A non-expanding transport distance for some structured equations
aa r X i v : . [ m a t h . A P ] F e b A non-expanding transport distance for some structured equations
Nicolas Fournier ∗ Benoˆıt Perthame †‡ February 9, 2021
Abstract
Structured equations are a standard modeling tool in mathematical biology. They are integro-differential equations where the unknown depends on one or several variables, representing the stateor phenotype of individuals. A large literature has been devoted to many aspects of these equationsand in particular to the study of measure solutions. Here we introduce a transport distance closelyrelated to the Monge-Kantorovich distance, which appears to be non-expanding for several (mainlylinear) examples of structured equations.
Mathematics Subject Classification:
Keywords and phrases:
Transport distances; Monge-Kantorovich distance; Coupling; Structured equa-tions; Mathematical biology.
Introduction
The subject of structured equations arises in several areas of biology and extends ordinary differen-tial equations by including parameters chosen because they bring some influence on the populationdynamics, see [11, 24, 30]. This leads to various integro-differential equations and partial differentialequation (P.D.E.) which also appear in many other areas as physics, communication science and in-dustry. Besides the interesting modeling issues, the questions which have been considered are aboutexistence of solutions, entropy properties and, mostly, long term convergence to steady states, withpossibly exponential rate of convergence. Another question concerns measure solutions, possibly afterrenormalization [2, 12]. Furthermore, in the context of a nonlinear neuroscience problem, convergenceof a particle system has recently been proved using transport costs with specific costs precisely adaptedto the coefficients [19].The present papers aims at showing that a simple variant of the Monge-Kantorovich transportdistance appears to be non-expanding along several structured equations. These include the renewalequation and a few other models listed below.We work in a state space that we denote by J , which can be [0 , ∞ ), [0 , ∞ ) × T with T a discretetorus, [0 , ∞ ) or [0 , ∞ ) × R d and we always use a cost function ̺ : J × J 7→ [0 , ∞ ) which satisfies ∗ Sorbonne Universit´e, CNRS, Laboratoire de Probabilit´e, Statistique et Mod´elisation, F-75005 Paris, France. Email:[email protected] † Sorbonne Universit´e, CNRS, Universit´e de Paris, Inria, Laboratoire Jacques-Louis Lions, F-75005 Paris, France.Email: [email protected]. ‡ B.P. has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020research and innovation programme (grant agreement No 740623). ( x, x ) = 0 and ̺ ( x, y ) = ̺ ( y, x ) > x = y and will typically take the form ̺ ( y, x ) = min( | x − y | , a )for some a > u , u ∈ P ( J ), the transport cost is defined as T ̺ ( u , u ) = inf v ∈H ( u ,u ) ZZ ̺ ( x, y ) v (d x, d y ) , H ( u , u ) = { v ∈ P ( J × J ) with marginals u and u } . (1)When ̺ is a distance on J , T ̺ is a distance on P ( J ) and, with a slight abuse of language, one refersto the Monge-Kantorovich distance. Recent accounts about the theory can be found in the books[32, 1, 31].Our approach relies on the coupling method, see the survey paper [20]. The first example of use of thecoupling method, to our knowledge, can be traced back to Dobrushin [14], where the nonlinear Vlasovequation is derived as mean-field limit of a deterministic system of interacting particles, making use ofsome transport cost. No P.D.E. is written for the coupling in [14], because everything may be expressedin terms of characteristics. See [21, Section 3] for a P.D.E. analogue to Dobrushin’s argument. Inthe same spirit, the Euler equation is derived from a deterministic system of interacting vortices inMarchioro-Pulvirenti [23, Section 5.3], using also a coupling argument, see also [22] for a result withthe strong transport distance d ∞ .The paper is organized as follows. We begin with the renewal equation in order to present in detailsthe results and method. Building on this, we extend the method to a system of renewal equations, somespace-age structured equation, the multi-time renewal equation, the growth-fragmentation equation,and to an age-size coupled model. All these equations are linear. We complete our study with amodel with sexual reproduction, which is quadratic, and generates new difficulties. Our setting isvery general and does not use uniqueness of solutions, therefore we complete them with a technicalappendix devoted to a uniqueness result by the Hilbert duality method when further regularity on thecoefficients is assumed. Our first example, also the simpler, is the general renewal equation. It allows to introduce the methodand to explain the choice of cost within the setting of the equation ∂u t ( x ) ∂t + ∂ [ g ( x ) u t ( x )] ∂x + d ( x ) u t ( x ) = b ( x ) N ( t ) , t ≥ , x ≥ ,u t ( x = 0) = 0 , N ( t ) = Z ∞ d ( x ) u t (d x ) , t ≥ . (2)When b = δ , the Dirac mass at 0, and g ≡
1, we find the classical renewal equation [17]. Themore general version at hand is motivated by various models proposed in mathematical neuroscience,[19, 26, 29]. We assume that g, d ∈ C ([0 , ∞ )) , b ∈ P ([0 , ∞ )) , g is non-increasing , g (0) ≥ , d ≥ . (3)We will further suppose that ∃ a > a ≤ inf | x − y |≤ a | x − y | max( d ( x ) , d ( y )) | d ( x ) − d ( y ) | . (4)2bserve that this last condition holds true with a = min( a ,
1) as soon as a >
0, where a = inf | x − y |≤ | x − y | max( d ( x ) , d ( y )) | d ( x ) − d ( y ) | . For example, d ( x ) = α + βx p satisfies such a condition, provided α > β ≥ p ≥
1, as well asany Lipschitz and uniformly positive function.
Theorem 1
Assume (3) - (4) . We consider the cost function on (0 , ∞ ) × (0 , ∞ ) defined by ̺ ( x, y ) = min( a, | x − y | ) . For any u , u ∈ P ([0 , ∞ )) , there exists a pair of weak measure solutions ( u t ) t ≥ , ( u t ) t ≥ ⊂ P ([0 , ∞ )) to (2) starting from u and u , i.e., such that for i = 1 , and all t ≥ , Z t Z ∞ d ( x ) u is (d x )d s < ∞ (5) and for all t ≥ , all ϕ ∈ C c ([0 , ∞ )) , Z ∞ ϕ ( x ) u it (d x ) = Z ∞ ϕ ( x ) u i (d x ) + Z t Z ∞ h g ( x ) ϕ ′ ( x ) + d ( x ) Z ∞ ( ϕ ( z ) − ϕ ( x )) b (d z ) i u is (d x )d s. (6) Moreover, for all t ≥ , we have T ̺ ( u t , u t ) ≤ T ̺ ( u , u ) . Notice that, even if we did not mention it, it follows from (6) that u ∈ C w ([0 , ∞ ); P ([0 , ∞ )). Also,the generality of this statement relies on the price that the solutions u t , u t may depend on the choiceof the pair ( u , u ). However, when g ∈ C b , the solutions are unique in distributional sense as provedin the appendix, and then the result is more standard. The regularity of g can certainly be loweredin view of the theory developed in [5, 6, 13]. Proof . We assume (3), fix u , u ∈ P ([0 , ∞ )) and consider any v ∈ H ( u , u ). There exists a family( v t ) t ≥ of probability measures on [0 , ∞ ) , starting from v , such that for all t ≥ Z t ZZ [ d ( x ) + d ( y )] v s (d x, d y )d s < ∞ , (7)and which weakly solves ∂v t ∂t + ∂ [ g ( x ) v t ] ∂x + ∂ [ g ( y ) v t ] ∂y + max( d ( x ) , d ( y )) v t = b ( x ) δ ( x − y ) ZZ min( d ( x ′ ) , d ( y ′ )) v t (d x ′ , d y ′ )+ b ( x ) Z (cid:0) d ( x ′ ) − d ( y ) (cid:1) + v t (d x ′ , y ) + b ( y ) Z (cid:0) d ( y ′ ) − d ( x ) (cid:1) + v t ( x, d y ′ ) . t ≥
0, all ϕ ∈ C c ([0 , ∞ ) ), ZZ ϕ ( x, y ) v t (d x, d y ) = ZZ ϕ ( x, y ) v (d x, d y ) + Z t ZZ h g ( x ) ∂ϕ ( x, y ) ∂x + g ( y ) ∂ϕ ( x, y ) ∂y i v s (d x, d y )d s + Z t ZZZ (cid:2) ϕ ( z, z ) − ϕ ( x, y ) (cid:3) min( d ( x ) , d ( y )) b (d z ) v s (d x, d y )d s + Z t ZZZ (cid:2) ϕ ( z, y ) − ϕ ( x, y ) (cid:3) ( d ( x ) − d ( y )) + b (d z ) v s (d x, d y )d s + Z t ZZZ (cid:2) ϕ ( x, z ) − ϕ ( x, y ) (cid:3) ( d ( y ) − d ( x )) + b (d z ) v s (d x, d y )d s. (8)Using (7) and then (8) with a function ϕ depending only on x , observing that v ∈ H ( u , u ) and thatmin( d ( x ) , d ( y )) + ( d ( x ) − d ( y )) + = d ( x ) , we deduce that the first marginal u t (d x ) = R y ∈ [0 , ∞ ) v t (d x, d y ) satisfies (5)-(6). Similarly, the secondmarginal u t (d y ) = R x ∈ [0 , ∞ ) v t (d x, d y ) satisfies (5)-(6). And it holds that v t ∈ H ( u t , u t ) for all t ≥ g, d are replaced by smooth and bounded functions, and from the following a priori tightness esti-mate. By the de la Vall´ee Poussin theorem, there exists a function h : [0 , ∞ ) → [0 , ∞ ) such thatlim x →∞ h ( x ) = ∞ and such that C := ZZ [ h ( x ) + h ( y )] v (d x, d y ) + Z h ( z ) b (d z ) < ∞ . One can moreover choose h smooth and satisfying 0 ≤ h ′ ≤
1. Applying (8) with ϕ ( x, y ) = h ( x )+ h ( y ),one immediately concludes that for all t ≥ ZZ [ h ( x ) + h ( y )] v t (d x, d y ) = ZZ [ h ( x ) + h ( y )] v (d x, d y ) + Z t ZZ [ g ( x ) h ′ ( x ) + g ( y ) h ′ ( y )] v s (d x, d y )d s + Z t ZZ h d ( x ) (cid:16) Z h ( z ) b (d z ) − h ( x ) (cid:17) + d ( y ) (cid:16) Z h ( z ) b (d z ) − h ( y ) (cid:17)i v s (d x, d y )d s ≤ C + 2 Ct + Z t ZZ h d ( x )( C − h ( x )) + d ( y )( C − h ( y )) i v s (d x, d y )d s, where C = sup x ≥ g ( x ) h ′ ( x ) is finite because g is continuous and non-increasing and because h ′ is[0 , d is continuous, non-negative and since h increases to infinity, there is L > d ( x )( C − h ( x )) ≤ L − d ( x ) h ( x ) /
2, whence finally ZZ [ h ( x ) + h ( y )] v t (d x, d y ) + 12 Z t ZZ (cid:2) d ( x ) h ( x ) + d ( y ) h ( y ) (cid:3) v s (d x, d y )d s ≤ C + 2 Ct + 2 Lt.
Since lim x →∞ h ( x ) = ∞ , this last a priori tightness estimate is sufficient to prove existence for (7)-(8).We next fix a >
0, set ρ ( x, y ) = min( | x − y | , a ) and we choose a coupling v ∈ H ( u , u ) such that RR ρ ( x, y ) v (d x, d y ) = T ρ ( u , u ). We apply (8) with ϕ = ρ (or more precisely, firstly to some smoothand compactly supported approximation ρ ε of ρ and then let ε → g ( x ) ∂ρ ( x, y ) ∂x + g ( y ) ∂ρ ( x, y ) ∂y = 1I {| x − y |≤ a } sgn( x − y )[ g ( x ) − g ( y )] ≤ g is non-increasing. Since b is a probability measure, we obtain ZZ ρ ( x, y ) v t (d x, d y ) ≤T ρ ( u , u ) − Z t ZZ ρ ( x, y ) max( d ( x ) , d ( y )) v s (d x, d y )d s + Z t ZZZ h ρ ( z, y )( d ( x ) − d ( y )) + + ρ ( x, z )( d ( y ) − d ( x )) + i b (d z ) v s (d x, d y )d s. Recalling that T ρ ( u t , u t ) ≤ RR ρ ( x, y ) v t (d x, d y ), in order to complete the proof, it is therefore sufficientto verify the inequality, for all x, y ≥ I ( x, y ) := (cid:0) d ( x ) − d ( y ) (cid:1) + max( d ( x ) , d ( y )) Z ̺ ( z, y ) b (d z ) + (cid:0) d ( y ) − d ( x ) (cid:1) + max( d ( x ) , d ( y )) Z ̺ ( x, z ) b (d z ) ≤ ̺ ( x, y ) . (9)Since ̺ ( x, y ) = min( a, | x − y | ) and since b is a probability measure, we have I ( x, y ) ≤ ( d ( x ) − d ( y ) (cid:1) + max( d ( x ) , d ( y )) a + (cid:0) d ( y ) − d ( x ) (cid:1) + max( d ( x ) , d ( y )) a = | d ( x ) − d ( y ) | max( d ( x ) , d ( y )) a ≤ min( | x − y | , a ) . Indeed, the last inequality is obvious if | x − y | ≥ a and follows from (4) otherwise. (cid:3) In mathematical biology, it is usual to describe the cell cycle, see [24, 30], using a system of renewalequations coupled at there boundaries as follows ∂u t ( x,i ) ∂t + ∂ [ g i ( x ) u t ( x,i )] ∂x + d i ( x ) u t ( x, i ) = 0 , t ≥ , x ≥ , i = 1 , ..., I,u t ( x = 0 , i ) = Z ∞ d i − ( x ) u t (d x, i − , t ≥ , i = 1 , ..., I, (10)with the convention u t ( x,
0) = u t ( x, I ). In terms of stochastic processes, a cell with age x in state i ages chronologically (when, say, g i ≡
1) until, with rate d i ( x ), it changes state to i + 1 where it startswith age x = 0. So the state space is J = [0 , ∞ ) × T, with T = { , ..., I } the torus, i.e., states 0 and I + 1 are identified to states I and 1 . We assume that for all i ∈ T , g i , d i ∈ C ([0 , ∞ )) , g i is non-increasing , g i (0) ≥ , d i ≥ , (11)and that ∃ a > a ≤ min ≤ i ≤ I inf | x − y |≤ a | x − y | max( d i ( x ) , d i ( y )) | d i ( x ) − d i ( y ) | . (12)This condition is satisfied if for all i ∈ T , there are α i > β i ≥ p i ≥ d i ( x ) = α i + β i x p i , or if all d i ’s are Lipschitz and uniformly positive. Theorem 2
Assume (11) - (12) . We consider the cost on J × J defined by ̺ ( x, i, y, j ) = min( a, | x − y | )1I { i = j } + a { i = j } . or any u , u ∈ P ( J ) , there exists a pair of weak measure solutions ( u t ) t ≥ , ( u t ) t ≥ ⊂ P ( J ) to (10) starting from u and u , i.e., such that for j = 1 , and all t ≥ Z t Z J d i ( x ) u js (d x, d i )d s < ∞ (13) and for j = 1 , , all t ≥ and all ϕ ∈ C c ( J ) , Z J ϕ ( x, i ) u jt (d x, d i ) = Z J ϕ ( x, i ) u j (d x, d i )+ Z t Z J h g i ( x ) ϕ ′ ( x, i ) + d i ( x )[ ϕ (0 , i + 1) − ϕ ( x, i )] i u js (d x, d i )d s. (14) Moreover, for all t ≥ , we have T ̺ ( u t , u t ) ≤ T ̺ ( u , u ) . Note that if u t ( x, i ) is a strong solution to (10), then u t (d x, d i ) := P k ∈ T u t ( x, k )d xδ k (d i ) is a weakmeasure solution to (10). Note also that the comments after Theorem 1 apply here too. Proof . We assume (11)-(12) and consider any coupling v ∈ H ( u , u ). There exists a family ( v t ) t ≥ of probability measures on J such that for all t ≥ Z t ZZ [ d i ( x ) + d j ( y )] v s (d x, d i, d y, d j )d s < ∞ , (15)and which weakly solves ∂v t ∂t + ∂ [ g i ( x ) v t ] ∂x + ∂ [ g j ( y ) v t ] ∂y + (cid:2) ( d i ( x ) ∨ d j ( y ))1I { i = j } + ( d i ( x ) + d j ( y ))1I { i = j } (cid:3) v t = δ (0 , ( x, y )1I { i = j } ZZ ( d i − ( x ′ ) ∧ d i − ( y ′ )) v t (d x ′ , i − , d y ′ , j − δ ( x )1I { j = i − } Z (cid:0) d i − ( x ′ ) − d j ( y ) (cid:1) + v t (d x ′ , i − , y, j )+ δ ( y )1I { i = j − } Z (cid:0) d j − ( y ′ ) − d i ( x ′ ) (cid:1) + v t ( x, i, d y ′ , j − δ ( x )1I { j = i − } Z d i − ( x ′ ) v t (d x ′ , i − , y, j ) + δ ( y )1I { i = j − } Z d j − ( y ′ ) v t ( x, i, d y ′ , j − . This means that for all ϕ ∈ C c ( J ), all t ≥ ZZ ϕ ( x, i, y, j ) v t (d x, d i, d y, d j ) = ZZ ϕ ( x, i, y, j ) v (d x, d i, d y, d j )+ Z t ZZ h g i ( x ) ∂ϕ ( x, i, y, j ) ∂x + g j ( y ) ∂ϕ ( x, i, y, j ) ∂y i v s (d x, d i, d y, d j )d s + Z t ZZ (cid:16)(cid:2) ϕ (0 , i + 1 , , j + 1) − ϕ ( x, i, y, j ) (cid:3) { i = j } (cid:0) d i ( x ) ∧ d j ( y ) (cid:1) + (cid:2) ϕ (0 , i + 1 , y, j ) − ϕ ( x, i, y, j ) (cid:3) { i = j } (cid:0) d i ( x ) − d j ( y ) (cid:1) + + (cid:2) ϕ ( x, i, , j + 1) − ϕ ( x, i, y, j ) (cid:3) { i = j } (cid:0) d j ( y ) − d i ( x ) (cid:1) + + (cid:2) ϕ (0 , i + 1 , y, j ) − ϕ ( x, i, y, j ) (cid:3) { i = j } d i ( x )+ (cid:2) ϕ ( x, i, , j + 1) − ϕ ( x, i, y, j ) (cid:3) { i = j } d j ( y ) (cid:17) v s (d x, d i, d y, d j )d s. (16)6sing (15) and (16) with a function ϕ depending only on ( x, i ) (or ( y, j )), we see that the first marginal u t (d x, d i ) = R ( y,j ) ∈J v t (d x, d i, d y, d j ) (or the second marginal u t (d y, d j ) = R ( x,i ) ∈J v t (d x, d i, d y, d j ))satisfies (13) and (14).As before, the existence for (15)-(16) follows from classical arguments, using an approximate problemwhere g i , d i are replaced by smooth and bounded functions, and from the following a priori tightnessestimate. By the de la Vall´ee Poussin theorem, there exists a function h : [0 , ∞ ) → [0 , ∞ ) such thatlim x →∞ h ( x ) = ∞ and such that C := ZZ [ h ( x ) + h ( y )] v (d x, d i, d y, d j ) < ∞ . One can moreover choose h smooth, satisfying h (0) = 0 and 0 ≤ h ′ ≤
1. Applying (16) with ϕ ( x, i, y, j ) = h ( x ) + h ( y ), one immediately concludes that for all t ≥ ZZ [ h ( x ) + h ( y )] v t (d x, d i, d y, d j ) = ZZ [ h ( x ) + h ( y )] v (d x, d i, d y, d j )+ Z t ZZ [ g i ( x ) h ′ ( x ) + g j ( y ) h ′ ( y )] v s (d x, d i, d y, d j )d s − Z t ZZ [ d i ( x ) h ( x ) + d j ( y ) h ( y )] v s (d x, d i, d y, d j )d s. Setting C = sup i ∈ T,x ≥ g i ( x ) h ′ ( x ), which is finite because g i is continuous, non-increasing and because h ′ is [0 , ZZ [ h ( x ) + h ( y )] v t (d x, d i, d y, d j ) + Z t ZZ (cid:2) d i ( x ) h ( x ) + d j ( y ) h ( y ) (cid:3) v s (d x, d i, d y, d j )d s ≤ C + 2 Ct.
This last a priori tightness estimate is sufficient to prove existence for (15)-(16).We now apply (16) with ϕ = ̺ , after regularization as before, where we recall that ̺ ( x, i, y, j ) =min( a, | x − y | )1I { i = j } + a { i = j } . Since g i is non-increasing, we have g i ( x ) ∂̺ ( x, i, y, j ) ∂x + g j ( y ) ∂̺ ( x, i, y, j ) ∂y = 1I { i = j } {| x − y |≤ a } sgn( x − y )[ g i ( x ) − g j ( y )] ≤ . Hence we find, choosing v such that RR ̺ ( x, i, y, j ) v (d x, d i, d y, d j ) = T ρ ( u , u ), ZZ ̺ ( x, i, y, j ) v t (d x, d i, d y, d j ) ≤ T ρ ( u , u ) + Z t ZZ (cid:16)(cid:2) ̺ (0 , i + 1 , , j + 1) − ̺ ( x, i, y, j ) (cid:3) { i = j } (cid:0) d i ( x ) ∧ d j ( y ) (cid:1) + (cid:2) ̺ (0 , i + 1 , y, j ) − ̺ ( x, i, y, j ) (cid:3) { i = j } (cid:0) d i ( x ) − d j ( y ) (cid:1) + + (cid:2) ̺ ( x, i, , j + 1) − ̺ ( x, i, y, j ) (cid:3) { i = j } (cid:0) d j ( y ) − d i ( x ) (cid:1) + + (cid:2) ̺ (0 , i + 1 , y, j ) − ̺ ( x, i, y, j ) (cid:3) { i = j } d i ( x )+ (cid:2) ̺ ( x, i, , j + 1) − ̺ ( x, i, y, j ) (cid:3) { i = j } d j ( y ) (cid:17) v s (d x, d i, d y, d j )d s. ̺ , the two last lines are non-positive. We thus arrive at ZZ ̺ ( x, i, y, j ) v t (d x, d i, d y, d j ) ≤ T ρ ( u , u ) − Z t ZZ { i = j } ∆ i ( x, y ) v s (d x, d i, d y, d j )d s, where ∆ i ( x, y ) =( d i ( x ) ∨ d i ( y ))( | x − y | ∧ a ) − ( d i ( x ) − d i ( y )) + a − ( d i ( y ) − d i ( x )) + a =( d i ( x ) ∨ d i ( y ))( | x − y | ∧ a ) − | d i ( x ) − d i ( y ) | a. Next, we check that ∆ i is always non-negative. If | x − y | ≥ a , this is obvious. If | x − y | ≤ a , thisfollows from (12). Since T ρ ( u t , u t ) ≤ RR ρ ( x, i, y, j ) v t (d x, d i, d y, d j ), the proof is complete. (cid:3) Next, we consider an example similar to that in the previous section, when the discrete parameter i is replaced by a continuous parameter z ∈ R d , which represents space or a physiological trait.The formalism makes the link with the heat equation through a standard physical process used inparticular to describe diffusion or anomalous diffusion, see recent analyses in [28, 8, 3]. We departfrom the equation ε ∂u t ( x, z ) ∂t + ∂u t ( x, z ) ∂x + d ( x ) u t ( x, z ) = 0 , t ≥ , x > , z ∈ R d ,u t ( x = 0 , z ) = Z ∞ Z R d d ( x ) u t (d x, z + εη ) k (d η ) , t ≥ , z ∈ R d . (17)This equation models particles characterized by their age x and position z . When in state x, z , theparticle’s age x grows linearly until there is a jump, at rate d ( x ), resulting in the particle moving from z to z − εη (with η chosen according to the probability density k ). At each jump, the age is reset to 0.Hence the state space is here J = [0 , ∞ ) × R d . Under a few assumptions, it is known that, for u ε thesolution to (17), U ε ( t, z ) = R ∞ u ε ( t, x, z )d x converges, as ε →
0, to the solution of the heat equationin R d .We assume that d ∈ C ([0 , ∞ )) , d ≥ , k ∈ P ( R d ) , (18)and, again, that ∃ a > a ≤ inf | x − y ] ≤ a | x − y | max( d ( x ) , d ( y )) | d ( x ) − d ( y ) | . (19) Theorem 3
Assume (18) - (19) and fix ε > . We consider the cost on J × J defined by ̺ ( x, z, y, r ) = min( a, | x − y | + | z − r | ) . For any u , u ∈ P ( J ) , there exists a pair of weak measure solutions ( u t ) t ≥ , ( u t ) t ≥ ⊂ P ( J ) to (17) starting from u and u , i.e., such that for i = 1 , and all t ≥ , Z t Z J d ( x ) u is (d x, d z )d s < ∞ (20)8 nd for i = 1 , , all t ≥ and all ϕ ∈ C c ( J ) , Z J ϕ ( x, z ) u it (d x, d z ) = Z J ϕ ( x, z ) u i (d x, d z )+ ε − Z t Z J h ∂ϕ ( x, z ) ∂x + d ( x ) Z R d [ ϕ (0 , z − εη ) − ϕ ( x, z )] k (d η ) i u is (d x, d z )d s. (21) Moreover, for all t ≥ , we have T ̺ ( u t , u t ) ≤ T ̺ ( u , u ) . Proof . We assume (18)-(19), consider u , u ∈ P ( J ) and a coupling v ∈ H ( u , u ). There exists afamily ( v t ) t ≥ of probability measures on J such that for all t ≥ Z t ZZ [ d ( x ) + d ( y )] v t (d x, d z, d y, d r ) < ∞ , (22)and which weakly solve ε ∂v t ∂t + ∂v t ∂x + ∂v t ∂y + ( b ( x ) ∨ b ( y )) v t = δ ( x ) δ ( y ) Z ∞ Z ∞ Z R d ( b ( x ′ ) ∧ b ( y ′ )) v t (d x ′ , z + εη, d y ′ , r + εη ) k ( η )d η + δ ( x ) Z J (cid:0) b ( x ′ ) − b ( y ) (cid:1) + v t (d x ′ , z + εη, y, r + εη ) k ( η )d η + δ ( y ) Z J (cid:0) b ( y ′ ) − b ( x ) (cid:1) + v t ( x, z + εη, d y ′ , r + εη ) k ( η )d η. This means that for all ϕ ∈ C c ( J ), all t ≥ ZZ ϕ ( x, z, y, r ) v t (d x, d z, d y, d r ) = ZZ ϕ ( x, z, y, r ) v (d x, d z, d y, d r )+ ε − Z t ZZ h ∂ϕ ( x, z, y, r ) ∂x + ∂ϕ ( x, z, y, r ) ∂y i v s (d x, d z, d y, d r )d s + ε − Z t ZZ (cid:16)(cid:2) ( d ( x ) ∧ d ( y )) Z R d [ ϕ (0 , z − εη, , r − εη ) − ϕ ( x, z, y, r )] k (d η )+ ( d ( x ) − d ( y )) + Z R d [ ϕ (0 , z − εη, y, r ) − ϕ ( x, z, y, r )] k (d η )+ ( d ( y ) − d ( x )) + Z R d [ ϕ ( x, z, , r − εη ) − ϕ ( x, z, y, r )] k (d η ) (cid:17) v s (d x, d z, d y, d r )d s. (23)Using (22) and (23) with a function ϕ depending only on ( x, z ) (or ( y, r )), we see that the first marginal u t (d x, d z ) = R ( y,r ) ∈J v t (d x, d z, d y, d r ) (and the second one u t (d y, d r ) = R ( x,z ) ∈J v t (d x, d z, d y, d r ))satisfies (20) and (21). 9he existence for (15)-(16) follows as usual from the following a priori tightness estimate. By thede la Vall´ee Poussin theorem, there is a function h : [0 , ∞ ) → [0 , ∞ ) such that lim x →∞ h ( x ) = ∞ and C := ZZ [ h ( x ) + h ( y ) + h ( | z | ) + h ( | r | )] v (d x, d z, d y, d r ) + Z h ( ε | η | ) k (d η ) < ∞ . One can moreover choose h smooth, satisfying h (0) = 0 and 0 ≤ h ′ ≤
1. Applying (23) with ϕ ( x, z, y, r ) = h ( x ) + h ( y ) + h ( | z | ) + h ( | r | ), one immediately concludes that for all t ≥ ZZ [ h ( x ) + h ( y ) + h ( | z | ) + h ( | r | )] v t (d x, d z, d y, d r ) ≤ C + ε − Z t ZZ [ h ′ ( x ) + h ′ ( y )] v s (d x, d z, d y, d r )d s + ε − Z t ZZ h d ( x ) Z [ h (0) + h ( | z − εη | ) − h ( x ) − h ( | z | )] k (d η )+ d ( y ) Z [ h (0) + h ( | r − εη | ) − h ( y ) − h ( | r | )] k (d η ) i v s (d x, d z, d y, d r )d s. Using that h ′ is [0 , h (0) = 0, we observe that Z (cid:2) h (0) + h ( | z − εη | ) − h ( x ) − h ( | z | ) (cid:3) k (d η ) ≤ Z h ( | εη | )(d η ) − h ( x ) ≤ C − h ( x ) . Using now that d ( x )[ C − h ( x )] ≤ L − d ( x ) h ( x ) / L >
0, we end up with ZZ [ h ( x )+ h ( y )+ h ( | z | )+ h ( | r | )] v t (d x, d z, d y, d r ) + 12 ε Z t ZZ [ d ( x ) h ( x ) + d ( y ) h ( y )] v s (d x, d z, d y, d r )d s ≤ C + 2 ε − t + 2 ε − Lt.
This a priori tightness estimate is sufficient, as usual, to prove existence for (22)-(23).We now apply (23) with ϕ = ̺ , where we recall that ̺ ( x, z, y, r ) = min( a, | x − y | + | z − r | ). Since ∂̺ ( x, z, y, r ) ∂x + ∂̺ ( x, z, y, r ) ∂y = 0 , we find, choosing v such that RR ̺ ( x, z, y, r ) v (d x, d z, d y, d r ) = T ρ ( u , u ), ZZ ̺ ( x, z, y, r ) v t (d x, d z, d y, d r ) ≤ T ρ ( u , u ) + ε − Z t ZZ ∆( x, z, y, r ) v s (d x, d z, d y, d r )d s, where ∆( x, z, y, r ) =( d ( x ) ∧ d ( y )) Z R d [ ̺ (0 , z − εη, , r − εη ) − ̺ ( x, z, y, r )] k (d η )+ ( d ( x ) − d ( y )) + Z R d [ ̺ (0 , z − εη, y, r ) − ̺ ( x, z, y, r )] k (d η )+ ( d ( y ) − d ( x )) + Z R d [ ̺ ( x, z, , r − εη ) − ̺ ( x, z, y, r )] k (d η ) . Since T ρ ( u t , u t ) ≤ RR ρ ( x, z, y, r ) v t (d x, d z, d y, d r ), it thus only remains to check that ∆ is alwaysnon-positive. But we have, assuming e.g. that d ( x ) ≥ d ( y ),∆( x, z, y, r ) = − d ( x )[( | x − y | + | z − r | ) ∧ a ] + d ( y )( | z − r | ∧ a )+ ( d ( x ) − d ( y )) Z R d [( | y | + | z − εη − r | ) ∧ a ] k (d η ) ≤ − d ( x )[( | x − y | + | z − r | ) ∧ a ] + d ( y )( | z − r | ∧ a ) + ( d ( x ) − d ( y )) a. | x − y | + | z − r | ≥ a , we have∆( x, z, y, r ) ≤ − d ( x ) a + d ( y ) a + ( d ( x ) − d ( y )) a = 0 . If | x − y | + | z − r | ≤ a , still assuming that d ( x ) ≥ d ( y ), we have∆( x, z, y, r ) ≤ − d ( x )( | x − y | + | z − r | ) + d ( y ) | z − r | + ( d ( x ) − d ( y )) a ≤ − d ( x ) | x − y | + ( d ( x ) − d ( y )) a ≤ | x − y | + | z − r | ≤ a implies that | x − y | ≤ a . Therefore, we always have∆( x, z, y, r ) ≤ (cid:3) Several applications use multi-time renewal equations to describe a population density subjected toaging or to time-evolution. Recently, for evaluating the efficiency of tracing softwares, it was used totake into account secondary infections, see [18]. In neuroscience, the interpretation is that neuroneskeep memory of their last spikes in the process of deciding when to fire again, see [10]. For two timesmemory, the equation reads ∂u t ( x ,x ) ∂t + ∂u t ( x ,x ) ∂x + ∂u t ( x ,x ) ∂x + d ( x , x ) u t ( x , x ) = 0 , x ≥ x ≥ , t ≥ ,u t ( x = 0 , x ) = R ∞ x d ( x , z ) u t ( x , d z ) , x ≥ , t ≥ . (24)An example of stochastic interpretation is that particles are individuals producing events at a ratedepending on the ages of its two last events. Here the variables x and x represent the ages of thetwo last events, so these ages increase linearly until, at rate d ( x , x ), they are reset to the values(0 , x ). Hence our state space is now J = { ( x , x ) ∈ [0 , ∞ ) : x > x } . We assume that d ∈ C ( J ) , d ≥ ∃ a > a ≤ inf | x − e x | + | x − e x |≤ a ( | x − e x | + | x − ˜ x | ) max( d ( x , x ) , d ( e x , e x )) | d ( x , x ) − d ( e x , e x ) | . (26)One can check that d ( x , x ) = α + βx p + γx p , with α > β ≥ γ ≥ p ≥ p ≥ Theorem 4
We assume (25) - (26) . We consider the cost on J × J defined by ̺ ( x , x , e x , e x ) = (cid:2) | x − e x | + | x − e x | (cid:3) ∧ a. For any u , u ∈ P ( J ) , there exists a pair of weak measure solutions ( u t ) t ≥ , ( u t ) t ≥ ⊂ P ( J ) to (24) starting from u and u , i.e., such that for i = 1 , and all t ≥ Z t Z J d ( x , x ) u is (d x , d x )d s < ∞ (27)11 nd for i = 1 , , all t ≥ and all ϕ ∈ C c ( J ) , Z J ϕ ( x , x ) u it (d x , d x ) = Z J ϕ ( x , x ) u i (d x , d x )+ Z t Z J h ∂ϕ ( x , x ) ∂x + ∂ϕ ( x , x ) ∂x + d ( x , x )[ ϕ (0 , x ) − ϕ ( x , x )] i u is (d x , d x )d s. (28) Moreover, for all t ≥ , we have T ̺ ( u t , u t ) ≤ T ̺ ( u , u ) . Proof . We assume (25)-(26), consider u , u ∈ P ( J ) and a coupling v ∈ H ( u , u ). There exists afamily ( v t ) t ≥ of probability measures on J such that for all t ≥ Z t ZZ [ d ( x , x ) + d ( e x , e x )] v t (d x , d x , d e x , d e x ) < ∞ (29)and which weakly solves, with zero flux boudary conditions at x = 0 and x = 0, ∂v t ∂t + ∂v t ∂x + ∂v t ∂x + ∂v t ∂ e x + ∂v t ∂ e x + ( d ( x , x ) ∨ d ( e x , e x )) v t = δ ( x ) δ ( e x ) ZZ (cid:0) d ( x , z ) ∨ d ( e x , e z ) (cid:1) v t ( x , d z, e x , d e z )+ δ ( x ) ZZ (cid:0) d ( x , z ) − d ( e x , e x ) (cid:1) + v t ( x , d z, e x , e x )+ δ ( e x ) ZZ (cid:0) d ( x , x ) − d ( e x , e z ) (cid:1) + v t ( x , x , e x , d e z ) . This means that for all ϕ ∈ C c ( J ), all t ≥ ZZ ϕ ( x , x , e x , e x ) v t (d x , d x , d e x , d e x ) = ZZ ϕ ( x , x , e x , e x ) v (d x , d x , d e x , d e x )+ Z t ZZ h ∂ϕ∂x + ∂ϕ∂x + ∂ϕ∂ e x + ∂ϕ∂ e x i ( x , x , e x , e x ) v s (d x, d z, d y, d r )d s + Z t ZZ (cid:16) ( d ( x , x ) ∧ d ( e x , e x ))[ ϕ (0 , x , , e x ) − ϕ ( x , x , e x , e x )]+ ( d ( x , x ) − d ( e x , e x )) + [ ϕ (0 , x , e x , e x ) − ϕ ( x , x , e x , e x )] (30)+ ( d ( e x , e x ) − d ( x , x )) + [ ϕ ( x , x , , e x ) − ϕ ( x , x , e x , e x )] (cid:17) v s (d x , d x , d e x , d e x )d s. Using (29) and (30) with a function ϕ depending only on ( x , x ) (or ( e x , e x )), we see that the marginalsof v satisfy (27) and (28).The existence for (29)-(30) follows as usual from the following a priori tightness estimate. By thede la Vall´ee Poussin theorem, there is a function h : [0 , ∞ ) → [0 , ∞ ) such that lim x →∞ h ( x ) = ∞ and C := ZZ [ h ( x ) + h ( x ) + h ( e x ) + h ( e x )] v (d x , d x , d e x , d e x ) < ∞ . h smooth, satisfying h (0) = 0 and 0 ≤ h ′ ≤
1, applying (30) with ϕ ( x , x , e x , e x ) = h ( x ) + h ( x ) + h ( e x ) + h ( e x ), one easily concludes as usual that for all t ≥ ZZ [ h ( x ) + h ( x ) + h ( e x ) + h ( e x )] v t (d x , d x , d e x , d e x )+ Z t ZZ (cid:0) d ( x , x ) h ( x ) + d ( e x , e x ) h ( e x ) (cid:1) v s (d x , d x , d e x , d e x )d s ≤ C + 4 t. Recalling that h increases to infinity and that h ( x ) ≤ h ( x ) for all ( x , x ) ∈ J , this a priori tightness estimate is enough to prove existence for (29)-(30).We now apply (23) with ϕ = ̺ , where ̺ ( x , x , e x , e x ) = min( a, | x − e x | + | x − e x | ). Since ∂ϕ∂x + ∂ϕ∂x + ∂ϕ∂ e x + ∂ϕ∂ e x = 0 , we find, choosing v such that RR ̺ ( x , x , e x , e x ) v (d x , d x , d e x , d e x ) = T ρ ( u , u ), ZZ ̺ ( x , x , e x , e x ) v t (d x , d x , d e x , d e x ) ≤ T ρ ( u , u ) + Z t ZZ ∆( x , x , e x , e x ) v s (d x , d x , d e x , d e x )d s, where ∆( x , x , e x , e x ) = − ( d ( x , x ) ∨ d ( e x , e x ))([2 | x − e x | + | x − e x | ] ∧ a )+ ( d ( x , x ) ∧ d ( e x , e x ))( | x − e x | ∧ a )+ (cid:0) d ( x , x ) − d ( e x , e x ) (cid:1) + ([2 | e x | + | x − e x | ] ∧ a )+ (cid:0) d ( e x , e x ) − d ( x , x ) (cid:1) + ([2 | x | + | e x − x | ] ∧ a ) . Since T ρ ( u t , u t ) ≤ RR ρ ( x , x , e x , e x ) v t (d x , d x , d e x , d e x ), it only remains to check that ∆ is alwaysnon-positive.When first 2 | x − e x | + | x − e x | ≥ a , this is obvious.When 2 | x − e x | + | x − e x | ≤ a , it suffices to verify that( d ( x , x ) ∨ d ( e x , e x ))[2 | x − e x | + | x − e x | ] ≥ ( d ( x , x ) ∧ d ( e x , e x )) | x − e x | + | d ( x , x ) − d ( e x , e x ) | a. This follows from the fact that( d ( x , x ) ∨ d ( e x , e x ))[ | x − e x | + | x − e x | ] ≥ | d ( x , x ) − d ( e x , e x ) | a by (26). (cid:3) The growth-fragmentation equation arises in several areas of biology. The variable represents forinstance the size of cells or the length of biopolymers. It also arises in communication science for TCP13onnections. A large literature is available on the subject and we refer for instance to [24, 12, 27, 4].The model combines growth with a rate g and fragmentation with a rate d and it is written ∂u t ( x ) ∂t + ∂ [ g ( x ) u t ( x )] ∂x + d ( x ) u t ( x ) = Z ∞ x d ( x ′ ) κ ( x, x ′ ) u t (d x ′ ) , t ≥ , x ≥ ,u t ( x = 0) = 0 , t ≥ . (31)Usual conditions on the fragmentation kernel are expressed through the identities κ ( x, x ′ ) = 0 for x > x ′ , Z x ′ κ ( x, x ′ )d x = 1 (32)which lead to the conservation law Z ∞ u t ( x )d x = Z ∞ u ( x )d x = 1 . To go further, we can specify κ ( x, x ′ ) = 1 x ′ β (cid:16) xx ′ (cid:17) for some β ∈ P ([0 , , (33)see the end of the section for a more general possible setting. We then assume that g, d ∈ C ([0 , ∞ )) , g is non-increasing , g (0) ≥ , d ≥ , (34)and ∃ a > a ≤ (cid:18) − Z rβ (d r ) (cid:19) inf | x − y ] ≤ a | x − y | max( d ( x ) , d ( y )) | d ( x ) − d ( y ) | . (35)If β is non-trivial in that R rβ (d r ) ∈ [0 , d ( x ) = α + βx p ,provided α > β ≥ p ≥ Theorem 5
We assume (33) - (34) - (35) . We choose again, on [0 , ∞ ) × [0 , ∞ ) the cost ̺ ( x, y ) = min( a, | x − y | ) . For any u , u ∈ P ([0 , ∞ )) , there exists a pair of weak measure solutions ( u t ) t ≥ , ( u t ) t ≥ ⊂ P ([0 , ∞ )) to (31) , starting from u and u , i.e., such that for i = 1 , , for all t ≥ , all A ≥ , Z t Z ∞ A +1 d ( x ) Z A/x β (d r ) u is (d x )d s < ∞ (36) and for all t ≥ , all ϕ ∈ C c ([0 , ∞ )) , Z ∞ ϕ ( x ) u it (d x ) = Z ∞ ϕ ( x ) u i (d x ) + Z t Z ∞ h g ( x ) ϕ ′ ( x ) + d ( x ) Z ( ϕ ( rx ) − ϕ ( x )) β (d r ) i u is (d x )d s. (37) Moreover, for all t ≥ , we have T ̺ ( u t , u t ) ≤ T ̺ ( u , u ) . ϕ ∈ C c ([0 , ∞ )) supported in [0 , A ], (cid:12)(cid:12)(cid:12) d ( x ) Z ( ϕ ( rx ) − ϕ ( x )) β (d r ) (cid:12)(cid:12)(cid:12) ≤ { x ≤ A +1 } k ϕ k ∞ sup [0 ,A +1] d + 1I { x>A +1 } k ϕ k ∞ d ( x ) Z A/x β (d r ) , where we used a rough upper bound by A + 1 to fit our assumption. Proof . Consider u , u ∈ P ([0 , ∞ )) and a coupling v ∈ H ( u , u ). There exists a family ( v t ) t ≥ ofprobability measures on [0 , ∞ ) satisfying, for all t ≥
0, all A ≥ Z t ZZ h { x ≥ A +1 } d ( x ) Z A/x β (d r ) + 1I { y ≥ A +1 } d ( y ) Z A/y β (d r ) i v s (d x, d y )d s < ∞ (38)and solving weakly ∂v t ∂t + ∂∂x (cid:2) g ( x ) v t (cid:3) + ∂∂y (cid:2) g ( y ) v t (cid:3) + ( d ( x ) ∨ d ( y )) v t = ZZ ( d ( x ′ ) ∧ d ( y ′ )) 1 x ′ y ′ β (cid:0) xx ′ (cid:1) δ (cid:0) xx ′ − yy ′ (cid:1) β (cid:0) xx ′ (cid:1) v t (d x ′ , d y ′ )+ Z ( d ( x ′ ) − d ( y )) + x ′ β (cid:0) xx ′ (cid:1) v t (d x ′ , y ) + Z ( d ( y ′ ) − d ( x )) + y ′ β (cid:0) yy ′ (cid:1) v ( x, d y ′ ) . This means that for all t ≥
0, all ϕ ∈ C c ([0 , ∞ ) ), ZZ ϕ ( x, y ) v t (d x, d y ) = ZZ ϕ ( x, y ) v (d x, d y ) + Z t ZZ h g ( x ) ∂ϕ ( x, y ) ∂x + g ( y ) ∂ϕ ( x, y ) ∂y i v s (d x, d y )d s + Z t ZZ ( d ( x ) ∧ d ( y )) Z (cid:2) ϕ ( rx, ry ) − ϕ ( x, y ) (cid:3) β (d r ) v s (d x, d y )d s + Z t ZZ ( d ( x ) − d ( y )) + Z (cid:2) ϕ ( rx, y ) − ϕ ( x, y ) (cid:3) β (d r ) v s (d x, d y )d s + Z t ZZ ( d ( y ) − d ( x )) + Z (cid:2) ϕ ( x, ry ) − ϕ ( x, y ) (cid:3) β (d r ) v s (d x, d y )d s. (39)One checks as usual that the two marginals of v t solve (36)-(37). Next, as in the previous sections,one finds that there is a function h : [0 , ∞ ) → [0 , ∞ ), strictly increasing to infinity, which taken as atest function gives ZZ [ h ( x ) + h ( y )] v t (d x, d y ) + Z t ZZ (cid:2) d ( x ) H ( x ) + d ( e x ) H ( e x ) (cid:3) v s (d x, d y )d s ≤ C + 2 Ct, where H ( x ) = R ( h ( x ) − h ( rx )) β (d r ) and C = sup x ≥ g ( x ) h ′ ( x ). This a priori estimate is sufficientto prove tightness and construct a solution to (36)-(37), because for any A ≥
1, any x ≥ A + 1, onehas Z A/x β (d r ) ≤ Z A/x h ( x ) − h ( rx ) h ( A + 1) − h ( A ) β (d r ) ≤ H ( x ) h ( A + 1) − h ( A ) . Applying now the above equation to ϕ = ̺ , where ̺ ( x, y ) = min( a, | x − y | ), one finds as usual, ifchoosing v correctly and using that g is non-increasing, that ZZ ̺ ( x, y ) v t (d x, d y ) ≤ T ̺ ( u , u ) + Z t ZZ ∆( x, y ) v s (d x, d y )d s, x, y ) = − ( d ( x ) ∨ d ( y )) ̺ ( x, y ) + ( d ( x ) ∧ d ( y )) Z ̺ ( rx, ry ) β (d r )+ ( d ( x ) − d ( y )) + Z ̺ ( rx, y ) β (d r ) + ( d ( y ) − d ( x )) + Z ̺ ( x, ry ) β (d r ) . (40)We finally show that ∆ is alway non-positive. If first | x − y | ≥ a , it is enough to check that a ( d ( x ) ∨ d ( y )) ≥ a ( d ( x ) ∧ d ( y )) + a ( d ( x ) − d ( y )) + + a ( d ( y ) − d ( x )) + , which is obvious. If next | x − y | ≤ a , it suffices to check that( d ( x ) ∨ d ( y )) | x − y | ≥ ( d ( x ) ∧ d ( y )) h Z rβ (d r ) i | x − y | + a ( d ( x ) − d ( y )) + + a ( d ( y ) − d ( x )) + , which follows from the fact that h − Z rβ (d r ) i ( d ( x ) ∨ d ( y )) | x − y | ≥ a | d ( x ) − d ( y ) | thanks to (35). (cid:3) A little study shows that the result still holds true if assuming, instead of (33), that the family offragmentation kernels ( κ ( · , x )) x ∈ R + ⊂ P ( R + ) satisfies κ ([0 , x ] , x ) = 1 for all x ≥ ∃ m ∈ [0 , , for all x, y ∈ R + , W ( κ ( · , x ) , κ ( · , y )) ≤ m | x − y | , where W is the usual Monge-Kantorovich distance on P ( R + ), and replacing (35) by ∃ a > a ≤ (1 − m ) inf | x − y ] ≤ a | x − y | max( d ( x ) , d ( y )) | d ( x ) − d ( y ) | . Indeed, it suffices to apply the usual strategy, starting from the coupling equation : ZZ ϕ ( x, y ) v t (d x, d y ) = ZZ ϕ ( x, y ) v (d x, d y ) + Z t ZZ h g ( x ) ∂ϕ ( x, y ) ∂x + g ( y ) ∂ϕ ( x, y ) ∂y i v s (d x, d y )d s + Z t ZZ ( d ( x ) ∧ d ( y )) Z x Z y (cid:2) ϕ ( x ′ , y ′ ) − ϕ ( x, y ) (cid:3) κ (d x ′ , d y ′ , x, y ) v s (d x, d y )d s + Z t ZZ ( d ( x ) − d ( y )) + Z x (cid:2) ϕ ( x ′ , y ) − ϕ ( x, y ) (cid:3) κ (d x ′ , x ) v s (d x, d y )d s + Z t ZZ ( d ( y ) − d ( x )) + Z y (cid:2) ϕ ( x, y ′ ) − ϕ ( x, y ) (cid:3) κ (d y ′ , y ) v s (d x, d y )d s, where for each x, y ∈ R + , κ ( · , · , x, y ) ∈ H ( κ ( · , x ) , κ ( · , y )) satisfies ZZ | x ′ − y ′ | κ (d x ′ , d y ′ , x, y ) = W ( κ ( · , x ) , κ ( · , y )) . Age and size structure
Models have been proposed which use several stucture variables. For instance, age or size only arenot enough to predict cell division. But a combination of both (or other physiological variables as sizeincrement) have been used, see [15, 16], leading to write ∂u t ( x, z ) ∂t + ∂u t ( x, z ) ∂x + ∂ [ g ( z ) u t ( x, z )] ∂z + d ( x, z ) u t ( x, z ) = 0 , t ≥ , x > , z > ,u t ( x, z = 0) = 0 , t ≥ , x > ,u t ( x = 0 , z ) = Z ∞ x =0 Z ∞ z ′ = z d ( x ′ , z ′ ) κ ( z, z ′ ) u t (d x ′ , d z ′ ) , t ≥ , z > J = [0 , ∞ ) . We assume that the coagulation kernel has the specific form (33),that g ∈ C ([0 , ∞ )) , g is non-increasing , g (0) ≥ , d ∈ C ( J ) , d ≥ . (42)and that ∃ a > a ≤ (cid:16) − Z rβ (d r ) (cid:17) inf | x − z | + | e x − e z |≤ a | x − e x | + | z − ˜ z || d ( x, z ) − d ( e x, e z ) | max( d ( x, z ) , d ( e x, e z )) . (43) Theorem 6
Assume (33) - (42) - (43) and consider the cost ̺ ( x, z, e x, e z ) = min( a, | x − e x | + | z − ˜ z | ) . For any u , u ∈ P ( J ) , there exists a pair of weak measure solutions ( u t ) t ≥ , ( u t ) t ≥ ⊂ P ( J ) to (41) ,starting from u and u , i.e., such that for i = 1 , and all t ≥ , all A ≥ , Z t Z J d ( x, z ) h { x ≥ A +1 } + 1I { x ≤ A +1 ,z ≥ A +1 } Z A/z β (d r ) i u is (d x, d z )d s < ∞ (44) and for all t ≥ , all ϕ ∈ C c ( J ) , Z J ϕ ( x, z ) u it (d x, d z ) = Z J ϕ ( x, z ) u i (d x, d z )+ Z t Z J h ∂ϕ ( x, z ) ∂x + g ( z ) ∂ϕ ( x, z ) ∂z + d ( x, z ) Z ( ϕ (0 , rz ) − ϕ ( x, z )) β (d r ) i u is (d x, d z )d s. (45) Moreover, for all t ≥ , we have T ̺ ( u t , u t ) ≤ T ̺ ( u , u ) . Here again, (45) makes sense thanks to (44): if ϕ ∈ C c ( J ) is supported in [0 , A ] , then d ( x, z ) (cid:12)(cid:12)(cid:12) Z ( ϕ (0 , rz ) − ϕ ( x, z )) β (d r ) (cid:12)(cid:12)(cid:12) ≤ { x ≤ A +1 ,z ≤ A +1 } k ϕ k ∞ sup [0 ,A +] d + 1I { x ≥ A +1 } k ϕ k ∞ d ( x, z )+ 1I { x ≤ A +1 ,z ≥ A +1 } k ϕ k ∞ d ( x, z ) Z A/z β (d r ) . roof . As before, we consider u , u ∈ P ( J ) and a coupling v ∈ H ( u , u ). There exists a family( v t ) t ≥ of probability measures on J such that, for all t ≥ Z t Z J (cid:16) d ( x, z ) h { x ≥ A +1 } + 1I { x ≤ A +1 ,z ≥ A +1 } Z A/z β (d r ) i + d ( e x, e z ) h { e x ≥ A +1 } + 1I { e x ≤ A +1 , e z ≥ A +1 } Z A/ e z β (d r ) i(cid:17) v s (d x, d z, d e x, d e z )d s < ∞ (46)and that weakly solves ∂∂t v t ( x, z, e x, e z ) + ∂v t ∂x + ∂v t ∂ e x + ∂∂z (cid:2) g ( z ) v t (cid:3) + ∂v t ∂ e z (cid:2) g ( e z ) v t (cid:3) + b ( x, z ) ∨ d ( e x, e z ) v t = δ ( x ) δ ( e x ) ZZ ( d ( x ′ , z ′ ) ∧ d ( e x ′ , e z ′ )) 1 z ′ e z ′ β (cid:0) zz ′ (cid:1) δ (cid:0) zz ′ − e z e z ′ (cid:1) v t (d x ′ , d z ′ , d e x ′ , d e z ′ )+ δ ( x ) Z ( d ( x ′ , z ′ ) − d ( e x, e z )) + z ′ β (cid:0) zz ′ (cid:1) v t (d x ′ , d z ′ , e x, e z )+ δ ( e x ) Z ( d ( e x ′ , e z ′ ) − d ( x, z )) + e z ′ β (cid:0) e z e z ′ (cid:1) v t ( x, z, d e x ′ , d e z ′ )This equation means that for all t ≥
0, for all ϕ ∈ C c ( J ), ZZ ϕ ( x, z, e x, e z ) v t (d x, d z, d e x, d e z ) = ZZ ϕ ( x, z, e x, e z ) v (d x, d z, d e x, d e z )+ Z t ZZ h ∂ϕ∂x + ∂ϕ∂ e x + g ( z ) ∂ϕ∂z + g ( e z ) ∂ϕ∂ e z i ( x, z, e x, e z ) v s (d x, d z, d e x, d e z )d s + Z t ZZ h ( d ( x, z ) ∧ d ( e x, e z )) Z [ ϕ (0 , rz, , r e z ) − ϕ ( x, z, e x, e z )] β (d r )+ ( d ( x, z ) − d ( e x, e z )) + Z [ ϕ (0 , rz, e x, e z ) − ϕ ( x, z, e x, e z )] β (d r ) (47)+ ( d ( e x, e z ) − d ( x, z )) + Z [ ϕ ( x, z, , r e z ) − ϕ ( x, z, e x, e z )] β (d r ) i v s (d x, d z, d e x, d e z )d s. The two marginals of v t solve (44)-(45). Next, one finds as usual that there is a function h : [0 , ∞ ) → [0 , ∞ ), strictly increasing to infinity, such that ZZ [ h ( x ) + h ( z ) + h ( e x ) + h ( e z )] v t (d x, d z, d e x, d e z )+ Z t ZZ [ d ( x, z ) H ( x, z ) + d ( e x, e z ) H ( e x, e z )] v s (d x, d z, d e x, d e z )d s ≤ C + 2(1 + C ) t, where H ( x, z ) = h ( x ) + R ( h ( x ) − h ( rx )) β (d r ) and C = sup x ≥ g ( x ) h ′ ( x ). This a priori estimate issufficient to construct a solution to (46)-(47), because for any A ≥ { x ≥ A +1 } + 1I { x ≤ A +1 ,z ≥ A +1 } Z A/z β (d r ) ≤ h ( x ) h ( A + 1) + Z h ( x ) − h ( rx ) h ( A + 1) − h ( A ) β (d r ) ≤ H ( x ) h ( A + 1) − h ( A ) . d ( x, z ) ∨ d ( e x, e z )) min( a, | x − e x | + | z − ˜ z | ) ≥ ( d ( x, z ) ∧ d ( e x, e z )) Z min( a, r | z − e z | ) β (d r ) + a | d ( x, z ) − d ( e x, e z ) | . When the min on the left hand side is achieved by a the inequality is obvious. Otherwise, we have tocheck that( d ( x, z ) ∨ d ( e x, e z )) (cid:2) | x − e x | + | z − ˜ z | (cid:3) ≥ Z rβ (d r ) ( d ( x, z ) ∧ d ( e x, e z )) | z − e z | + a | d ( x, z ) − d ( e x, e z ) | . This again is satisfied if (cid:16) − Z rβ (d r ) (cid:17) ( d ( x, z ) ∨ d ( e x, e z )) (cid:2) | x − e x | + | z − ˜ z | (cid:3) ≥ a | d ( x, z ) − d ( e x, e z ) | , which is the condition (43), recall that we are in the case where | x − e x | + | z − ˜ z | ≤ a . (cid:3) Here a female of type x ′ mates with a male of type x ′∗ , chosen with the probability u t ( · ), the newbornis distributed with type x according to the law K ( x ; x ′ , x ′∗ ). As often in this theory, we assumethe distribution of males and females are identical, and rely on the formalism which can be foundin [7, 25, 9] for instance. The (homogeneous) model reads ∂ t u t ( x ) + u t ( x ) = ZZ R d K ( x ; x ′ , x ′∗ ) u t (d x ′ ) u t (d x ′∗ ) , t ≥ , x ∈ R d . (48)For keeping the total population constant, the kernel K ≥ ZZ R d K (d x ; x ′ , x ′∗ ) = 1 . For instance, we can think of two extreme cases of either a Dirac concentration or a uniform distribu-tion, K ( x ; x ′ , x ′∗ ) = δ θx ′ +(1 − θ ) x ′∗ ( x ) , θ ∈ (0 , , or K ( x ; x ′ , x ′∗ ) = 1 | x ′ − x ′∗ | { x ∈ ( x ′ ,x ′∗ ) } . (49)These distributions can be generalized to the form K (d x ; x ′ , x ′∗ ) = Z δ x ′ σ + x ′∗ (1 − σ ) ( x ) h (d σ ) , (50)with h a probability distribution on [0 ,
1] such that R σh (d σ ) = θ ∈ (0 , heorem 7 With the notations and assumptions above, we choose, for some p ≥ , ̺ ( x, y ) = | x − y | p . For any u , u ∈ P p ( R d ) , there exists a pair of weak measure solutions ( u t ) t ≥ , ( u t ) t ≥ ⊂ P p ( R d ) to (48) , starting from u and u , i.e., such that for i = 1 , , all t ≥ and all ϕ ∈ C c ( R d ) , Z R d ϕ ( x ) u it (d x ) = Z R d ϕ ( x ) u i (d x )+ Z t ZZZ (cid:2) ϕ ( x ) − θϕ ( x ′ ) − (1 − θ ) ϕ ( x ′∗ ) (cid:3) K (d x ; x ′ , x ′∗ ) u is (d x ′ , d x ′∗ )d s. (51) Moreover, for all t ≥ , we have T ̺ ( u t , u t ) ≤ T ̺ ( u , u ) . Proof . We use a coupling K ( x, y ; x ′ , x ′∗ , y ′ , y ′∗ ), to be chosen later, with the property Z R d K ( x, d y ; x ′ , x ′∗ , y ′ , y ′∗ ) = K ( x ; x ′ , x ′∗ ) , Z R d K (d x, y ; x ′ , x ′∗ , y ′ , y ′∗ ) = K ( y ; y ′ , y ′∗ ) . Then, we introduce the coupling equation ∂ t v t ( x, y ) + v t ( x, y ) = ZZZZ K ( x, y ; x ′ , x ′∗ , y ′ , y ′∗ ) v t (d x ′ , d y ′ ) v t (d x ′∗ , d y ′∗ ) . This means that for all t ≥
0, for all ϕ ∈ C c ( R d ), Z R d ϕ ( x, y ) v t (d x, d y ) = Z R d ϕ ( x, y ) v (d x, d y )+ Z t ZZZZ (cid:2) ϕ ( x, y ) − θϕ ( x ′ , y ′ ) − (1 − θ ) ϕ ( x ′∗ , y ′∗ ) (cid:3) K (d x, d y ; x ′ , x ′∗ , y ′ , y ′∗ ) v s (d x ′ , d y ′ ) v s (d x ′∗ , d y ′∗ )d s, (52)where θ ∈ (0 ,
1) is the same as in (50). For existence, we have to check the tightness in P p ( R d ). Bythe de la Vall´ee Poussin theorem, there is a function h : R d → [0 , ∞ ) such that lim | x |→∞ h ( x ) = ∞ and C := ZZ [ h ( x ) | x | p + h ( y ) | y | p ] v (d x, d y ) < ∞ . One can moreover choose h smooth, satisfying h (0) = 0 and such that x h ( x ) | x | p convex. Choosing h ( x ) | x | p + h ( y ) | y | p as a test function in (52), we conclude the bound, for all t ≥ ZZ [ h ( x ) | x | p + h ( y ) | y | p ] v t (d x, d y ) ≤ C, because, by convexity, the second term in the right hand side of (52) is non-positive.For the non-expansion property, we just have to show that the right hand side is non-positive for ̺ ( x, y ) = | x − y | p (arguing again after truncation, regularization), if choosing as coupling kernel K (d x, d y ; x ′ , x ′∗ , y ′ , y ′∗ ) = Z h (d σ ) δ σx ′ +(1 − σ ) x ′∗ ( x ) δ σy ′ +(1 − σ ) y ′∗ ( y ) . Z (cid:12)(cid:12) σx ′ + (1 − σ ) x ′∗ − σy ′ − (1 − σ ) y ′∗ (cid:12)(cid:12) p h (d σ ) ≤ θ | x ′ − y ′ | p + (1 − θ ) | x ′∗ − y ′∗ | p which, by convexity, is immediate. (cid:3) A Uniqueness of measure solutions
The coupling method is most powerful when the measure solutions are unique. This uniquenessproblem, in particular for coefficients with low regularity, can lead to several deep developments,[2, 12]. Here, we consider regular coefficients so that the Hilbert Uniqueness Method can be appliedwithout difficulty both to the Structured Equations under consideration and to the coupled equations.We treat in details the example of the renewal equation, i.e., (2) when b = δ . We assume that d ∈ C ([0 , ∞ )) , g ( x ) ∈ C b ( R + ) , g (0) ≥ , (53)We define the weak solutions (or distributional solutions), as follows. Definition 8
A function ( u t ) t ≥ ⊂ P (0 , ∞ ) satisfies the renewal equation (2) in the distributionsense, if for all T > and all test function ψ ∈ C (cid:0) [0 , T ] × [0 , ∞ [ (cid:1) such that ψ ( x, T ) ≡ , we have − Z T Z ∞ (cid:20) ∂ψ ( x, t ) ∂t + g ( x ) ∂ψ ( x, t ) ∂x − d ( x ) ψ ( x, t ) + d ( x ) ψ (0 , t ) (cid:21) u t (d x ) d t = Z ∞ ψ ( x, u (d x ) . Theorem 9 (Well posednesss)
We assume (53) . There is a unique weak solution of the renewalequation (2) . For the existence part, we refer to [2, 12] where more elaborate equations are treated. For uniqueness,need to study the inhomogeneous dual problem. We introduce a source term S ( x, t ) on a given timeinterval [0 , T ] and ( − ∂∂t ψ ( x, t ) − g ( x ) ∂∂x ψ ( x, t ) + d ( x ) ψ ( x, t ) = ψ (0 , t ) d ( x ) + S ( x, t ) ,ψ ( x, T ) = 0 . (54)This problem is backward in t and x , therefore it does not use a boundary condition at x = 0. Lemma 10 (Existence for the dual problem)
Assume (53) , S ∈ C (cid:0) [0 , T ) × R + (cid:1) and d ∈ C (0 , ∞ ) , then there is a unique C solution to the dual equation (54) . Moreover ψ ( x, t ) vanishes for x ≥ R > for some R depending on the data and T , and the bound holds sup ≤ t ≤ T, x ∈ R + | ψ ( x, t ) | ≤ C ( T ) k S k ∞ . roof . We use the method of characteristics based on the solution of the differential systemparametrized by the Cauchy data ( x, t ) which is fixed ( dds X s = g ( X s ) , ≤ s ≤ T,X t = x ≥ . It is well-posed thanks to the Cauchy-Lipschitz theorem and X s ≥ g (0) ≥ X s depends on ( x, t ) and thus the notation X s ≡ X s ( x, t ).Then, we set e ψ ( s ; x, t ) = ψ ( s, X s ) e R ts d ( σ,X σ )d σ , e d ( s ; x, t ) = d ( s, X s ) e R ts d ( σ,X σ )d σ , e S ( s ; x, t ) = S ( s, X s ) e R ts d ( σ,X σ )d σ , and ignore the parameter ( x, t ) when the statements are clear enough. We rewrite equation (54) as dds e ψ ( s ) = (cid:2) ∂∂t ψ + g ∂∂x ψ − d ψ (cid:3) e R ts d ( σ,X σ )d σ (cid:12)(cid:12)(cid:12) ( s,X s ) = − ψ (0 , s ) e d ( s ) − e S ( s ) , Next, we integrate between s = t and s = T , use the Cauchy data at t = T and the identity e ψ ( t ) = ψ ( x, t ), and we obtain ψ ( x, t ) = Z Tt (cid:2) ψ (0 , s ) e d ( s ; x, t ) + e S ( s ; x, t ) (cid:3) d s. (55)This integral equation can be solved first for x = 0. Then, equation (55) is reduced to the Volterraequation ψ (0 , t ) = Z Tt (cid:2) ψ (0 , s ) e d ( s ; 0 , t ) + e S ( s ; 0 , t ) (cid:3) d s, ≤ t ≤ T which, thanks to the (backward) Cauchy-Lipschitz theorem, has a unique solution that vanishes for t = T . By the C regularity of the data, we also have ψ (0 , t ) ∈ C ([0 , T ]).Since ψ (0 , t ) is now known, formula (55) gives us the explicit form of the solution for all ( x, t ).Notice that, in the compact support statement, e ψ ( x, t ) vanishes for x ≥ R where R denotes the sizeof the support of S in x , plus T k g k ∞ . The uniform bound on ψ also follows from formula (55), (cid:3) Proof . [Uniqueness for the renewal equation.] With the help of the dual problem, we can use theHilbert Uniqueness Method. The idea is simple: when the coefficients d , g satisfy the assumptions ofLemma 10, we can use the solution ψ of (54) as a test function in the weak formulation of Definition 8.For the difference u = u − u between two possible solutions u , u with the same initial data, wearrive at Z T Z ∞ (cid:20) ∂ψ ( x, t ) ∂t + g ( x ) ∂ψ ( x, t ) ∂x − d ( x ) ψ ( x, t ) + d ( x ) ψ (0 , t ) (cid:21) u t (d x )d t = 0 . ψ ( · , · ) ∈ C which is the case when d ∈ C . Then, taking into account (54), we arrive at Z T Z ∞ S ( x, t ) u t (d x )d t = 0 , for all T > S ∈ C , and this implies u ≡ d is merely continuous, we consider a regularized family d p → d where the convergence holdslocally uniformly. Then, for a given function S ∈ C , we solve (54) with d p in place of d and call ψ p its solution (which is uniformly bounded with compact support). Inserting it in the definition ofweak solutions, we obtain Z T Z ∞ S ( x, t ) u t (d x )d t = R p ,R p = Z T Z ∞ [ d p − d ( x )][ ψ p ( x, t ) − ψ p (0 , t )] u t (d x ) d t, and using that ψ p is uniformly bounded, we deduce that | R p | ≤ T k ψ k ∞ k d p − d k ∞ −−−−→ p →∞ . Therefore, we have recovered the identity Z T Z ∞ S ( x, t ) u t (d x )d t = 0 , for all functions S ∈ C , andthis implies again u ≡ (cid:3) References [1]
L. Ambrosio, N. Gigli, and G. Savar´e , Gradient flows in metric spaces and in the space ofprobability measures , Lectures in Mathematics ETH Z¨urich, Birkh¨auser Verlag, Basel, second ed.,2008.[2]
V. Bansaye, B. Cloez, and P. Gabriel , Ergodic behavior of non-conservative semigroups viageneralized Doeblin’s conditions , Acta Appl. Math., 166 (2020), pp. 29–72.[3]
H. Berry, T. Lepoutre, and A. M. Gonz´alez , Quantitative convergence towards a self-similar profile in an age-structured renewal equation for subdiffusion , Acta Appl. Math., 145(2016), pp. 15–45.[4]
J. Bertoin and A. R. Watson , The strong Malthusian behavior of growth-fragmentation pro-cesses , Ann. H. Lebesgue, 3 (2020), pp. 795–823.[5]
S. Bianchini and M. Gloyer , An estimate on the flow generated by monotone operators ,Comm. Partial Differential Equations, 36 (2011), pp. 777–796.[6]
F. Bouchut, F. James, and S. Mancini , Uniqueness and weak stability for multi-dimensionaltransport equations with one-sided Lipschitz coefficient , Ann. Sc. Norm. Super. Pisa Cl. Sci. (5),4 (2005), pp. 1–25. 237]
R. B¨urger , The mathematical theory of selection, recombination, and mutation , Wiley Series inMathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2000.[8]
V. Calvez, P. Gabriel, and A. Mateos Gonz´alez , Limiting Hamilton-Jacobi equation forthe large scale asymptotics of a subdiffusion jump-renewal equation , Asymptot. Anal., 115 (2019),pp. 63–94.[9]
V. Calvez, J. Garnier, and F. Patout , Asymptotic analysis of a quantitative genetics modelwith nonlinear integral operator , J. ´Ec. polytech. Math., 6 (2019), pp. 537–579.[10]
J. Chevallier, M. J. C´aceres, M. Doumic, and P. Reynaud-Bouret , Microscopic ap-proach of a time elapsed neural model , Math. Models Methods Appl. Sci., 25 (2015), pp. 2669–2719.[11]
J. M. Cushing , An introduction to structured population dynamics , vol. 71 of CBMS-NSF Re-gional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics(SIAM), Philadelphia, PA, 1998.[12]
T. Debiec, M. Doumic, P. Gwiazda, and E. Wiedemann , Relative entropy method for mea-sure solutions of the growth-fragmentation equation , SIAM J. Math. Anal., 50 (2018), pp. 5811–5824.[13]
R. J. DiPerna and P.-L. Lions , Ordinary differential equations, transport theory and Sobolevspaces , Invent. Math., 98 (1989), pp. 511–547.[14]
R. L. Dobruˇsin , Vlasov equations , Funktsional. Anal. i Prilozhen., 13 (1979), pp. 48–58, 96.[15]
M. Doumic, M. Hoffmann, N. Krell, and L. Robert , Statistical estimation of a growth-fragmentation model observed on a genealogical tree , Bernoulli, 21 (2015), pp. 1760–1799.[16]
M. Doumic, A. Olivier, and L. Robert , Estimating the division rate from indirect measure-ments of single cells , Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), pp. 3931–3961.[17]
W. Feller , On the integral equation of renewal theory , Ann. Math. Statistics, 12 (1941), pp. 243–267.[18]
L. Ferretti, C. Wymant, M. Kendall, L. Zhao, A. Nurtay, L. Abeler-D¨orner,M. Parker, D. Bonsall, and C. Fraser , Quantifying sars-cov-2 transmission suggests epi-demic control with digital contact tracing , Science, 368 (2020).[19]
N. Fournier and E. L¨ocherbach , On a toy model of interacting neurons , Ann. Inst. HenriPoincar´e Probab. Stat., 52 (2016), pp. 1844–1876.[20]
N. Fournier and B. Perthame , Transport distances for pdes: the coupling method , EMS Surv.Math. Sci., 7 (2020), pp. 1–31.[21]
F. Golse, C. Mouhot, and T. Paul , On the mean field and classical limits of quantummechanics , Comm. Math. Phys., 343 (2016), pp. 165–205.[22]
M. Hauray , Wasserstein distances for vortices approximation of Euler-type equations , Math.Models Methods Appl. Sci., 19 (2009), pp. 1357–1384.2423]
C. Marchioro and M. Pulvirenti , Mathematical theory of incompressible nonviscous fluids ,vol. 96 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994.[24]
J. A. J. Metz and O. Diekmann , The dynamics of physiologically structured populations ,vol. 68 of Lecture Notes in Biomath., Springer, Berlin, 1986.[25]
S. Mirrahimi and G. Raoul , Dynamics of sexual populations structured by a space variableand a phenotypical trait , Theoretical Population Biology, 84 (2013), pp. 87–103.[26]
S. Mischler and Q. Weng , Relaxation in time elapsed neuron network models in the weakconnectivity regime , Acta Appl. Math., 157 (2018), pp. 45–74.[27]
P. Monmarch´e , On H and entropic convergence for contractive PDMP , Electron. J. Probab.,20 (2015), pp. Paper No. 128, 30.[28] S. Nordmann, B. Perthame, and C. Taing , Dynamics of concentration in a population modelstructured by age and a phenotypical trait , Acta Appl. Math., 155 (2018), pp. 197–225.[29]
K. Pakdaman, B. Perthame, and D. Salort , Dynamics of a structured neuron population ,Nonlinearity, 23 (2010), pp. 55–75.[30]
B. Perthame , Transport equations in biology , Frontiers in Mathematics, Birkh¨auser Verlag,Basel, 2007.[31]
F. Santambrogio , Optimal transport for applied mathematicians , vol. 87 of Progress in Nonlin-ear Differential Equations and their Applications, Birkh¨auser/Springer, Cham, 2015.[32]