CONTROL OF THE CONTINUITY EQUATION WITH A NON LOCAL FLOW

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1 ESAIM: COCV DOI: 1.151/cocv/217 ESAIM: Control, Optimisation and Calculus of Variations CONTROL OF THE CONTINUITY EQUATION WITH A NON LOCAL FLOW Rinaldo M. Colombo 1, Michael Herty 2 and Magali Mercier 3 Abstract. This paper focuses on the analytical properties of the solutions to the continuity equation with non local flow. Our driving examples are a supply chain model and an equation for the description of pedestrian flows. To this aim, we prove the well posedness of weak entropy solutions in a class of equations comprising these models. Then, under further regularity conditions, we prove the differentiability of solutions with respect to the initial datum and characterize this derivative. A necessary condition for the optimality of suitable integral functionals then follows. Mathematics Subject Classification. 35L65, 49K2, 93C2. Received February 13, 29. Revised September 3, 29. Published online March 24, Introduction We consider the scalar continuity equation in N space dimensions { t ρ +divρvρ = t, x R + R N ρ,x=ρ x x R N 1.1 with a non local speed function V. This kind of equations appears in numerous examples, a first one being 1 the supply chain model introduced in [3,4], where V ρ = v, ρxdx see Section 3. Another example comes from pedestrian traffic, in which a reasonable model can be based on 1.1 with the functional V ρ = vρ η vx, see Section 4. Throughout, our assumptions are modeled on these examples. Other analytically similar situations are found in [5], where a kinetic model for pedestrians is presented, and in [23], which concerns the Keller-Segel model. In the following we postulate assumptions on the function V which are satisfied in the cases of the supply chain model and of the pedestrian model, but not for general functions. In particular, we essentially require below that V is a non local function, see 2.1. The first question we address is that of the well posedness of 1.1. Indeed, we show in Theorem 2.2 that 1.1 admits a unique local in time weak entropy solution on a time interval I. For all t in I, wecalls t the nonlinear Keywords and phrases. Optimal control of the continuity equation, non-local flows. 1 Department of Mathematics, Brescia University, Via Branze 38, Brescia, Italy. rinaldo@ing.unibs.it 2 RWTH Aachen University, Templergraben 55, 5256 Aachen, Germany. 3 Université delyon,universitélyon1,école Centrale de Lyon, INSA de Lyon, CNRS UMR 528, Institut Camille Jordan, 43 blvd. du 11 Novembre 1918, Villeurbanne Cedex, France. Article published by EDP Sciences c EDP Sciences, SMAI 21

2 354 R.M. COLOMBO ET AL. local semigroup that associates to the initial condition ρ the solution S t ρ of 1.1 attimet. As in the standard case, S t turns out to be L 1 -Lipschitz. Then, we look for the Gâteaux differentiability of the map ρ S t ρ, in any direction r and for all t I. AweakGâteaux differentiability of the semigroup generated by 1.1 is proved along any solution ρ C I;W 1,1 W 1, R N ; R and in any direction r L 1 R N ; R. Full differentiability follows under stronger assumptions. Moreover, the Gâteaux derivative of S t at ρ in the direction r is uniquely characterized as weak entropy solution to the following linear non-local Cauchy problem, that can be formally obtained by linearizing 1.1: { t r +divrv ρ+ρdvρr = t, x I R N r,x=r x x R N 1.2 where ρt, x =S t ρ x. Thus, also the well posedness of the nonlocal problem 1.2 needed to be proved, see Proposition 2.8. Remark that, in both 1.1 and1.2, solutions are constructed in C I; L 1 R N ; R. Therefore, we mostly refer to Kružkov solutions, see [21], Definition 1. Indeed, this definition of solutions is more demanding than that of weak solutions. Besides, it allows us to apply the results in [15], used in the subsequent part concerning the differentiability of solutions with respect to the initial datum. However, we note that in the case of the standard transport equation 5.2 and with the regularity conditions assumed below, the two notions of solution coincide, see Lemma 5.1. We recall here the well known standard i.e. local situation: the semigroup generated by a conservation law is in general not differentiable in L 1, not even in the scalar 1D case, see [9], Section 1. To cope with these issues, an entirely new differential structure was introduced in [9],andfurtherdevelopedin[6,1], also addressing optimal control problems, see [11,13]. However, the mere definition of the shift differential in the scalar 1D case takes alone about a page, see [13], p We refer to [7,8,18,24,25] for further results and discussions about the scalar one-dimensional case. Then, we introduce a cost function J : C I,L 1 R N ; R R and, using the differentiability property given above, we find a necessary condition on the initial data ρ in order to minimize J along the solutions to 1.1 associated to ρ. We stress that the present necessary conditions are obtained within the functional setting typical of scalar conservation laws, i.e. within L 1 and L. No reflexivity property is ever used. The paper is organized as follows. In Section 2, we state the main results of this paper. The differentiability is proved in Theorem 2.1 and applied to a control in supply chain management in Theorem 3.2. Sections 3 and 4 provide examples of models based on 1.1, and in Section 5 we give the detailed proofs of our results. 2. Notation and main results Denote R + =[, + [, R + =], + [ andbyi, respectively I ex, the interval [,T[, respectively [,T ex [, for T,T ex >. The open ball in R N centered at with radius δ is denoted by B,δ. Furthermore, we introduce the norms: v L = ess sup vx, v W 1,1 = v L 1 + x v L 1, x R N v W 2, = v L + x v L + 2 x v L, v W 1, = v L + x v L Existence of a weak entropy solution to 1.1 Let V : L 1 R N ; R C 2 R N ; R N be a functional, not necessarily linear. A straightforward extension of [21], Definition 1, yields the following definition of weak solutions for 1.1.

3 CONTROL OF THE CONTINUITY EQUATION WITH A NON LOCAL FLOW 355 Definition 2.1. Fix ρ L R N ; R. A weak entropy solution to 1.1 oni is a bounded measurable map ρ C I; L 1 loc RN ; R which is a Kružkov solution to { t ρ +divρwt, x = ρ,x=ρ x where wt, x =V ρt x. Here, differently from [21], Definition 1, we require the full continuity in time. Introduce the spaces X =L 1 L BVR N ; R and X α =L 1 BV R N ;[,α] for α> both equipped with the L 1 distance. Obviously, X α L R N ; R for all α>. We pose the following assumptions on V, all of which are satisfied in the examples on supply chain and pedestrian flow as shown in Section 3 and Section 4, respectively. V1: There exists a function C C R + ; R + such that for all ρ L 1 R N, R, V ρ L R N ; R N, x V ρ L R N ;R N N C ρ L R N ;R, x V ρ L 1 R N ;R N N C ρ L R N ;R, 2 x V ρ L1 R N ;R N N N C ρ L R N ;R. There exists a function C C R + ; R + such that for all ρ 1,ρ 2 L 1 R N, R V ρ 1 V ρ 2 L R N ;R N C ρ 1 L R N ;R ρ 1 ρ 2 L1 R N ;R, 2.1 x V ρ 1 x V ρ 2 L1 R N ;R N N C ρ 1 L R N ;R ρ 1 ρ 2 L1 R N ;R. V2: There exists a function C C R + ; R + such that for all ρ L 1 R N, R, 2 x V ρ L R N ;R N N N C ρ L R N ;R. V3: V : L 1 R N ; R C 3 R N ; R N and there exists a function C C R + ; R + such that for all ρ L 1 R N, R, 3 x V ρ L R N ;R N N N N C ρ L R N ;R. Condition 2.1 essentially requires that V be a non local operator. Note that V3 implies V2. Existence of a solution to 1.1 at least locally in time can be proved under only assumption V1, see Theorem 2.2. The stronger bounds on V ensure additional regularity of the solution which is required later to derive the differentiability properties, see Proposition 2.5. Theorem 2.2. Let V1 hold. Then, for all α, β > with β>α, there exists a time T α, β > such that for all ρ X α, problem 1.1 admits a unique weak entropy solution ρ C [,Tα, β]; X β in the sense of Definition 2.1. Moreover, 1 ρt L β for all t [,Tα, β]. 2 There exists a function L C R + ; R + such that for all ρ,1,ρ,2 in X α, the corresponding solutions satisfy, for all t [,Tα, β], ρ 1 t ρ 2 t L 1 Lt ρ,1 ρ,2 L 1.

4 356 R.M. COLOMBO ET AL. 3 There exists a constant L = Lβ such that for all ρ X α, the corresponding solution satisfies for all t [,Tα, β] TV ρt TV ρ +Lt ρ L e Lt and ρt L ρ L e Lt. The above result is local in time. Indeed, as β grows, Lβ may well grow, even unboundedly. Hence, both the total variation and the L norm of the solution may well blow up in finite time. To ensure global existence in time we need additional conditions on V : A: V is such that for all ρ L 1 R N ; R andallx R N,divVρ x. B: The function C in V1 is bounded, i.e. C L R + ; R +. Note that in the supply chain model discussed in Section 3, condition A applies. On the contrary, in the case of the pedestrian model in Section 4, iterating Theorem 2.2, we obtain the existence of solution up to time i T α i,α i+1. The latter turns out to be a convergent series, see Remark 5.5. Lemma 2.3. Assume all assumptions of Theorem 2.2. Let also A hold. Then, for all α>, thesetx α is invariant for 1.1, hence if the initial datum ρ is such that ρ L R N ;R α, then ρt L R N ;R α as long as the solution ρt exists. Condition B, although it does not guarantee the boundedness of the solution, does ensure the global existence of the solution to 1.1. Theorem 2.4. Let V1 hold. Assume moreover that A or B hold. Then, there exists a unique semigroup S : R + X X with the following properties: S1: For all ρ X, the orbit t S t ρ is a weak entropy solution to 1.1. S2: S is L 1 -continuous in time, i.e. for all ρ X,themapt S t ρ is in C R + ; X. S3: S is L 1 -Lipschitz with respect to the initial datum, i.e. for a suitable positive L C R + ; R +, for all t R + and all ρ 1,ρ 2 X, S t ρ 1 S t ρ 2 L1 R N ;R Lt ρ 1 ρ 2 L1 R N ;R. S4: There exists a positive constant L such that for all ρ X and all t R +, TV ρt TV ρ +Lt ρ L R N ;R e Lt. Higher regularity of the solutions of 1.1 can be proved under stronger bounds on V. Proposition 2.5. Let V1 and V2 hold. With the same notations as in Theorem 2.2, ifρ X α,then ρ W 1,1 L R N ; R = t [,Tα, β], ρt W 1,1 R N ; R, ρ W 1, R N ; R = t [,Tα, β], ρt W 1, R N ; R, and there exists a positive constant C = Cβ such that Furthermore, if V also satisfies V3, then ρt W 1,1 e 2Ct ρ W 1,1 and ρt W 1, e 2Ct ρ W 1,. ρ W 2,1 L R N ;[α, β] = t [,Tα, β], ρt W 2,1 R N ; R and for a suitable non-negative constant C = Cβ, we have the estimate The proofs are deferred to Section 5. ρt W 2,1 e Ct 2e Ct 1 2 ρ W 2,1.

5 2.2. Differentiability CONTROL OF THE CONTINUITY EQUATION WITH A NON LOCAL FLOW 357 This section is devoted to the differentiability of the semigroup S defined in Thm. 2.2 with respect to the initial datum ρ, according to the following notion. Recall first that a map F : L 1 R N ; R L 1 R N ; R is strongly L 1 Gâteaux differentiable in any direction at ρ L 1 R N ; R if there exists a continuous linear map DF ρ : L 1 R N ; R L 1 R N ; R such that for all r L 1 R N ; R and for any real sequence h n withh n, F ρ + h n r F ρ h n n DF ρ r strongly in L 1. Besides proving the differentiability of the semigroup, we also characterize the differential. Formally, a sort of first order expansion of 1.1 with respect to the initial datum can be obtained through a standard linearizing procedure, which yields 1.2. Now, we rigorously show that the derivative of the semigroup in the direction r is indeed the solution to 1.2 with initial condition r. To this aim, we need a forth and final condition on V. V4: There exists a function K C R + ; R + such that for all ρ L 1 R N ; R +, for all r L 1 R N ; R, V ρ + r V ρ DV ρr W 2, K ρ L + ρ + r L r 2 L 1, DV ρr W 2, K ρ L r L 1. Consider now system 1.2, where ρ C I ex, X is a given function. We introduce a notion of solution for 1.2 and give conditions which guarantee the existence of a solution. Definition 2.6. Fix r L R N ; R. A function r C I; L 1 loc RN ; R +, bounded and measurable, is a weak solution to 1.2 if for any test function ϕ C c I R N ; R T R N [r t ϕ + rat, x x ϕ div bt, xϕ] dx dt =where a=v ρ b=ρ DV ρr 2.2 and r = r a.e. in R N. We now extend the classical notion of Kružkov solution to the present non local setting. Definition 2.7. Fix r L R N ; R +. A function r C I; L 1 loc RN ; R +, bounded and measurable, is a Kružkov solution to the nonlocal problem 1.2 if it is a Kružkov solution to { t r +divrat, x+bt, x = r,x=r x where a = V ρ andb = ρ DV ρr. In other words, r is a Kružkov solution to 1.2 if for all k R and for any test function ϕ Cc I R N ; R + T [r k t ϕ +r kv ρ x ϕ div kv ρ+ρdvρr ϕ]sgnr kdxdt R N and rt r dx = forallδ>. lim t + B,δ Condition V4 ensures that if ρ W 1,1 R N ; R, then DV ρr C 2 R N ; R N and hence for all t, the map x ρt, xdv ρt rt, x is in W 1,1 R N ; R, so that the integral above is meaningful. Proposition 2.8. Let V1, V4 hold. Fix ρ C I ex ; L 1 R N ; R such that ρt W 1, W 1,1 R N ; R for all t I ex. Then, for all r L 1 L R N ; R there exists a unique weak entropy solution to 1.2

6 358 R.M. COLOMBO ET AL. in C I ex ; L 1 R N ; R and for all time t I ex, with C = C K ρ L [,t] R N ;R as in V4 ρ L [,t] R N ;R as in V1 and K = rt L 1 e Kt ρ L I;W 1,1 e Ct r L 1 rt L e Ct r L + Kte 2Ct e Kt ρ L I;W 1,1 ρ L I;W 1, r L 1. If V2 holds, ρ L I ex ;W 1, W 2,1 R N ; R and r W 1,1 L R N ; R, then for all t I ex, rt W 1,1 R N ; R and rt W 1,1 1 + C te 2C t r W 1,1 + Kt1 + Cte 4C t r L 1 ρ L I;W 2,1 where C =max { C, K ρ L I ex;w 1,1 R N ;R}. Under V1 and V4, the following mild differentiability result can be proved. Proposition 2.9. Let V1 and V4 hold. Let ρ W 1, W 1,1 R N ; R and r X 1.Then,thereexist h > and T = T ρ L, such that for all h [,h ] the solution ρ to 1.1 and the solution ρ h to { t ρ h +divρ h V ρ h = t, x R + R N ρ h,x=ρ x+hr x x R N 2.3 are defined for all t [,T ]. Moreover, if there exists an r L 1 [,T ] R N ; R such that ρ h ρ/h h r, then r is a distributional solution to 1.2, i.e. it satisfies 2.2. Below, we consider the following stronger hypothesis, under which we derive a result of strong Gâteaux differentiability and uniqueness of the derivative. V5: There exists a function K C R + ; R + such that for all ρ, ρ L 1 R N ; R div V ρ V ρ DV ρ ρ ρ L 1 K ρ L + ρ L ρ ρ L 1 2 and the map r div DV ρr is a bounded linear operator on L 1 R N ; R L 1 R N ; R, i.e. for all ρ, r L 1 R N ; R div DV ρr L 1 R N ;R ρ K L R N ;R r L 1 R N ;R. Theorem 2.1. Let V1, V3, V4 and V5 hold. Let ρ W 1, W 2,1 R N ; R, r W 1,1 L R N ; R, and denote T ex thetimeofexistenceofthesolutionof1.1 with initial condition ρ.then,forall time t I ex the local semigroup defined in Theorem 2.2 is strongly L 1 Gâteaux differentiable in the direction r. The derivative DS t ρ r of S t at ρ in the direction r is DS t ρ r =Σ ρ t r where Σ ρ t I ex. is the linear application generated by the Kružkov solution to 1.2, whereρ = S t ρ, then for all 2.3. Necessary optimality conditions for problems governed by 1.1 Aiming at necessary optimality conditions for non linear functionals defined on the solutions to 1.1, we prove the following chain rule formula.

7 CONTROL OF THE CONTINUITY EQUATION WITH A NON LOCAL FLOW 359 Proposition Let T > and I =[,T[. Assume that f C 1,1 R; R +, ψ L I R N ; R and that S : I L 1 L R N ; R L 1 L R N ; R is strongly L 1 Gâteaux differentiable. For all t I, let Jρ = f S t ρ ψt, xdx. 2.4 R N Then, J is strongly L Gâteaux differentiable in any direction r W 1,1 L R N ; R. Moreover, DJρ r = f S t ρ Σ ρ t r ψt, x dx. R N Proof. Since fρ h fρ f ρρ h ρ Lipf ρ h ρ 2,wehave Jρ + hr Jρ h f S t ρ DS t ρ r ψt, x dx R N f S t ρ S t ρ + hr S t ρ DS t ρ r R h ψt, x dx N +Lipf 1 h R N S t ρ + hr S t ρ 2 ψt, x dx. The strong Gâteaux differentiability of S t in L 1 then yields f S t ρ S t ρ + hr S t ρ DS t ρ r R h ψt, x dx = o1 as h N thanks to S t ρ L and to the local boundedness of f.furthermore, S t ρ,s t ρ + hr L 1 h S tρ + hr S t ρ h DS t ρ r pointwise a.e. S t ρ + hr S t ρ h pointwise a.e. the Dominated Convergence Theorem ensures that the higher order term in the latter expansion tend to as h. The above result can be easily extended. First, to more general non linear functionals Jρ =JS t ρ, with J : X R + such that for all ρ X there exists a continuous linear application DJ ρ: X Rsuch that for all ρ, r X: J ρ + hr Jρ h DJ ρr. h Secondly, to functionals of the type T Jρ = fs t ρ ψt, x dx dt or Jρ = R N T J S t ρ dt. This generalization, however, is immediate and we omit the details. Once the differentiability result above is available, a necessary condition of optimality is straightforward.

8 36 R.M. COLOMBO ET AL. Proposition Let f C 1,1 R; R + and ψ L I ex R N ; R. Assume that S : I L 1 L R N ; R L 1 L R N ; R is strongly L 1 Gâteaux differentiable. Define J as in 2.4. Ifρ L 1 L R N ; R solves the problem find min J ρ subject to {ρ is solution to 1.1} ρ then, for all r L 1 L R N ; R R N f S t ρ Σ ρ t r ψt, x dx = Demand tracking problems for supply chains Recently, Armbruster et al. [4], introduced a continuum model to simulate the average behavior of highly re-entrant production systems at an aggregate level appearing, for instance, in large volume semiconductor production line. The factory is described by the density of products ρt, x at stage x of the production at a time t. Typically, see [1,4,2], the production velocity V is a given smooth function of the total load 1 ρt, xdx, for example 1 vu =v max /1 + u and V ρ =v ρt, sds. 3.1 Thefullmodel,givenby withn = 1, fits in the present framework. Proposition 3.1. Let v C 1 [, 1]; R. Then, the functional V defined as in 3.1 satisfies A, V1, V2, V3. Moreover, if v C 2 [, 1]; R, thenv satisfies also V4 and V5. The proof is deferred to Section 5.4. The supply chain model with V given by 3.1 satisfiesv1 to V5 and A. Therefore, Theorem 2.4 applies and, in particular, the set [, 1] is invariant yielding global well posedness. By Theorem 2.1, the semigroup S t ρ is Gâteaux differentiable in any direction r and the differential is given by the solution to 1.2. Note that the velocity is constant across the entire system at any time. In fact, in a real world factory, all parts move through the factory with the same speed. While in a serial production line, speed through the factory is dependent on all items and machines downstream, in a highly re-entrant factory this is not the case. Since items must visit machines more than once, including machines at the beginning of the production process, their speed through factory is determined by the total number of parts both upstream and downstream from them. Such re-entrant production is characteristic for semiconductor production lines. Typically, the output of the whole factory over a longer timescale, e.g. following a seasonal demand pattern or ramping up or down a new product, can be controlled by prescribing the inflow density to a factory ρt, x = = λt. The influx should be chosen in order to achieve either of the following objective goals [4]: 1 Minimize the mismatch between the outflow and a demand rate target dt overafixedtimeperiod demand tracking problem. This is modelled by the cost functional 1 T 2 dt ρ1,t2 dt. 2 Minimize the mismatch between the total number of parts that have left the factory and the desired total number of parts over a fixed time period dt. The backlog of a production system at a given time t is defined as the total number of items that have been demanded minus the total number of items that have left the factory up to that time. Backlog can be negative or positive, with a negative backlog corresponding to overproduction and a positive backlog corresponding to a shortage. This problem is modeled by 1 T t 2 2 dτu ρ1,τdτ dt. In both cases we are interested in the influx λt. A numerical integration of this problem has been studied in [22]. In order to apply the previous calculus we reformulate the optimization problem for the influx density

9 CONTROL OF THE CONTINUITY EQUATION WITH A NON LOCAL FLOW 361 λt =ρ t whereρ is the solution to a minimization problem for J 1 ρ = 1 2 J 2 ρ = dx S T ρ x 2 dx x 2 dξ S T ρ ξ dξ dx, 3.2 respectively, where S t ρ is the solution to 1.1 and3.1. Clearly, J 1 and J 2 satisfy the assumptions imposed in the previous section. The assertions of Proposition 2.12 then state necessary optimality conditions, which we summarize in the theorem below. Theorem 3.2. Let T> be given. Let the assumptions of Proposition 3.1 hold. Let ρ W 1, W 2,1 R; R be a minimizer of J 1 as defined in 3.2, withs being the semigroup generated by Then, for all r W 1, W 2,1 R; R we have t r + x 1 v max dx ρt,x rt,xdx =, r 1 ρ dx + ρ 1 r dx =, ρ dx where r,x=r x. The latter Cauchy problem is in the form 1.2 and Proposition 2.8 proves its well posedness. The latter proof is deferred to Section A model for pedestrian flow Macroscopic models for pedestrian movements are based on the continuity equation, see [12,14,16,19], possibly together with a second equation, as in [17]. In these models, pedestrians are assumed to instantaneously adjust their vector speed according to the crowd density at their position. The analytical construction in Section 2 allows to consider the more realistic situation of pedestrian deciding their speed according to the local mean density at their position. We are thus led to consider 1.1 with V ρ =vρ η v 4.1 where η C 2 c R 2 ;[, 1] has support spt η B, 1 and η L 1 =1, 4.2 so that ρ ηx is an average of the values attained by ρ in Bx, 1. Here, v = vx is the given direction of the motion of the pedestrian at x R 2. Then, the presence of boundaries, obstacles or other geometric constraint can be described through v, see[16]. Note that condition A does not allow any increase in the crowd density. Hence, it may not hold when describing, say, the evacuation through a narrow exit. Therefore, in general, for this example we have only a local in time solution by Theorem 2.2. As in the preceding example, first we state the hypotheses that guarantee assumptions V1 to V5. Proposition 4.1. Let V be defined in 4.1 and η be as in If v C 2 R; R and v C 2 W 2,1 R 2 ; S 1,thenV satisfies V1 and V2. 2 If moreover v C 3 R; R, v C 3 R 2 ; R 2 and η C 3 R 2 ; R then V satisfies V3. 3 If moreover v C 4 R; R, v C 2 R 2 ; R 2 and η C 2 R 2 ; R, thenv satisfies V4 and V5.

10 362 R.M. COLOMBO ET AL. The proof is deferred to Section 5.4. A typical problem in the management of pedestrian flows consists in keeping the crowd density ρt, x below a given threshold, say ˆρ, in particular in a sensible compact region Ω. To this aim, it is natural to introduce a cost functional of the type T Jρ = f S t ρ x ψt, xdxdt 4.3 R N where f: f C 1,1 R; R +, fρ =forρ [, ˆρ], fρ > andf ρ > forρ>ˆρ. ψ: ψ C [,T] R N ;[, 1], with spt ψt = Ω, is a smooth approximation of the characteristic function of the compact set Ω, with Ω. Section 2.3 then applies, yielding the following necessary condition for optimality. Theorem 4.2. Let T> and the assumptions of 1 3 in Proposition 4.1 hold, together with f and ψ. Let ρ W 1, W 2,1 R; R be a minimizer of J as defined in 4.3, withs being the semigroup generated by Then,forallr W 1, W 2,1 R; R, ρ satisfies 2.5. The proof is deferred to Section 5.4. Consider the problem of evacuating a meeting room. Then, the optimal ρ corresponds to the best distribution of people during meetings, so that the room is evacuated in the minimal time in case of need. Below, we denote by W N the Wallis integral 5. Detailed proofs W N = 5.1. A lemma on the transport equation π/2 In what follows, a key role will be played by the transport equation cos α N dα. 5.1 { t r +divrwt, x = Rt, x r,x=r x. 5.2 The next lemma is similar to other results in recent literature, see for instance [2]. It provides the existence and uniqueness of solutions to 5.2 with the present regularity conditions, showing also that the concepts of weak and Kružkov solution here coincide, see also [26]. Lemma 5.1. Let T>, so that I =[,T[, andw be such that w C I R N ; R N wt C 1 R N ; R N t I w L I R N ; R N x w L I R N ; R N N. Assume that R L I; L 1 R N ; R L I R N ; R and r L 1 L R N ; R. Then, 1 the function r defined by t rt, x =r X; t, x exp div w τ,xτ; t, x dτ t t + R τ,xτ; t, x exp div w u, Xu; t, x du dτ, τ

11 CONTROL OF THE CONTINUITY EQUATION WITH A NON LOCAL FLOW 363 where t Xt; t,x is the solution to the Cauchy problem dχ = wt, χ dt 5.5 χt =x is such that r C I; L 1 R N ; R ; 2 the function r, as defined by 5.4, isakružkov solution to 5.2; 3 any weak solution to 5.2 coincides with r as defined in 5.4. Recall that, in the present case, [21], Definition 1, amounts to define Kružkov solution to 5.2 a function r L I; L 1 loc RN ; R, continuous from the right in time, such that for all k R, for all test function ϕ Cc ],T[ RN ; R + T [r k t ϕ + w x ϕ+r kdiv wϕ]sgnr kdx dt 5.6 R N and such that for almost every t [,T], for any δ>, lim rt, x r x dx =. 5.7 t B,δ On the other hand, by weak solution to 5.3 wemeanamapr C I; L 1 R N ; R L I R N ; R that satisfies 5.3 in distributional sense. Remark that as an immediate corollary of Lemma 5.1, we obtain that any weak solution to 5.2 isalsoakružkov solution and is represented by 5.4. Note that the expression 5.4 is formally justified integrating 5.2 along the characteristics 5.5 and obtaining d r t, χt + r t, χt div w t, χt = R t, χt. dt Recall for later use that the flow X = Xt; t,x generated by 5.5 can be used to introduce the change of variable y = X; t, x, so that x = Xt;,y, due to standard properties of the Cauchy problem 5.5. Denote by Jt, y =det y Xt;,y the Jacobian of this change of variables. Then, J satisfies the equation djt, y dt =divw t, Xt;,y Jt, y 5.8 t with initial condition J,y= 1. HenceJt, y = exp div w τ,xτ;,y dτ which, in particular, implies Jt, y > for all t I, y R N. Proof of Lemma Let R n L I; C 1 R N ; R and r,n C 1 R N ; R approximate R and r in the sense R n R L I;L 1 R N ;R and r,n r n L 1 R N ;R as n +. Call r n the corresponding quantity as given by 5.4. Then, by 5.4, also r n r L I;L 1 R N ;R as n + so that r L I; L 1 R N ; R. Concerning continuity in time, simply note that, by 5.4, r n C I; L 1 R N ; R and r is the uniform limit of the r n.

12 364 R.M. COLOMBO ET AL. 2 The boundedness requirement follows from 5.4: rt L R N ;R r L R N ;R + t R L I R N ;R e t divw L I R N ;R Let k R and ϕ Cc I R N ; R +. Then, according to [21], Definition 1, we prove 5.6 forr given as in 5.4. By 5.8, the semigroup property of X and denoting Rt, y = t R τ,xτ;,y Jτ,ydτ, so that R is W 1,1 in time, we get + [ r y Rt, y [r k t ϕ + w x ϕ+r k div wϕ]sgnr kdx dt = + R N R Jt, y Jt, y k N t ϕ t, Xt;,y + w t, Xt;,y x ϕ t, Xt;,y ] +R t, Xt;,y k div w t, Xt;,y ϕ t, Xt;,y = r y sgn Jt, y [ + Rt, y + Jt, y k Jt, ydy dt r y d R dt ϕ t, Xt;,y kjt, y d ϕ t, Xt;,y dt N ] kϕt, Xt;,y d dt Jt, y+ d dt Rt, yϕ t, Xt;,y sgn r y+rt, y kjt, y dy dt + d = R dt r y+rt, y kjt, y ϕ t, Xt;,y N sgn r y+rt, y kjt, y dy dt + d = R dt r y+rt, y kjt, y ϕ t, Xt;,y dy dt N =. Finally, 5.7 holds by the continuity proved above. 3 Let r beaweaksolutionto5.3. It is enough to consider the case R =andr =, thanks to the linearity of 5.3. Then, fix τ ],T], choose any ϕ C 1 I; Cc 1RN ; R and let β ε C 1 I; R such that β ε t =1fort [ε, τ ε], β εt [, 2/ε] fort [,ε], β εt [ 2/ε, ] for t [τ ε, τ] andβt = for t [τ,t]. By the definition of weak solution, τ ε τ = r t ϕ + rw ϕ β ε dx dt + β ε rϕdxdt + rϕβ ε dx dt R N R N τ ε R N = τ β ε r t ϕ + rw ϕdxdt + R N τ + β εt rt, x ϕt, xdxdt. τ ε R N 1 εβ ε εt R N rεt, x ϕεt, xdx dt Consider the three terms separately. As ε, the first one converges to R N τ r tϕ + w ϕdt dx. Consider the second summand, by the Dominated Convergence Theorem, it tends to, since r =.

13 CONTROL OF THE CONTINUITY EQUATION WITH A NON LOCAL FLOW 365 τ τ ε Passing to the latter term, note that: β ε R t τ rt, x ϕt, xdxdt + rτ,xϕτ,xdx = N R N τ ε β ε R t rt, x ϕt, x rτ,xϕτ,x dx dt N as ε by the continuity of r in time and the smoothness of ϕ. Therefore, choose any η C 1 c RN ; R and define ϕ as the backward solution to t ϕ + w ϕ =with ϕτ =η. Then, τ = r t ϕ + w ϕdtdx rτ,xϕτ,xdx R N R N = rτ,xεxdx R N proving that rτ, and hence r, vanishes identically Proof of Theorem 2.2 Corollary 5.2. In the same setting of Lemma 5.1, ifr =then: ρ for a.e. x ρt for a.e. x and for all t I, div w for a.e. t, x ρt L ρ L for all t I, ρt L R N ;R ρ L R N ;R t exp div w L [,t] R N ;R. 5.1 The proof follows directly from Lemma 5.1 using in particular 5.4. Lemma 5.3. In the same setting of Lemma 5.1, ifr =and also div w L I; L 1 R N ; R and x div w L I; L 1 R N ; R N 5.11 then, setting κ = NW N 2N +1 x w L I R N ;R N N and κ =2N x w L I R N ;R N N we have the following bound on the total variation: TV ρt TV ρ e κt t + NW N e R κt s e s divw L x div ws, x dx ds ρ L N 5.12 where x div ws, x is the usual Euclidean norm of a vector in R N.

14 366 R.M. COLOMBO ET AL. Let ρ 1, ρ 2 be the solutions of 5.2 associated to w 1, w 2 with R 1 = R 2 =and with initial conditions ρ 1,, ρ 2, in X.Then where ρ 1 ρ 2 t L 1 e κt ρ 1, ρ 2, L 1 + eκt e κt κ κ TV ρ 1, w 1 w 2 L t e κt s e κt s + NW N e s divw κ κ L x div w 1 s, x dx ds 5.13 R N ρ 1, L w 1 w 2 L + t e κt s e s divw L R N div w 1 w 2 s, x dx ds ρ 1, L, κ = NW N 2N +1 x w 1 L I R N ;R N N and κ =2N x w 1 L I R N ;R N N. Proof. The bound 5.12 follows from [15], Theorem 2.5, the hypotheses on w being satisfied thanks to 5.3 and More precisely, we do not have here the C 2 regularity in time as required in [15], Theorem 2.5, but going through the proof of this result, we can see that only the continuity in time of the flow function ft, x, r = rwt, x is necessary. Indeed, time derivatives of f appear in the proof of [15], Theorem 2.5, when we bound the terms J t and L t,see[15], between 4.18 and However, the use of the Dominated Convergence Theorem allows to prove that J t and L t converge to zero when η goes to without any use of time derivatives. The continuity in times follows from [15], Remark 2.4, thanks to 5.11 ofw and the bound on the total variation. Similarly, the stability estimate 5.13 is based on [15], Theorem 2.6. Indeed, we use once again a flow that is only C instead of C 2 in time. Besides, in the proof of [15], Theorem 2.6, the L bound into the integral term in [15], Theorem 2.6, can be taken only in space, keeping time fixed. With this provision, the proof of 5.13 is exactly the same as that in [15], so we do not reproduce it here. The same estimate is thus obtained, except that the L bound of the integral term is taken only in space. Lemma 5.4. In the same setting of Lemma 5.1 if R =and 5.11 holds, together with 2 x w L I R N ;R N N N C for a suitable C>, where we assume that C> x w L I R N ;R N N,then { ρt W 1,1 R N ; R for all t I ρ W 1,1 R N ; R ρt W 1,1 e 2Ct ρ W 1,1, { ρt W 1, R N ; R for all t I ρ W 1, R N ; R ρt W 1, e 2Ct ρ W 1,. If moreover 3 x w L I R N ;R N N N N C then ρ W 2,1 R N ; R implies that ρt W 2,1 R N ; R for all t I and ρt W 2,1 1 + Ct 2 e 3Ct ρ W 2,

15 CONTROL OF THE CONTINUITY EQUATION WITH A NON LOCAL FLOW 367 Proof. The W 1,1 and W 1, bounds follow from the representation 5.4 with R =, noting that x X L e Ct. Indeed, x Xt;,x=Id + x Xt;,x t t t x wτ; Xτ;,x x Xτ;,xdτ, hence x wτ; Xτ;,x x Xτ;,x dτ C x Xτ;,x dτ and a direct application of Gronwall Lemma gives the desired bound. Hence, we obtain ρt L e 2Ct e Ct ρ L +e 2Ct ρ L and consequently ρt W 1, e 2Ct ρ W 1,. Similarly, the W 1,1 estimate also comes from 5.4. To obtain the W 2,1 bound 5.14, apply 5.4 and Gronwall Lemma to get 2 x X L e 2Ct e Ct. Using the estimates above, together with 2 ρt L 1 2e 2Ct 3e Ct +1e Ct ρ L 1 +3e Ct 1e 2Ct ρ L 1 +e 3Ct 2 ρ L 1, we obtain ρt W 2,1 2e Ct 1 2 e Ct ρ W 2,1, concluding the proof. We use now these tools in order to obtain the existence of a solution for 1.1. Proof of Theorem 2.2. Fix α, β > withβ>α.lett α, β =lnβ/α /Cβ, with C as in V1. Define the map C Q : I; X β C I; X β σ ρ where I =[,Tα, β[ and ρ is the Kružkov solution to { t ρ +divρw= ρ,x=ρ x with w = V σ ρ X α Assumptions V1 imply the hypotheses on w necessary in Corollary 5.2 and Lemma 5.3. Therefore, a solution ρ to 5.15 exists, is unique and in X. Note that by 5.1, by the choice of T α, β andbyv1, wehave ρt L β and hence Q is well defined. Fix σ 1,σ 2 in C I; X β. Call w i = V σ i andρ i the corresponding solutions. With the same notations of [15], Theorem 2.6, we let κ = NW N 2N +1 x w 1 L I R N ;R N N, κ =2N xw 1 L I R N ;R N N. Note that by 5.1 κ κ N π/2 1 2 N π x dx = π 1 1 3π 2 2N +1 8 > 1 hence κ >κ. Then, by V1, we obtain a bound on κ. Indeed, x V σ 1 L I R N ;R N N σ C 1 L I R N ;R,

16 368 R.M. COLOMBO ET AL. and since σ 1 C I; X β, finally κ NW N 2N +1Cβ. Let us denote C = Cβ and C = NW N 2N +1Cβ Again, V1 implies the following uniform bounds on all σ 1,σ 2 C I; X β : 2 x V σ 1 L I;L 1 R N ;R N N N C, V σ 1 V σ 2 L I R N ;R N C σ 1 σ 2 L I;L 1 R N ;R, div V σ 1 V σ 2 L I;L 1 R N ;R C σ 1 σ 2 L I;L 1 R N ;R. Thus, we can apply [15], Theorem 2.6. We get, for all t I, ρ 1 ρ 2 t L 1 Cte C t TV ρ σ 1 σ 2 L [,t];l 1 + C 2 NW N e Ct ρ L σ 1 σ 2 L [,t];l 1 + C e Ct ρ L t t e C t s σ 1 σ 2 s L 1 ds. t se C t s ds Therefore, we obtain the following Lipschitz estimate: Qσ 1 Qσ 2 L I;L 1 T [ TV ρ CTeC +NW N CT +1e CT ρ L ] σ1 σ 2 L I;L 1. Here we introduce the strictly increasing function ft =CTe C T [ TV ρ +NW N CT +1e CT ] ρ L and we remark that ft whent. Choose now T 1 > sothatft 1 =1/2. Banach Contraction Principle now ensures the existence and uniqueness of a solution ρ to 1.1on[, T ] in the sense of Definition 2.1, with T =min{t α, β,t 1 }.Infact,ifT 1 <Tα, β, we can prolongate the solution until time T α, β. Indeed, if we take ρ T 1 as initial condition, we remark that ρ T 1 L ρ L e CβT1. Consequently, the solution of 5.15 on[t 1,Tα, β] instead of I satisfy, thanks to 5.1 ρt L ρ T 1 L e Cβt T1 ρ L e CβT1 e Cβt T1 ρ L e CβT α,β, which is less than β thanks to the definition of T α, β and since ρ X α. Now, we want to define a sequence T n such that T n T α, β forn sufficiently large. Let assume that the sequence is constructed up to time T n with T n <Tα, β. For T n+1 ]T n,tα, β], we obtain the contraction estimate Qσ 1 Qσ 2 L [T n,t n+1];l 1 CT n+1 T n e C T [ n+1 T n ] TV ρt n +NW N CT n+1 T n +1e CTn ρ L σ 1 σ 2 L [T n,t n+1];l 1 [ ] TV ρ e C T n + C T n e C T n +NW N CT n+1 T n +1e CTn ρ L C T n+1 T n e C T n+1 T n σ 1 σ 2 L [T n,t n+1];l 1 where we used the bounds on TV ρt n and ρt n L R N ;R provided by Lemma 5.3 and Corollary 5.2 associated to V1. We may thus extend the solution up to time T n+1,wherewetaket n+1 >T n such that [ ] TV ρ e C T n + CT n e C T n +NW N CT n+1 T n +1e CTn ρ L CT n+1 T n e C T n+1 T n = 1 2 If T n+1 >Tα, β, then we are done.

17 CONTROL OF THE CONTINUITY EQUATION WITH A NON LOCAL FLOW 369 If we assume that the sequence T n defined by induction as above remains less than T α, β, in particular it is bounded. Consequently the left hand side above tends to, whereas the right hand side is taken equal to 1/2 >. Hence, the sequence T n is unbounded. In particular, for n large enough, T n is larger than T α, β; thus the solution to 1.1 is defined on all I. The Lipschitz estimate follows by applying the same procedure as above, in the case when the initial conditions are not the same. The L and TV bounds follow from 5.1 and from Lemma 5.3. The proof of Lemma 2.3 directly follows from Lemma 5.3. Proof of Theorem 2.4. We consider the assumptions A and B separately. A: Let T>, so that I =[,T[, and fix a positive α. As in the proof of Theorem 2.2, we define the map Q : C I; X α C I; X α σ ρ where ρ is the Kružkov solution to 5.15 withρ X α. The existence of a weak entropy solution for 5.15 in C I,L 1 R N ; R is given by Lemma 5.1, the set of assumptions V1 allowing to check the hypotheses on w necessary to apply Lemma 5.3. Note that furthermore A, thanks to Corollary 5.2, givesanl bound on ρ, so that for all t I, ρt [,α], a.e. in x. Fix σ 1,σ 2 in C I; X α, call w i = V σ i andletρ 1, ρ 2 be the associated solutions. With the same notations of [15], Theorem 2.6, we let as in the proof of Theorem 2.2, κ = NW N 2N +1 x w 1 L I R N ;R N N, κ =2N xw 1 L I R N ;R N N so that κ >κ. Then, by V1 we have: x V σ 1 L I R N ;R N N σ C 1 L I R N ;R N N, and since σ 1 C I; X α, we have σ 1 L α so that κ NW N 2N +1Cα. Denote C = NW N 2N +1Cα and C = Cα The following bounds are also available uniformly for all σ 1,σ 2 C R + ; X α, by V1: 2 x V σ 1 L I;L 1 R N ;R N N N C, V σ 1 V σ 2 L I R N ;R C σ 1 σ 2 L I;L 1 R N ;R, div V σ 1 V σ 2 L I;L 1 R N ;R C σ 1 σ 2 L I;L 1 R N ;R. Applying [15], Theorem 2.6, we get ρ 1 ρ 2 t L 1 Cte C t TV ρ σ 1 σ 2 L [,t];l 1 t + C 2 NW N t se C t s ds σ 1 σ 2 L [,t];l 1 + t C e C t s σ 1 σ 2 s L 1 ds. So that Qσ 1 Qσ 2 L I;L 1 CTeC T [TV ρ +NW N CT +1] σ 1 σ 2 L I;L 1. Here we introduce the function ft =CTe C T [TV ρ +NW N CT + 1] and we remark that ft when T. Choose now T 1 > sothatft 1 = 1 2. Banach Contraction Principle now ensures the existence and uniqueness of a solution to 1.1 on[,t 1 ] in the sense of Definition 2.1.

18 37 R.M. COLOMBO ET AL. Let assume that we have defined T 1 <T 2 <...<T n.lett n+1 >T n,then Qσ 1 Qσ 2 L [T n,t n+1];l 1 CT n+1 T n e C T n+1 T n [TV ρt n + NW N CT n+1 T n +1] σ 1 σ 2 L [T n,t n+1];l 1 [ ] TV ρ e C T n + C T n e C T n + NW N CT n+1 T n +1 C T n+1 T n e C T n+1 T n σ 1 σ 2 L [T n,t n+1];l 1 where we used the bounds on ρt n L R N ;R and TV ρt n provided by Corollary 5.2 and Lemma 5.3 associated to the conditions A and V1. We may thus extend the solution up to time T n+1, that we define implicitly by [ ] TV ρ e C T n + CT n e C T n + NW N CT n+1 T n +1 CT n+1 T n e C T n+1 T n = 2 1 Note that this equation defines a unique T n+1, the function T [ ] TV ρ e C T n + CT n e C T n + NW N CT T n +1 CT T n e C T T n, being increasing, taking value in T = T n and going to + when T goes to. If the sequence T n is bounded, then the left hand side above tends to, whereas the right hand side is taken equal to 1/2 >. Hence, the sequence T n is unbounded and the solution to 1.1 isdefinedonallr +. S4 follows from Lemma 5.1 and V1. S3 is proved as S1. Note that the Lipschitz constant so obtained is dependent on time. The bound S2 follows from Lemma 5.3, thatgives TV ρt TV ρ e C t + NW N Cte C t ρ L. B: Repeat the proof of Theorem 2.2 and, with the notation therein, note that if we find a sequence α n such that n T α n,α n+1 =+ where T α, β = [ln β/α] /Cβ, then the solution is defined on the all R +. It is immediate to check that B implies that k T α n,α n+1 n=1 C L R +;R + 1 ln αk + as k + completing the proof. Proof of Proposition 2.5. The bounds on ρ in W 1, and W 1,1 follow from Lemma 5.4, thanks to V2. The W 2,1 bound comes from 5.14 in Lemma 5.4, thanks to V Gâteaux differentiability First of all, if r L L 1 R N ; R andρ L I ex ;W 1,1 W 1, R N ; R, we prove that the equation 1.2 admits a unique solution r C I ex ; L 1 R N ; R. Proof of Proposition 2.8. We use here once again Lemma 5.1 in order to get an expression of the Kružkov solution for 5.2. We assume now that ρ C I ex ;W 1, W 1,1 R N ; R and we define w = V ρ; we also set, for all s C I ex ; L 1 R N ; R, R =divρdv ρs. Thanks to the assumptions on ρ and V4, we obtain

19 CONTROL OF THE CONTINUITY EQUATION WITH A NON LOCAL FLOW 371 R L I ex ; L 1 R N ; R L I ex R N ; R. Let ε I ex. Then, on [,T ex ε] we can apply Lemma 5.1 giving the existence of a Kružkov solution to t r +divrw =R, rx, = r L L 1 R N ; R. Let T [,T ex ε] andi =[,T[. We denote Q the application that associates to s C I; L 1 R N ; R the Kružkov solution r C I; L 1 R N ; R of 5.2 with initial condition r L L 1 R N ; R, given by Lemma 5.1. Thatistosay Q: s rt, x =r X; t, x exp t t div V ρτ,xτ; t, x dτ div ρdv ρs τ,xτ; t, x exp t τ div V ρu, Xu; t, x du dτ. Let us give some bounds on r. The representation of the solution 5.4 allows indeed to derive a L bound on r. For all t I, thanks to V1 and V4 we get, with C = C ρ L [,T ex ε] R N ;R, rt L r L e Ct + te Ct ρ L [,t];w 1, DV ρ W 1, s L [,t],l 1. The same expression allows also to derive a L 1 bound on rt rt L 1 r L 1e Ct + te Ct ρ L [,t];w 1,1 DV ρ W 1, s L [,t],l 1. Now, we want to show that Q is a contraction. We use once again the assumption V4. For all s 1,s 2 L I;L 1 BVR N ; R continuous from the right, we have Thus, we get: div ρdv ρs 1 s 2 L 1 R N ;R C ρ W 1,1 R N ;R s 1 s 2 L 1 R N ;R. Qs 1 Qs 2 L I;L 1 C ρ L I;W 1,1 s 1 s 2 L I;L 1 T e CT 1 ρ L [,T ex ε];w 1,1 s 1 s 2 L I;L 1. exp T τ div V ρ L dτ Then, for T small enough, can apply the Fixed Point Theorem, that gives us the existence of a unique Kružkov solution to the problem. Furthermore, as the time of existence does not depend on the initial condition, we can iterate this procedure to obtain existence on the interval [,T ex ε]. Finally, as this is true for all ε I ex,we obtain the same result on the all interval I ex. The L 1 bound follows from 5.4. Let T I ex and t I, then for a suitable C = C ρ L I R N ;R t rt L 1 r L 1e Ct + ρ L I;W 1,1 div DV ρ L rτ L 1 dτ. AuseofV4 and an application of Gronwall Lemma gives rt L 1 e Ct e K ρ L I,W 1,1 t r L 1, where K = K ρ L I R N ;R is as in V4.

20 372 R.M. COLOMBO ET AL. The L bound comes from the same representation formula. Indeed, for T I ex and t I we have t rt L e Ct r L + ρ L I;W 1, div DV ρ L rτ L 1 dτ. Then, the last rτ L 1 is bounded just as above. We get rt L e Ct r L + Kte 2Ct e K ρ L I,W 1,1 t r L 1 ρ L I,W 1,. Finally, we get a W 1,1 bound using the expression of the solution given by Lemma 5.1. Indeed, assuming in addition V2 and V4, we get rt L 1 R N ;R e2ct r L 1 + Cte 2Ct r L 1 + K1 + Cte 2Ct ρ L I;W 2,1 e 2Ct r L 1 + Cte 2Ct r L 1 + Kt1 + Cte 3Ct e K ρ L I,W 1,1 t r L 1 ρ L I;W 2,1. Hence, denoting C =max{c, K ρ L I,W 1,1 },weobtain t rτ L 1 dτ rt W 1,1 r W 1,11 + C te 2C t + Kt1 + Cte 4C t r L 1 ρ L I;W 2,1 concluding the proof. Now, we address the weak Gâteaux differentiability of the semigroup generated by 1.1. Proof of Proposition 2.9. Let α = max{ ρ L, 1} and β > α. Fix h [,h ]withh small enough so that β > α1 + h. Note that ρ,r X α. By Theorem 2.2, 1.1 admits the weak entropy solutions ρ C [,Tα, β]; X β, and 2.3 admits the solution ρ h C [,Tα1 + h,β]; X β. Note that T α1 + h,β= lnβ/α1 + h Cβ = T α, β ln1 + h Cβ T α, β and T α1 + h,β tends to T α, β ash goes to. In particular, both solutions are defined on the interval [,T ], where T = T α1 + h,β. Write now the definition of weak solution for ρ, ρ h.letϕ Cc [,T ] R N ; R ρ t ϕ +ρv ρ x ϕdx dt =, R + R N ρ h t ϕ +ρ h V ρ h x ϕdx dt =, R N R + use V4 and write, for a suitable function ε = ερ, ρ h, V ρ h =Vρ +DV ρρ h ρ +ερ, ρ h, with 2. ερ, ρ h L R N ;R ρ K2β h ρ L 1 R ;R Then, N ρv ρ ρ h V ρ h =ρ ρ h V ρ+ρdv ρρ ρ h +ρ ρ h DV ρρ ρ h ρ h ερ, ρ h.

21 CONTROL OF THE CONTINUITY EQUATION WITH A NON LOCAL FLOW 373 Consequently, [ ρ ρh ρ ρh t ϕ + V ρ+ρdv ρ h h R + R N ρ ρh h + ρ ρ ] h ερ, ρ h DV ρρ ρ h ρ h x ϕ dx dt =. h h Using V4, ρt X β and the estimate on ε we obtain for all t [,T ]: x ϕ ρ ρ h ερ, ρ h DV ρρ ρ h ρ h R h h dx N Cβ ρ ρ h + βk2β ρ ρ h L 1 ρ ρ h R h N L 1 x ϕ dx, as h, by the Dominated Convergence Theorem, since ρ h ρ h we get That is to say, r is a distributional solution to 1.2. R is bounded in L 1 [,T ] R N ; R. Then, if ρ h ρ/h h r, R N [r t ϕ +rv ρ+ρdv ρr x ϕ]dx dt =. As this is true for all h small enough, finally we obtain a solution on the all interval [,Tα, β[. Hence, we observe that if ρ C I ex R N ; R, then r is defined on all I ex. Assume now that V4 and V5 are satisfied by V. We want to show that with these hypotheses, we have now strong convergence in L 1 to the Kružkov solution of 1.2. Proof of Theorem 2.1. Let α, β > withβ>α,andh [,h ]withh small enough so that β>α1 + h. Let us denote T h =T α1 + h,βforh [,h ] the time of existence of the solution of 1.1 givenby Theorem 2.2. Fix ρ W 1, W 2,1 R N ;[,α], r L W 1,1 R N ;[,α]. Let ρ, respectively ρ h,betheweak entropy solutions of 1.1 given by Theorem 2.2 with initial condition ρ, respectively ρ + hr. Note that these both solutions are in C [,Th ]; L 1 R N ; R. Furthermore, under these hypotheses for ρ and r,we get thanks to Proposition 2.5 that the corresponding solutions ρ and ρ h of 1.1 areinc [,Th ]; W 1, W 2,1 R N ;[,β], condition V3 being satisfied. Hence, we can now introduce the Kružkov solution r C [,Th [ ; L 1 R N ; R of 1.2, whose existence is given in this case by Proposition 2.8. Note that, r being in W 1,1 R N ; R andρ L [,Th ]; W 2,1 R N ; R and V2, V4 being satisfied, rt isalsoin W 1,1 R N ; R for all t [,Th [ thanks to the W 1,1 bound of Proposition 2.8. Let us denote z h = ρ+hr. We would like to compare ρ h and z h thanks to [15], Theorem 2.6. A straightforward computation shows that z h is the solution to the following problem, { t z h +divz h V ρ+hdv ρr = h 2 div r DV ρr, z h = ρ + hr X α1+h. Note that the source term being in C [,Th [ ; L 1 R N ; R, and the flow being regular, we can apply to this equation Lemma 5.1 that gives existence of a Kružkov solution. As in the proof of Lemma 5.1, wemakeheretheremarkthat[15], Theorem 2.6, can be used with the second source term in C [,Th [ ; L 1 R N ; R and the flow C 2 in space and only C in time. Besides, we

22 374 R.M. COLOMBO ET AL. also use the same slight improvement as in the proof of Lemma 5.3, takingthel norm in the integral term only in space, keeping the time fixed. We get, with κ = NW N 2N +1 x V ρ h L [,T h ] R N ;R and κ =2N x V ρ h L [,T h ] R N ;R,forsomeT [,Th ] and with I =[,T], ρ h z h L I;L 1 T eκt TV ρ + hr V ρ h V ρ hdv ρr L [,T h ] R N ;R N T + NW N T te R κt t ρ h t L x div V ρ h dx dt N V ρ h V ρ hdv ρr L [,T h ] R N ;R N T + h 2 e R κt t div rdv ρr dx dt N + T e κt t R N div V ρ h V ρ hdv ρr dx dt max t [,T ] { ρ ht L, z h t L }. Then, setting C = Cβ andk = K2β, we use: the bound of ρ and ρ h in L given by Corollary 5.2 ρt L ρ L e Ct β and ρ h t L ρ + hr L e Ct β ; the properties of V given in V1 to get x div V ρ h L R + R N ;R C and x div V ρ h L I;L 1 R N ;R C ; the property V4, respectively V5, toget V ρ h V ρ hdv ρr L I R N ;R ρ K h ρ 2 L I;L 1 R N ;R + ρ h z h L I;L 1 R N ;R, respectively div V ρ h V ρ hdv ρr L I;L 1 R N ;R ρ K h ρ 2 L I;L 1 R N ;R + ρ h z h L I;L 1 R N ;R ; the property V4 to get div rdv ρr L 1 K r W 1,1 r L 1. Gathering all these estimates, denoting C = NW N 2N +1C, weobtain ρ h z h L I;L 1 T ec T TV ρ + hr +NW N CTβ K ρ h ρ 2 L I;L 1 + ρ h z h L I;L 1 Then, dividing by h and introducing + h 2 KTe C T r L I;W 1,1 r L I;L 1 + β + h sup rt L t I T e C T K ρ h ρ 2 L I;L 1 + ρ h z h L I;L 1. F h T =KTe C T [TV ρ +h TV r +NW N CTβ + β + h rt L ],

23 CONTROL OF THE CONTINUITY EQUATION WITH A NON LOCAL FLOW 375 we obtain ρ h z h h L I;L 1 [ F h T ρ h ρ ρ h ρ L I;L 1 h + ρ h z h L I;L 1 h L I;L 1 ] + hkt e C T r L I;W 1,1 r L I;L 1. Note that F h is a function that vanishes in T = and that depends also on ρ, r and h. Hence, we can find T T h small enough such that F h T 1/2. Furthermore, for all T T h, h F h T isincreasing and hence h h implies F h T F h T. Noting moreover that ρ h ρ h L has a uniform bound M in h I;L 1 by 2 in Theorem 2.2, wegetfort T 1 ρ h ρ 2 r = 1 ρ h z h h 2 h L I;L 1 L I;L 1 M 2 ρ h ρ L I;L 1 + hkt ec T r L I,W 1,1 r L I;L 1. The right side above goes to when h, so we have proved the Gâteaux differentiability of the semigroup S for small time. Finally, we iterate like in the proof of Theorem 2.2 in order to have existence on the all interval [,Th ]. Let T 1 be such that F h T 1 =1/2 and assume T 1 <Th. If we assume the Gâteaux differentiability is proved until time T n T h, we make the same estimate on [T n,t n+1 ], T n+1 being to determine. We get ρ h z h L [T n,t n+1];l 1 T n+1 T n e C T n+1 T n TV ρ h T n + NW N CT n+1 T n β K ρ h ρ 2 L [T n,t n+1];l 1 + ρ h z h L [T n,t n+1];l 1 + h 2 KT n+1 T n e C T n+1 T n r L [T n,t n+1];w 1,1 r L [T n,t n+1];l 1 + β + h sup rt L T n+1 T n e C T n+1 T n [T n,t n+1] K ρ h ρ 2 L [T n,t n+1];l 1 + ρ h z h L [T n,t n+1];l 1. Then, we divide by h and we introduce, for T T n [ F h,n T =KT T n e C T T n TV ρ +htv r e CTn + βc T n e C T n ] + NW N CT T n β + β + h sup [T n,t n+1] rt L. We define T n+1 >T n such that F h,n T n+1 = 1 2. ThisispossiblesinceF h,n vanishes in T = T n and increases to infinity when T. Hence, as long as T n+1 T h, we get ρ h ρ h r L [T n,t n+1];l 1 KM ρ h ρ L [T n,t n+1];l 1 +2hKT n+1 T n e C T n+1 T n r L [T n,t n+1],w 1,1 r L [T n,t n+1];l 1. The next question is to wonder if T n goesuptot h. We assume that it is not the case: then necessarily, F h,n T n+1 n, since T n+1 T n. This is a contradiction to F h,n T n+1 =1/2.

24 376 R.M. COLOMBO ET AL. Consequently, T n n and the Gâteaux differentiability is valid for all time t [,Th ]. Then, making h goes to, we obtain that the differentiability is valid on in the interval [,Tα, β[. It remains to check that the Gâteaux derivative is a bounded linear operator, for t and ρ fixed. The linearity is immediate. Additionally, due to the L 1 estimate on the solution r of the linearized equation 1.2 givenby Proposition 2.8, weobtain DS t ρ r L 1 = rt L 1 e Kt ρ L I;W 1,1 e Ct r L 1, so that the Gâteaux derivative is bounded, at least for t T<T ex Proofs related to Sections 3 and 4 Proof of Proposition 3.1. Note that vρ is constant in x, hence div V ρ =, and A is satisfied. Besides, we easily obtain x V ρ L R;R =, xv ρ L1 R;R =, 2 x V ρ L 1 R;R =and V ρ 1 V ρ 2 L R;R v L R;R ρ 1 ρ 2 L 1 R;R, x V ρ 1 x V ρ 2 L 1 R;R =, so that V1 is satisfied. Similarly, 2 xv ρ =and 3 xv ρ = imply easily that V2 and V3 are satisfied. We consider now V4: isv is C 2 then, for all A, B R, vb =va+v AB A+ 1 v sb +1 sa1 sb A 2 ds. Choosing A = 1 ρξdξ and B = 1 ρξdξ, weget v ρξdξ v ρξdξ v ρξdξ ρ ρξdξ 1 L 2 v L ρ ρ 2 L 1 and we choose K = 1 2 v L,DV ρr =v 1 ρξdξ 1 rξdξ. Condition V4 is then satisfied since there is no x-dependence, so V ρ V ρ DV ρ ρ ρ W 2, = V ρ V ρ DV ρ ρ ρ L 1 2 v L ρ ρ 2 L 1. Similarly, DV ρr W 2, = DV ρr L v L r L 1. Finally, consider V5: L [v div ρξdξ v ρξdξ v ρξdξ ρ ρξdξ] =, L div v ρξdξ rξdξ =. 1 Concluding the proof. Proof of Proposition 4.1. The proof exploits the standard properties of the convolution.