Memory effects in measure transport equations

Transport equations with a nonlocal velocity field have been introduced as a continuum model for interacting particle systems arising in physics, chemistry and biology. Fractional time derivatives, given by convolution integrals of the time-derivative with power-law kernels, are typical for memory effects in complex systems. In this paper we consider a nonlinear transport equation with a fractional time-derivative. We provide a well-posedness theory for weak measure solutions of the problem and an integral formula which generalizes the classical push-forward representation formula to this setting.


Introduction
The measure-valued formulation of nonlinear transport equations have received an increasing interest both from a theoretical perspective [2,3] and in various applications, as a continuum model for interacting particle systems (e.g. in crowd motion, population dynamics, bacterial chemotaxis, kinetic theory, social systems, etc., see [4,6] and reference therein).
Recently, anomalous transport problems describing processes deviating from the usual Gaussian behaviour have been observed in different fields (see [9]). In this phenomena, the standard diffusive behaviour is replaced by a subdiffusive one, in which the mean square displacement of the diffusing particles is of order t β with β < 1. Corresponding models leads to the study of differential equations where the time-derivative is replaced by a fractional one (see for example [1,7,12]). The stochastic dynamics driven by fractional differential equations is usually referred to motion in a non-homogeneous medium, the particles can speed up or slow down according to a random clock which is the inverse to a stable subordinator (time-change of processes) and the dynamic is not Markovian, see [10]. Fractional derivatives are also considered as a typical approach to add a memory effect to a complex system. The memory term introduced by the convolution operator in time finds applications in demography, viscoelastic and biomaterial materials, biological processes and, in general, in the study constitutive relations depending on the history of the state variables (see [8,15]).
In this paper we consider a measure solution approach to the nonlinear transport equation where ∂ β (0,t] · is a nonlocal time-differential operator given by the Caputo fractional derivative of order β ∈ (0, 1) Since a fractional derivative at time t depends on the values of the time-derivative in the interval (0, t) with a progressively decreasing weight on the past history, the problem is non local in time. The coefficient v[µ](x) is a nonlinear velocity field which depends on the solution itself, for example v[µ](x) = R d K(x − y)dµ(y). An appropriate choice of kernel K allows to describes different phenomena of the physical model such as aggregation, repulsion and diffusion ( [4,3,6]). The datum µ 0 is a given positive measure on R d representing the initial distribution of the population. Observe that measure solutions allow to describe within an unified approach the evolution both for continuous and discrete populations. For β = 1, the fractional derivative ∂ β (0,t] · coincides with the derivative ∂ t ·. In this case the measure solution of problem (1.1) is given by the push-forward µ t = Φ t #µ 0 of the initial datum µ 0 by means of the flow map Φ t associated to the velocity field v. For β ∈ (0, 1), we prove that a solution of (1.1) is given by the integral formula where m µ is the solution of the linear transport equation with the velocity field given byṽ µ (x, s) = ∞ 0 v[µ r ]g β (r, s)dr and Φ µ the associated flow. The kernel h β (s, t), g β (s, t) are the probability density functions (PDF in short) associated to the time-increasing, continuous stochastic processes D t , E t , which are a β-stable subordinator and its inverse ( [5,9,10]). The paper is organized as follows. In Section 2, we review definitions and some basic properties of the subordinator process, its inverse and of the associated PDF g β , h β . In Section 3, we study a linear transport equation with Caputo time-derivative. In Section 4 we prove the well-posedness of weak measure solutions to problem (1.1).

Preliminary definitions and properties
Throughout the paper, we always assume that β ∈ (0, 1]. For f : [0, T ] → R, the Riemann-Liouville fractional integral of order β is defined by and the Caputo fractional derivative of order β by (for a complete account of the theory of fractional derivatives, we refer to [13]). Fractional derivatives arise in the study of differential equations satisfied by probability density functions of a class of Lévy processes, called stable subordinators ( [5,10,11]). For β ∈ (0, 1), a β-stable subordinator is a one-dimensional, non-decreasing Lévy process D t starting at 0 which is self-similar , i.e. {D t , t ≥ 0} has the same finite dimensional distribution as {t 1/β D 1 , t ≥ 0}, and such that the Laplace transform of D 1 is given by E(e −sD 1 ) = e −s β for s ≥ 0. The inverse stable process {E t } t≥0 , defined as the first passage time of the process D t over the level t, i.e.
has sample paths which are continuous, non-decreasing and such that and has finite γ-moment for any γ > 0, given by for some positive constant C(γ, β). In the next proposition, we give two crucial properties of h β we will exploit in the following (see [10,11]).
We introduce a space of measures with an appropriate norm where we consider solutions to the measure-valued transport equation (see [6]). For f : R d → R, we define the norm · BL as where |f | L is the Lipschitz seminorm, and we consider the Banach space (BL(R d ), · BL ). Denoted by M(R d ) the space of finite measure on R d , we define a norm on this space by where

Linear transport equations with Caputo time derivative
In order the explain the construction of a solution to (1.1) in a simpler setting, in this section we consider the case of a linear transport equation with a Caputo time-derivative We introduce a notion of measure solution to (3.1).
We assume that v : Associated to (3.1), we consider the linear problem with standard time-derivative where the vector fieldṽ : (the process D t is the β-stable subordinator defined in Section 2 and g β the corresponding PDF). Since for any alsoṽ satisfies assumption (H1) and we can define the flow corresponding to the vector fieldṽ by The flow Φ t gives the trajectory issuing from the point x at time 0 and arriving at the point Φ t (x, 0) at time t. It is well known that the unique measure solution to [4,14]). Because of the well posedness in measure theoretic sense of the problem (3.2) we get the corresponding result for problem (3.1).

4)
where m ∈ C(R + , M + (R d )) is the solution of the linear transport problem (3.2). Moreover, let µ 1 , µ 2 be two solutions of (3.1) corresponding to the initial data µ 1 0 , µ 2 0 . Then Proof. By assumption (H1), the flow Φ s exists for any s ∈ R + and the push-forward m s = Φ s #µ 0 is globally well defined. Therefore also formula (3.4) is well defined for any t ∈ [0, T ]. Moreover µ t , for any t ∈ R + , is a finite measure on R d since we have We claim that (3.4) defines a weak solution to (3.1).
Moreover, since D Et = t for any t and recalling that h β (·, t), g β (·, t) are the PDFs of E t , D t , we have Replacing the previous identities in Def. 3.1 we get that µ is a solution if Recalling that m is a weak solution to (3.2), i.e. for any f ∈ C ∞ c (R d ) and for almost any we get the claim.
To prove that µ is continuous with respect to t, we estimate (3.6) By the estimate for r, r ′ ∈ R + , we get Replacing the previous estimates in (3.6), we obtain for the arbitrariety of f We finally prove (3.5). For any f ∈ BL(R d ) such that f BL ≤ 1, we have Remark 3.3. If µ 0 = δ x 0 , then the solution of (3.2) is given by δ Φt (x 0 ), while the solution of (3.1) by µ t = E δ Φ E t (x 0 ) . We can interpret this formula in the following way: for the single particle the standard time t is replaced by an internal clock E t . The sample of the process E t can be constant on some interval, corresponding to trapping events in the motion, and assume arbitrarily large values, but with a probability decaying exponentially to 0 (see (2.3)). The solution of the transport equation (3.1) is obtained by averaging with respect to the internal clock E t .

Remark 3.4. Assume that the velocity v is a given vector field in L
Because of the weak regularity of v, it is natural to consider the characteristic flow Φ associated to v in the sense of Filippov. In [14] it is proved that the push-forward of the initial datum by means of the Filippov flow Φ gives a unique measure solution to the corresponding transport problem with a standard time-derivative. Since the velocityṽ defined in (3.3) satisfies the same assumptions of v, included the OSL condition, the solution to problem (3.2) is well defined for any s ∈ R + . Therefore, also under in this weaker setting, formula (3.4) gives the solution to (3.1).

Nonlinear transport equations with Caputo time derivative
In this section we consider the Cauchy problem for a nonlinear transport equation with Caputo time-derivative, i.e. Also in this case, we consider solution in the measure theoretic sense.
Definition 4.1. A solution to (4.1) is a map µ ∈ C(R + , M + (R d )) such that µ t=0 = µ 0 and for any f ∈ C ∞ c (R d ) and for almost any t ∈ R + ∂ β Along this section we assume that (H2) v is bounded by V 0 > 0 and Lipschitz continuous, i.e. there exists L > 0 such that for any µ 2 )) .
(H3) µ 0 ∈ M + (R d ) and, for given constants C k , Fixed µ ∈ C(R + , M + (R d )), we consider the linear problem where the linear velocity fieldṽ µ : Observe thatṽ µ is bounded by V 0 . Moreover, we have By the self-similarity of D t it follows g β (s, Henceṽ µ is Lipschitz continuous in x and continuous in t. Defined the flow associated to the vector fieldṽ µ by we have that the measure solution to (4.2) is given by m µ s = Φ µ s #µ 0 .
where m µ ∈ C(R + , M + (R d )) is the solution of the linear transport problem (4.2). Moreover, let µ 1 , µ 2 be two solutions of (4.1) corresponding to the initial data µ 1 0 , µ 2 0 . Then, there exists a constant C = C(T ) such that In the next lemma, we prove existence of a solution to (4.1) when the initial datum µ 0 has compact support and the velocity is null outside a given ball.  B(0, 2R), for some positive constant R. Then the Cauchy problem (4.1) admits a solution µ ∈ C(R + , M + (R d )) implicitly defined by the integral formula (4.5).
Proof. Consider the map Ψ : C(R + , M + (R d )) → C(R + , M + (R d )) which associates to a measure ν the measure µ = Ψ(ν) defined, for any t ∈ R + , by µ t (dx) = ∞ 0 m ν r (dx)h β (r, t)dr where m ν is the solution of the linear transport equation v ν is defined as in (4.3) with ν in place of µ and Φ ν is the associated flow. We prove that the map Ψ admits a fixed point which satisfies the integral formula (4.5).
Let us check that Ψ is well-defined and continuous from the set into itself, for an appropriate constant R 0 to be chosen later. Given ν ∈ C, we set µ = Ψ(ν). Since m ν r = Φ ν r #µ 0 (dx), we have To estimate the first order moment, observe that since v[ν](x) ≡ 0 in R d \ B(0, 2R), theñ v ν (x, s) ≡ 0 for all (x, s) ∈ (R d \ B(0, 2R)) × R + . Therefore, if x ∈ B(0, R), it follows that Φ ν r (x, 0) ∈ B(0, 2R) for all r ∈ R + . Hence, for k = 1, 2, we get (4.7) We estimate d BL (µ t , µ t ′ ) for 0 ≤ t ′ < t. Arguing as in the estimate (3.7), for f ∈ BL(R d ) such that f BL ≤ 1 we have (4.8) By (4.7) and (4.8), it follows that the map Ψ is well defined. Moreover if R 0 in the definition of C is greater than the constants appearing in the estimates (4.7) and (4.8), then Ψ maps C into itself.
To prove that Ψ is continuous, we first estimate the distance between the flows corresponding to different measures. Given µ, ν ∈ C(R + , M + (R d )), by (H2) we have Hence, Therefore, by Gronwall's inequality, we have the estimate Let ν k ∈ C converge to some ν for k → ∞ and set µ k = Ψ(ν k ), µ = Ψ(ν). For f ∈ BL(R d ) such that f BL ≤ 1, taking into account that if x ∈ B(0, R), then Φ ν k r (x, 0), Φ ν r (x, 0) ∈ B(0, 2R) for all r ∈ R + we have for any t > 0 Therefore the convergence of µ k to µ and the continuity of Ψ. We conclude by Schauder Theorem that the map ν → µ = Ψ(ν) has a fixed point µ in C which indeed satisfies the integral formula (4.5). As in Prop. 3.2, it is possible to show that formula (4.5) defines a weak solution of the problem and therefore it follows the existence of a solution to (4.1) in R + .
Proof of Prop. 4.2. Given R > 0, we consider a sequence of initial data given by µ R 0 = χ B(0,R) (x) · µ 0 , where χ B(0,R) is the characteristic function of the set B(0, R), and a sequence of velocity fields v R [µ](x) = v[µ](x) · σ R (x) where σ R : R d → R is a smooth, non negative function such that σ R (x) = 1 for x ∈ B(0, 2R − 1), σ R (x) = 0 for x ∈ R d \ B(0, 2R) and |Dσ R | ≤ 1. By Lemma 4.3, for any R > 1, there exists a solution µ R to the Cauchy problem (4.1) given by the formula where m R is the solution of (4.1) corresponding to the velocityṽ R (x, s) = ∞ 0 v R [µ R s ](x)g β (r, s)ds and Φ R the associated flow. For any T > 0, we consider the restriction of the sequence {µ R } to the interval [0, T ]. We estimate the first and second order moment of µ R t for t ∈ [0, T ], uniformly with respect to R. By (4.4), we have Moreover estimate (4.8), which is independent of R, gives that the sequence µ R is also equicontinuous with respect to t. We conclude that µ R is tight in [0, T ] and therefore, see [2], there exists a measure µ ∈ C([0, T ], M + (R d )) such that sup [0,T ] d BL µ R t , µ t tends to 0 for R → ∞, up to a subsequence. By a diagonal argument, we can extend µ ∈ C(R + , M + (R d )).
Since inequality (4.13) holds for any f and all the terms on the right hand side tend to 0 for R → ∞, we get the convergence of ∞ 0 m R s (dx)h β (s, t)dr to ∞ 0 m µ s (dx)h β (s, t)dr for R → ∞.