Φ , T and m , however it will be especially important to keep track the precise dependenceof them on m (in particular we use these estimates also in the limiting procedure when m → + ∞ ). Tohighlight this dependence, we denote the constants as C ( m ). Lemma 3.1.
For any m ∈ (1 , + ∞ ] , τ > and any k ∈ { , . . . , N − } one has (3.6) 12 τ N − X k =0 W ( ρ τk +1 , ρ τk ) ≤ F m ( ρ τ ) + G ( ρ τ ) − F m ( ρ τN ) − G ( ρ τN ) , and (3.7) k ρ ,τk +1 + ρ ,τk +1 k L m (Ω) ≤ C ( m ) , where (3.8) C ( m ) := ( (cid:16) (2 m − (cid:0) M k Φ k L ∞ (Ω) + M k Φ k L ∞ (Ω) (cid:1) + k ρ + ρ k mL m (Ω) (cid:17) /m , if m < ∞ , , if m = ∞ . Proof.
The proofs of both inequalities are immediate by the optimality of ρ τk +1 w.r.t. ρ τk in (MM m ). Sowe omit them. (cid:3) Corollary 3.2.
Hypotheses (H Φ ) and (H mρ ) imply that τ N − X k =0 (cid:16) W ( ρ ,τk , ρ ,τk − ) + W ( ρ ,τk , ρ ,τk − ) (cid:17) ≤ C ( m ) , where (3.9) C ( m ) := ( m − k ρ + ρ k mL m + 4 M k Φ k L ∞ + 4 M k Φ k L ∞ , if m ∈ (1 , + ∞ ) , M k Φ k L ∞ + 4 M k Φ k L ∞ , if m = + ∞ . is independent of τ. Lemma 3.3 (Bounds for ρ i,τ , v i,τ and E i,τ ) . Assume that we constructed the discrete densities ρ i,τk for τ > , k ∈ { , . . . , N } and i = 1 , . Let ρ i,τ be the continuous interpolations and let v i,τ and E i,τ be theassociated velocity field and momentum variables respectively. Then (1) ρ i,τ is bounded in AC ([0 , T ]; ( P M i (Ω) , W )) uniformly in τ > ; (2) v i,τ is bounded in L ([0 , T ]; L ρ i,τ (Ω; R d )) uniformly in τ > ; (3) E i,τ and e E i,τ are bounded in M d ([0 , T ] × Ω) uniformly in τ > .Proof. For τ > , by construction ρ i,τ is a constant speed geodesic interpolation with the correspondingvelocity field v i,τ . This implies that ˆ T k v i,τt k L ρi,τt d t = ˆ T | ( ρ i,τ ) ′ | W ( t )d t = N − X k =1 τ W ( ρ i,τk − , ρ i,τk ) ≤ C ( m ) . Now, by Corollary 3.2 we obtain that (1)-(2) hold true.To estimate the total variation of E i,τ we write | E i,τ | ([0 , T ] × Ω) = ˆ T ˆ Ω | v i,τt | ρ i,τt d x d t ≤ ˆ T (cid:18) ˆ Ω | v i,τt | ρ i,τt d x (cid:19) (cid:18) ˆ Ω ρ i,τt d x (cid:19) d t ≤ p M i √ T ˆ T ˆ Ω | v i,τt | ρ i,τt d x d t ! ≤ p M i T C ( m ) . In the last inequality we used the previously obtained bound on v i,τ . The bound on e E i,τ rely on the sameargument. (cid:3) Lemma 3.4 (Bounds on ˜ p τ ) . Let us consider the piecewise constant interpolation [0 , T ] ∋ t ˜ p τt of thepressure variables defined in (3.5) . Then ˜ p τ is bounded L ([0 , T ]; H (Ω)) independently of τ > . Proof.
The proof is similar to the ones in [25, Proposition 6.13] and [36, Lemma 3.6]. We sketch it below.Let us use the fact that ∇ p τk = −∇ ϕ ik /τ − ∇ Φ i , ρ i,τk +1 − a . e . , for all k ∈ { , . . . , N − } where ϕ ik is anoptimal Kantorovich potential in the transport of ρ i,τk +1 onto ρ i,τk . First, let us compute ˆ Ω |∇ p τk | ρ ik +1 d x ≤ τ ˆ Ω |∇ ϕ ik | ρ ik +1 d x + 2 ˆ Ω |∇ Φ i | ρ i d x = 2 τ W ( ρ i,τk +1 , ρ i,τk ) + 2 k∇ Φ i k L ∞ (Ω) , then adding up the two inequalities for i = 1 , p τk is supported on { ρ k +1 + ρ k +1 = 1 } ),one finds ˆ Ω |∇ p τk | d x = ˆ Ω |∇ p τk | ( ρ k +1 + ρ k +1 )d x ≤ X i =1 (cid:18) τ W ( ρ i,τk +1 , ρ i,τk ) + 2 k∇ Φ i k L ∞ (Ω) (cid:19) . Integrating in time k∇ ˜ p τt k L (Ω) using Corollary 3.2 one has that ∇ ˜ p τ ∈ L ([0 , T ] × Ω) . Using the fact that L d ( { ˜ p τt = 0 } ) ≥ L d ( { ˜ ρ ,τt + ˜ ρ ,τt < } ) ≥ L d (Ω) − M − M > t ∈ [0 , T ], one concludes by asuitable version of Poincar´e’s inequality that ˜ p τ uniformly bounded in L ([0 , T ]; H (Ω)) as desired. (cid:3) We show now gradient estimates, derived from the optimality conditions (2.11).
Theorem 3.5.
Let m ∈ (1 , + ∞ ) . Then for the piecewise constant interpolation ˜ ρ i,τ ( i = 1 , ) introducedin (3.4) one has (3.10) k∇ (˜ ρ ,τ + ˜ ρ ,τ ) m − / k L ([0 ,T ] × Ω) ≤ C ( m ) , k (˜ ρ ,τ + ˜ ρ ,τ ) m − / k L ([0 ,T ] × Ω) ≤ C ( m ) , and (3.11) k ˜ ρ ,τ + ˜ ρ ,τ k L q ([0 ,T ]; L m (Ω)) ≤ C ( q, m ) , ∀ q ≥ , where C ( m ) , C ( m ) , C ( q, m ) > are constants independent of τ > . The first two bounds imply inparticular that (3.12) k (˜ ρ ,τ + ˜ ρ ,τ ) m − / k L ([0 ,T ]; H (Ω)) ≤ (cid:0) C ( m ) + C ( m ) (cid:1) / . Proof.
We use the inequality (2.12), writing for ( ρ ,τk +1 , ρ ,τk +1 ), i.e. ˆ Ω (cid:12)(cid:12)(cid:12) ∇ ( ρ ,τk +1 + ρ ,τk +1 ) m − / (cid:12)(cid:12)(cid:12) d x ≤ m − / m X i =1 τ W ( ρ i,τk +1 , ρ i,τk ) + X i =1 M i k∇ Φ i k L ∞ ! . Since the curves ˜ ρ i,τ ( i = 1 ,
2) are piecewise constant interpolations, i.e. ˜ ρ i,τt = ρ i,τk +1 for t ∈ ( kτ, ( k + 1) τ ] , one has ˆ T ˆ Ω (cid:12)(cid:12)(cid:12) ∇ (˜ ρ ,τt + ˜ ρ ,τt ) m − / (cid:12)(cid:12)(cid:12) d x d t = τ N − X k =0 ˆ Ω (cid:12)(cid:12)(cid:12) ∇ ( ρ ,τk +1 + ρ ,τk +1 ) m − / (cid:12)(cid:12)(cid:12) d x ≤ m − / m X i =1 N − X k =0 τ W ( ρ i,τk +1 , ρ i,τk )+ 2( m − / m τ X i =1 N − X k =0 M i k∇ Φ i k L ∞ ≤ m − / m C ( m ) + T X i =1 M i k∇ Φ i k L ∞ ! =: C ( m ) , which implies (3.10), with(3.13) C ( m ) := √ m − / m C ( m ) + T X i =1 M i k∇ Φ i k L ∞ ! / . Similarly, using the estimations from Theorem 2.4 and (3.7), we can write ˆ T k (˜ ρ ,τt + ˜ ρ ,τt ) m − / k L (Ω) d t = τ N − X k =0 k ( ρ ,τk +1 + ρ ,τk +1 ) m − / k L (Ω) ≤ τ N − X k =0 (cid:16) C Ω k∇ ( ρ ,τk +1 + ρ ,τk +1 ) m − / k L (Ω) + k ρ ,τk +1 + ρ ,τk +1 k m − / L m (Ω) L d (Ω) / (2 m ) − / (cid:17) ≤ τ N − X k =0 (cid:16) C k∇ ( ρ ,τk +1 + ρ ,τk +1 ) m − / k L (Ω) + k ρ ,τk +1 + ρ ,τk +1 k m − L m (Ω) L d (Ω) /m − (cid:17) ≤ (cid:16) C C ( m ) + T C ( m ) m − L d (Ω) /m − (cid:17) Thus, the second estimation in (3.10) holds true with(3.14) C ( m ) := √ (cid:16) C C ( m ) + T C ( m ) m − L d (Ω) /m − (cid:17) / . Using (3.7), for any q ≥ ˆ T k ˜ ρ ,τt + ˜ ρ ,τt k qL m (Ω) d t = τ N − X k =0 k ρ ,τk +1 + ρ ,τk +1 k qL m (Ω) ≤ T C ( m ) q . So defining C ( q, m ) := T /q C ( m ) , one obtains the last estimation (3.11) (cid:3) In what follows – using a refined version of the Aubin-Lions lemma – we prove a strong compactnessresult for ˜ ρ ,τ + ˜ ρ ,τ where ˜ ρ ,τ and ˜ ρ ,τ are the piecewise constant interpolations. Proposition 3.6.
Let m ∈ (1 , + ∞ ) . Then the sequence of curves defined as ˜ ρ ,τ n + ˜ ρ ,τ n (for any sequence ( τ n ) n ≥ of positive reals that converges to 0) is strongly pre-compact in L m − ([0 , T ] × Ω) . Proof.
We will use a refined version of the classical Aubin-Lions lemma to prove this result (see [40] andTheorem B.1). Then we will argue as in in [19].Let us set B := L m − (Ω) , F : L m − (Ω) → [0 , + ∞ ] defined as F ( ρ ) := ( k ρ m − / k H (Ω) , if ρ ∈ H (Ω) ∩ P M + M (Ω) , + ∞ , otherwiseand g : L m − (Ω) × L m − (Ω) → [0 , + ∞ ] defined as g ( µ, ν ) := ( W ( µ, ν ) , if µ, ν ∈ P M + M (Ω) , + ∞ , otherwise . In this setting, (cid:0) ˜ ρ ,τ n + ˜ ρ ,τ n (cid:1) n ≥ and F satisfy the assumptions of Theorem B.1. Indeed, from Theorem3.5 one has in particular that ˆ T k (˜ ρ ,τt + ˜ ρ ,τt ) m − / k H (Ω) d t ≤ C ( m ) + C ( m ) . The injection H (Ω) ֒ → L (Ω) is compact, the injection i : η η m − is continuous from L (Ω) to L m − (Ω) and the sub-levelsets of ρ
7→ k ρ m − / k H (Ω) are compact in L m − (Ω).Moreover, by Corollary 3.2, Lemma A.3 and by the fact that g defines a distance on D ( F ), one has that g also satisfies the assumptions from Theorem B.1, hence the implication of the theorem holds and onehas that (cid:0) ˜ ρ ,τ n + ˜ ρ ,τ n (cid:1) n ≥ is pre-compact in M (0 , T ; L m − ) . Finally, the uniform bound (3.7) impliesthe strong pre-compactness of (cid:0) ˜ ρ ,τ n + ˜ ρ ,τ n (cid:1) n ≥ in L m − ([0 , T ] × Ω) . (cid:3) The limit systems as τ ↓ . We proceed with the final step of our scheme, i.e. as the time stepsize goes to zero, we show that along a subsequence the discrete solutions converge to yield a very weaksolution of the PDE systems, in the sense of Definition 3.1. We use the convention of L m − ([0 , T ] × Ω) = L ∞ ([0 , T ] × Ω) whenever m = + ∞ . Proposition 3.7.
Let m ∈ (1 , + ∞ ] and let us consider any sequence ( τ n ) n ≥ which converges to zero.Then, along a subsequence the following holds: (1) There exists ρ i ∈ AC ([0 , T ]; ( P M i (Ω) , W )) ∩ L m − ([0 , T ] × Ω) ( i = 1 , ) s.t. ρ i,τ n → ρ i and ˜ ρ i,τ n → ρ i as n → + ∞ uniformly on [0 , T ] w.r.t. W , in particular weakly − ⋆ in P M i (Ω) for all t ∈ [0 , T ] . (2) There exists E i ∈ M d ([0 , T ] × Ω) ( i = 1 , ) s.t. E i,τ n ⋆ ⇀ E i and e E i,τ n ⋆ ⇀ E i as n → + ∞ . Proof.
In Lemma 3.3 we obtained uniform bounds on the metric derivative of the continuous interpolations ρ i,τ n ( i = 1 , , T ] ∋ t ρ it ∈ P M i (Ω) ,i = 1 , τ n ) ρ i,τ n t → ρ it uniformly on [0 , T ]w.r.t. W as n → + ∞ , in particular weakly- ⋆ in P M i (Ω) for all t ∈ [0 , T ] . The other interpolation ˜ ρ i,τ n coincides with ρ i,τ n at every node point kτ, hence it is straightforwardthat (up to a subsequence taken for τ n ) it converges to the same curve ρ i uniformly on [0 , T ] w.r.t. W . Lemma 3.3 states also that E i,τ n and e E i,τ n are uniformly bounded sequences in M d ([0 , T ] × Ω) , hencethere exist E i ∈ M d ([0 , T ] × Ω) such that (up to a subsequence taken for τ n ) E i,τ n ⋆ ⇀ E i and e E i,τ n ⋆ ⇀ E i ( i = 1 ,
2) in M d ([0 , T ] × Ω) as n → + ∞ . The convergence of E i,τ n and e E i,τ n to the same limit E i followsfrom the same argument as in the proof of [36, Theorem 3.1]. (cid:3) These convergences imply that one can pass to the limit in the weak formulation (3.3) as τ ↓ ρ i , E i ) solves as well the continuity equation(3.15) ∂ t ρ i + ∇ · E i = 0on [0 , T ] × Ω (with initial condition ρ i (0 , · ) = ρ i ) in the same weak sense.In particular, by Lemma 3.3 one has that the sequence (cid:0) B ( ρ i,τ n , E i,τ n ) (cid:1) n ∈ N is uniformly boundedfor any positive vanishing sequence ( τ n ) n ∈ N , where B denotes the Benamou-Brenier action functional(see its precise definition and properties in Appendix A). In particular, by the lower semicontinuity ofthis functional, there exists v i such that v it ∈ L ρ i (Ω; R d ) for a.e. t ∈ [0 , T ] and at the limit (as τ ↓ E i = v i · ρ i . This implies further that the equation (3.15) has the form(3.16) ∂ t ρ i + ∇ · ( v i ρ i ) = 0 . Precise form of the limit systems.
Now we shall work with the piecewise constant interpolations˜ ρ i,τ , i = 1 , e E i,τ , i = 1 , Theorem 3.8.
Let m ∈ (1 , + ∞ ) and let ˜ ρ i,τ , i = 1 , be the piecewise constant interpolations betweenthe densities ( ρ i,τk ) Nk =0 and e E i,τ , i = 1 , the corresponding momentum variables. Taking any sequence ( τ n ) n ≥ that goes to zero, the following holds along a subsequence: (1) (cid:0) ˜ ρ ,τ n + ˜ ρ ,τ n (cid:1) n ≥ converges strongly in L m − ([0 , T ] × Ω) to ρ + ρ ; (2) (cid:16) e E ,τ n + e E ,τ n (cid:17) n ≥ , i = 1 , converges in the sense of distributions to −∇ ( ρ + ρ ) m − ∇ Φ ρ −∇ Φ ρ , i.e., (3.17) v ρ + v ρ = −∇ ( ρ + ρ ) m − ∇ Φ ρ − ∇ Φ ρ in the sense of distributions on [0 , T ] × Ω . Proof.
Let us show (1). Proposition 3.6 implies already that ˜ ρ ,τ n + ˜ ρ ,τ n (up to some subsequencethat we do not relabel) converges strongly in L m − ([0 , T ] × Ω) . Also, by Proposition 3.7 we have that˜ ρ i,τ n t ⋆ ⇀ ρ it as n → ∞ for all t ∈ [0 , T ] . Hence the limit of (˜ ρ ,τ n + ˜ ρ ,τ n ) n ≥ is precisely ρ + ρ and ρ i ∈ L m − ([0 , T ] × Ω) ∩ AC ([0 , T ]; ( P M i (Ω) , W )) , i = 1 , . We show now (2) . By definition of e E i,τ on has that e E ,τ + e E ,τ = −∇ (˜ ρ ,τ + ˜ ρ ,τ ) m − ∇ Φ ˜ ρ ,τ − ∇ Φ ˜ ρ ,τ . Since by (1) ˜ ρ ,τ n + ˜ ρ ,τ n → ρ + ρ strongly in L m − ([0 , T ] × Ω) as n → + ∞ , one has that (˜ ρ ,τ n +˜ ρ ,τ n ) m → ( ρ + ρ ) m strongly in L − m ([0 , T ] × Ω) as n → + ∞ . This, together with the weak − ⋆ convergence of (˜ ρ i,τ n ) n ≥ to ρ i implies the first part of the statement. On the other hand one hasobtained already that e E i,τ n ⇀ E i = v i ρ i as n → + ∞ , thus (3.17) follows as well. (cid:3) Remark 3.3. (1)
Let us underline the fact that it is unclear whether we could show a strongerversion of Theorem 3.8(2), i.e. the convergence (up to passing to a subsequence) of (cid:16) e E i,τ n (cid:17) n ≥ to − mm − ρ i ∇ ( ρ + ρ ) m − − ∇ Φ i ρ i i = 1 , , which is necessary in order to obtain the weakformulation of the PDE system at the limit . (2) In Theorem 3.8 if in addition (cid:0) ˜ ρ i,τ n (cid:1) n ≥ either for i = 1 or i = 2 converges a.e. in [0 , T ] × Ω then both sequences ( i = 1 , ) converge strongly in L m − ([0 , T ] × Ω) to ρ i and the correspondingmomentum (cid:16) e E i,τ n (cid:17) n ≥ ( i = 1 , ) converge in the sense of distributions to − mm − ρ i ∇ ( ρ + ρ ) m − −∇ Φ i ρ i . The study of a stable scenario when this holds true is the subject of Section 4.Proof of Remark 3.3(2).
First, clearly the pointwise convergence of (cid:0) ˜ ρ i,τ n (cid:1) n ≥ and the strong convergenceof (cid:0) ˜ ρ ,τ n + ˜ ρ ,τ n (cid:1) n ≥ in L m − ([0 , T ] × Ω) by a suitable version of Vitali’s convergence theorem imply thestrong convergence of (cid:0) ˜ ρ i,τ n (cid:1) n ≥ in L m − ([0 , T ] × Ω) to ρ i . Since one of the terms in the sum of thesesequences and the sum itself converges strongly, so does the other term as well. Let us recall the formula e E i,τ n = − mm − ρ i,τ n ∇ (˜ ρ ,τ n + ˜ ρ ,τ n ) m − − ∇ Φ ˜ ρ ,τ n . The strong convergence of (cid:0) ˜ ρ i,τ n (cid:1) n ≥ implies that the second term of e E i,τ n , i.e. −∇ Φ i ˜ ρ i,τ n (since ∇ Φ i is in L ∞ (Ω; R d )) converges strongly in L m − ([0 , T ] × Ω; R d ) to −∇ Φ i ρ i . Thus in particular weakly- ⋆ in M d ([0 , T ] × Ω) . The first term of e E i,τ n can be written as − mm − ρ i,τ n ∇ (˜ ρ ,τ n + ˜ ρ ,τ n ) m − = − mm − (cid:2) ∇ (˜ ρ ,τ n + ˜ ρ ,τ n ) m − (cid:3) (˜ ρ ,τ n + ˜ ρ ,τ n ) / ˜ ρ i,τ n (˜ ρ ,τ n + ˜ ρ ,τ n ) / , and notice furthermore that − mm − (cid:2) ∇ (˜ ρ ,τ n + ˜ ρ ,τ n ) m − (cid:3) (˜ ρ ,τ n + ˜ ρ ,τ n ) / = − mm − / ∇ (˜ ρ ,τ n + ˜ ρ ,τ n ) m − / . Theorem 3.5 implies that the sequence (cid:16) − mm − / (˜ ρ ,τ n + ˜ ρ ,τ n ) m − / (cid:17) n ≥ is uniformly bounded in thespace L ([0 , T ]; H (Ω)) , hence there exists a subsequence (not relabeled) and some ξ ∈ L ([0 , T ]; H (Ω))such that (cid:16) − mm − / (˜ ρ ,τ n + ˜ ρ ,τ n ) m − / (cid:17) n ≥ is converging weakly to ξ as n → + ∞ . In particular, − mm − / (˜ ρ ,τ n + ˜ ρ ,τ n ) m − / ⇀ ξ weakly in L ([0 , T ] × Ω) and − mm − / ∇ (˜ ρ ,τ n + ˜ ρ ,τ n ) m − / ⇀ ∇ ξ weakly in L ([0 , T ] × Ω; R d ) as n → + ∞ . By Proposition 3.6 one has that (cid:0) ˜ ρ ,τ n + ˜ ρ ,τ n (cid:1) n ≥ converges strongly to ρ + ρ in L m − ([0 , T ] × Ω) , which implies in particular that (cid:0) (˜ ρ ,τ n + ˜ ρ ,τ n ) m − / (cid:1) n ≥ converges strongly in L ([0 , T ] × Ω) to ( ρ + ρ ) m − / . This together with the above weak convergences implies that ∇ ξ = − mm − / ∇ ( ρ + ρ ) m − / ∈ L ([0 , T ] × Ω; R d ) . Now, by the strong convergence of ˜ ρ i,τ n to ρ i in L m − ([0 , T ] × Ω) one has that˜ ρ i,τ n (˜ ρ ,τ n + ˜ ρ ,τ n ) / → ρ i ( ρ + ρ ) / as n → + ∞ pointwisely a.e. in [0 , T ] × Ω . Moreover, since the densities are non-negative one has ˜ ρ i,τn (˜ ρ ,τn +˜ ρ ,τn ) / ≤ (˜ ρ i,τ n ) and (˜ ρ i,τ n ) converges to ( ρ i ) strongly in L m − ([0 , T ] × Ω) . Hence Lebesgue’s dominatedconvergence theorem implies that ˜ ρ i,τ n (˜ ρ ,τ n + ˜ ρ ,τ n ) / → ρ i ( ρ + ρ ) / as n → + ∞ , strongly in L m − ([0 , T ] × Ω) . Gluing together the two previous results, one obtains that − mm − ∇ (˜ ρ ,τ n + ˜ ρ ,τ n ) m − ˜ ρ i,τ n convergesweakly to − mm − ∇ ( ρ + ρ ) m − ρ i in L r ([0 , T ] × Ω; R d ) as n → + ∞ , where + m − + r = 1 , i.e. r = m − m − >
1. So in particular the convergence is weakly- ⋆ in M d ([0 , T ] × Ω), which together withthe strong convergence of the term ∇ Φ i ˜ ρ i,τ n implies the thesis. (cid:3) Theorem 3.9.
Let m = + ∞ and let and let us consider ˜ ρ i,τ , i = 1 , the piecewise constant interpolationsbetween the densities ( ρ i,τk ) Nk =0 and e E i,τ , i = 1 , the corresponding momentum variables. Let us considermoreover p τ the piecewise constant interpolations between the pressure variables ( p τk ) N − k =0 . Let us takeany positive sequence ( τ n ) n ≥ such that τ n ↓ as n → + ∞ , and let us consider the weak limit p of ( p τ n ) n ≥ in L ([0 , T ]; H (Ω)) and ρ i the limit of ( ρ i,τ n ) n ≥ in L ∞ ([0 , T ]; ( P M i (Ω) , W )) (up to passingto a subsequence that we do not relabel). Then we have the following: p (1 − ( ρ + ρ )) = 0 a . e . in [0 , T ] × Ω . Proof.
First notice that by Lemma 3.4 and Proposition 3.7 the weak limits p and ρ i ( i = 1 ,
2) exist.Furthermore, (1) follows from the previously mentioned results, Lemma A.3 and [36, Lemma 3.5]. (cid:3)
Remark 3.4.
In Theorem 3.9 if ˜ ρ ,τ n and ˜ ρ ,τ n are such that L d ( { ˜ ρ ,τ n t > } ∩ { ˜ ρ ,τ n t > } ) = 0 for all t ∈ [0 , T ] and for all n ∈ N , then e E i,τ n ⋆ ⇀ −∇ p − ∇ Φ i ρ i , as n → + ∞ , in M d ([0 , T ] × Ω) , i = 1 , . If moreover L d ( { ρ t > } ∩ { ρ t > } ) = 0 for a.e. t ∈ [0 , T ] , then ∇ p = ρ i ∇ p, a . e . in [0 , T ] × Ω , i = 1 , . Proof.
To show the remark let us recall the form of the momentum variables, i.e. e E i,τ n = ˜ v i,τ n ˜ ρ i,τ n = −∇ p τ n ˜ ρ i,τ n − ∇ Φ i ˜ ρ i,τ n = −∇ p τ n − ∇ Φ i ˜ ρ i,τ n , where the last equality holds true since p τ n (1 − (˜ ρ ,τ n + ˜ ρ ,τ n )) = 0 for all t ∈ [0 , T ] and a.e. in Ω andby the assumption L d ( { ˜ ρ ,τ n t > } ∩ { ˜ ρ ,τ n t > } ) = 0 for all t ∈ [0 , T ], one has { ˜ ρ ,τ n t + ˜ ρ ,τ n t = 1 } = a . e . { ˜ ρ ,τ n t = 1 }∪{ ˜ ρ ,τ n t = 1 } for all t ∈ [0 , T ] and the two sets are disjoint a.e. in Ω. By the weak convergencesof ( p τ n ) n ≥ to p and (˜ ρ i,τ n ) n ≥ to ρ i ( i = 1 ,
2) one can easily conclude that e E i,τ n ⋆ ⇀ −∇ p − ∇ Φ i ρ i , as n → + ∞ , in M d ([0 , T ] × Ω) , i = 1 , . Also, ∇ p = ρ i ∇ p, i = 1 , , T ] × Ω follows easily from (1) and the assumption L d ( { ˜ ρ t > } ∩ { ˜ ρ t > } ) = 0 for all t ∈ [0 , T ], as desired. (cid:3) Segregation of the densities.
As mentioned in the introduction, it seems natural to look forinitial configurations of the system (PME m ) where there is no mixing of the densities, to strengthen ourconvergence results in the continuum limit. We shall describe such initial configurations in one spacedimension in the next section. Here we describe some properties of the time-discrete solutions (obtainedby the JKO scheme) which hold for all dimensions. In particular, we show that when Φ = Φ + C (forsome C ∈ R ), then the densities stay segregated if initially they were so. We derive also some propertiesof the mixed region, when the two initial densities are mixed in a special way. Still, these statements areonly true for time-discrete solutions, and we cannot rule out the possibility that the limiting densities endup mixed, for instance, due to “fingering” phenomena (see also the numerical observations in [28] whichdisplays fingering phenomena when a system, similar to (2.4), has unstable combination of diffusionconstants and source terms). It seems that additional geometric property is required to preserve thesegregation property in the continuum limit. Proposition 3.10.
Let m ∈ (1 , + ∞ ] . Let us assume moreover that Φ and Φ are such that Φ = Φ + C on Ω for some C ∈ R , with the hypothesis (H Φ ) fulfilled. Let ( ρ , ρ ) ∈ P M (Ω) × P M (Ω) satisfy (H mρ ) and let ( ρ , ρ ) be the minimizers in (MM m ) constructed with the help of ( ρ , ρ ) . Then the followingstatements hold true. (1) If L d (cid:0) { ρ > } ∩ { ρ > } (cid:1) = 0 , then L d (cid:0) { ρ > } ∩ { ρ > } (cid:1) = 0 . (2) Let us define the Borel measurable sets A := { ρ > } ∩ { ρ > } and B := { ρ > } ∩ { ρ > } .Let us suppose that L d ( A ) > and L d ( B ) > . If there exists r > such that rρ ≤ ρ a.e. in A , then rρ ≤ ρ a.e. in B .Proof. Let us use the notation ∇ Φ := ∇ Φ = ∇ Φ . Using (2.11), the optimal transport maps T i ( i = 1 , ρ i onto ρ i (see Lemma 2.2-2.3) can be written ( ρ i − a . e . ) as T i = ( id + τ (cid:16) mm − ∇ ( ρ + ρ ) m − + ∇ Φ (cid:17) , if m ∈ (1 , + ∞ ) , id + τ ( ∇ p + ∇ Φ) , if m = ∞ . In particular, observe that T = T a.e. in { ρ > } ∩ { ρ > } . We show (1). Suppose that the Borel measurable set B := { ρ > } ∩ { ρ > } has positive Lebesguemeasure. For any x ∈ B such that x is a Lebesgue point of ρ , ρ and T , T and T ( x ) = T ( x ) is aLebesgue point for both ρ and ρ , one has (since T ( x ) = T ( x )) that T ( x ) = T ( x ) ∈ { ρ > } ∩{ ρ > } . In particular the positive mass of each ρ i ( i = 1 ,
2) on B is transported onto { ρ > }∩{ ρ > } .On the other hand, since both ρ and ρ are absolutely continuous w.r.t. L d , this mass cannot besupported on an L d -null set, which is a contradiction to the assumption L d (cid:0) { ρ > } ∩ { ρ > } (cid:1) = 0.We show (2). First observe that one can write two Jacobian equation in a weak sense, i.e.(3.18) det( DT i ) = ρ i ρ i ◦ T i , ρ i − a . e .. Since the measures ρ i , i = 1 , T i are differentiable ρ i − a . e . and the previous equation holds true pointwisely ρ i − a . e . (see for instance [17, Theorem 3.1]). Let us choose x ∈ B such that it is a Lebesgue point of both ρ and ρ and itis a point of differentiability of both T and T (in particular ρ ( x ) > ρ ( x ) > ρ and ρ , one may assume that T ( x ) := T ( x ) = T ( x ) is a Lebesgue point of both ρ and ρ . The Jacobian equation (3.18) yields that ρ ( x ) /ρ ( x ) = ρ ( T ( x )) /ρ ( T ( x )) ≥ r , which concludes the proof. (cid:3) Segregated weak solutions in 1D
In this section we study the local segregation property of the supports for the time-discrete solutions.As a consequence we show the existence of segregated weak solutions of the systems (PME m ) and (PME ∞ )in one spacial dimension.4.1. Separation of the supports and ordering property in one space dimension.Framework Hyp-1D.
We set the following geometric framework (see also Figure 1 below for illustra-tion). (1) d = 1 , Ω a bounded open interval, the potentials Φ i , i = 1 , are semi-convex and C (Ω) ; (2) The drifts are ‘ordered’, in the sense that ∂ x Φ ( x ) ≥ ∂ x Φ ( x ) for all x ∈ Ω . This means inparticular that Φ − Φ is increasing; (3) ρ and ρ are two densities such that for a.e. x ∈ { ρ > } and y ∈ { ρ > } one has that y < x .We refer to this last property as “ordering of the supports” of the initial densities. This impliesin particular that L ( { ρ > } ∩ { ρ > } ) = 0 . Let us point out that the assumptions from Hyp-1D immediately imply with reasoning parallel toProposition 3.10 that L ( { ρ > } ∩ { ρ > } ) = 0 for one-step minimizers ( ρ , ρ ) given by (MM m ).To see this, suppose { ρ > } ∩ { ρ > } = a.e B for some Borel measurable set B such that L ( B ) > . As in the proof of Proposition 3.10, The optimal transport map T i ( i = 1 ,
2) in the transport of ρ i onto ρ i is given by T i = ( id + τ (cid:16) mm − ∂ x ( ρ + ρ ) m − + ∂ x Φ i (cid:17) , if m ∈ (1 , + ∞ ) , id + τ ( ∂ x p + ∂ x Φ i ) , if m = ∞ . The above formula and the assumption ∂ x Φ − ∂ x Φ ≥ T ( x ) ≥ T ( x ) a.e. in B , whichcontradicts the ordering property of the initial data.Still, this separation property is not enough to iterate over time steps unless the ordering property ofthe initial configuration is preserved for ( ρ , ρ ). This is what we prove next. Proposition 4.1.
Let m ∈ (1 , + ∞ ] and suppose the assumptions in (Hyp-1D) and the hypotheses (H Φ ) - (H mρ ) are in place. Let us denote by ( ρ , ρ ) the one-step time discrete solutions given by (MM m ) for k = 0 . Then the ordering property from (Hyp-1D) holds true for { ρ > } and { ρ > } . ρ − ∂ x Φ ρ − ∂ x Φ Figure 1.
Ordering of the supports of the initial data
Proof.
Suppose the contrary, i.e. there exist B ⊆ { ρ > } and B ⊆ { ρ > } with L ( B ) > L ( B ) > x ∈ B and y ∈ B x < y (see Figure 2 for illustration). Claim: there exist E i ⊆ B i , i = 1 , θ > δ > L ( E ) = L ( E ) > E = E + θ and ρ i ≥ δ a.e. on E i , i = 1 , Proof of the claim.
Let us take x ∈ B , y ∈ B Lebesgue points. This means in particular that ρ ( x ) > , ρ ( y ) > r ↓ B r ( x ) (cid:12)(cid:12) ρ ( x ) − ρ ( x ) (cid:12)(cid:12) d x = 0 , lim r ↓ B r ( y ) (cid:12)(cid:12) ρ ( x ) − ρ ( y ) (cid:12)(cid:12) d x = 0 . Now let us take r > δ := min (cid:8) ρ ( x ) / , ρ ( y ) / (cid:9) . Let us considermoreover the measurable sets ˜ E ⊆ B ∩ B r ( x ) and ˜ E ⊆ B ∩ B r ( y ) defined as ˜ E i := (cid:8) ρ i ≥ δ (cid:9) , i =1 , . By construction, for r > L ( B r ( x ) \ ˜ E ) / L ( B r ( x )) ≤ / L ( B r ( y ) \ ˜ E ) / L ( B r ( y )) ≤ / . Indeed, one has B r ( x ) (cid:12)(cid:12) ρ ( x ) − ρ ( x ) (cid:12)(cid:12) d x ≥ L ( B r ( x )) ˆ B r ( x ) \ ˜ E (cid:12)(cid:12) ρ ( x ) − ρ ( x ) (cid:12)(cid:12) d x ≥ ρ ( x )2 L ( B r ( x ) \ ˜ E ) L ( B r ( x )) , and by (4.1) the l.h.s. tends to 0 as r ↓ , so for r > L ( B r ( x ) \ ˜ E ) / L ( B r ( x )) ≤ / ρ and ˜ E . Fix such an r > . Furthermore, set θ := y − x and define E := ˜ E ∩ ( ˜ E − θ ) and E := E + θ. Thus, L ( E ) L ( B r ( y )) = L ( E ) L ( B r ( x )) ≥ − L ( B r ( x ) \ ˜ E ) L ( B r ( x )) − L ( B r ( y ) \ ˜ E ) L ( B r ( y )) = 13 . This finishes the proof of the claim, since r > ρ , ˜ ρ ) in (MM m ) which has less energy (we refer to Figure 2 forthe illustration) than ( ρ , ρ ), yielding the contradiction. Define ˜ ρ and ˜ ρ as˜ ρ = ρ , in Ω \ ( E ∪ E ) ,ρ − δ, in E ,δ, in E , and ˜ ρ = ρ , in Ω \ ( E ∪ E ) ,δ, in E ,ρ − δ, in E . We construct corresponding transport maps (not necessarily optimal ones), ˜ T between ˜ ρ and ρ and˜ T between ˜ ρ and ρ as˜ T = (cid:26) T , in Ω \ E ,T ( · − θ ) , in E , and ˜ T = (cid:26) T , in Ω \ E ,T ( · + θ ) , in E . By construction ˜ T i ˜ ρ i = ρ i , i = 1 ,
2. Let us use the notation E := T ( E ) and E := T ( E ), these areBorel measurable sets and subsets of { ρ > } and { ρ > } respectively. ρ ρ x E ρ y E θδ ρ E ρ E Figure 2.
Ordering property for { ρ > } and { ρ > } (on the right). This is isviolated by { ρ > } and { ρ > } (on the left)Notice that by construction ˜ ρ + ˜ ρ = ρ + ρ in Ω, hence(4.2) F m (˜ ρ , ˜ ρ ) = F m ( ρ , ρ ) . Now let us see how the other two energy terms in (MM m ) change by considering (˜ ρ , ˜ ρ ) as competitors.Let us use the notation h ( x ) := ∂ x Φ ( x ) − ∂ x Φ ( x ). First, E G := G (˜ ρ , ˜ ρ ) − G ( ρ , ρ ) = ˆ Ω Φ ˜ ρ d x + ˆ Ω Φ ˜ ρ d x − ˆ Ω Φ ρ d x − ˆ Ω Φ ρ d x = δ (cid:18) ˆ E Φ ( x )d x − ˆ E Φ ( x )d x (cid:19) + δ (cid:18) ˆ E Φ ( x )d x − ˆ E Φ ( x )d x (cid:19) = ˆ E δ [(Φ ( x + θ ) − Φ ( x + θ )) − (Φ ( x ) − Φ ( x ))] d x = ˆ E δθ [ ∂ x Φ ( ξ x,θ ) − ∂ x Φ ( ξ x,θ )] d x where in the last equality we used the mean value theorem and ξ x,θ is some point in ( x, x + θ ). Wecompute now the change in the W terms. Recall the structure of the transport maps ˜ T i , i = 1 , E W := 12 τ W (˜ ρ , ρ ) + 12 τ W (˜ ρ , ρ ) − τ W ( ρ , ρ ) − τ W ( ρ , ρ ) ≤ τ ˆ E | x − T ( x − θ ) | δ d x + 12 τ ˆ E | x − T ( x + θ ) | δ d x − τ ˆ E | x − T ( x ) | δ d x − τ ˆ E | x − T ( x ) | δ d x = 12 τ ˆ E (cid:0) | x + θ − T ( x ) | − | x − T ( x ) | (cid:1) δ d x + 12 τ ˆ E (cid:0) | x − θ − T ( x ) | − | x − T ( x ) | (cid:1) δ d x = δθτ ˆ E ( T ( x + θ ) − T ( x ))d x where η i = δ · L E i and η i = T i η i , i = 1 , . Now, it is easy to see that E G + E W <
0. Indeed, by the assumptions (2) from (Hyp-1D) one has that ∂ x Φ − ∂ x Φ nonpositive, thus(4.3) E G + E W ≤ δθ ˆ E (cid:26) [ ∂ x Φ ( ξ x,θ ) − ∂ x Φ ( ξ x,θ )] + 1 τ [ T ( x + θ ) − T ( x )] (cid:27) d x is negative since by the assumption (3) from (Hyp-1D) T ( x + θ ) − T ( x ) < E G + E W <
0, which together with (4.2) imply that (˜ ρ , ˜ ρ ) is a bettercompetitor than ( ρ , ρ ). This is clearly a contradiction to the uniqueness of the minimizer in (MM m ).Thus the ordering property for { ρ > } and { ρ > } follows. (cid:3) Discussion on possibly mixed initial data.
Extending the above proposition to more generalcases seems to be challenging, due to possible presence of the mixing zone { ρ > } ∩ { ρ > } . The mainissue, for instance to localize our argument, would be to ensure the finite propagation of mixing zone.The only available result in this direction arises in the case of the stiff pressure limit, m = ∞ , Φ i = c i x ,and with full saturation, that is when we have the constraint ρ + ρ = 1. In this case Otto ([38]) showedin one dimensional setting that there is a unique description of the mixing zone that propagates withfinite speed generated by the entropy solution of a conservation law. While we are not sure whether thesame uniqueness results hold for our undersaturated case, we believe that the mixing zone should travelwith finite speed at least in one dimension.4.3. Existence of a solution for (PME m ) supposing (Hyp-1D) .Theorem 4.2. Let us suppose that m ∈ (1 , + ∞ ] and the setting of (Hyp-1D) takes place. Let us consider ( ρ , ρ ) to be any subsequential limit (uniformly in time w.r.t. W ) of the piecewise constant interpolationcurves (˜ ρ ,τ n , ˜ ρ ,τ n ) when τ n ↓ , with the initial densities ( ρ , ρ ) . Then ( ρ , ρ ) satisfies L (cid:0) { ρ t > } ∩ { ρ t > } (cid:1) = 0 , ∀ t ∈ [0 , T ] and the sets { ρ t > } and { ρ t > } are ordered in the sense of (Hyp-1D) for all t ∈ [0 , T ] . Proof.
First, let us recall that the ordering of { ρ > } and { ρ > } in (Hyp-1D) is such that { ρ > } is to the left of { ρ > } . Second, let us underline that by Proposition 3.7 (1) ρ i is obtained as the uniform limit in time w.r.t. W (as τ ↓
0) of the piecewise constant interpolation curves ˜ ρ i,τ ( i = 1 , τ > I ,τ , I ,τ : [0 , T ] → Ωdefined as I ,τ ( t ) := inf n x : x ∈ Leb (cid:16) { ˜ ρ ,τt > } (cid:17)o and I ,τ ( t ) := sup n x : x ∈ Leb (cid:16) { ˜ ρ ,τt > } (cid:17)o . These functions are well-defined, since Ω is bounded and in particular Proposition 4.1 implies that I ,τ ( t ) ≤ I ,τ ( t ) for all t ∈ [0 , T ] and for any τ >
0. Also, by the boundedness of Ω, these functions areuniformly bounded in t and τ .Let us take a sequence ( τ n ) n ≥ , s.t. τ n ↓ n → + ∞ and sup t ∈ [0 ,T ] W (˜ ρ i,τt , ρ it ) → n → + ∞ ,( i = 1 , (cid:0) I i,τ n ( t ) (cid:1) n ≥ is a bounded sequence for each t ∈ [0 , T ], so up to passing to a subsequence (thatwe do not relabel), it has a poitwise limit as n → + ∞ that we denote by I i ( t ) for t ∈ [0 , T ] and i = 1 , Claim: (1) ρ t ( y ) = 0 for a.e. y > I ( t ) and (2) ρ t ( x ) = 0 for a.e. x < I ( t ). Proof of the claim.
Let us suppose that the claim is false, i.e. the first statement fails to be true (theproof of (2) is parallel). Then there exits r > δ > ˆ I ( t )+2 rI ( t )+ r ρ t ( x )d x > δ > . But, for n ∈ N large enough such that (cid:12)(cid:12) I ,τ n ( t ) − I ( t ) (cid:12)(cid:12) < r/ W (˜ ρ ,τ n t , ρ t ) ≥ ( r/ ˆ I ( t )+2 rI ( t )+ r ρ t ( x )d x = ( r/ δ, which yields a contradiction to the fact that W (˜ ρ ,τt , ρ t ) → n → + ∞ . A similar argument can beperformed to show (2), thus the claim follows.Now, since I ,τ n ( t ) ≤ I ,τ n ( t ) for all n ∈ N and t ∈ [0 , T ] , after passing to subsequences if necessary,one has that I ( t ) ≤ I ( t ) for any limit points I ( t ) , I ( t ) and for all t ∈ [0 , T ] . This together with theClaim imply that L (cid:0) { ρ t > } ∩ { ρ t > } (cid:1) = 0 , ∀ t ∈ [0 , T ] and that the sets { ρ t > } and { ρ t > } areordered in the sense of (Hyp-1D) for all t ∈ [0 , T ] . The result follows. (cid:3)
Remark 4.1.
When m ∈ (1 , + ∞ ) , the above result allows to obtain the strong convergence result of thedensity sequences (˜ ρ i,τ n ) n ≥ , i = 1 , separately. When m = + ∞ , together with Proposition 4.1 this resultis crucial to fulfill the hypotheses in Remark 3.4, which will lead to the precise weak form of the (PME ∞ ) system. Theorem 4.3.
Let us suppose that m ∈ (1 , + ∞ ) and the setting of (Hyp-1D) takes place. Consider thepiecewise constant interpolations ˜ ρ i,τ n ( i = 1 , ) for some ( τ n ) n ≥ such that τ n ↓ as n → + ∞ . Then upto passing to a subsequence with ( τ n ) n ≥ , (cid:0) ˜ ρ i,τ n (cid:1) n ≥ ( i = 1 , ) converges strongly in L m − ([0 , T ] × Ω) ,in particular pointwise a.e. in [0 , T ] × Ω . Proof.
Let us show first that (˜ ρ i,τ n ) n ≥ (up to passing to a subsequence) converges strongly to ρ i ( i =1 ,
2) in L ([0 , T ] × Ω). We pass to subsequences if necessary (that we do not relabel) to ensure that(˜ ρ ,τ n + ˜ ρ ,τ n ) n ≥ converges strongly to ρ + ρ in L m − ([0 , T ] × Ω) and (˜ ρ i,τ n ) n ≥ converges to ρ i ( i = 1 ,
2) weakly in L m − ([0 , T ] × Ω) as n → + ∞ . We compute k ˜ ρ i,τ n − ρ i k L ([0 ,T ] × Ω) = ˆ [0 ,T ] × Ω | ˜ ρ i,τ n − ρ i | d t ⊗ d x = ˆ { ρ i > } | ˜ ρ i,τ n − ρ i | d t ⊗ d x + ˆ ([0 ,T ] × Ω) \{ ρ i > } ˜ ρ i,τ n d t ⊗ d x = ˆ { ρ i > } | ˜ ρ ,τ n + ˜ ρ ,τ n − ( ρ + ρ ) | d t ⊗ d x + ˆ ([0 ,T ] × Ω) \{ ρ i > } ˜ ρ i,τ n d t ⊗ d x ≤ ˆ [0 ,T ] × Ω | ˜ ρ ,τ n + ˜ ρ ,τ n − ( ρ + ρ ) | d t ⊗ d x + ˆ T ˆ ([0 ,T ] × Ω) \{ ρ i > } ˜ ρ i,τ n d t ⊗ d x → , as n → + ∞ , where in the third equality we used the facts (see Theorem 4.2) that ρ i = 0 a.e. in { ρ i +1 > } and˜ ρ i,τ n = 0 a.e. in { ρ i +1 > } , with the convention i + 1 = 1 , when i = 2. Moreover, both terms in the lastsum converge to 0. Indeed, the convergence of the first term is a consequence of the strong convergenceof (˜ ρ ,τ n + ˜ ρ ,τ n ) n ≥ to ρ + ρ in L m − ([0 , T ] × Ω) as n → + ∞ . The convergence to 0 of the last termis a consequence of the weak convergence of ˜ ρ i,τ n to ρ i in L m − ([0 , T ] × Ω) . This together with Theorem 3.5 and Proposition 3.6 imply that (up to passing to a subsequence) (cid:0) ρ i,τ n (cid:1) n ≥ converges strongly in L m − ([0 , T ] × Ω). (cid:3)
We state now the results on the existence of weak solutions of the PDE systems (PME m ) and (PME ∞ ). Theorem 4.4.
Let us assume that m ∈ (1 , + ∞ ) , the hypotheses (H mρ ) and (H Φ ) are fulfilled and thesetting in (Hyp-1D) takes place. Then the system (PME m ) has a weak solution ( ρ , ρ ) in the sense of (Weak) such that ρ i ∈ L m − ([0 , T ] × Ω) ∩ AC ([0 , T ]; ( P M i (Ω) , W )) , i = 1 , and ( ρ + ρ ) m − / ∈ L ([0 , T ]; H (Ω)) . In addition, ρ i ∈ L q ([0 , T ] × Ω) for all ≤ q ≤ m and E i := − mm − ∂ x ( ρ + ρ ) m − ρ i − ∂ x Φ i ρ i belongs to L r ([0 , T ] × Ω; R d ) for some ≤ r < with uniform bounds in m . Lastly, if m → + ∞ ,q can be arbitrary large and r can be chosen arbitrary close to 2.Proof. By Theorem 3.8 one has that the limit densities ρ and ρ belong to L m − ([0 , T ] × Ω) ∩ AC ([0 , T ]; ( P M i (Ω) , W )) . The same theorem establishes the convergence of (˜ ρ i,τ , e E i,τ ) and the precise form of the limit. By thefact that (˜ ρ i,τ , e E i,τ ) and ( ρ i,τ , E i,τ ) converge weakly as measures to the same limit ( ρ i , E i ) and by thefact that this latter pair solves the continuity equation (3.15) in the weak sense (3.3), so does the preciselimit of (˜ ρ i,τ , e E i,τ ) developed in Remark 3.3(2) (notice that by Theorem 4.3 the assumptions in Remark3.3(2) are fulfilled). This means in particular that the limit equation reads (for i = 1 ,
2) as ∂ t ρ i − ∂ x (cid:18) mm − ∂ x ( ρ + ρ ) m − ρ i + ∂ x Φ i ρ i (cid:19) = 0 , that has to be understood in the weak sense (Weak) with no-flux boundary condition.Finally, let us obtain the uniform (w.r.t m ) bounds on ρ i and E i . First, by Theorem 3.5 (3.11) one hasthat the limit curves are bounded in L p ([0 , T ]; L m (Ω)) for all p ≥ p = m and any 1 ≤ q ≤ m . Then H¨older’s inequality yields(4.4) k ρ i k L q ([0 ,T ] × Ω) ≤ ( T L (Ω)) m − qqm k ρ i k L m ([0 ,T ] × Ω) ≤ T q L (Ω) m − qqm C ( m ) . Second, let us write E i = mm − / ∂ x ( ρ + ρ ) m − / ρ i ( ρ + ρ ) / + ∂ x Φ i ρ i . Notice that by (3.10) (Theorem 3.5) the L bound for ∂ x ( ρ + ρ ) m − / remains the same after passingto the limit with the time step τ . Also, by the previous bound on ρ i , ρ i ( ρ + ρ ) / is bounded uniformly in L q ([0 , T ] × Ω) (since ρ i ( ρ + ρ ) / ≤ ( ρ i ) / a.e.). These observations, together with the fact that ∂ x Φ i isuniformly bounded let us conclude by H¨older’s inequality that(4.5) k E i k L r ≤ mm − / k ∂ x ( ρ + ρ ) m − / k L k ρ i k / L r/ (2 − r ) + k ∂ x Φ i k L ∞ k ρ i k L r , provided 1 ≤ r < n r − r , r o ≤ q. Thus the thesis of the theorem follows. (cid:3)
Lemma 4.5.
Let m ∈ (1 , + ∞ ) and let us consider ( ρ , ρ ) the solution of (PME m ) supposing (Hyp-1D) with given initial data ( ρ , ρ ) . We assume – similarly to the hypotheses (H mρ ) in the m = + ∞ case –that the measure of Ω is large enough, i.e. (4.6) L (Ω) > ( M + M ) , where M i denotes the total mass of ρ i . Then – uniformly in m – we have the following regularity estimates (1) ( ρ + ρ ) m ∈ L ([0 , T ]; C ,α (Ω)) for some < α < / and in particular it is uniformly boundedin L r ([0 , T ] × Ω) for some < r < ; (2) ( ρ + ρ ) m − / ∈ L ([0 , T ]; C , / (Ω)) and in particular it is uniformly bounded in L ([0 , T ] × Ω) .Proof. We show (1). Using the notations from Theorem 4.4, one has that E + E = − ∂ x ( ρ + ρ ) m − ∂ x Φ ρ − ∂ x Φ ρ . By the estimations from Theorem 4.4 we know that the quantities E + E and ∂ x Φ ρ + ∂ x Φ ρ arebounded uniformly in L r ([0 , T ] × Ω; R d ) for some 1 < r < ∂ x ( ρ + ρ ) m is uniformly bounded in L r ([0 , T ] × Ω; R d ). Furthermore, thePoincar´e-Wirtinger inequality yields that(4.7) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( ρ + ρ ) m − T L (Ω) ˆ T ˆ Ω ( ρ + ρ ) m d x d t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L r ([0 ,T ] × Ω) ≤ (cid:13)(cid:13) ∂ x ( ρ + ρ ) m (cid:13)(cid:13) L r ([0 ,T ] × Ω) . So the l.h.s. is uniformly bounded. Let us show that the average of ( ρ + ρ ) m is uniformly bounded.Let us fix 0 < ε < − ( M + M ) / L (Ω). Claim 1. For every t ∈ [0 , T ] there exists r > and x ∈ Ω such that ρ t + ρ t ≤ − ε a.e. in B r ( x ) . Fix t ∈ [0 , T ]. Let us suppose that the claim is not true. Then in every ball B r ( x ), ρ t + ρ t > − ε a.e. Since Ω is a bounded interval, this in particular means that ρ t + ρ t > − ε a.e. in Ω. Furthermore, ˆ Ω ( ρ t + ρ t )d x > (1 − ε ) L (Ω) > M + M , and this is clearly a contradiction (by the choice of ε ) to fact that ´ Ω ( ρ t + ρ t )d x = M + M , thus theclaim follows. Claim 2. ( ρ + ρ ) m ∈ L ([0 , T ]; C ,α (Ω)) for some < α < / . In particular, for a.e. t ∈ [0 , T ] , ( ρ t + ρ t ) m has bounded oscillation uniformly in m . Notice that for f : [0 , T ] × Ω → R measurable such that ∂ x f ∈ L r ([0 , T ] × Ω) for some r > a, b ∈ Ω , a < b defining osc [ a,b ] f t := sup x ∈ [ a,b ] f t ( x ) − inf x ∈ [ a,b ] f t ( x ) , one has the estimate ˆ T (osc [ a,b ] f t )d t ≤ ˆ T ˆ ba | ∂ x f t | d x d t ≤ ˆ T ˆ ba | ∂ x f t | r d x d t ! r ( T | a − b | ) r ′ ≤ ˆ T ˆ Ω | ∂ x f t | r d x d t ! r ( T | a − b | ) r ′ , where 1 /r + 1 /r ′ = 1. Since the integrand on the l.h.s. of the previous inequality is non-negative, thisimplies that osc [ a,b ] f t ≤ C | a − b | r ′ for a.e. t ∈ [0 , T ] , hence in particular f t has bounded oscillation, with a constant that depends only on k ∂ x f k L r and T . Applying this reasoning to ( ρ + ρ ) m , one obtains the statement of the claim.Now Claim 1 and Claim 2 imply that ( ρ t + ρ t ) m is uniformly bounded for a.e. t ∈ T . This meansfurthermore that the average T L (Ω) ´ T ´ Ω ( ρ + ρ ) m d x d t is uniformly bounded, which together with(4.7) implies (1).The proof of (2) follows the same lines. The bound k ∂ x ( ρ ,m + ρ ,m ) m − / k L ([0 ,T ] × Ω) ≤ C ( m ) in(3.10) from Theorem 3.5 remains uniform, since C ( m ) remains bounded uniformly when m → + ∞ . Thisbound is enough to perform the same analysis as in (1), thus we can conclude the same way. (cid:3) Theorem 4.6.
Let us assume that m = + ∞ , the hypotheses (H mρ ) and (H Φ ) are fulfilled and the settingin (Hyp-1D) takes place. Then the system (PME ∞ ) has a weak solution ( ρ , ρ , p ) in the sense of (Weak) such that ρ i ∈ L ∞ ([0 , T ] × Ω) ∩ AC ([0 , T ]; ( P M i (Ω) , W )) , i = 1 , and p ∈ L ([0 , T ]; H (Ω)) . One hasmoreover ρ + ρ ≤ a.e. in [0 , T ] × Ω , p ≥ , p (1 − ρ i ) = 0 a.e. in { ρ i > } , i = 1 , .Proof. Let us take a positive vanishing sequence of time steps ( τ n ) n ≥ and consider the piecewise con-stant and continuous interpolations of density curves (˜ ρ i,τ n ) n ≥ , ( ρ i,τ n ) n ≥ and momenta ( e E i,τ n ) n ≥ ,( E i,τ n ) n ≥ . By Proposition 3.7 (up to passing to subsequences) these objects converge (to ρ i and E i respectively) in the appropriate weak senses and one has a limit system as in (3.15)-(3.16). To identify aprecise form of the system, we use the fact that the momentum sequences ( e E i,τ n ) n ≥ and ( E i,τ n ) n ≥ andthe curve sequences (˜ ρ i,τ n ) n ≥ and ( ρ i,τ n ) n ≥ converge to the same limit.Now observe that the setting in (Hyp-1D) implies that Theorem 4.2 can be applied, so the assumptionsof Remark 3.4 are fulfilled. This implies that the limit momenta have the form E i = − ∂ x p − ∂ x Φ i ρ i = − ∂ x pρ i − ∂ x Φ i ρ i , i = 1 , . Here p ∈ L ([0 , T ]; H (Ω)) is the weak limit of ( p τ n ) n ≥ obtained in Theorem 3.9, so in particular p ≥ p (1 − ( ρ + ρ )) = 0 a.e. in [0 , T ] × Ω . These imply that the limit system has the form ∂ t ρ i − ∂ x (cid:0) ∂ x pρ i + ∂ x Φ i ρ i (cid:1) = 0 , i = 1 , , which has to be understood in the weak sense with no-flux boundary conditions.At last, since Theorem 4.2 implies in particular that L ( { ρ t > } ∩ { ρ t > } ) = 0 for all t ∈ [0 , T ],the relation p (1 − ( ρ + ρ )) = 0 a.e. in [0 , T ] × Ω reads as p (1 − ρ i ) = 0 a.e. in { ρ i > } , i = 1 , (cid:3) It is not hard to verify that, for a fixed τ >
0, the functionals in (MM m ) Γ-convergence as m → ∞ tothe functional where F m is replaced by F ∞ . Thus, it is natural to pose the question about the convergenceof the corresponding gradient flow solutions in the spirit of Sandier and Serfaty (see [41]). Unfortunately,one cannot use these kinds of results directly, and obtain the convergence of the continuum solutions of(PME m ) to the solutions of (PME ∞ ), mainly due to the lack of uniqueness. Hence, it is necessary toproceed by studying the convergence of the continuum solutions at the PDE level. This will be addressedin the next section.4.4. Passing to the limit as m → + ∞ . We will show that solutions of (PME m ) converge, along asubsequence as m → + ∞ , to a solution of (PME ∞ ).We suppose that the initial data satisfy k ρ + ρ k L ∞ ≤ . Let us recall (see Definition 3.1) that a triple of nonnegative functions ( ρ , ∞ , ρ , ∞ , p ∞ ), such that ρ i, ∞ ∈ AC ([0 , T ]; P M i (Ω)), k ρ , ∞ + ρ , ∞ k L ∞ ([0 ,T ] × Ω) ≤
1, and p ∞ ∈ L ([0 , T ]; H (Ω)) , is a weaksolution of (PME ∞ ) if for any φ ∈ C ([0 , T ] × Ω) and 0 < s < t ≤ T we have − ˆ ts ˆ Ω ρ i, ∞ ∂ t φ d x d τ − ˆ ts ˆ Ω ( ∂ x p + ∂ x Φ i ) ρ i, ∞ · ∂ x φ d x d τ = ˆ Ω ρ i, ∞ ( s, x ) φ ( s, x )d x − ˆ Ω ρ i, ∞ ( t, x ) φ ( t, x )d x, and p ∞ (1 − ρ , ∞ − ρ , ∞ ) = 0 a.e. in [0 , T ] × Ω. Theorem 4.7.
Let ( ρ ,m , ρ ,m ) be a weak solution to (PME m ) in the setting of (Hyp-1D) with initialdata satisfying k ρ + ρ k L ∞ ≤ , where now we have noted m as a parameter. We assume moreover thatthe geometric condition (4.6) holds true for the domain Ω .Then, there exist ρ i, ∞ ∈ L ∞ ([0 , T ] × Ω) ∩ AC ([0 , T ]; ( P M i (Ω) , W )) and p ∞ ∈ L r ([0 , T ]; W ,r (Ω)) for all r ∈ (1 , , such that along a subsequence when m → + ∞ , ρ i,m ⇀ ρ i, ∞ weakly in L q ([0 , T ] × Ω) for any q ≥ , ( ρ ,m + ρ ,m ) m ⇀ p ∞ in L r ([0 , T ]; W ,r (Ω)) and E i,m ⇀ ∂ x p ∞ ρ i, ∞ + ∂ x Φ i ρ i, ∞ weakly in L r ([0 , T ] × Ω) .Moreover ( ρ , ∞ , ρ , ∞ , p ∞ ) is a weak solution of (PME ∞ ) .Proof. Let us recall the weak formulation of the system (PME m ).(4.8) − ˆ ts ˆ Ω ρ i,m ∂ t φ d x d τ − ˆ ts ˆ Ω E i,m · ∂ x φ d x d τ = ˆ Ω ρ i,ms ( x ) φ ( s, x )d x − ˆ Ω ρ i,mt ( x ) φ ( t, x )d x, for all 0 ≤ s < t ≤ T and φ ∈ C ([0 , T ] × Ω) , where E i,m := − (cid:16) mm − ∂ x ( ρ ,m + ρ ,m ) m − + ∂ x Φ i (cid:17) ρ i,m . First, by the assumption k ρ + ρ k L ∞ ≤
1, the bounds for ρ i,m in AC ([0 , T ]; ( P M i (Ω) , W )) (seeLemma 3.3 and (3.9)) are uniform in m , so clearly up to passing to a subsequence with m , ( ρ i,m ) m> converges weakly- ⋆ to some ρ i, ∞ ∈ AC ([0 , T ]; ( P M i (Ω) , W )). In particular this convergence is uniformin time w.r.t. W . By the uniform estimation (4.4), it follows that along a subsequence ( ρ i,m ) m> converges weakly to ρ i, ∞ in L q ([0 , T ] × Ω) for all q ≥
1. In particular, these weak convergences allow usto obtain that in (4.8) the first term on the l.h.s., and both terms on the r.h.s. pass to the limit.Second, Lemma 4.5 ensures the uniform boundedness of ( ρ ,m + ρ ,m ) m in L r ([0 , T ]; W ,r (Ω)) forsome r ∈ (1 ,
2) (where r can be chosen arbitrarily close to 2 for m large enough) hence there exists p ∞ ∈ L r ([0 , T ]; W ,r (Ω)) such that up to passing to a subsequence in m , ( ρ ,m + ρ ,m ) m ⇀ p ∞ weakly in L r ([0 , T ]; W ,r (Ω)). In particular, one has also that ∂ x ( ρ ,m + ρ ,m ) m ⇀ ∂ x p ∞ weakly in L r ([0 , T ] × Ω).Notice that the convergence E i,m ⇀ ∂ x p ∞ ρ i, ∞ + ∂ x Φ i ρ i, ∞ weakly in L r ([0 , T ] × Ω) is much moredelicate, since both terms in the product − mm − ∂ x ( ρ ,m + ρ ,m ) m − ρ i,m (in the definition of E i,m ) convergeonly weakly.We shall provide the convergence E i,m ⇀ ∂ x p ∞ ρ , ∞ + ∂ x Φ i ρ i, ∞ only for i = 2, the other case isanalogous. Observe that by the uniform estimation (in m ) on E ,m in L r ([0 , T ] × Ω) (for some 1 < r < E , ∞ ∈ L r ([0 , T ] × Ω) such that up to passing to a subsequence, E ,m ⇀ E , ∞ weakly in L r ([0 , T ] × Ω) as m → + ∞ . Now let us identify the limit E , ∞ . Let us fix a subsequence (thatfor simplicity of notation we denote by m ), such that ( ρ ,m ) m> converges weakly to ρ , ∞ and ( E ,m ) m> converges weakly to E , ∞ as m → + ∞ in the previously described spaces.Let us fix 0 ≤ s < t ≤ T . For all τ ∈ [ s, t ], we define I ,m ( τ ) be the “right-most point” of the supportof ρ ,mτ , i.e. I ,m ( τ ) := sup (cid:8) x : x ∈ Leb (cid:0) { ρ ,mτ > } (cid:1)(cid:9) and let us define the set I = (cid:8) ( τ, x ) ∈ [ s, t ] × Ω : ∃ ( τ n , x n ) n ≥ , s.t. x n = I ,m n ( τ n ) , and ( τ n , x n ) → ( τ, x ) as n → ∞ (cid:9) , where ( m n ) n ≥ is a subsequence of the previously chosen subsequence. Then I is a closed subset of[ s, t ] × Ω. Let I ( τ ) := I ∩ ( { τ } × Ω). Note that in particular I ( τ ) is the collection of all subsequentiallimits of I ,m ( τ ) . Observe that if for some τ ∈ ( s, t ), y ∈ Ω lies to the left of I ( τ ), i.e. if y < x for any x ∈ I ( τ ), then( τ, y ) lies in the complement of { ρ ,m > } for sufficiently large m , and thus defining J − := { ( τ, y ) ∈ ( s, t ) × Ω : y < x for any x ∈ I ( τ ) } , one has that when restricted to J − , E ,m = − ∂ x ( ρ ,m ) m − ∂ x Φ ρ ,m , in the sense of distributions for sufficiently large m . Similarly, defining J + := { ( τ, y ) ∈ ( s, t ) × Ω : y > x for any x ∈ I ( τ ) } , one has that ρ ,m = 0 a.e. on J + , hence when restricted to J + , E ,m = 0 a.e. for sufficiently large m. Clearly, J − , I , J + are Lebesgue measurable and one can write ( s, t ) × Ω = J − ∪ I ∪ J + , L [0 , T ] ⊗ L Ω − a.e. Thus, we write furthermore ˆ ts ˆ Ω E ,m · ∂ x φ d x d τ = ˆ J − E ,m · ∂ x φ d τ ⊗ d x + ˆ I E ,m · ∂ x φ d τ ⊗ d x + ˆ J + E ,m · ∂ x φ d τ ⊗ d x = ˆ J − E ,m · ∂ x φ d τ ⊗ d x + ˆ I E ,m · ∂ x φ d τ ⊗ d x. Moreover, the very same decomposition remains valid for the weak limit E , ∞ as well. Claim 1. ´ I E , ∞ · ∂ x φ d τ ⊗ d x can be made arbitrarily small for any smooth test function φ . If ( L ⊗ L )( I ) = 0, then this is obvious. So one can suppose that this set has positive measure. Toshow the claim, let us define the width of I ( τ ), i.e. W ( τ ) := max {| x − y | : x, y ∈ I ( τ ) } = x τ − x τ , where x τ := max { x : x ∈ I ( τ ) } and x τ := min { x : x ∈ I ( τ ) } and these are well-defined since I ( τ ) iscompact. Let T n := (cid:8) τ ∈ ( s, t ) : W ( τ ) ≥ n (cid:9) . Then
I ⊂ A n ∪ B n , where A n := [ τ ∈ T n (cid:0) { τ } × ( x τ + 1 /n, x τ ) (cid:1) and B n = I \ A n . Note that A n and B n are Lebesgue measurable and the measure of B n in [0 , T ] × Ω is at most 2
T /n ,which goes to zero as n → ∞ . t x Figure 3.
The sets I (with orange) and A n (with blue)We know that W ( ρ i,mτ , ρ , ∞ τ ) → m → + ∞ , uniformly in τ . This implies in particular that thesequence ( ρ ,mτ ) m are Cauchy w.r.t. W uniformly in τ , which means that for any n ∈ N , there exists N ( n ) > τ ∈ [0 , T ](4.9) W ( ρ ,m k τ , ρ ,m l τ ) ≤ n , ∀ k, l > N ( n ) , where m k and m l denote elements of the sequence denoted by m . Now, let us define A n ( τ ) := A n ∩{ τ }× Ω . Let us show that for all τ ∈ ( s, t ) on A n ( τ ) all elements of the sequence ( ρ ,mτ ) m have small mass.Indeed, on the one hand, for any point x ∈ A n ( τ ) ∩ I ( τ ) (if the intersection is ∅ , then there is nothingto show), there is a subsequence ˜ m of m such that I , ˜ m ( τ ) → x as ˜ m → + ∞ . On the other hand, since x τ ∈ I ( τ ) , there exists another subsequence m of m such that I ,m ( τ ) → x τ as m → + ∞ . The inequality(4.9) implies that 1 n ˆ A n ( τ ) ρ , ˜ mt d x ≤ W ( ρ , ˜ mτ , ρ ,mτ ) ≤ n for n large enough and ˜ m and m larger than N ( n ). The first inequality holds because of the fact that x − x τ ≥ /n . Thus for all τ ∈ ( s, t ) ˆ A n ( τ ) ρ ,mτ d x ≤ /n, for any m large enough. Considering any smooth test function φ supported in A n ∪ J + , the weak formulation (4.8) togetherwith the fact that E ,m and ρ ,m vanish on J + yield (cid:12)(cid:12)(cid:12)(cid:12) ˆ A n E ,m · ∂ x φ d t ⊗ d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup A n | ∂ t φ | Tn + sup A n | φ | n , for sufficiently large m . This together with the fact that the measure of B n is at most 2 T /n and
I ⊂ A n ∪ B n implies that ´ I E ,m · ∂ x φ d t ⊗ d x is arbitrary small provided m is large enough, whichimplies in particular that ´ I E , ∞ · ∂ x φ d t ⊗ d x is as small as we would like.The above claim shows that one needs to describe the weak limit E , ∞ only on the set J − . Whenrestricted to J − , one can write E ,m = − ∂ x ( ρ ,m ) m − ∂ x Φ ρ ,m =: − ∂ x p m − ∂ x Φ ρ ,m , and the second term on the r.h.s. of the previous formula passes to the limit due to the weak convergenceof ( ρ ,m ) m> . Let us consider a smooth test function φ compactly supported in J − . Then one haslim m → + ∞ − ˆ J − ∂ x p m · ∂ x φ d τ ⊗ d x = lim m → + ∞ ˆ J − p m ∂ xx φ d τ ⊗ d x = lim m → + ∞ ˆ J − ( ρ ,m ) m ∂ xx φ d τ ⊗ d x = lim m → + ∞ ˆ ts ˆ Ω ( ρ ,m + ρ ,m ) m ∂ xx φ d x d τ = ˆ ts ˆ Ω p ∞ ∂ xx φ d x d τ = − ˆ ts ˆ Ω ∂ x p ∞ ∂ x φ d x d τ = − ˆ J − ∂ x p ∞ · ∂ x φ d τ ⊗ d x Hence when restricted to J − , E , ∞ = − ∂ x p ∞ − ∂ x Φ ρ , ∞ . Note that ρ , ∞ = 0 as well in I ∪ J + , so wecan write E , ∞ = − ∂ x p ∞ χ { ρ , ∞ > } − ∂ x Φ ρ , ∞ . Below we will show that p ∞ vanishes in ρ , ∞ <
1. This allows us to write E , ∞ = − ∂ x p ∞ ρ , ∞ − ∂ x Φ ρ , ∞ . Similar reasoning yields the concrete form of E , ∞ as well.Let us show that k ρ i, ∞ k L ∞ ≤
1. By Lemma 4.5 we know that for a.e. t ∈ [0 , T ], ˆ Ω ( ρ i,mt ) m d x ≤ C ,where the constant C is independent of m . Thus for any δ >
0, on the set where ρ i,mt ≥ δ a.e. wehave by Chebyshev’s inequality that(4.10) (1 + δ ) m L ( { ρ i,mt ≥ δ } ) ≤ ˆ { ρ i,mt ≥ δ } ( ρ i,mt ) m d x ≤ ˆ Ω ( ρ i,mt ) m d x ≤ C. This implies L ( { ρ i,mt ≥ δ } ) ≤ C/ (1 + δ ) m → m → ∞ , and thus by the arbitrariness of δ > ρ i, ∞ ≤ , T ] × Ω.At last, it remains to show that p ∞ is supported in the region { ρ , ∞ + ρ , ∞ = 1 } . Notice that sinceTheorem 4.2 yields that L ( { ρ , ∞ t > } ∩ { ρ , ∞ t > } ) = 0 for all t ∈ [0 , T ], it is enough to show that p ∞ (1 − ρ i, ∞ ) = 0 a.e. in { ρ i, ∞ > } i = 1 ,
2. We show this property only in the case of i = 2, the othercase is analogous. Let us use the notations p m := ( ρ ,m + ρ ,m ) m = a . e . ( ρ ,m ) m , in { ρ , ∞ > } ⊆ a . e . J − ˜ p m := ( ρ ,m + ρ ,m ) m − / = a . e . ( ρ ,m ) m − / = ( p m ) − m , in { ρ , ∞ > } ⊆ a . e . J − Claim 2. When m → + ∞ , ˆ T ˆ Ω p m (1 − ρ ,m )d x d t and ˆ T ˆ Ω ˜ p m (1 − ρ ,m )d x d t are arbitrary small. Let δ > , T ] ∋ t ´ Ω χ { ρ ,mt ≥ δ } d x is a measurablefunction and by (4.10), [0 , T ] ∋ t ´ Ω χ { ρ ,mt ≥ δ } d x is integrable. Similar properties are valid for other characteristic functions of the (sub)level sets. Thus, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ T ˆ Ω p m (1 − ρ ,m )d x d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ˆ T ˆ { ρ ,mt < − δ } p m (1 − ρ ,m )d x d t + ˆ T ˆ { − δ ≤ ρ ,mt ≤ δ } p m | − ρ ,m | d x d t + ˆ T ˆ { ρ ,mt > δ } p m ( ρ ,m − x d t ≤ (1 − δ ) m T L (Ω) + δC + k p m k L q ([0 ,T ] × Ω) k ρ ,m k L q ([0 ,T ] × Ω) ˆ T ˆ { ρ ,mt > δ } x d t ! /q ≤ (1 − δ ) m T L (Ω) + δC + (cid:18) T C (1 + δ ) m (cid:19) /q , where q + q + q = 1, q < r, q ≤ m and C is independent of m . Here we used also the uniformestimations from Lemma 4.5(1) and Theorem 4.4. Now the last sum is as small as desired by choosing δ > m large enough, which shows the first part of the claim. A similar reasoning and Lemma4.5(2) yield the second part of the claim.Last, one can use the same arguments as in [30, Lemma 3.4, Step 3. in the proof of Theorem 2.1.]to conclude that p ∞ (1 − ρ , ∞ ) = 0 a.e. in J − . Indeed, we know that ˜ p m is uniformly bounded in L ([0 , T ]; H (Ω)) and by Lemma 4.5(2) ρ ,mt is uniformly bounded for a.e. t ∈ [0 , T ]. These togetherwith Claim 2 imply in particular that mentioned results from [30] can be applied to obtain that˜ p ∞ (1 − ρ , ∞ ) = 0 , a.e. in J − , where ˜ p ∞ is the weak limit of (˜ p m ) m> in L ([0 , T ]; H (Ω)) as m → + ∞ . It remains to show onlythat p ∞ = ˜ p ∞ a.e. in J − . This is straight forward. Indeed, notice that ˜ p m = ( p m ) − m a.e. in J − .Lemma 4.5 implies that for a.e. t ∈ [0 , T ] both p m ( t, · ) and ˜ p m ( t, · ) are (uniformly) H¨older continuousand converge uniformly (up to passing to a subsequence). This means that p ∞ t and ˜ p ∞ t are the uniformlimits for a.e. t ∈ [0 , T ] and p ∞ t = ˜ p ∞ t for a.e. t ∈ [0 , T ]. The result follows. (cid:3) Characterization of the pressure in the case of m = + ∞ . Here we establish the optimalregularity of the pressure, which is Lipschitz continuity for a.e. time. The pressure can be discontinuousin time even in the single density case, when two components of the congested density zone merge intoone (see the discussion in [23] for instance.)
Proposition 4.8.
Let ( ρ , ∞ , ρ , ∞ , p ∞ ) be a solution of the system (PME ∞ ) in the setting of (Hyp-1D) .Then p ∞ ( t, · ) is uniformly Lipschitz continuous for a.e. t ∈ [0 , T ] . Moreover, p ∞ ( t, · ) is as smooth as Φ i in the interior of the sets { ρ i, ∞ = 1 } , i = 1 , for a.e. t ∈ [0 , T ] . Let us remark first that p ∞ may have positive boundary data on ∂ { ρ , ∞ = 1 } ∩ ∂ { ρ , ∞ = 1 } . Also,notice that the set { ρ i, ∞ = 1 } may have empty interior. Proof.
Considering the sum of the two weak equations tested against smooth test functions supported inthe interior of { ρ , ∞ t + ρ , ∞ t = 1 } one obtains that(4.11) − ∂ xx p ∞ ( t, · ) = ∂ x ( ∂ x Φ ρ , ∞ + ∂ x Φ ρ , ∞ ) , in the interior of { ρ , ∞ t + ρ , ∞ t = 1 } , for a.e. t ∈ [0 , T ] with homogeneous Dirichlet boundary data (this is because of the fact that p ∞ ( t, · ) isH¨older continuous a.e. t ∈ [0 , T ]). Since L (cid:16) { ρ , ∞ t > } ∩ { ρ , ∞ t > } (cid:17) = 0 and the sets { ρ , ∞ t > } and { ρ , ∞ t > } are ordered in the sense of (Hyp-1D), one gets(4.12) − ∂ xx p ∞ ( t, · ) = ∂ xx Φ i , in the interior of { ρ i, ∞ t = 1 } . Therefore p ∞ ( t, · ) is as smooth as Φ i in the interior of the sets { ρ i, ∞ t = 1 } , i = 1 ,
2, for a.e. t ∈ [0 , T ].Let us remark also that since p ∞ ( t, · ) is H¨older continuous, the set { p ∞ ( t, · ) > } ⊆ { ρ , ∞ t + ρ , ∞ t = 1 } is open. Thus p ∞ ( t, · ) is as smooth as Φ i in { p ∞ ( t, · ) > } ∩ int { ρ i, ∞ t = 1 } . Let us show that p ∞ ( t, · ) is Lipschitz continuous for a.e. t ∈ [0 , T ]. Notice that by the previousarguments, p ∞ ( t, · ) fails to be differentiable in at most countably many points of Ω, and except thesepoints it is smooth. By the fact that p ∞ ( t, · ) is a Sobolev function, we know that it is absolutely continuousfor a.e t ∈ [0 , T ]. Moreover | ∂ x p ∞ ( t, · ) | ≤ C, a . e . in Ω , where C > {k ∂ x Φ i k L ∞ : i = 1 , } . This together withthe absolute continuity imply that p ∞ ( t, · ) Lipschitz continuous for a.e. t ∈ [0 , T ] . (cid:3) Patch solutions.Proposition 4.9.
Let ( ρ , ∞ , ρ , ∞ , p ∞ ) be a solution of the system (PME ∞ ) in the setting of (Hyp-1D) .Let us suppose that ρ , ∞ and ρ , ∞ are patches , i.e. ρ i, ∞ = χ A i for open intervals A i in Ω , i = 1 , . Wesuppose moreover that the drifts − ∂ x Φ i , i = 1 , are compressive, meaning that ∂ xx Φ i ≥ on Ω . Then ρ i, ∞ t , i = 1 , is a patch for all t ∈ [0 , T ] , i.e. there exists { A i ( t ) } t ∈ [0 ,T ] : a family of open intervals suchthat ρ i, ∞ t = χ A i ( t ) . Remark 4.2.
While we believe our argument can be extended to measurable sets instead of open intervals,we do not pursue this generalization for simplicity.Proof.
Let us recall that the sets { ρ , ∞ t > } and { ρ , ∞ t > } are ordered for all t ∈ [0 , T ] in the sense of(Hyp-1D), with ρ , ∞ supported to the left of ρ , ∞ . We show the proposition only for ρ , ∞ , the case of ρ , ∞ is analogous.We define an extension ˜ p of p ∞ to the right of the support of ρ , ∞ . For a.e. t where p ∞ satisfies (4.12),let x ( t ) := sup Leb (cid:16) { ρ , ∞ t > } (cid:17) . If p ∞ ( t, x ( t )) = 0 then we let ˜ p ( t, · ) = p ∞ ( t, · ). If p ( t, x ( t )) >
0, thismeans x ( t ) lies in the interior of { ρ , ∞ t + ρ , ∞ t = 1 } . In this case let us define ˜ p to be a C extension of p ∞ to the right of x ( t ) such that − ∂ xx ˜ p = ∂ xx Φ .Let us consider the test function φ , as the solution of the transport equation ∂ t φ t − v ε ∂ x φ = 0with initial condition φ (0 , · ) = χ A , where we define v ε := v⋆η ε with v = ( ∂ x ˜ p + ∂ x Φ i ) and η ε is a standardmollifier (the mollification being performed only w.r.t. the space variable). Then k v ε ( t, · ) − v ( t, · ) k L q ≤ Cε in the set { ρ , ∞ t > } for any q > t ∈ [0 , T ], where C depends on the Lipschitz constant of p ∞ and Φ . Then, from the weak expression we have (cid:12)(cid:12)(cid:12)(cid:12) ˆ Ω ( ρ , ∞ φ )( t, · )d x − ˆ Ω ( ρ , ∞ φ )(0 , · )d x (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ˆ Ω ( v ε − v ) ρ , ∞ ∂ x φ d x d τ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cε.
Since φ solves a transport equation with spatially smooth velocity which is integrable in time, this equationis well-posed and φ can be represented using the method of characteristics. In particular φ ( t, · ) = χ A t for some measurable set A t for each time t >
0. Hence the l.h.s. of the above equation is(4.13) ˆ Ω ( ρ , ∞ φ )( t, · )d x − ˆ Ω ( ρ , ∞ φ )(0 , · )d x = ˆ A t ρ , ∞ t d x − ˆ A x. Now we claim that(4.14) ∂ x v ε ≥ . This is true because ˜ p satisfies − ∂ xx ˜ p = ∂ xx Φ (or ˜ p = p ∞ ) to the right of x ( t ) as well as in any interiorpoint of { ρ , ∞ t = 1 } , and to the left of x ( t ) we have ˜ p = p ∞ = max { p ∞ , } , which makes p ∞ convex atany boundary point of the set { ρ , ∞ = 1 } to the left of x ( t ). Thus we conclude that ∂ x v = ∂ xx p ∞ + ∂ xx Φ ≥ −∞ , x ( t )) ∩ Ω , in the distributional sense, which yields ∂ x v ε ≥ Due to (4.14) we have L ( A t ) ≤ L ( A ). This, the fact that ρ , ∞ t ≤ ˆ A t (1 − ρ , ∞ t )d x ≤ ˆ A t (1 − ρ , ∞ t )d x + L ( A ) − L ( A t ) ≤ Cε.
We still need to let ε ↓ A is an interval then A t is an interval(that depends on ε >
0) with uniformly bounded velocity with respect to ε , hence along a subsequencethe endpoints (as a function of t they are equicontinuous) uniformly converge to limiting endpoints( a ( t ) , b ( t )) { t> } as ε ↓
0. Let A ( t ) := ( a ( t ) , b ( t )). Here ρ , ∞ t should be identically one (because of theprevious inequality). But this means that along this subsequence, v was incompressible except in a smallset in A t , so that makes L ( A t ) very close to L ( A ) and | b ( t ) − a ( t ) | = L ( A ). Since ρ , ∞ preservesmass over time, this means that ρ , ∞ t = χ A ( t ) . (cid:3) It remains to describe the evolution of the patches { ρ i, ∞ = 1 } . As we see in [24] in [35], the evolutionlaws are different depending on whether there are regions of the densities with values between zero andone. In the above Proposition, we have patch solutions supported on an interval, and the continuity ofthe densities over time in W -distance yields that each patch { ρ i, ∞ = 1 } evolves continuously in time.Therefore it follows that the space-time interior of those sets taken at time t equals spatial interior attime t . Thus from (4.12) we have p ∞ ( t, · ) ∈ C at every time in the interior of { ρ , ∞ t + ρ , ∞ t = 1 } . Remark 4.3.
With the aforementioned regularity of p ∞ and { ρ i, ∞ = 1 } at hand, one can verify withtest functions in the weak formulation that the following holds: the velocity law on one-phase boundarypoints is given by (4.15) V = ν ix ( − ∂ ix p ∞ − ∂ x Φ i ) on ∂ { ρ i, ∞ = 1 } , where ν x is the outward normal of the set { ρ i, ∞ = 1 } , V is the normal velocity of the interface and ∂ ix denotes the x -derivative taken from the interior of the set { ρ i, ∞ = 1 } . This yields the flux matchingacross different densities, (4.16) ∂ x p ∞ x + ∂ x Φ = ∂ x p ∞ x + ∂ x Φ on ∂ { ρ , ∞ = 1 } ∩ ∂ { ρ , ∞ = 1 } . The equations (4.12) , (4.15) and (4.16) corresponds to a generalized two-phase Hele-Shaw flow evolvingby the pressure variable p ∞ where different drift potentials are present for each phase ρ i, ∞ . Appendix A. Optimal transport toolbox
Lemma A.1.
Let f : [0 , + ∞ ) → R be a C convex function that is superlinear at + ∞ . Let
M > . Weconsider F : P M (Ω) → R ∪ + ∞ defined as F ( ρ ) = ˆ Ω f ( ρ ( x ))d x, if ρ ≪ L d , + ∞ , otherwise . Let ν ∈ P M (Ω) be given. Then there exists a solution ̺ ∈ P ac ,M (Ω) of the minimization problem min ρ ∈ P M (Ω) (cid:26) F ( ρ ) + 12 W ( ρ, ν ) (cid:27) . If in addition ν ≪ L d or if f is strictly convex, then ̺ is unique.Moreover, ∃ C ∈ R such that for a suitable Kantorovich potential ϕ in the optimal transport of ̺ onto ν one has the following first order necessary optimality condition fulfilled (A.1) (cid:26) f ′ ( ̺ ) + ϕ = C, ̺ − a . e .,f ′ ( ̺ ) + ϕ ≥ C, on { ̺ = 0 } . If f ′ (0) is finite, then one can express the above condition as f ′ ( ̺ ) = max { C − ϕ, f ′ (0) } . Proof.
The proof of the previous results can be found in [11] or [43, Chapter 7]. (cid:3) It turns out that ( P M (Ω) , W ) is a geodesic space and constant speed geodesics (and absolutelycontinuous curves in general) can be characterized by special solutions of continuity equations. Since thischaracterization is true for any M > , we simply set M = 1 in the theorem below. Theorem A.2 (see [3, 43]) . (1) Let Ω ⊂ R d compact and ( µ t ) t ∈ [0 ,T ] be an absolutely continuouscurve in ( P (Ω) , W ) . Then for a.e. t ∈ [0 , T ] there exists a vector field v t ∈ L µ t (Ω; R d ) s.t. • the continuity equation ∂ t µ t + ∇ · ( v t µ t ) = 0 is satisfied in the weak sense; • for a.e. t ∈ [0 , T ] , one has k v t k L µt ≤ | µ ′ | W ( t ) , where | µ ′ | W ( t ) := lim h → W ( µ t + h , µ t ) | h | denotes the metric derivative of the curve [0 , T ] ∋ t µ t w.r.t. W , provided the limit exists. (2) Conversely, if ( µ t ) t ∈ [0 ,T ] is a family of measures in P (Ω) and for each t one has a vector field v t ∈ L µ t (Ω; R d ) s.t. ´ T k v t k L µt d t < + ∞ and ∂ t µ t + ∇ · ( v t µ t ) = 0 in the weak sense, then [0 , T ] ∋ t µ t is an absolutely continuous curve in ( P (Ω) , W ) , with | µ ′ | W ( t ) ≤ k v t k L µt fora.e. t ∈ [0 , T ] and W ( µ t , µ t ) ≤ ´ t t | µ ′ | W ( t )d t. If moreover | µ ′ | ∈ L (0 , T ) , then we say that µ belongs to the space AC ([0 , T ]; ( P (Ω) , W )) . (3) For curves ( µ t ) t ∈ [0 , that are geodesics in ( P (Ω) , W ) one has the equality W ( µ , µ ) = ˆ | µ ′ | W ( t )d t = ˆ k v t k L µt d t. (4) For µ , µ ∈ P ac (Ω) , a constant speed geodesic connecting them is a curve ( µ t ) t ∈ [0 , such that W ( µ s , µ t ) = | t − s | W ( µ , µ ) for any t, s ∈ [0 , . One can compute this constant speed geodesicusing McCann’s interpolation, i.e. µ t := ( T t ) µ , for all t ∈ [0 , , where T t := (1 − t )id + tT with T µ = µ the optimal transport map between µ and µ . Moreover, the velocity field in thecontinuity equation is given by v t := ( T − id) ◦ ( T t ) − . Let us introduce the
Benamou-Brenier functional B : M ([0 , T ] × Ω) × M d ([0 , T ] × Ω) → R ∪ { + ∞} defined as B ( µ, E ) := ˆ T ˆ Ω | v t | d µ t ( x )d t, if E = E t ⊗ d t, µ = µ t ⊗ d t and E t = v t · µ t , + ∞ , otherwise . It is well-known (see for instance [43, Proposition 5.18]) that B is jointly convex and lower semicon-tinuous w.r.t. the weak − ⋆ convergence. In particular if ( µ, E ) solves ∂ t µ + ∇ · E = 0 in the weak sensewith B ( µ, E ) < + ∞ , implies that t µ t is a curve in AC ([0 , T ]; ( P (Ω) , W )) . The following comparison result appears to be well-known but we write it here for completeness.
Lemma A.3.
Let µ , ν ∈ P M (Ω) and µ , ν ∈ P M (Ω) . Then the following inequality holds true (A.2) W ( µ + µ , ν + ν ) ≤ W ( µ , ν ) + W ( µ , ν ) . Remark A.1.
Note that with the abuse of notation, W on the l.h.s. of (A.2) denotes the − Wassersteindistance on P M + M (Ω) , while on the r.h.s. W denotes the corresponding distances on P M (Ω) and P M (Ω) respectively.Proof of Lemma A.3. The quantity on the l.h.s. of (A.2) is realized by an optimal plan γ ∈ Π M + M ( µ + µ , ν + ν ) i.e. W ( µ + µ , ν + ν ) = ˆ Ω × Ω | x − y | d γ. Similarly the quantities on the r.h.s. can be written with the help of some optimal plans γ i ∈ Π M i ( µ i , ν i ) ,i = 1 , , i.e. W ( µ i , ν i ) = ˆ Ω × Ω | x − y | d γ i . Now set ˜ γ := γ + γ . Clearly since ( π x ) ˜ γ = µ + ν and ( π y ) ˜ γ = µ + ν one has ˜ γ ∈ Π M + M ( µ + µ , ν + ν ). Hence W ( µ + µ , ν + ν ) = ˆ Ω × Ω | x − y | d γ ≤ ˆ Ω × Ω | x − y | d˜ γ = ˆ Ω × Ω | x − y | d γ + ˆ Ω × Ω | x − y | d γ ≤ W ( µ , ν ) + W ( µ , ν ) . Therefore, inequality (A.2) follows. (cid:3)
Appendix B. A refined Aubin-Lions lemma
In [40] the authors present the following version of the classical Aubin-Lions lemma (see [4]):
Theorem B.1. [40, Theorem 2]
Let B be a Banach space and U be a family of measurable B -valuedfunction. Let us suppose that there exist a normal coercive integrand F : (0 , T ) × B → [0 , + ∞ ] , meaningthat (1) F is B (0 , T ) ⊗ B ( B ) -measurable, where B (0 , T ) and B ( B ) denote the σ -algebgras of the Lebesguemeasurable subsets of (0 , T ) and of the Borel subsets of B respectively; (2) the maps v F t ( v ) := F ( t, v ) are l.s.c. for a.e. t ∈ (0 , T ) ; (3) { v ∈ B : F t ( v ) ≤ c } are compact for any c ≥ and for a.e. t ∈ (0 , T ) , and a l.s.c. map g : B × B → [0 , + ∞ ] with the property [ u, v ∈ D ( F t ) , g ( u, v ) = 0] ⇒ u = w, for a . e . t ∈ (0 , T ) . If sup u ∈U ˆ T F ( t, u ( t ))d t < + ∞ and lim h ↓ sup u ∈U ˆ T − h g ( u ( t + h ) , u ( t ))d t = 0 , then U is relatively compact in M (0 , T ; B ) . Conflict of Interest –
The authors declare that they have no conflict of interest.
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Department of Mathematics, UCLA, California, USA
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