On Continuity Equations in Space-time Domains

In this paper we consider a class of continuity equations that are conditioned to stay in general space-time domains, which is formulated as a continuum limit of interacting particle systems. Firstly, we study the well-posedness of the solutions and provide examples illustrating that the stability of solutions is strongly related to the decay of initial data at infinity. In the second part, we consider the vanishing viscosity approximation of the system, given with the co-normal boundary data. If the domain is spatially convex, the limit coincides with the solution of our original system, giving another interpretation to the equation.


Introduction
Let Ω T be a given space-time domain in R d × (0, T ), denoted by In this domain we consider a continuity equation of the form: where v(x, t) = −( ∇µ/µ + ∇V + ∇W * µ)(x, t) with ≥ 0, in the space of probability measures, with the constraint that the support of µ lies in the closure of Ω T . When > 0, this constraint yields the co-normal boundary data on the lateral boundary of Ω T (1.5).
The first-order system, = 0, will be formulated using a projection operator (1.2): we will show that this system can be obtained as the vanishing viscosity limit as → 0.
The above system describes the density of moving particles which are confined to some region and flow with a velocity field v inside of the domain. One part of the velocity field is generated from interactions between different particles represented by the interaction potential W , given by (∇W * µ)(x, t) := ∇W (x − y)dµ(y, t).
This type of problem arises in many applications with various interaction kernel W , such as in swarming models with W (x) = −Ce −|x| , W (x) = −Ce −|x| 2 and in models of chemotaxis with W (x) = 1 2π log |x|, see [3,4] for more references. At the same time, the particles are subject to an external potential V (x). Both V, W are assumed to be smooth and λ-convex. More assumptions will be presented in section 3.1 and 4.1. For the diffusion term, the model takes into account random movements of the particles.
In the first part of this paper, we consider = 0. Let c(x, t) be the speed of the boundary (with positive sign if the boundary is expanding) and n(x, t) be the unit space outer normal for x ∈ ∂Ω(t). We set c(x, t) = 0, n(x, t) = 0 if (x, t) are not on the boundary. For simplicity we may omit the dependencies and write c, n. For each (x, t) ∈ Ω T , we define a projection operator P x,t : R d → R d as follows (1.1) Here µ 0 is a probability measure on Ω(0). First we assume it is compactly supported, and later we will also consider probability measures with exponential decay properties. In both cases µ 0 has finite second moment.
Our main contributions are two-fold. The first part of the results are mainly motivated by the previous work by Carrillo, Slepcev and Wu [4], where they show the well-posedness of equation (1.2) in stationary, non-convex domains with compactly supported initial data. We generalize the well-posedness result to general space-time domains and allow non-compactly supported initial data. Second, we show that (1.2) can be obtained as the limit as → 0 of the diffusion equation (1.5) given with the co-normal boundary data, imposing the additional condition that the domain is bounded and spatially convex. This result is significant since it provides a natural justification for the first-order system (1.2).
1.1. First Main Result. As in [4], we use particle approximations. The hard part is to show the limit of particle approximating solutions is indeed a weak solution to (1.2) due to the fact that the projecting operator P x,t (v) is discontinuous with respect to x, t on the boundary. So instead we show that the limit is a gradient flow solution by taking limit of the "curve of maximal slope" (see inequality (3.9)) which is then a weak solution.
The novelty in this paper is that, comparing with [4], for space-time domains an extra term (Ẽ below or see (3.9)) appears in the curve of maximal slope, where w(µ)(x, r) := −(∇V + ∇W * µ)(x, r). Intuitively this extra termẼ comes from the moving boundary constraints and the possible situation that particles attempt to move out of the domain with potential velocity w but end up moving with the boundary with velocity P x,t w. According to the definition of P x,t , it is not hard to see thatẼ(µ) is only nontrivial if µ is singular with mass concentrating on the boundary. Alternatively if the domain is stationary, this term vanishes, since if the boundary speed c(x, t) = 0, P x,t at boundary points are projections onto the tangential plane of (x, t).
We need more careful analysis because of this term. To be specific, the key point is to show the lower semi-continuity of E(µ) in µ. This can be proved ifẼ(w) + 1 2 |P x,t w| 2 is convex in w which is the case when we have expanding boundary. However it is not convex if the boundary speed is negative. This difficulty can be resolved by observing the two facts: the sequence w n we take limit of converges uniformly in (x, t) if W is C 2 ; E(w)(x, t) is lower semi-continuous in x, t. These two also guarantee the lower semi-continuity of E in µ (see Lemma 3.5).
The energy associated to (1.2) contains potential energy and interaction energy: Here µ(·, t) will only be supported in Ω(t).
For non-compactly supported initial data with exponential decay property (see condition (R) in section 3.4), existence of solutions can be done by particle approximation as well as truncation method. Uniqueness of solutions satisfying exponential decay is proved by a stability estimate (1.4). We will provide examples in Theorem 3.3 showing that the requirement below that p < 1, as well as the exponential decay condition, are essential. Now we summarize the main theorem in part one. We use d W (·, ·) to denote the 2-Wasserstein distance between probability measures. The Wasserstein metric and the notion of weak solutions will be discussed in section 2.

Second Main Result.
In the second part, we consider the case > 0: on Ω(0). (1.5) Here ∂ l Ω T := Ω T \Ω T is the lateral boundary of Ω T ; c is the speed of the boundary. The co-normal boundary condition above gives the mass preservation. The associated energy φ is given by In the first term, u is the probability density of µ if µ is absolutely continuous with respect to Euclidean measure. We set φ (µ) = ∞ if µ is not absolutely continuous.
With the convergence of → 0 in mind, we show the existence of solutions by discrete-time gradientflow (JKO) solutions (see [14]). For this purpose, technically we further require V, W to be C 2 and bounded below. In time-dependent domain, the scheme is slightly different from the standard version that we minimize each movement among probability measures with support contained in Ω T (see (4.2)). To obtain the continuum time limit of the discrete-time solutions, we show the uniform boundedness of the second moment and the boundedness of φ along solutions from the discrete scheme. This is one part that the analysis for problems on stationary domains that cannot be directly carried over for the time-dependent domains. The problem is solved in Proposition 4.3, the proof of which is inspired by the work of Di Marino, Maury and Santambrogio [9] who encountered the same problem. Also let us mention that solutions obtained in this way inherit the gradient flow structure which will be important later.
By a gradient flow argument, we have uniqueness of solutions in bounded and spatially convex moving domains, see Remark 4.6. For non-convex bounded stationary domain, we give a uniqueness proof based on an L 2 stability estimate, see Theorem 4.2.
After establishing the well-posedness of weak solutions, we send → 0. It will be proved that if the domain is bounded and spatially convex, equations (1.5) are indeed the vanishing viscosity approximation of the first order equation (1.2) in the first part. This convergence justifies the formulation of equation (1.2), in addition to the derivation via particle system.
We use a Gronwall type argument. By the gradient flow theory (mainly Sections 8,10,11 [1]), the time derivative of the 2-Wasserstein distance between µ and µ is related to the Fréchet subdifferentials of their energy φ, φ at µ, µ respectively. We want to use the convexity of the energies to finish the argument and to do so we also need to consider φ (µ). A serious problem arises that the value of U at µ can be infinity. This is because, in general even with smooth initial data, µ can concentrate mass in finite time as discussed in [2].
To overcome this problem, we develop a new modification method. We select aμ =ũdx which is close to µ, andũ is bounded point-wisely by −α for some 0 < α < 1. Using that the domain is bounded, we obtain U (μ) → 0 as → 0. Then by the variational inequality (4.17), it turns out that we need µ,μ to be close not only in 2-Wasserstein metric but also in Pseudo-Wasserstein distance with base µ (the definition will be given in section 2). Finding such a modification of µ is one of the most technical part of this paper since the information we have about the base µ at each time is limited. Simple convolution with a bump function won't give the expectedμ, see Appendix A. The modification is done for general absolutely continuous base measure in Lemma 4.9. We need the convexity of the domain and we will use generalized geodesics in probability measure space with Pseudo-Wasserstein metric and Brunn-Minkowski inequality. Now let us give the main theorem of the second part of this paper.
Theorem B. Assume conditions (C1)-(C4)(O1)(O2) hold (see sections 3.1, 4.1 for details), µ 0 is absolutely continuous (with respect to Lebesgue measure) probability measures supported in Ω(0) with finite second moments, and φ (µ 0 ) < ∞ for some > 0. Then for any fixed T > 0 (a) (Theorem 4.1) There exists a weak solution µ (·) to equation (1.5) and for each t ≥ 0, µ (t) is absolutely continuous with respect to Lebesgue measure. (b) (Theorem 4.3) Suppose Ω(t) is bounded and convex for all t ∈ [0, T ]. Let µ (·) be the weak solution to equation (1.5) and µ(·) be the weak solution to equation (1.2) with the same initial data. Then there exist constants c, C that Lastly let us mention that in [6], the vanishing viscosity limit problem in the whole domain was studied in the case when V = 0 and −W is the Newtonian potential. Their proof heavily relies on the specific choice of kernel W , and also the fact that the domain is R d which eliminates the task of determining the limiting boundary condition. letting y s ∈ argmin ys∈∂Ω(t+s) |y s − x|, then c(x, t)n(x, t) = lim s→0 y s − x s .
Throughout the paper, we fix a time T > 0 which is assumed to be large. We say a constant is universal if it only depends on T and constants in conditions (O1)(O2)(C1)-(C4) (λ, r p , L and bounds about c, V, W ). We denote by C a constant which may depend on universal constants and µ 0 , possibly changing from one estimate to another.
A spatial ball in R d centered at x with radius r is denoted by B r (x), and we may simply write B r if x is the origin. Given S ⊆ R d , we use the notation vol{S} as the Lebesgue measure of S.
Given a probability measure µ, we write m 2 (µ) = R d |x| 2 dµ as the second moment of µ. The set of all probability measures on Ω with finite second moment is denoted by P 2 (Ω). The set of absolutely continuous (with respect to Lebesgue measure) probability measures with finite second moment is written as P a 2 . For µ ∈ P a 2 , we usually write µ = uL d where u is its density. For probability measures supported in Ω, we will think of them as measures in R d , extended by 0 outside Ω. Now we give the definition of weak (measure) solutions to equations (1.2) and (1.5).
Finally let us recall the Brunn-Minkowski theorem.
3. Part One. Nonlocal First Order Equations 3.1. Settings and Assumptions. We study equation (1.2) in the first part of this paper. Suppose S ⊂ R d is open and the boundary is C 1 . Then we say S is r-prox-regular if for any point x ∈ ∂S, y ∈ S we have n(x), y − x ≤ |y − x| 2 r where n(x) is the unit normal at x (see [5] for more results). This is the same as: for any boundary point x, there is a ball of radius r that intersects S at exactly x.
Now we list the assumptions below.
(O1) For each t, Ω(t) is a non-empty open subset of R d which is always r p -prox-regular for some r p > 0. The lateral boundary ∪ t (∂Ω(t) × {t}) is C 1 in both time and space direction, particularly the boundary speed c(x, t) is continuous if restricted to the boundary. In addition, we require we require W ∈ C 2 (R d ).
(C3) There exists some λ ∈ R such that V, W are λ-convex in R d .

Particle Approximations.
As stated in the introduction, we use particle approximations. Consider: We look for a solution of the form µ(t) = N i=1 m i δ xi(t) . By the weak formulation, equation (1.2) becomes ẋ i (t) = P x,t (w(µ)(x i (t), t)) a.e. for t > 0 The ODE can be solved by a differential inclusions argument; the proof is in the appendix.
be the finite sum of delta masses. Then the ODE system (3.1) has a locally absolutely continuous solution (for each i, x i (·) is absolutely continuous).
By direct computations, (C1),(C2) and the fact that j m j = 1 This provides us a uniform bound for m 2 (µ(t)) which only depends on T, m 2 (µ 0 ). And then This shows the linear growth of x i in time that And this illustrates that the solutions µ(t) are always compactly supported in finite time. Also for particles starting outside B CR , they will be outside B R for t ∈ [0, T ]. This will be used in Theorem 3.2. Let k(x, t)n = w(x, t) − P x,t w(x, t) and so k actually depends on µ. Then From the above which gives the 1 2 -Hölder continuity of µ(·) in Wasserstein distance. Now let us recall the metric derivative of an absolutely continuous curve in P 2 (R d ), h .

Y ZHANG
Then we can show the following proposition regarding the well-posedness of solutions for the projected discrete systems.
By (3.7), µ(t) is an absolutely continuous curve with respect to Wasserstein metric. By the previous Stability of Discrete Solutions. The following proposition gives the stability result of solutions in the discrete case. The proof is similar to the one in Proposition 5.1 [4]. The only difference is the movement of the boundary which can be controlled by condition (O1).
Suppose µ 1 , µ 2 are solutions with discrete type initial measures µ 1 0 , µ 2 0 as in Proposition 3.2. Then there exists a constant C depending on the support of the initial data, the conditions and T , such that ) be defined as in Proposition 3.1 and let Estimate (3.5) shows that for t ∈ [0, T ], µ i (t) are compactly supported. By (3.6), |k i (x,t)| rp ≤ C in the support of µ i (t). Let γ t be an optimal transport plan between µ 1 (t) and µ 2 (t), then By Theorem 8.4.7 from [1]: Since W is λ-convex and even, we get Here Remark 3.4. If we were more careful on the dependence of the constant on T and the support of the initial data (suppose supp(µ i 0 ) ⊂ B R ), we would find out that the constant in the above proposition can be bounded by C exp(CRT exp(CT )) where C is a universal constant.

3.3.
Compactly Supported Initial Data. Suppose µ 0 ∈ P 2 (Ω(0)) and consider a sequence of delta masses µ n 0 = k(n) i=1 m i δ x n i converging to µ 0 in Wasserstein metric. Without loss of generality, we assume that µ n 0 are supported in a compact set for all n. Suppose µ n (t) is a solution to (1.2) given by Proposition 3.2 with initial value µ n 0 . Proposition 3.3 shows that for each t, µ n (t) is a Cauchy sequence once µ n 0 is Cauchy. So we can write the limit as µ(t) which again is compactly supported. Now we need to show that the limit is the tangent velocity field of µ(t) by simply letting n goes to infinity due to the discontinuity of P x,t in (x, t), which is also explained in Remark 3.2 [4]. To overcome this problem, we use gradient flow method. Let us start with the following definition.
Definition 3.1. Let µ(·) be an absolutely continuous curve in P 2 (R d ) with compact support in Ω T . We say that µ(·) is a curve of maximal slope with respect to φ in a time-dependent domain, if for all 0 ≤ s < t < T : Here as before, w(x, r) = w(µ)(x, r) = (−∇V − W * µ)(x, r). As mentioned in the introduction, the last term of (3.9) appears because of the time-dependence of the domain. For 0 ≤ s < t ≤ T we denote We need the following lemma which shows the lower semi-continuity of E(·) as µ n → µ. We require that W ∈ C 2 if the domain is compressing locally.
Lemma 3.5. Notations are as above and suppose µ n (t) converges to µ(t) in Wasserstein distance uni- Proof. We consider the integrals on the boundary of c(x, t) ≥ 0 and c(x, t) < 0 separately. Let us still write t for time dependence and w (or w i ) are continuous vector fields in Ω T . For each (x, t) ∈ Ω T , we define and so E(w) is lower semi-continuous in x, t if c ≥ 0. Then as did in Lemma 3.7 [4], by Proposition 6.42 [12] that for each t, there are two families of countable many bounded and continuous functions and similarly for w n (x, t). Then for each N ∈ N and t, Then take sup over i ∈ N and use Lebesgue's monotone convergence theorem. We see If W is C 2 , we can show the lower semi-continuity of E without non-negative assumptions on c. We want to show that E(w)(x, t) is lower semi-continuous on w, x, t. Continuity on w is clear by definition and we have uniform continuity for all (x, t) ∈ supp(µ(t)) ⊆ Ω T since the support is compact and c is continuous. Let γ(y, y ) be the optimal transport plan between µ n and µ. By (C2), |D 2 W | is bounded in compact sets. By definition Take → 0 in (3.13) and integrate in time, we proved Now we give the main theorem: Suppose µ 0 is a probability measure compactly supported in Ω(0) and {µ n (·)} are the discrete solutions as stated above. Then µ n (·) converges to µ(·) which is an absolutely continuous curve in P 2 (R d ) of maximal slope with respect to φ (see definition 3.9). And µ(·) satisfies the equation (1.2) in the weak sense. Suppose we have two such solutions µ i (·), i = 1, 2 with initial data µ i 0 ∈ P 2 (R d ), and µ i (·) are compactly supported for t ∈ [0, T ]. Then there exists a constant C that Here C depends on universal constants and the support of Thus φ (µ n (t)) is absolutely continuous in t since µ n (·) is absolutely continuous curves and {m 2 (µ n )} are uniformly bounded. We do not need µ n to be compactly supported here.
Then we show that the discrete solutions µ n are curves of maximal slope. By direct computation In the above we used the notation (3.8). By Proposition 3.2 |v n | 2 dµ n a.e. in time, we deduce that µ n (·) is a curve of maximal slope according to the definition (3.9): (3.14) Now we want to show that µ(t) is also a curve of maximal slope. By Theorem 3.6 [4], Because µ n (t), µ(t) are compactly supported locally uniformly in time, W, V ∈ C 1 (R d ) and µ n (t) → µ(t) in d W , we have for each t, lim n→∞ φ(µ n (t)) = φ(µ(t)). (3.16) Recall the notation k(x, t)n(x, t) = w(x, t) − P x,t w(x, t).
(3.17) Then by (3.14) (3.15) and Lemma 3.5, sending n to infinity shows This gives Next because µ(·) is an absolutely continuous curve in P 2 (R d ), by Theorem 8.3.1 [1], there exists a unique tangent vector fieldv(x, t) ∈ L 2 (µ(t), R d ) such that the continuity equation holds The goal is to show v =v.
We claim the following chain rule that for a.e. t > 0 A similar result is proved in Theorem 1.5 [4]. For the convenience of readers, we give a sketch of proof below. For τ > 0, select We used (C2)(C3) (λ-convexity of V, W and symmetry of W ) in the above inequality and the constants C, C only depend on λ. Then applying that µ(t) is absolutely continuous, for a.e. t > 0 By Proposition 8.4.6 [1], Ω(t) y−· τ dr τ (y) converges tov(·, t) weakly in L 2 (µ(t)) where r τ (y) is the disintegration of γ τ t with respect to µ(t). Note µ(t) is absolutely continuous, therefore for a.e. t > 0 Similarly, we have Again since µ(t) is absolutely continuous, as done in the discrete case (see Theorem 3.1) we have t → φ(µ(t)) is absolutely continuous. Thus, we can conclude with the chain rule.
Then using the notation (3.17) Recall r τ as defined above. We have n(x, r) · y − x τ dr τ (y)du(x, r).
Note in the above equation, y ∈ Ω(r + τ ), x ∈ ∂Ω(r). Take x (x, r, τ ) to be one of the closest point to x on ∂Ω(r + τ ). So for τ small enough, (x −x) τ = c(x, r)n + O(τ ). Also since c(x, r) is continuous and µ(r) is compactly supported, we get Now by r p -prox regularity of Ω(r), within the support of µ, C k here is the bound of k(x, r) for (x, r) ∈ Ω T and it depends on the support of µ 0 . By the dominated convergence theorem the last term of (3.21) In view of the fact that µ(·) is an absolutely continuous curve with respect to the Wasserstein metirc, we have the above expressions vanish. So Apply this to (3.20), we get Finally compare (3.18) with (3.23) and make use of (3.19), we find in For the stability result, considering that µ 1 , µ 2 are compactly supported, the proof is essentially the same as the one in Proposition 3.3. Remark 3.6. As can be seen in the proof, for Theorem 3.1 we only need λ-convexity of V, W locally. Also we can weaken the condition (O1) to be local prox-regularity: for every ball B R , ∂Ω(t) ∩ B R is r p -prox-regular for some r p (R) > 0.  This condition requires some exponentially decay of measures which is slightly more general than compact supported ones. We say a curve of measures {µ(t), t ≥ 0} satisfy condition (R) locally uniformly if for each t ∈ [0, T ], µ(t) satisfies the condition for some c r (T ).
Here t p , ε only depend on p, c r and universal constants.
Proof. For existence, we use the particle approximation method as before: let µ n (·) = be solutions to equation (1.2) with discrete initial data µ n 0 → µ 0 . First let us assume the convergence of µ n (·) and show the limit is a solution. Expressions or estimates (3.12) (3.14) (3.15) still hold. But we need to be careful on And similar linear bounds also hold for k(x, t), P x,t w(x, t). So for any small > 0, we can choose R large enough such that for all n. Then we only need to considerμ n (t) = j m j δ {xj (t),|xj (0)|≤R} . The contribution for particles starting outside B R will be under control by (3.5) again. Now since the integration is inside a compact set, the proof will then follows from Lemma 3.5. The proof of (3.22) is similar.
For any L >> 1, let C be the constant as given in (3.5) and L := C L. Then the above By (3.6), This shows . Select t 0 = cr C where c r comes from (R). By the condition, δ(L, µ 0 ) exp(c r L) can be any small if L is large. Recall that {µ n 0 } is Cauchy and we further take d W (µ n 0 , µ m 0 ) to be small, thus {µ n (t)} is also Cauchy for 0 ≤ t ≤ t 0 . Then we can consider each time interval: [0, t 0 ], [t 0 , 2t 0 ]... inductively and we proved that µ n (t) → µ(t) for all t ≤ T .
For any 0 < p < 1, write ε := d W (µ n 0 , µ m 0 ) and choose L = −2 log ε cr , t 0 as above. For t ≤ t 0 (1 − p), Let ε be small enough and L is then large enough. By (R), Notice if µ(·) solves the equation (1.2), then m 2 (µ(t)) ≤ C for t ∈ [0, T ]. The second claim about the stability of solutions satisfying condition (R) follows from the above argument for the discrete type solutions. This shows that solutions satisfying condition (R) are unique.
Remark 3.9. We comment on several situations where the condition (R) can (or possibly) be dropped. (i) As in Theorem 1.9 [4], if Ω(t) is convex for all t and µ i (t)(i = 1, 2) are solutions with general initial data µ i 0 ∈ P 2 (Ω(t)), then there exists a universal constant C such that d W µ 1 (t), µ 2 (t) ≤ Cd W (µ 1 0 , µ 2 0 ). Here we do not need any assumptions on the decay of solutions. The proof follows from the observation that in (3.26) k(x, t)n(x, t), x − y ≥ 0 by the convexity. And as a corollary we have the uniqueness result. (ii) If ∇V, ∇W and c(x, t) are uniformly bounded, we can conclude the same as in (i). (iii) We guess that for the non-local term if W is compactly supported, there is a better stability result.

Examples and Stability of Solutions.
In this section, we will show that the stability result in Theorem 3.2 cannot be improved to d W µ 1 (t), µ 2 (t) ≤ Cd W (µ 1 0 , µ 2 0 ) as long as the domain is unbounded and non-convex. Moreover we give examples showing that without condition (R), the stability of solutions is even weaker than that in Theorem 3.2. All these suggest that the stability of solutions to (1.2) is strongly related to the decay of initial data at infinity. Theorem 3.3. There exists V, W, Ω satisfy conditions (C1)-(C3)(O1) such that the following holds for any t 0 > 0. (i) There is µ 0 ∈ P 2 (Ω) and µ n 0 → µ 0 (write µ n (t) as a solution to equation (1.2) with initial data µ n 0 ) that µ(·), µ n (·) satisfy (R) locally uniformly and (ii) For any p > 1 2 , there is µ 0 ∈ P 2 (Ω) and µ n 0 → µ 0 that lim inf Here P is the projection operator defined in (1.1) with zero boundary speed.

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Let us start by solving the equation with initial data µ 0 = δ (x0,cos(2πx0)) where x 0 is close to some large integer. Suppose δ (xt,yt) is a solution, then by simple calculations (x t , y t ) = P (−∇V (x t , y t ) = P (−y t , −x t ).
It is not hard to see that if x t is large enough, the delta mass moves along the boundary. So P (−y t , −x t ) = P (− cos(2πx t ), −x t ) and Notice the equation x t = ϑ t is an autonomous system. For each large integer N , there are two equilibrium points in [N − 1 2 , N + 1 2 ] namely the solutions of 2πx = cot(2πx). One equilibrium point is close to N − 1 2 which is stable and the other is close to N which is unstable. We denote the stable one as (N ) 1 * and the other as (N ) 2 * . We can show (N ) 1 Also we construct a family of µ n 0 → µ 0 . Denote j α := j + (j) −α with 0 < α < 1 and let m j δ (j,cos(2πj)) + ∞ j=n m j δ (jα,cos(2πjα)) .
If selecting α = 1 − , β = 3 − , then p can be any close to 1 2 . This shows that in Theorem 3.2 if without (R), p cannot be greater than 1 2 . So at least we can claim that the stability is weaker.

Part Two. Second Order Equations
In the second part of this paper we show the well-posedness of the second order continuity equation (1.5) and then we send the diffusion term to 0. If Ω(t) is bounded and convex for each t, we will prove that (1.5) is indeed the vanishing viscosity approximation of (1.2).

4.1.
Assumptions and JKO Scheme. We make the following assumptions. We will make a remark about several generalizations later.
(O2) The lateral boundary of Ω T is uniformly C 1 in space. There exists L > 0 that Here d H is the Hausdorff distance. From this we know that there exists r p > 0 such that both ∂Ω(t), ∂Ω(t) C are r p -prox regular for all t ∈ [0, T ]. For V, W , we assume (C1)-(C3) hold and furthermore we assume: Recall that the associated energy φ is defined in (1.6). We define the proper domain of functional φ is Notice there is no difference between µ ∈ Dom(φ 1 , t) and µ ∈ Dom(φ , t) for some > 0. Next as a convention, Without loss of generality, we only prove well-posedness for = 1. We have the following equation: Suppose µ 0 ∈ Dom(φ 1 , 0) and conditions (C1)-(C4)(O1)(O2) hold. We use the following variant of the celebrated JKO scheme. Fix a small time step τ > 0, define J τ,t : P a 2 (Ω(t)) → P a 2 (Ω(t + τ )) by is bounded below. And we can find a sequence of measures whose energy converges to the infimum and they all belong to P a 2 due to the internal energy. Then lower semi-continuity of φ 1 and compactness guarantee the existence of the limit. Details can be found in section 2.1 in [1] or Lemma 4.2 of [17]. Actually if {Ω(t)} is convex for all t, we have the uniqueness of the minimizer. However, here we only need the existence result.

Some Estimates.
First we prove a technical lemma which is enlightened by Corollary 2.6 in [9]. It will be used to compare µ(t) with J τ,t (µ(t)) whose support is different. Proposition 4.1. Suppose the domain satisfies condition (O2). Then for t ∈ [0, T − τ ] and µ ∈ P a 2 (Ω(t)), there exists a Lipschitz continuous map t : Here C is some constant independent of t, τ which only depends on the geometry of Ω T .
Proof. By (O2), ∂Ω(t) is uniformly C 1 for all t. So there exists r p > 0 such that for each t ∈ [0, T ], there is a unique Lipschitz map q t : ∂Ω(t) → Ω(t) such that Without loss of generality, we can assume ∂Ω(t) C is r p -prox regular. So for any x near the boundary with distance ≤ rp 2 to the boundary, there exists a unique y ∈ ∂Ω(t) such that d(x, y) = d (x, ∂Ω(t)). We denote such y = p t (x). So and q t • p t = id on ∂Ω r t /∂Ω(t). Now define a continuous map Q t : Ω(t) → Ω(t), Note by the assumption made on the boundary of the domain It is not hard to see that DQ t exists almost everywhere and |DQ t | is uniformly bounded. Define Then estimate (4.4) is satisfied. We want to show that t : Ω(t) → Ω(t+τ ). Otherwise if there exists x ∈ Ω(t) such that t(x) = Ω(t+τ ), by (O2) and from the geometry for any z ∈ ∂B(0, rp 3 ), the line segment connecting x and Q t (x) + z lies in Ω(t). Then In view of the fact that z is arbitrary with length ≤ rp 3 , we end up with a contradiction to the Lipschitz variation of Ω(·).
Finaly for (4.3) , As a corollary, we have the Lemma 4.2 below. Notice that the hypotheses (4.5) is weaker than (4.3) which is made to allow possible weaker assumptions (than (O2)) on the domain (see Remark 4.5).
Y ZHANG Proposition 4.3. Assume (C4) and under the assumption of Lemma 4.2. For fixed µ 0 ∈ Dom(φ 1 , 0), T , if τ is small enough and nτ < T , then there exists C > 0 independent of τ, k, n such that Proof. By Lemma 4.2, 1 2τ To give a bound to m 2 (µ), we use the trick as in proposition 4.1 [14]. Since , and by (4.7) and the lower bound of V, W , we see Here q < 1. Considering that C is independent of τ, n (which only depends on T, µ 0 , Ω T ) and n can be any positive integer such that nτ < T , the above shows m 2 (µ n τ ) is uniformly bounded. According to (4.7) And in view of (4.6), {φ 1 (µ k τ )} are uniformly bounded. 4.3. Convergence of Discrete Solutions. We define a discrete type solution with time step τ as (4.8) As proved in [14], is compact in P a 2 (R d ). Then according to Proposition 4.3, {µ τ (t), t ≤ T, τ > 0} is a compact subset in P a 2 (R d ). We connect every pair of consecutive discrete values (µ k−1 τ , µ k τ ) with a constant speed geodesic parametrized in each interval [(k − 1)τ, kτ ] byμ Here t is an optimal transport map from µ k−1 τ to µ k τ . Again by Proposition 4.3, {μ τ , τ > 0} are Hölder continuous curves. Ascoli-Arzela Theorem yields the relative compactness ofμ τ in C 0 [0, T ]; P 2 (R d ) .
Remark 4.4. In stationary domain, the rate at which solutions to the discrete gradient flow converge to solutions of the gradient flow was studied in [1,7]. But we cannot prove the exponential formula in the case that the domain is time-dependent. This can be an interesting direction for future research.

Uniqueness Result.
We state two uniqueness results of equation (4.1). In the first one, we study the L 2 stability of solutions in a stationary domain. The proof is postponed to the appendix. The second one is stated in the remark below where we require the space-time domain to be convex. Theorem 4.2. Suppose the domain Ω ⊂ R d is stationary and bounded with (C1)-(C3)(O2) hold. Suppose µ 0 ∈ P a 2 (Ω) and its density u 0 ∈ L 2 (Ω). Then there exists a unique weak solution µ(·) to equation (4.1) with density u(t) ∈ L 2 (Ω) for each t.
If µ i (t)(i = 1, 2) are two solutions with initial data µ i 0 ∈ P a 2 (Ω) with their densities u i 0 ∈ L 2 (Ω), then there is C depending on the domain and universal constants such that for a.e. t ∈ [0, T ] Assuming Ω(t) is uniformly bounded and convex for all t ≤ T , then if µ i (t)(i = 1, 2) are two solutions with initial data µ i 0 ∈ P a 2 (Ω(0)), there is a universal C(T ) such that . This claim can be proved in a similar way as done in Theorem 11.1.4 [1] or section 2 of [10], as long as the domain is convex at any fixed time.

Convergence to the First Order Equation.
We consider equations (1.2) and (1.5) in bounded, convex domain in this section. Let µ be the weak solution to (1.5) and µ be the weak solution to (1.2). We want to show that µ converges to µ in Wasserstein metric as → 0.
Recall (1.6) and write φ(µ), φ (µ ) as the energies. The internal energy is denoted as The metric slope of functional φ for µ ∈ Dom(φ , t) at time t is Now we give two lemmas. The proof of the first lemma is standard (see Proposition 10.4.13 [1]), but we still need to be careful since the domain is time-dependent. We postpone the proof in the appendix.
Proof. By the Euclidean Logarithmic Sobolev inequality (see [13] [8]) and the fact that u (t) is supported in Ω(t), Write U (t) := U (µ (t)). Then We used Lemma 4.7 and the regularity of V, W in the last inequality. Now for small enough, assume 2 e N ≥ N holds for all N ≥ − 1 2 . Thus which finishes the proof.
The following lemma is one important ingredient to the proof of the convergence. Note it is possible that µ / ∈ the proper domain of φ 1 (or equivalently of φ ), the plan is to regularize it and replace it by aμ ∈ P 2 a . As explained in the introduction, we look for aμ with density function uniformly bounded by −α for some 0 < α < 1. Additionally we need d µ (µ,μ) to be small where d µ (·, ·) is the Pseudo-Wasserstein metric with base µ . As a remark, this is stronger than requiring d W (µ,μ) to be small. Lemma 4.9. Given any µ ∈ P 2 (Ω), v ∈ P a 2 (Ω) where Ω is a bounded, convex subset of R d . For any s > 0 small enough, there exists µ s ∈ P 2 a (Ω) such that d v (µ, µ s ) ≤ Cs and The constant C only depends on the diameter and the volume of Ω.
Proof. Without loss of generality, suppose Ω has volume 1 in Euclidean measure. Let e be the Euclidean measure restricted in Ω and then e ∈ P 2 a (Ω). Since v is absolutely continuous, t e v and t µ v exist and t e v is one to one on Ω outside a v zero measure subset. Let be the generalized geodesic joining µ, e with base v, which is defined as in Definition 9.2.2 [1]. Due to the convexity of the domain, we have µ s ∈ P 2 (Ω). By Proposition 2.6.4 [7], the generalized geodesic is of constant speed in the sense that Since the domain is bounded, d v (µ, e) is uniformly bounded for all probability measures v, µ, e. We deduce that d v (µ, µ s ) ≤ Cs. Now we show the pointwise boundedness of µ s . Let ϕ = χ Br (x) which equals 1 in B r and 0 outside. Thus S} . Now we apply Brunn-Minkowski inequality (Lemma 2.1) to find the above By (4.16), for any ϕ = χ Br (x) we find out This shows that u s is an L ∞ function in Ω with bound s −d .
Now we give our main theorem in the second part of this paper. Proof. For any ω 1 ∈ P a 2 (Ω(t)), let µ s := (st ω1 µ + (1 − s)i) # µ with µ = µ (t). The convexity of the domain implies µ s ∈ P a 2 (Ω(t)). For any Fréchet subdifferential of φ at µ (see section 10 By (C3), φ isλ−convex forλ = min{λ, 3λ}. So by the Characterization by Variational inequalities and monotonicity in 10.1.1 [1], Then we take s → 0 and find By the JKO scheme, µ is a gradient flow solution and we can choose ξ = −v , the tangent velocity field of µ . Similarly since µ is a gradient flow solution, ξ := P x,t (−∇V − ∇W * µ) = −v is one Fréchet subdifferential of φ at µ and then for any ω 2 ∈ P 2 (Ω(t)) For each t we use Lemma 4.9 to modify µ. Take v = µ , s = 1 d+2 and letμ = µ s ∈ P a 2 (Ω(t)) with Plug in ω 1 =μ in (4.17), Let γ be an optimal transport plan between µ, µ . The above Next by Hölder's inequality By Lemma 4.7, Ω T |ξ | 2 dµ dt is uniformly bounded and For small , geometrically µ 2 is just a small perturbation of µ 1 . This shows that a little shift may cause a large difference in Pseudo-Wasserstein metric. And so it is possible that the convolution of µ with 1 d ϕ( · ) (ϕ is a bump function and is a small positive value) is far away from µ in view of the Pseudo-Wasserstein metric.
Appendix B. Proof of Proposition 3.1 To solve this ODE, we cite the following result from [11] about the existence of differential inclusions.
Also assume that F : R dN × [0, T ] → nonempty convex compact subsets of R dN satisfies: Then for any x 0 ∈ Ω(0) N , the following sweeping process with perturbation has at least one absolutely continuous (in supremum norm) solution x(t). Here N Ω(t) N , x(t) denotes the normal cone at x(t) if x(t) is on the boundary, otherwise it is an empty set.
In our case of C 1 boundary, the normal cone simply means the collection of all outer normal vectors. Then we prove the following proposition.
Proof. (of Proposition 3.1) To apply Theorem B.1, we need to verify all the conditions. Proposition 2.5 in [4] showed that if Ω(t) is r-prox-regular then so is Ω(t) N . The absolute continuity of Ω(t) N follows from condition (O1). Also the upper semi-continuity of w(µ)(x, t) follows from the definition (3.2). We may write w(x, t) for abbreviation. For each N and all t, the linear growth of |w(·, t)| can be proved by definition as well as estimate (3.3).
In all, the assumptions in Theorem B.1 are satisfied, and thus there exists x(t) = (x 1 , ..., x N )(t) absolutely continuous such that x 1 , t), ..., w(x N , t)) , a.e. t ≥ 0, Then we show the solutions above are the solutions for the projected systems similarly as did in Lemma 2.4 [4].
So we can write k i (x, t)n(x i , t) = h i (x, t) such thaṫ Set k i (x, t) = 0 if x i (t) / ∈ ∂Ω(t) and we have k i ≥ 0. Claim w(x i , t) − k i (x, t)n(x i , t) = P x,t (w(x i , t)) a.e. in time.
Since x i (·) is absolutely continuous in R, we only need to consider all the t ∈ [0, T ] where x i (t) are differentiable. Also we only need to consider the case when x i (t) ∈ ∂Ω(t). Because x i (t) is supported in Ω(t),ẋ i (t) · n(x i , t) ≤ c(x i , t). Ifẋ i · n = c, we have w(x i , t) · n(x i , t) ≥ c(x i , t). By definition of P x,t , we only need to check the equality in the normal direction that P x,t (w(x i , t)) · n(x i , t) = c(x i , t) = (w(x i , t) − k i (x, t)n(x i , t)) · n(x i , t).
Ifẋ i (t) · n(x i , t) < c(x i , t), in view of the continuity of c(x, t), x i (t ) is in the interior of Ω(t) a.e. for t close to t. Then by continuity, k i (x, t) = 0. We also have at t P x,t (w(x i , t)) = w(x i , t) = w(x i , t) − k i (x, t)n(x i , t).
So (3.1) is satisfied a.e. for all t.
Since ρ ι (·, t) → ρ(·, t) in L 2 (Ω) a.e.dt, by Gronwall we have the stability result which also shows the uniqueness of solutions in L 2 norm.
Let us define v τ (t) := v ,k τ if t ∈ ((k − 1)τ, kτ ]. Then by (4.8), (D.1), v τ ∈ L 2 (u τ dxdt, R d ) with a bound independent of both τ and . Recall µ τ → µ in Wasserstein metric uniformly for all t ∈ [0, T ], then by Theorem 5.4.4 [1] v τ converges weakly to some w ∈ L 2 (u dxdt, R d ). From the previous discussion and the equation, we know that v τ converges weakly to v = ∇u u + ∇V + ∇W * µ . It is not hard to see that such limit is unique, so we have v = w a.e. dµ dt. This finishes the proof with C independent of