1 Introduction

In recent literature,mainly scalar Bellman equations which are coupled with a Fokker-Planck equation are studied and we refer to starting paper[9]or to a survey[6],see also[7].These equations model a Nash game with a large number of players behaving similarly,so that the decision can be approximate by a single decision make(a representative agent).So we have a scalar Bellman equation.The present paper considers a model suggested by Bensoussan,accompanied by co-authors(see[3]),where the decision of finite number of N players of large population of Nash-game-playersare approximated by N representative agents.So the Bellman system is a system of backward parabolic equations coupled with a forward mean field equation.

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In principle,the dependence of the coefficients on the data in the nonlinearities of the equation may be a functional one.But in order to have the first insight in the difficulties and in order to simplify the presentation,we confine to a point-wise dependence of Hamiltonians with respect to the mean field variable here.Generally,obtaining the existence of solution of the Bellman mean field dependent system with growth with respect to the mean field variable in nonlinearities is the critical subject.Without additional assumptions only a poor growth behaviour is permitted.For the case of scalar Bellman equations,for obtaining the global solvability,[10–11]gave a quite exhaustive analysis of the growth conditions for the Hamiltonians concerning the dependence on∇u(u is a generalized value function)and the mean field m,which appears in the pay off functional.

In comparison,in the present paper,we restrict ourselves to Hamiltonians which grow quadratically with respect to∇u,which is from the point of view of PDE analysis the most interesting case.In a related paper[1],a mean field dependence of the pay off functional is assumed,but no such a dependence of m in the dynamics of the system is considered.Also in[1],thegrowth propertiesof the data with respect to m,werecrucial to obtain global solvability of the problem.

In this paper we go much beyond the scalar theory and the theory developed in[1]and obtain the existence result for much broader class of problems.As a key tool for the existence of a solution we use the method of sub and super solution used in the context of Bellman systems in[2].

2 Derivation of the System

In this paper we study a system of partial differential equations which arises as a necessary condition of a Nash-Point-problem for Vlasov-McKean-functionals

where(0,T)is a given time interval,Ω ⊂ R d is a cube(0,1)d and Q:=(0,T)×Ωis a space-time cylinder.The function v:=(v1,···,v N)with N ∈ N is the vector of the control functions,i.e.,

is the control of the i-th player,where i=1,···,N and M ∈ N is given.For every i=1,···,N the function

is the so-called pay off function of the i-th player.Finally,the function

is the so-called mean field variable.This means that for a given v,it is a weak nonnegative solution of the following parabolic equation(“mean field equation”):

Then using also the assumption(2.21),we see that it satisfies

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对照组采用常规的方法拔出尿管：在拔管前采用间歇性夹管方式放尿，夹闭导尿管，每2～3小时开放1次，训练膀胱功能。观察组运用循证护理，具体如下：

is a given mapping and we postpone the discussion about its structure to the end of the section.

For a bounded v,under natural assumptions on the mapping g,the problem(2.2)–(2.3)has a unique solution

which allows us to define(2.1).Since the mean field variable depends on the choice of v,we will frequently also write m=m(v),which should not be understood as an algebraic relation,whenever the couple(m,v)solves(2.2).

In(2.4)we use the standard notation for Bochner,Sobolev and Lebesgue spaces,and the subscript“per” indicates the periodicity with respect to Ω and this notation will be kept through the whole paper.In what follows,we also omit writing the dependence of function on(t,x)explicitly to shorten all formulae,i.e.,we use the following abbreviation f(m,v)for f(t,x,m(t,x),v(t,x))or for f(·,m,v)in what follows,where f:=(f 1,···,f N).

Having the mean field variable m and the control function v,we can define a “pre-version”of the so-called Bellman system for a further function via the backward parabolic system

Similarly,wereplace v in themean field equation(2.2)and obtain thebackward-forward system

We call this system the “pre-Bellman equation” since v has not been replaced by a feed back formula

yet and we introduce the meaning of(2.6)in the next subsection.Moreover,L=(L 1,···,L N)are the so-called modified Lagrangians1 We use here the word modified since they diff er from standard Lagrangians in Bellman systems,which is however here caused by the fact that f and g depend on m..

It is evident,that(2.2)and(2.5)do not form a closed problem and one needs to connect v with m and∇u via some relationship.It will be shown in the next subsection that the necessary condition for classical Nash–Point problem may serve as such constraint.In addition assuming further certain qualitative properties of f,we will be able to give a good meaning to the feed back formula(2.6)and thus to avoid the presence of the control variable v in the analysis.The main goal of the paper is to introduce certain structural assumptions on f and g such that they describe very general mean field dependent Bellman system on one hand,and for which we can establish the existence of a weak solution on the other hand.

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2.1 Derivation of the full system

Finally,we need to close the problem(2.2)and(2.5)by an algebraic condition which is necessary condition for the Nash-point of functionals J’s.The classical Nash-point problem reads:For given u T,find v∈L∞(0,T;L∞(Ω;R M N))such that

for all z∈ L∞(0,T;L∞(Ω;R M))and corresponding m’s,the solutions2 We may define this also for other L p(L q)spaces once the uniqueness of m(v)is guaranteed. to(2.2)with v replaced by

The classical version treats the case where f i and g do not depend on m.In that case,the problem(2.7)is purely analytical(meaning stochastic free)formulation of a stochastic differential game driven by the dynamics

In recent years,interest came up to study cases with m-dependence of the pay-off f and/or the dynamics g.From PDE’s point of view,this leads to new interesting version of the Bellman system.

Here

Although,it is not known whether the problem(2.1)admits a Nash-point,we derive in what follows certain necessary conditions that must be fulfilled by the hypothetical Nash-point,which finally allow us to connect the pre-Bellman system(2.5)with the mean field equation(2.2)via the feed back formula(2.6)or its “equivalent”.

Under natural assumptions on the data(see Subsection 2.2)and,say,v∈L∞(0,T;L∞(Ω;R N M)),it is easy to see that the Gateaux derivatives of the J i and of m(v)exist.For

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with arbitrary smoothΩ-periodic functionwe obtain that

with i=1,···,N satisfies

and is completed by the initial condition

Here,we use the subscript to abbreviate the notion of partial derivative,i.e.,g v(v,m):=∂v g(v,m)and g m(v,m):= ∂m g(v,m).Furthermore,assuming that v is the Nash-equilibrium,we have for all i=1,···,N that

for arbitrary smoothΩ-periodic function z.Notice that here m:=m(v),i.e.,m solves(2.2)with v.To evaluate the terms not involving explicitly z,we use the equations(2.5)and(2.8).First,multiplying(2.8)by u i,integrating over Q and using integration by parts(note that M i(0)=0),we deduce

Next,multiplying the i-th equation in(2.5)by M i,we observe

Finally,subtracting(2.11)from(2.10),we obtain the following identity:

Thus,using this relation in the necessary condition(2.9),we see that

for all i=1,···,N and all smooth Ω-periodic z’s.This consequently leads to the necessary compatibility condition

Thus,now we have a closed system of equations.Namely,(2.2),(2.5)and(2.13)forms a well-defined problem for which we want to establish our analytical result.Indeed,the first one will deal just with(2.2),(2.5)and(2.13)and leads to the uniform a prioriestimate for(m,v,u).

However,to obtain also the existence result,we shall require that for a given(m,∇u),we can find a unique v solving(2.13).For such a solution we define the feed back formula(2.6)as

and replacing v in(2.5)by the feed back formula,we obtain

which is supposed to be satisfied in Q,equipped with theΩ-periodic boundary conditions and completed by the initial condition

which does not contain a control function v.Nevertheless,this system is equivalent to(2.2)and(2.5)provided that v is defined such that it satisfies(2.13).The second result of the paper will be therefore established for(2.15),provided that the feed back formulaωis well defined.

2.2 Structural assumptions on f and g

We keep the notation from the introduction here.Through the whole text,we assume that all partial derivatives f v,g v,f m and g m with respect to v and m,respectively,exist and together with f and g are Carathéodory mappings,i.e.,are measurable with respect to(t,x)for all(v,m)and for almost all(t,x)they are continuous with respect to(v,m).This assumption will not be mentioned explicitly in what follows but we rather assume it implicitly in all statements.In what follows we also assume that K and C i are positive constants and the same for constants r,s≥0 which will be used to denote certain powers.Furthermore,we will frequently use C to denote a generic constant that may change from line to line but will depend only on data.In case,it will depend on some important quantity,and it will be clearly denoted in the text.

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First,we state the assumptions forWe assume that f and f v satisfy the following growth conditions:

Please,observe here that if g is given by(2.23),then(2.13)reduces to

and also one sided sum coerciveness,i.e.,we assume

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Concerning the structural assumptions on f we just assume that

We also prescribe the behaviour of f at 0,i.e.,for all i=1,···,N,we assume that

and finally the coerciveness ofi.e.,we assume that

Next,we focus on the assumptions on g.The standard ones are related to the growth estimates,which will be supposed to be given by

The forthcoming structural assumptions on g and also on f v are in fact the key restrictions of the paper.First,to simplify the further analysis,we shall assume that3 The linearity of g with respect to v is in fact not a necessary assumption.A more delicate here is just behaviour of g with respect to m.

where b0 does not depend on(t,x).The matrices are given functions of(t,x), with i=1,···,d and k=1,···,M and the meaning of A j v j is

Hence,as it is assumed we have a d-dimensional vector function g=(g1,···,g d).The inhomogeneities b0:R→R d and b1:R→R are given.Concerning the assumptions on functions b1 and b0,we require that

and for matrices A j we assume that

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Furthermore,one of the essentially required properties of A j is that they have the same range,i.e.,we assume that for all i,j=1,···,N,almost all(t,x)and all z ∈ R N d there holds

For derivatives of b0,we need that

Finally,for the derivative of b1 with respect to m,we introduce a certain“smallness assumption”:There existsδ∈ [0,1)such that for all b∈ R+and all v∈ R N M there holds

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Although the above assumption seems to be complicated,it naturally appears in the first a priori estimate of the problem.In the next subsection,we shall show that in many cases,the assumption(2.28)still allows very general behaviour of all quantities with respect to m,namely,the case when one can dominate the function g in certain sense.In addition,since(2.28)may seem to be complicated,we can replace it by

since then(2.28)follows easily from the previous assumptions.

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for all v and all m≥0.In addition,we assume the following one sided estimate for f i:There exists α ∈ [0,1)such that for all i=1,···,N there holds

which must be valid for all i=1,···,N and all j=1,···,M.

The above assumptions are sufficient for establishing formal a priori estimates.However,for getting also the existence of a weak solution,we need to give a well meaning to the feed back formula,or in other words,we need to guarantee the unique solvability of(2.30)for given m and∇u.There can be introduced many assumptions that would lead to such a goal but we follow here the most standard one,which is the monotone operator approach.For a given m,we define the mapping T:R N M→R N M by

and assume that it is continuous with respect to(m,v)and measurable with respect to(t,x)and the strictly monotone,i.e.,for all

that is supposed to be satisfied in the space-timecylinder Q,fulfills spatially periodic boundary conditions(with respect to the unit cubeΩ)and is completed by the initial data

and from the standard monotone operator theory,we can conclude the existence of a unique v solving(2.30)and also consequently,we obtain that the feed back law(2.6)is well defined.

2.3 Prototypical examp le

Next,we state the second main theorem of the paper,which is the existence result.

where B i’s are arbitrary bounded measurable matrices.Next,for b1 we consider

and for b0 and matrices A i,we just require(2.25)–(2.27).With such a choice,all assumptions(2.16)–(2.27)and also(2.31)are satisfied and we shall just to show what is the meaning of(2.28).A very direct computation leads to the necessary condition

which is surely satisfied whenever

2.4 Statement of main results

The main result of the paper is twofold.First,we give the uniform a priori estimate result which holds for all sufficiently smooth solutions provided that parameters satisfy the assumptions stated above.

Theorem 2.1 Let g and f satisfy(2.16)–(2.28).Then any sufficiently regular solution(m,v,u)to(2.2),(2.5)and(2.13)satisfies the following estimate:

where

provided that

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and

Our prototype example is the following.For f,we assume a structure

Theorem 2.2 Let g and f satisfy(2.16)–(2.28)and(2.31).Assume that(2.38)–(2.39)is fulfilled.Then for arbitrary u T ∈ L∞ and nonnegative m0∈ Lσ(Ω)with σ fulfilling(2.37)there exists a weak solution satisfying the estimate(2.36)and fulfilling or almost all t∈(0,T),

and completed by the relationship between v and(m,∇u),

To illustrate the power of the result,we just consider our prototypical example.First in case that s0=0,we see that the only restriction is

In the opposite extreme case,i.e.,if

then

3 Algebraic Estimates for Lagrangians and Hamiltonians

In this section,we derive basic algebraic inequalities that are satisfied for Lagrangians and consequently also for Hamiltonians provided that f and g satisfy the assumption introduced in Subsection 2.2,namely the assumptions(2.16)–(2.28).The key observation is that under these assumptions,the Lagrangians satisfy the lower sum coerciveness and the proper upper estimates.It will be also evident from the estimates below why we require 2s≤r in main results of the paper.

Lemma 3.1 Let f and g satisfy(2.16)–(2.28).Then there exists a constant C>0 and ansuch that for almost all(t,x),all(m,v,∇u)fulfilling(2.13)and all i=1,···,N there hold:

(i)Sum coerciveness

(ii)Upper bound

(iii)Global bound

Proof We start with the proof of(3.1).Using the definition of L in(2.5)we get the identity

Hence,using(2.18)for the first part and(2.22)for the second part,we observe that

where for the second estimate we used the Young inequality.This finishes the proof of(3.1).

Next,we look for(3.2).Using the definition of Lagrangians,we directly obtain for arbitrary ε>0 the following identity:

Next,using(2.17)–(2.18),(2.22)and the Young inequality,we have

It remains to estimate the term with f i.We use the convexity of f i with respect to v i and the assumption(2.20).Then to evaluate we also use the constraint(2.13),which however in our case reduces to(2.30)due to the structural assumptions(2.23).Doing so,we get

Then we estimate terms involving Aℓ∇u i with the help of(2.24),(2.26)and(2.30)as follows

Finally,we focus on estimate of v i.It follows from(2.21),(2.24),(2.26)and(2.30)that

where for the last inequality,we used the Young inequality and the structural constraint(2.30).Hence absorbing the corresponding term to the left-hand side,using(2.16)and the Young inequality,we get

Next,using the Young inequality in(3.5),we find

Finally,we substitute(3.6)into(3.7)to estimate term with v i and with the help of the assumption(2.16),we obtain

Consequently,(3.8)combined with(3.4)directly implies

Thus,if we choosesuch that

which is always possible and the maximal value of suchε0 depends only on the generic constant C,we obtain the desired estimate(3.2).

The estimate(3.3)is then a simple combination of(3.1)–(3.2).The proof is finished.

4 Uniform a Priori Estimates—Proof of Theorem 2.1

This section is devoted to the estimates for solution(m,u,v)of(2.2),(2.5)and(2.13)that depend only on data of the problem provided that the assumptions(2.16)–(2.28)are satisfied.Here,we proceed rather formally,considering that the solution is sufficiently regular.The justification of such a procedure then will be provided in the proof of the existence result,i.e.,in the proof of Theorem 2.2.

4.1 Estimates for m

We start with estimates for m.First,if m0≥0 almost everywhere inΩ,then the standard minimum principle for parabolic equations implies that(sufficiently smooth)solution satisfies m≥0 almost everywhere in Q as well.Next,settingϕ:=1 in(2.40),we get with the help of nonnegativity of m that

Consequently,we have

Next,in order to obtain estimates on(u,v)we need to improve the information about m since the Lagrangians(or Hamiltonians)depend heavily on m.Our goal is to prove the starting point inequality

where the constant C depends only on data given by assumptions on f and g.

Proof of(4.2)We first setϕ :=(m+1)σ−1 in(2.40),where σ is given by(2.37),i.e.,

With such a choice ofϕ,we get the identity

where the second equality follows from the structural assumption on g(see(2.23)).Clearly,the second term on the right-hand side vanishes due to the integration by parts and spatially periodic boundary conditions.For the first integral we use the Young inequality to conclude

Thus,using the bounds for b1 and A j,namely(2.24)–(2.25),we obtain after integration over(0,T)

Next,we setϕ:=u i in(2.40)and z:=m in the i-th equation in(2.41),integrate the result with respect to time and use integration by parts to obtain the following two identities:

and

Subtracting(4.4)from(4.5)we arrive at identity

where for the last inequality we used the Hölder inequality and the uniform bound(4.1).Next,we evaluate the last integral.Using the structural assumption(2.23)and the growth assumptions(2.24)–(2.27),we see that

To estimate I2 we use(2.26)and the constraint(2.30)to get

For the term I1 we use the Young and the Hölder inequalities to obtain for arbitraryε>0,

Finally,substituting these estimates into(4.6),summing the result over i=1,···,N and using(2.28),we obtain

Consequently,sinceδ<1 we can absorb the last term by the left-hand side.Therefore,we can use the sum coerciveness,i.e.,the assumption(2.18),to deduce that

Finally,using the estimate(4.8)in(4.3),we see that(recalling(2.37))

Thus,settingε>0 sufficiently small,we obtain the desired inequality(4.2).

Our next goal is to improve the information coming from(4.2)such that the estimate on E depends only on u and not on m.It means that we want to show that

provided that

The estimate(4.9)will be shown by a certain interpolation of(4.1)–(4.2).

Proof of(4.9)First,we recall the following parabolic interpolation inequality:

Next,we apply(4.12)onto the function withσgiven by(2.37)to get

Finally,we also use the a priori estimate(4.1)and the interpolation inequality

which is valid for allto obtain

where for the last inequality we used(4.13).

Next,we use(4.14)to handle the right-hand side of(4.2).Assuming that

we get from(4.14),

provided that

which by using of(2.37)can be shown to be equivalent to(4.10).Notice that(4.17)directly implies the validity of(4.15).Hence,we can absorb the first term on the right-hand side of(4.2)to get

In case that

we can use(4.1)to conclude(4.9)directly.If(4.19)is not true,we again use(4.14)to get from(4.18)

which after using the Young inequality leads to(4.9),provided that

Note that(4.20)is a stronger assumption that(4.19)and it can be shown by using(2.37)that(4.20)is equivalent to(4.11).Hence the proof of(4.9)is complete.

4.2 Estimates for u

This subsection is devoted to the uniform estimates on u,which will still depend on m.To be more precise,we want to show that for arbitrary p>d+2 we have the estimate

Proof of(4.21)We start with the estimates for below for the quantity

It is not difficult to observe from(2.5),(3.1)and the fact that 2s≤r,that w satisfies almost everywhere in Q,

Next,let us consider w1 a solution to

completed by zero initial condition,i.e.,w1(T)=0.Then by a standard parabolic estimate,we obtain that

whenever p>d+2.Then subtracting(4.23)from(4.22),we obtain

Hence from the theory for subsolutions to parabolic equation(see[8]),we obtain

which together with(4.24)and the assumption that u(T)∈L∞ leads to the final estimate from below

which is valid for arbitrary p>d+2.

Next,we focus on estimates from above.Keeping the notation for w,we can derive from(2.5)that

Hence,using(3.2),we get

where the second inequality follows from the assumption 2s≤r.Thus,we can repeat step by step the procedure for w and using the fact that u(T)∈ L∞(Ω),we obtain

Finally,we derive the uniform bound(4.21).First,summing(4.28)over i=1,···,N,we have

Since,we can combine this estimate with(4.26)to get

Consequently,it follows from(4.28)that

Finally to obtain also estimate from below for u i,we use(4.29)–(4.30)and get

which together with(4.30)implies the desired estimate(4.21).Hence the proof is complete.

4.3 Uniform L∞bounds

This subsection is devoted to the uniform bound for u,which directly implies the part of uniform estimates stated in(2.36).Here,we combine(4.14),(4.9)and(4.21)to obtain the final bound.We go back to(4.21)and estimate the right-hand side.Although we need to choose p>d+2,we formally provide all computation for p=d+2 and in the final restriction on the size of s0(orσ)we just use the strict inequality sign.Hence,we need to estimate the term on the right-hand side of(4.21),i.e.,the integral

If 2s0(d+2)<1 then the integral on the right-hand side of(4.21)is bounded due to(4.1)and therefore we immediately get

Hence,in what follows,we assume the opposite case.Assuming that

we can use(4.14)with q:=2s0(d+2)and we deduce

and it follows from(4.21)that

Inserting this estimate into the right-hand side of(4.9),we also deduce

Hence,in case

we can absorb the right-hand side by left-hand side to obtain

which implies a part of(2.36).Notice that(4.37)is a stronger assumption than(4.20)and therefore all needed assumptions,i.e.,the assumptions(4.17)and(4.37),are already encoded in(2.38).Furthermore,using(4.2)and(4.9),we obtain also the bound for m and∇m stated in(2.36).Finally,from(4.8),we deduce the bound for term with m(m r+1)|v|2 in(2.36).

4.4 Uniform estimates for∇u

This subsection is devoted to the last remaining part of(2.36),i.e.,the part of the estimate with∇u.Keeping the notation from the previous sections,we start with estimates for∇w.Using(3.1)and the fact that 2s≤r we have

Next,we multiply(4.39)byintegrate over Q and use integration by parts to obtain

Thus,we see that we can absorb the term with∇w by the left-hand side and due to the L∞bound for u(see(4.38)),and L 2s0(d+2)bound for m(see(4.34)),we deduce from(4.40)that

Notice that the first in(4.41)together with(4.8)leads to the estimate(2.36)for term involving|v|2.

Next,we use the inequality(4.27),i.e.,

which we multiply byand integrate over Q.Repeating step by step the procedure(4.40)–(4.41)and using uniform bounds on u and m,we get

Finally,combining(4.41)and(4.43),we have for all i=1,···,N,

which finishes the proof of(2.36).

5 Existence of Solution—Proof of Theorem 2.2

This section is devoted to the proof of the existence of a solution to(2.2),(2.5)and(2.30).Notice that due to the assumption(2.31)–(2.32),we know that(2.32)is equivalent to

with a Carathéodory mapping ω.Therefore we can omit(2.30)and replace v by ω(m,∇u)in(2.2)and(2.5)and solve the problem only for unknowns(m,u).In fact,this is also the way how one can get the existence of a solution to an approximative problem.Nevertheless,for the sake of simplicity and to simplify the notation,we keep writing v in what follows.

Second,we do not provide the complete and rigorous proof here.We rather emphasize those steps that are different from the known procedure for Bellman systems.To be more specific,we provide here the proof of weak sequential stability,which is the key property of the system of equations we have in mind.It means that we shall consider a sequence of(m n,u n,v n)of smooth solutions to(2.2),(2.5)and(2.30)(which is however equivalent to(2.15),once the mappingωis well defined)corresponding sequence of initial data

with nonnegativeOur goal is to show that

strongly inwhere the triple(m,u,v)solves again(2.2),(2.5)and(2.30)with initial data(u T,m0).Indeed such a result suggests that the rigorous existence proof is doable.Indeed,approximating Hamiltonians H by a sequence of bounded functions and similarly g by a sequence of boundedone may consider that for a such approximative system it is classical to obtain the existence of solution(m n,u n,v n)and the only remaining part of the proof is then the weak sequential stability.For details,how one can approximate Hamiltonians properly,we refer to[2].

5.1 Uniform a priori estimates

In this part we just use the result of Theorem 2.1,which holds for sufficiently smooth solutions.Indeed,we may assume that

such that(m n,u n,v n)satisfies for all

for all

and almost everywhere in Q,

with g given as

and L(m n,v n,u n)given as

Next,we focus on the estimate for the time derivatives.First,using(3.3)and uniform bounds(2.36),we see that

Consequently,we can deduce from(5.5)and also from(5.3)that for some q>d,

Similarly,using(2.22)and(5.3),we see that for some q∈ (1,∞),

5.2 Limit n→∞

Having(5.3),(5.1)and(5.10)–(5.11)and using the Aubin-Lions lemma,we can find subsequences that we do not relabel,such that for some q>d,

where M(K)denotes the space of Radon measures on K.Having(5.12)–(5.20),we can use the theory for parabolic equations with L 1 or measure right-hand side(see[4–5])to conclude that

Consequently,since the operator T in(2.31)–(2.32)is strictly monotone,it follows from(5.17)and(5.21)that

which in particular also implies(note that L and g are Carathéodory)

Thus,the convergence results(5.12)–(5.24)allow us to let n → ∞ in(5.4)–(5.7)to obtain that(m,v,u)solves(2.40),(2.41)and

for allThus,to finish the proof of Theorem 2.2,we need to show that

Indeed,having(5.26),we can read from the equation that the time derivative of u is better,i.e.,that

for some q>d and integrating by parts with respect to the time variable t in(5.25),we find that(2.41)holds true.

5.3 Identification of L and the proof of(5.26)

This last part is devoted to the proof of(5.26).We follow the procedure developed in[2]with the necessary changes due to the presence of the mean field variable m.We also proceed here slightly formally and refer the interested reader to[2],where the very similar procedure is made rigorously.Defining

it follows from(5.5)that in the sense of distributions we have

Next,we multiply(5.27)by

and obtain

Consequently,multiplying the resulting identity by arbitrary nonnegativeand integrating the result over Q,we get by using integration by parts that

Next,we letThanks to(5.12)–(5.14),it is easy to deduce that

Next,using(3.1),we get that

Consequently,we see that the right-hand side of(5.29)is bounded from below by a strongly convergent function,so using the point-wise convergence result(5.23)and the Fatou lemma,we get

Hence,combining(5.30)–(5.31),we see that for all nonnegative there holds

Finally,takingϕ:=e2Cwψwith arbitrarywe deduce that4 In fact we must mollify the test function with respect to the time variable and then to pas to the limit.Since such a procedure was explained in details in[2],we do not provide the complete proof here.

which implies that in the sense of distributions

Similarly,we deduce the opposite type inequalities.It follows from(5.5)that for all i=1,···,N,

Denoting z n:=(u i)n−ε0w n and multiplying(5.35)by e2Cz n we get

Hence,using(3.2),we see that the right-hand side is bounded by an convergent sequence and therefore we can proceed similarly as before by using the Fatou lemma to obtain

Thus summing with respect to i=1,···,N and dividing by(1 − ε0)we see that

which combined with(5.34)gives

Consequently also we obtain from(5.37)that

Finally using(5.38)–(5.39),we get

Hence,(5.39)implies that

and(2.41)follows.This completes the proof of Theorem 2.2.

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