
A possible example of a stationary solution to (3) with
In this paper we introduce the notion of fuzzy dynamic system in b -metric-like space. By applying this, discuss some new refinements of the F-fuzzy Suzuki-type fixed point results for the fuzzy operators are presented. Also, establish the concept fuzzy dynamic system instead of the Piscard iterative sequence, which improves the existing results for such analysis as those presented here. Includes some tangible instances and an application are given to highlight the usability and validity of the theoretical results.
Citation: Amjad Ali, Eskandar Ameer, Suhad Subhi Aiadi, Muhammad Tariq, Muhammad Arshad, Nabil Mlaiki, Wasfi Shatanawi. New extension to fuzzy dynamic system and fuzzy fixed point results with an application[J]. AIMS Mathematics, 2023, 8(1): 1208-1229. doi: 10.3934/math.2023061
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In this paper we introduce the notion of fuzzy dynamic system in b -metric-like space. By applying this, discuss some new refinements of the F-fuzzy Suzuki-type fixed point results for the fuzzy operators are presented. Also, establish the concept fuzzy dynamic system instead of the Piscard iterative sequence, which improves the existing results for such analysis as those presented here. Includes some tangible instances and an application are given to highlight the usability and validity of the theoretical results.
The mathematical modelling of the collective motion through aggregation/diffusion phenomena has raised a lot of interest in the recent years and it has been deeply studied for its application in several areas, such as biology [9, 39, 47, 48], ecology [35, 42, 43], animal swarming [3, 4, 40, 46] sociology and economics, [49, 50, 51]. One of the common idea in this modelling approach is that a certain population composed by agents evolves according to long-range attraction and short-range repulsion forces between agents. We are interested in modelling the problem of predator-prey interactions, namely we consider two populations that attract (prey) and repel (predators) each others. The pioneering works for this problem are the ones by Lotka, [36] and Volterra[54], which describe the predator-prey interaction via reaction terms in a set of differential equations, possibly combined with diffusion terms, see [41] and the references therein.
As in [21], in this paper we model predator-prey interactions via a transport terms rather than a reaction ones as follows: consider
{˙Xi(t)=−N∑k=1mkX(∇Sr1(Xi(t)−Xk(t))+∇Sa1(Xi(t)−Xk(t)))−M∑h=1mhY∇K(Xi(t)−Yh(t)),˙Yj(t)=−M∑h=1mhY(∇Sr2(Xi(t)−Xk(t))+∇Sa2(Yj(t)−Yh(t)))+αM∑k=1mkX∇K(Yj(t)−Xk(t)), | (1) |
with
The formal limit when the number of particles tends to infinity leads to the following system of partial differential equations
{∂tρ=div(ρ∇(d1ρ−Sa1∗ρ−K∗η)),∂tη=div(η∇(d2η−Sa2∗η+αK∗ρ)), | (2) |
where
The goal of this paper is to show that the model above catches one of the main features that occur in nature, namely the formation of spatial patterns where the predators are surrounded of empty zones and the prey aggregates around, that is usually observed in fish schools or in flock of sheeps, see [32, 38]. In the fully aggregative case, namely system (2) with
Existence theory for solutions to system of the form (2) can be performed in the spirit of [10, 20]. In particular, system (2) should be framed in the context of non symmetrizable systems, for which the Wasserstein gradient flow theory developed in [1] and adapted to systems in [22] does not work. In [10, 20, 22], the authors consider different choices for the diffusion part (no diffusion in [22], diagonal nonlinear diffusion in [10] and cross-diffusion with dominant diagonal in [20]), and the existence of solutions is proved via an implicit-explicit version of the JKO scheme [33].
In the following, we reduce our analysis to the one-dimensional setting
{∂tρ=∂x(ρ∂x(d1ρ−Sρ∗ρ−K∗η))∂tη=∂x(η∂x(d2η−Sη∗η+αK∗ρ)). | (3) |
We are interested in the existence of stationary solutions to (3), which are solutions to the following system
{0=(ρ(d1ρ−Sρ∗ρ−K∗η)x)x,0=(η(d2η−Sη∗η+αK∗ρ)x)x, | (4) |
as well as their properties, e.g., symmetry, compact support, etc. We shall assume throughout the paper that
{0=(ρ(θρ−˜Sρ∗ρ−˜K∗η)x)x,0=(η(θη−Sη∗η+˜α˜K∗ρ)x)x. |
The stationary equation for the one species case, i.e.,
∂tρ=∂x(ρ∂x(θρ−S∗ρ)) |
is studied several papers, see [2, 7, 11, 16] and therein references. In [7] the Krein-Rutman theorem is used in order to characterise the steady states as eigenvectors of a certain non-local operator. The authors prove that a unique steady state with given mass and centre of mass exists provided that
In this paper we apply the aforementioned approach in order to show that stationary solutions to (3) are composed of multiple Barenblatt profiles. Let us introduce, for fixed
M={(ρ,η)∈(L∞∩L1(R))2:ρ,η≥0,‖ρ‖L1=zρ,‖η‖L1=zη}. |
Definition 1.1. We say that a pair
●
●
supp(ρ)=Nρ⋃i=1Iiρandsupp(η)=Nη⋃i=1Iiη, |
respectively and
where, for
Example 1.1. A possible example of steady states as defined above is a steady state
Remark 1.1. We remark that one should be careful with finding a shape of steady state. More precisely, one should choose the numbers
In order to simplify the notations, in the following we will denote with
(A1)
(A2)
(A3)
Note that assumption (A2) together with the sign in front of the nonlocal terms
The main result of the paper is the following
Theorem 1.1. Assume that
(i)
Bil=Nl∑j=1S′l(cmil−cmjl)zjl+αlNh∑j=1K′(cmil−cmjh)zjh=0, | (5) |
(ii) the following quantities
Dil=−Nl∑j=1S″l(cmil−cmjl)zjl−αlNh∑j=1K″(cmil−cmjh)zjh, | (6) |
are strictly positive, for all
where
where
● each interval
●
● the solutions
zρ=Nρ∑i=1ziρandzη=Nη∑i=1ziη, |
respectively.
The paper is structured as follows. In Section 2 we recall the basics notions on optimal transport and we introduce the
We start collecting preliminary concepts on the Wasserstein distance. Let
Pp(Rn)={μ∈P(Rn):mp(μ)=∫Rn|x|pdμ(x)<∞}. |
For a measure
∫Rkf(y)dT#μ(y)=∫Rnf(T(x))dμ(x)forallfBorelfunctionsonRk. |
We endow the space
Wpp(μ,ν)=infγ∈Γ(μ,ν){∫Rn×Rn|x−y|pdγ(x,y)}, | (7) |
where
Since we are working in a 'multi-species' structure, we consider the product space
Wpp(μ,ν)=Wpp(μ1,ν1)+Wpp(μ2,ν2), |
with
uμ(z)≐inf{x∈R:μ((−∞,x])>z},z∈[0,1], | (8) |
see [14].
In the Section 1 we mentioned that the well-posedness of (3) can be stated according to the results in [10, 20] in an arbitrary space dimension
F[μ,ν](ρ,η)=θ2∫Rnρ2+η2dx−12∫RnρSρ∗ρdx−12∫RnηSη∗ηdx−∫RnρK∗μdx+α∫RnηK∗νdx, |
for a fixed reference couple of measures
Definition 2.1. A curve
(i)
(ii) for almost every
ddt∫Rnϕρdx=−θ∫Rnρ∇ρ⋅∇ϕdx+∫Rnρ(∇Sρ∗ρ+∇K∗η)∇ϕdx,ddt∫Rnφηdx=−θ∫Rnη∇η⋅∇φdx+∫Rnη(∇Sη∗η−α∇K∗ρ)∇ϕdx. |
Theorem 2.1. Assume that (A1)-(A3) are satisfied. Let
F[μ0](μ0)<+∞. |
Then, there exists a weak solution to (3) in the sense of Definition 2.1 with
As already mentioned the proof of Theorem 2.1 is a special case of the results in [10, 20] and consists in the following main steps:
1. Let
μk+1τ∈argminμ∈P2(Rn)2{12τW22(μkτ,μ)+F[μkτ](μ)}. | (9) |
For a fixed
μτ(t)=(ρτ(t),ητ(t))=μkτt∈((k−1)τ,kτ],k∈N, |
with
2. There exists an absolutely continuous curve
3. There exist a constant
∫T0[||ρτ(t,⋅)||2H1(Rn)+||ητ(t,⋅)||2H1(Rn)]dt≤C(T,μ0), | (10) |
and the sequence
L2((0,T)×Rn)×L2((0,T)×Rn). |
The estimate in (10) can be deduced by using the so called Flow-interchange Lemma introduced in [37], see also [25]. In order to deduce the strong convergence we use the extended version of the Aubin-Lions Lemma in [44].
4. The approximating sequence
μϵ=(ρϵ,ηϵ)=(Pϵ#ρk+1τ,ηkτ), | (11) |
where
0≤12τ[W22(μk+1τ,μϵ)−W22(μkτ,μk+1τ)]+F[μkτ](μϵ)−F[μkτ](μϵ). | (12) |
After some manipulations, sending first
The existence of weak solutions to the purely non-local systems, i.e.,
{∂tρ=div(ρ∇(Sρ∗ρ+Kρ∗η)),∂tη=div(η∇(Sη∗η+Kη∗ρ)), | (13) |
with generic cross-interaction kernels
{∂tρ=div(ρ∇(Sρ∗ρ+K∗η)),∂tη=div(η∇(Sη∗η−αK∗ρ)), | (14) |
and its relation with the particle system
{˙Xi(t)=−N∑k=1mkX∇Sρ(Xi(t)−Xk(t))−M∑k=1mkY∇K(Xi(t)−Yk(t)),˙Yj(t)=−M∑k=1mkY∇Sη(Yj(t)−Yk(t))+αN∑k=1mkX∇K(Yj(t)−Xk(t)). | (15) |
It is proved that stationary states of system (14) are linear combinations of Dirac's deltas, namely
(ˉρ,ˉη)=(N∑k=1mkXδˉXk(x),M∑h=1mhYδˉYh(x)). | (16) |
where
{0=N∑k=1∇Sρ(ˉXk−ˉXi)mkX+M∑h=1∇K(ˉYh−ˉXi)mhY0=M∑h=1∇Sη(ˉYh−ˉYj)mhY−αN∑k=1∇K(ˉXk−ˉYj)mkX. | (17) |
for
Cα=αN∑i=1miXXi−M∑j=1mjYYj | (18) |
that is a conserved quantities, and therefore one would like to produce a unique steady state once the quantity
As a preliminary result, we now prove the existence of one possible shape of steady state, that will be a prototype example for the general case. The steady state is what we can call a mixed steady state, that identifies the case in which the predators can catch the prey, see Figure 2.
The proof of the existence of such steady state follows by using the strong version of the Krein-Rutman theorem, see [26].
Theorem 2.2 (Krein-Rutman). Let
(i) the spectral radius
(ii)
As pointed out in [6], using this strategy we can only obtain existence of stationary states for a diffusion coefficient that depends on the support, without providing any one-to-one correspondence between the diffusion constant (eigenvalue) and the support. Even if non completely satisfactory, the following results give a useful insight on the possible conditions we can expect in order to get existence of steady states, see Remark 3.1.
Let us first introduce a proper definition for mixed steady states as in Figure 2.
Definition 2.2. Let
Iρ:=supp(ρ)=[−Rρ,Rρ],andIη:=supp(η)=[−Rη,Rη]. |
Let us now assume that
{θρ(x)−Sρ∗ρ(x)−K∗η(x)=Cρx∈Iρθη(x)−Sη∗η(x)+αK∗ρ(x)=Cηx∈Iη. | (19) |
where
{θρx=∫Rρ−RρSρ(x−y)ρy(y)dy+∫Rη−RηK(x−y)ηy(y)dyx∈Iρθηx=∫Rη−RηSη(x−y)ηy(y)dy−α∫Rρ−RρK(x−y)ρy(y)dyx∈Iη. | (20) |
By symmetry properties of the kernels
θρx=∫Rρ0(Sρ(x−y)−Sρ(x+y))ρy(y)dy+∫Rη0(K(x−y)−K(x+y))ηy(y)dy,θηx=∫Rη0(Sη(x−y)−Sη(x+y))ηy(y)dy−α∫Rρ0(K(x−y)−K(x+y))ρy(y)dy. | (21) |
In order to simplify the notations, we set
˜G(x,y):=G(x−y)−G(x+y),forG=Sρ,Sη,K. |
Notice that
{θp(x)=∫Rρ0˜Sρ(x,y)p(y)dy+∫Rη0˜K(x,y)q(y)dyθq(x)=∫Rη0˜Sη(x,y)q(y)dy−α∫Rρ0˜K(x,y)p(y)dy. | (22) |
Proposition 2.1. Assume that
C<∫Rη0˜Sη(x,y)h(y)dy∫Rρ0˜K(x,y)k(y)dy, | (23) |
for any
α<min{C,−S′η(Rη)zη−R2ηK″(0)zρ}, |
where
Proof. Let us first introduce the following Banach space
XRρ,Rη={(p,q)∈C1([0,Rρ])×C1([0,Rη]):p(0)=q(0)=0}, |
endowed with the
TRρ,Rη[P]:=(f,g)∈C1([0,Rρ])×C1([0,Rη]), |
where
f(x)=∫Rρ0˜Sρ(x,y)p(y)dy+∫Rη0˜K(x,y)q(y)dyforx∈[0,Rρ],g(x)=∫Rη0˜Sη(x,y)q(y)dy−α∫Rρ0˜K(x,y)p(y)dyforx∈[0,Rη]. |
The assumptions on the kernels ensure that the operator
KRρ,Rη={P∈XRρ,Rη:p≥0,q≥0}. |
It can be shown that this set is indeed a solid cone in
HRρ,Rη={P∈KRρ,Rη:p′(0)>0,p(x)>0forallx∈(0,Rρ),andq′(0)>0,q(x)>0forallx∈(0,Rη)}⊂∘KRρ,Rη, |
where
∫Rη0˜Sη(x,y)q(y)dy−α∫Rρ0˜K(x,y)p(y)dy>0, | (24) |
if and only if
ddx|x=0(∫Rη0˜Sη(x,y)q(y)dy−α∫Rρ0˜K(x,y)p(y)dy)=∫Rη0˜Sη,x(0,y)q(y)dy−α∫Rρ0˜Kx(0,y)p(y)dy=∫Rη0(S′η(−y)−S′η(y))q(y)dy−α∫Rρ0(K′(−y)−K′(y))p(y)dy=−2∫Rη0S′η(y)q(y)dy+2α∫Rρ0K′(y)p(y)dy:=A. |
Now, we need to find the condition on
−2Rη∫Rη0S′η(y)q(y)dy=−2Rη∫Rη0S″η(y)η(y)dy≥(1Rη∫Rη0−S″η(y)dy)(2Rη∫Rη0η(y)dy)=−S′η(Rη)zηR2η. |
Using the concavity assumption of
−2∫Rρ0K′(y)p(y)dy=−2∫Rρ0K″(y)ρ(y)dy<−K″(0)zρ. |
Thus,
α<−S′η(Rη)zη−R2ηK″(0)zρ. | (25) |
As a consequence,
TRρ,Rη[P]=θP, |
with an eigenspace generated by one given nontrivial element
Remark 2.1. Motivated from the purely aggregative case (13), we actually expect a more rich behavior for the possible steady states configurations, such as the separated stationary state in Figure 3. This is expected as a possible winning strategy for the prey, since it corresponds to prey finding a safe distance from the predators. Unfortunately, the symmetrization argument used in the previous proof seems to fail, since in this case we need to require the symmetry around
In this Section we prove the existence and uniqueness of a multiple bumps steady state in the sense of Definition 1 fixing masses and a small diffusion coefficient. Following the approach in [5, 6], we first formulate the problem in terms of the pseudo-inverse functions and then we use the Implicit Function Theorem (cf. [19, Theorem 15.1]).
We start rewriting our stationary system in terms of pseudo-inverse functions. Let
{0=(ρ(θρ−Sρ∗ρ−αρK∗η)x)x,0=(η(θη−Sη∗η−αηK∗ρ)x)x. | (26) |
where
∫Rxρ(x)dx=cmρ,∫Rxη(x)dx=cmη. |
Remember that the only conserved quantity in the evolution, together with the masses, is the joint centre of mass
CMα=αcmρ−cmη, | (27) |
that we can consider fixed. Define the cumulative distribution functions of
Fρ(x)=∫x−∞ρ(x)dx,Fη(x)=∫x−∞η(x)dx. |
Let
ul(z)=inf{x∈R:Fl(x)≥z},l∈{ρ,η}, |
supported on
supp(ul)=[0,zl]:=Jl,l∈{ρ,η}. |
For
∫ρi(x)dx=ziρ,∫ηj(x)dx=zjη,i=1,2,…,Nρ,j=1,2,…,Nη, |
and the centres of masses accordingly,
∫xρi(x)dx=cmiρ,∫xηj(x)dx=cmjη,i=1,2,…,Nρ,j=1,2,…,Nη, |
and we can always assume that the centres of masses are ordered species by species, i.e.
{Nρ∑j=1S′ρ(cmiρ−cmjρ)zjρ+Nη∑j=1K′(cmiρ−cmjη)zjη=0,i=1,…,Nρ,Nη∑j=1S′η(cmiη−cmjη)zjη−αNρ∑j=1K′(cmiη−cmjρ)zjρ=0,i=1,…,Nη, | (28) |
coupled with the conservation of the joint centre of mass
ul(z)=Nl∑i=1uil(z)1Jil(z),l∈{ρ,η}, |
where
supp(ul)=[0,zl]=Jl=Nl⋃i=1[i∑k=1zk−1l,i∑k=1zkl]:=Nl⋃i=1[ˆzil,˜zil]:=Nl⋃i=1Jil,l∈{ρ,η}, |
with
{θ2∂z((∂zuρ(z))−2)=∫JρS′ρ(uρ(z)−uρ(ξ))dξ+αρ∫JηK′(uρ(z)−uη(ξ))dξ,z∈Jρ,θ2∂z((∂zuη(z))−2)=∫JηS′η(uη(z)−uη(ξ))dξ+αη∫JρK′(uη(z)−uρ(ξ))dξ,z∈Jη. | (29) |
The restriction to
θ2∂z((∂zuil(z))−2)=Nl∑j=1∫JjlS′l(uil(z)−ujl(ξ))dξ+αlNh∑j=1∫JjhK′(uil(z)−ujh(ξ))dξ,z∈Jil. | (30) |
Similar to [5, 6], we suggest the linearization
uil=cmil+δvili=1,2,…,Nl,andl∈{ρ,η}, |
with
δ2∂z((∂zvil(z))−2)=Nl∑j=1∫JjlS′l(cmil−cmjl+δ(vil(z)−vjl(ξ)))dξ+αlNh∑j=1∫JjhK′(cmil−cmjh+δ(vil(z)−vjh(ξ)))dξ. | (31) |
Multiplying (31) by
δ2∂zvil(z)=Nl∑j=1∫JjlSl(cmil−cmjl+δ(vil(z)−vjl(ξ)))dξ+αlNh∑j=1∫JjhK(cmil−cmjh+δ(vil(z)−vjh(ξ)))dξ+Ail,z∈Jil, | (32) |
where
Ail=−Nl∑j=1∫JjlSl(cmil−cmjl+δ(λil−vjl(ξ)))dξ−αlNh∑j=1∫JjhK(cmil−cmjh+δ(λil−vjh(ξ)))dξ. | (33) |
Next, we set
δ3z=Nl∑j=1∫JjlGl(cmil−cmjl+δ(vil(z)−vjl(ξ)))dξ+αlNh∑j=1∫JjhH(cmil−cmjh+δ(vil(z)−vjh(ξ)))dξ+Ailδvil(z)+βil,z∈Jil. | (34) |
Let us denote with
βil=δ3ˉzil−Nl∑j=1∫JjlGl(cmil−cmjl−δvjl(ξ))dξ−αlNh∑j=1∫JjhH(cmil−cmjh−δvjh(ξ))dξ. | (35) |
As a consequence of all above computations, we get the following relation for
δ3(z−ˉzil)=Nl∑j=1∫JjlGl(cmil−cmjl+δ(vil(z)−vjl(ξ)))−Gl(cmil−cmjl−δvjl(ξ))dξ−δvil(z)Nl∑j=1∫JjlSl(cmil−cmjl+δ(λil−vjl(ξ)))dξ+αlNh∑j=1∫JjhH(cmil−cmjh+δ(vil(z)−vjh(ξ)))−H(cmil−cmjh−δvjh(ξ))dξ−δvil(z)αlNh∑j=1∫JjhK(cmil−cmjh+δ(λil−vjh(ξ)))dξ. | (36) |
If we define, for
Fil[p;δ](z)=ˉzil−z+δ−3[Nl∑j=1∫JjlGl(cmil−cmjl+δ(vil(z)−vjl(ξ)))−Gl(cmil−cmjl−δvjl(ξ))dξ−δvil(z)Nl∑j=1∫JjlSl(cmil−cmjl+δ(λil−vjl(ξ)))dξ+αlNh∑j=1∫JjhH(cmil−cmjh+δ(vil(z)−vjh(ξ)))−H(cmil−cmjh−δvjh(ξ))dξ−δvil(z)αlNh∑j=1∫JjhK(cmil−cmjh+δ(λil−vjh(ξ)))dξ],z∈Jil, | (37) |
we have that (30) reduces to the equation
∫JjlGl(cmil−cmjl+δ(vil(z)−vjl(ξ)))−Gl(cmil−cmjl−δvjl(ξ))dξ=[Sl(cmil−cmjl)δvil(z)+δ22S′l(cmil−cmjl)(vil(z))2+δ36S″l(cmil−cmjl)(vil(z))3]|Jjl|+∫Jjlδ32S″l(cmil−cmjl)(vjl(ξ))2vil(z)dξ+R(S‴l,δ4), | (38) |
where we used the fact that
−δvil(z)∫JjlSl(cmil−cmjl+δ(λil−vjl(ξ)))dξ=[−Sl(cmil−cmjl)δvil(z)−δ2S′l(cmil−cmjl)λilvil(z)−δ32S″l(cmil−cmjl)(λil)2vil(z)]|Jjl|−∫Jjlδ32S″l(cmil−cmjl)(vjl(ξ))2vil(z)dξ+R(S‴l,δ4). | (39) |
Summing up the contributions in (38) to (39), we get that the self-interaction part in (37) reduces to
δ3[δ−12S′l(cmil−cmjl)vil(z)(vil(z)−2λil)+16S″l(cmil−cmjl)((vil(z))3−3vil(z)(λil)2)]|Jjl|+R(S‴l,δ4). | (40) |
Similarly, for the cross-interaction part we obtain
δ3[δ−12K′(cmil−cmjh)vil(z)(vil(z)−2λil)+16K″(cmil−cmjh)((vil(z))3−3vil(z)(λil)2)]|Jjh|+R(K‴,δ4). | (41) |
Putting together the contributions of (40) and (41) in the functional equation (37), we get
Fil[p;δ](z)=(ˉzil−z)+Dil6(3vil(z)(λil)2−(vil(z))3)+δ−1Bil2vil(z)(vil(z)−2λil)+R(S‴l,K‴,δ4), | (42) |
where we used the notations introduced in (6) and (5), namely
Dil=−Nl∑j=1S″l(cmil−cmjl)|Jjl|−αlNh∑j=1K″(cmil−cmjh)|Jjh|, |
and
Bil=Nl∑j=1S′l(cmil−cmjl)|Jjl|+αlNh∑j=1K′(cmil−cmjh)|Jjh|. |
Note that since the values
(ˉzil−z)+Dil6(3vil(z)(λil)2−(vil(z))3)=0,z∈Jil, | (43) |
that gives a unique solution once the value of
Λil[p;δ]=Fil[p;δ](˜zil). | (44) |
Performing Taylor expansions similar to the ones in (38) and (39) we get that
Λil[p;0]=(ˉzil−˜zil)+Dil3(λil)3, | (45) |
and we are now in the position to solve
{(ˉzil−z)+Dil6(3vil(z)(λil)2−(vil(z))3)=0,(ˉzil−˜zil)+Dil3(λil)3=0. | (46) |
The second equation in (46) admits a solution once we have that
λil=(3(˜zil−ˉzil)Dil)1/3. | (47) |
By construction
ˉρi(x)=Diρ2((λiρ)2−(x−cmiρ)2)1Iiρ(x),i=1,…,Nρ,ˉηh(x)=Dhη2((λhη)2−(x−cmhη)2)1Ihη(x),h=1,…,Nη, | (48) |
where the intervals
likk=cmikk−λikk,rikk=cmikk+λikk,ik=1,…,Nk,k=ρ,η. |
We are now ready to reformulate (36) as a functional equation on a proper Banach space. Consider the spaces
Ωil={v∈L∞([ˉzil,˜zil))|vincreasing,v(ˉzil)=0},i=1,…,Nl,l∈{ρ,η}, | (49) |
endowed with the
Ωl=Nl×i=1Ωil,forl∈{ρ,η}. |
We now introduce the space
Ω=Ωρ×RNρ×Ωη×RNη, | (50) |
with elements
|||ω|||=Nρ∑i=1(‖viρ‖L∞+|λiρ|)+Nη∑i=1(‖viη‖L∞+|λiη|). | (51) |
For
|||ω|||γ=|||ω|||+∑l∈{ρ,η}Nl∑i=1supz∈˜Jil|λil−vil(z)|(˜zil−z)γ, | (52) |
and set
T[ω;δ](z):=(Fρ[ω;δ](z)Λρ[ω;δ]Fη[ω;δ](z)Λη[ω;δ]), | (53) |
where for
Fl[ω;δ](z)=(F1l[ω;δ](z),…,FNll[ω;δ](z)),Λl[ω;δ](z):=(Λ1l[ω;δ],…,ΛNll[ω;δ]). | (54) |
The operator
DT[ω;δ]=(DvρFρ(δ)DλρFρ(δ)DvηFρ(δ)DληFρ(δ)DvρΛρ(δ)DλρΛρ(δ)DvηΛρ(δ)DληΛρ(δ)DvρFη(δ)DλρFη(δ)DvηFη(δ)DληFη(δ)DvρΛη(δ)DλρΛη(δ)DvηΛη(δ)DληΛη(δ)), | (55) |
where the components are actually matrices defined by
DvhFl(δ)=(∂Fil[ω;δ]∂vjh(νjh))Nl,Nhi,j=1,DλhFl(δ)=(∂Fil[ω;δ]∂λjh(ajh))Nl,Nhi,j=1DvhΛl(δ)=(∂Λil[ω;δ]∂vjh(νjh))Nl,Nhi,j=1,DλhΛl(δ)=(∂Λil[ω;δ]∂λjh(ajh))Nl,Nhi,j=1, |
where
∂Fil[ω;δ]∂vil(νil)=−δ−2∫Jilνil(ξ)[Sl(δ(vil(z)−vil(ξ)))−δvil(z)S′l(δ(λil−vil(ξ)))−Sl(−δvil(ξ))]dξ+δ−2νil(z)Nl∑j=1∫Jjl[Sl(cmil−cmjl+δ(vil(z)−vjl(ξ)))−Sl(cmil−cmjl+δ(λil−vjl(ξ)))]dξ+δ−2νil(z)αlNh∑j=1∫Jjh[K(cmil−cmjh+δ(vil(z)−vjh(ξ)))−K(cmil−cmjh+δ(λil−vjh(ξ)))]dξ. |
A Taylor expansion around
∂Fil[ω;0]∂vil(νil)=Dil2((λil)2−(vil(z))2)νil(z). |
Concerning the other terms in
∂Fil[ω;δ]∂vjl(νjl)=−δ−2Nl∑j=1∫Jjlνjl(ξ)[Sl(cmil−cmjl+δ(vil(z)−vjl(ξ)))−Sl((cmil−cmjl−δvjl(ξ))−δvil(z)S′l(cmil−cmjl+δ(λil−vjl(ξ)))]dξ, |
that all vanish in the limit
∂Fil[ω;δ]∂λil(ail)=−δ−1vil(z)ail[Nl∑j=1∫JjlS′l(cmil−cmjl+δ(λil−vjl(ξ)))dξ+αlNh∑j=1∫JjhK′(cmil−cmjh+δ(λil−vjh(ξ)))dξ]. |
Then, Taylor expansion w.r.t.
∂Fil[ω;0]∂λil(ail)=Dilλilvil(z)ail. |
Since all the entrances in the matrix
∂Fil[ω;δ]∂vjh(νjh)=−δ−2αlNh∑j=1∫Jjhνjh(ξ)[K(cmil−cmjh+δ(vil(z)−vjh(ξ)))−K(cmil−cmjh−δvjh(ξ))−δvil(z)K′(cmil−cmjh+δ(λil−vjh(ξ)))]dξ, |
that vanish in the limit
∂Λil[ω;δ]∂vjl(νjl)=−δ−2Nl∑j=1∫Jjlνjl(ξ)[Sl(cmil−cmjl+δ(λil(z)−vjl(ξ)))−Sl((cmil−cmjl−δvjl(ξ))−δλilS′l(cmil−cmjl+δ(λil−vjl(ξ)))]dξ. |
The terms in
∂Λil[ω;δ]∂vjh(νjh)=−δ−2αlNh∑j=1∫Jjhνjh(ξ)[K(cmil−cmjh+δ(λil−vjh(ξ)))−K(cmil−cmjh−δvjh(ξ))−δλilK′(cmil−cmjh+δ(λil−vjh(ξ)))]dξ, |
and the usual Taylor expansions around
∂Λil[ω;δ]∂λil(ail)=−δ−1ailNl∑j=1∫JjlλilS′l(cmil−cmjl+δ(λil−vjl(ξ)))dξ−δ−1ailαlNh∑j=1∫JjhλilK′l(cmil−cmjh+δ(λil−vjh(ξ)))dξ. |
The last Taylor expansion gives
∂Λil[ω;0]∂λil(ail)=Dil(λil)2ail. |
We have proved that
DT[ω;0]=(El1dg(Diρλiρviρaiρ)000dg(Diρ(λiρ)2aiρ)0000El2dg(Diηλiηviηaiη)000dg(Diη(λiη)2aiη)), | (56) |
where
El1=dg(Diρ2((λiρ)2−(viρ)2)νiρ),El2=dg(Diη2((λiη)2−(viη)2)νiη), |
and
Lemma 3.1. For
Proof. Thanks to the previous computations it is easy to see that
sup|||ω|||1/2≤11(˜zil−z)|∂Fil∂vil[⋅,δ](νil)−∂Λil∂vil[⋅;δ](νil)−(∂Fil∂vil[⋅;0](νil)−∂Λil∂vil[⋅;0](νil))+∂Fil∂λil[⋅,δ](ail)−∂Λil∂λil[⋅;δ](ail)−(∂Fil∂λil[⋅;0](ail)−∂Λil∂λil[⋅;0](ail))+∂Fil∂vjh[⋅,δ](νjh)−∂Λil∂vjh[⋅;δ](νjh)−(∂Fil∂vjh[⋅;0](νjh)−∂Λil∂vjh[⋅;0](νjh))+∂Fil∂λjh[⋅,δ](ajh)−∂Λil∂λjh[⋅;δ](ajh)−(∂Fil∂λjh[⋅;0](ajh)−∂Λil∂λjh[⋅;0](ajh))|↘0, | (57) |
as
1(˜zil−z)[∂Fil∂vjh[⋅,δ](νjh)−∂Λil∂vjh[⋅;δ](νjh)−(∂Fil∂vjh[⋅;0](νjh)−∂Λil∂vjh[⋅;0](νjh))]=−αl(vil(z)−λil)2˜zil−zNh∑j=1∫JjhK″(cmil−cmjh+δ(λil−vjh(ξ)))νjh(ξ)2dξ−δαl(vil(z)−λil)3˜zil−zNh∑j=1∫JjhK‴(˜x(ξ))νjh(ξ)6dξ=−δαl(vil(z)−λil)2˜zil−zNh∑j=1∫JjhK‴(ˉx(ξ))(λil−vjh(ξ))νjh(ξ)2dξ−δαl(vil(z)−λil)3˜zil−zNh∑j=1∫JjhK‴(˜x(ξ))νjh(ξ)6dξ.=−αl((vil(z)−λil)2˜zil−z+(vil(z)−λil)3˜zil−z)O(δ), |
where in the first equality we did a Taylor expansion around the point
1(˜zil−z)[∂Fil∂λjh[⋅,δ](ajh)−∂Λil∂λjh[⋅;δ](ajh)−(∂Fil∂λjh[⋅;0](ajh)−∂Λil∂λjh[⋅;0](ajh))]=0. |
The first two rows in (57) can be treated as follows,
1(˜zil−z)[∂Fil∂vil[⋅,δ](νil)−∂Λil∂vil[⋅;δ](νil)−(∂Fil∂vil[⋅;0](νil)−∂Λil∂vil[⋅;0](νil))+∂Fil∂λil[⋅,δ](ail)−∂Λil∂λil[⋅;δ](ail)−(∂Fil∂λil[⋅;0](ail)−∂Λil∂λil[⋅;0](ail))]=δ(˜zil−z)[Nl∑j=1∫Jjlδ−2(vil(z)−λil)(νil(z)−ail)S′l(cmil−cmjl+δ(λil−vjl(ξ)))+δ−112(vil(z)−λil)2(νil(z)−νjl(ξ))S″l(cmil−cmjl+δ(λil−vjl(ξ)))+16(vil(z)−λil)3(νil(z)−νjl(ξ))S‴l(˜x1(ξ))dξ+αlNh∑j=1∫Jjhδ−2(vil(z)−λil)(νil(z)−ail)K′(cmil−cmjh+δ(λil−vjh(ξ)))+δ−112(vil(z)−λil)2(νil(z)−νjl(ξ))K″(cmil−cmjh+δ(λil−vjh(ξ)))+16(vil(z)−λil)3(νil(z)−νjl(ξ))K‴(˜x2(ξ))dξ]=(2(νil(z)−ail)(vil(z)−λil)(˜zil−z)+(2νil(z)−1)((vil(z)−λil)3(˜zil−z)+(vil(z)−λil)2(˜zil−z)))O(δ). |
Since the functions
\frac{\lambda_l^i-v_l^i(z)}{(\tilde{z}_l^i - z)^{1/2}}, |
are uniformly bounded in
Lemma 3.2. For any
Proof. Given
\begin{equation} D\mathcal{T}[\omega_0;0] \omega = w, \end{equation} | (58) |
admits a unique solution
\begin{equation*} ||\omega||_{1/2}\leq C ||w||_{1}. \end{equation*} |
The determinant of the matrix in (56) is given by
\begin{equation} \det D\mathcal{T} = \left(\prod\limits_{i = 1}^{N_{\rho}}\frac{(D_{\rho}^i)^2}{2} \Big( (\lambda_{\rho}^i)^2 - (v_{\rho}^i)^2 \Big) (\lambda_{\rho}^i)^2 \right) \cdot \left(\prod\limits_{i = 1}^{N_{\eta}}\frac{(D_{\eta}^i)^2}{2} \Big( (\lambda_{\eta}^i)^2 - (v_{\eta}^i)^2 \Big) (\lambda_{\eta}^i)^2 \right) , \end{equation} | (59) |
that is always different from zero under the condition
\begin{equation*} \begin{aligned} \nu_l^i(z) & = \frac{- 2 \sigma_l^i(z)}{D_l^i \big( (v_l^i(z))^2 - (\lambda_l^i)^2 \big)} + \frac{2 \lambda_l^i v_l^i(z) a_l^i}{ (v_l^i(z))^2 - (\lambda_l^i)^2 }, \\ a_l^i & = \frac{k_l^i}{D_l^i(\lambda_l^i)^2}, \end{aligned} \end{equation*} |
that implies
\begin{equation*} \frac{a_l^i-\nu_l^i(z)}{(\tilde{z}_l^i - z)^{1/2}} , \end{equation*} |
is uniformly bounded since
We are now in the position of proving the main result of the paper, namely Theorem 1.1, that we recall below for convenience.
Theorem 3.1. Assume that
(i)
\begin{equation} B_{l}^i = \sum\limits_{j = 1}^{N_l} S'_l( cm_l^i - cm_l^j ) z_l^j+ \alpha_l\sum\limits_{j = 1}^{N_h} K'( cm_l^i - cm_h^j ) z_h^j = 0, \end{equation} | (60) |
(ii) the following quantities
\begin{equation} D_{l}^i = - \sum\limits_{j = 1}^{N_l} S''_l(cm_l^i - cm_l^j) z_l^j -\alpha_l \sum\limits_{j = 1}^{N_h} K''(cm_l^i - cm_h^j ) z_h^j, \end{equation} | (61) |
are strictly positive, for all
where
where
● each interval
●
● the solutions
z_\rho = \sum\limits_{i = 1}^{N_\rho} z_\rho^i \;\mathit{and}\; z_\eta = \sum\limits_{i = 1}^{N_\eta} z_\eta^i, |
respectively.
Proof. Consider
\mathcal{T}[\omega;\delta](z) = 0, |
admits a unique solution
\omega = ( v_\rho^1(z),\ldots,v_\rho^{N_\rho}(z),\lambda_\rho^1,\ldots,\lambda_\rho^{N_\rho},v_\eta^1(z),\ldots,v_\eta^{N_\eta}(z),\lambda_\eta^1,\ldots,\lambda_\eta^{N_\eta} )\,, |
for
Remark 3.1 Note that conditions (ii) in Theorem 3.1 turn to be conditions on the positions of the centres of masses and on the value of
\alpha < \min\limits_{i = 1,\ldots,N_\eta}\frac{ \sum\limits_{j = 1}^{N_\eta} S''_\eta(cm_\eta^i - cm_\eta^j) |J_\eta^j|}{ \sum\limits_{j = 1}^{N_\rho} K''(cm_\eta^i - cm_\rho^j ) |J_\rho^j| }. |
Note that the above conditions are comparable to the one we got in the proof of Section 2 using the Krein-Rutman approach.
In this section, we study numerically solutions to system (3) using two different methods, the finite volume method introduced in [12, 13] and the particles method studied in [23, 28]. We validate the results about the existence of the mixed steady state and the multiple bumps steady states. Moreover, we perform some examples to show the variation in the behaviour of the solution to system (3) under different choices of initial data and the parameter
\begin{equation} \begin{cases} \partial_t u_{\rho}(z) = &-\frac{\theta}{2}\partial_{z}\Big(\big(\partial_{z}u_{\rho}(z)\big)^{-2}\Big) + \int_{J_{\rho}}S_{\rho}^{\prime}\big(u_{\rho}(z)-u_{\rho}(\xi)\big)d\xi \\ &+ \alpha_{\rho}\int_{J_{\eta}}K^{\prime}\big(u_{\rho}(z)-u_{\eta}(\xi)\big)d\xi,\; \; z\in J_{\rho} \\ \partial_t u_{\eta}(z) = &-\frac{\theta}{2}\partial_{z}\Big(\big(\partial_{z}u_{\eta}(z)\big)^{-2}\Big) + \int_{J_{\eta}}S_{\eta}^{\prime}\big(u_{\eta}(z)-u_{\eta}(\xi)\big)d\xi \\ &+ \alpha_{\eta}\int_{J_{\rho}}K^{\prime}\big(u_{\eta}(z)-u_{\rho}(\xi)\big)d\xi,\; \; z\in J_{\eta}, \end{cases} \end{equation} | (62) |
as the following: Let
\begin{equation} \begin{cases} \partial_t X_{\rho}^i(t) = & \frac{\theta}{2N} \Big(\big(\rho^{i-1}(t)\big)^{-2} - \big(\rho^{i}(t)\big)^{-2} \Big) + \frac{1}{N}\sum\limits_{j = 1}^N S_{\rho}^{\prime}\big(X_{\rho}^i(t)-X_{\rho}^j(t)\big)d\xi \\ & + \frac{\alpha_{\rho}}{N}\sum\limits_{j = 1}^N K^{\prime}\big(X_{\rho}^i(t)-X_{\eta}^j(t)\big)d\xi, \qquad i = 1,\cdots N, \\ \partial_t X_{\eta}^i(t) = & \frac{\theta}{2N} \Big(\big(\eta^{i-1}(t)\big)^{-2} - \big(\eta^{i}(t)\big)^{-2} \Big) + \frac{1}{N}\sum\limits_{j = 1}^N S_{\eta}^{\prime}\big(X_{\eta}^i(t)-X_{\eta}^j(t)\big)d\xi \\ & + \frac{\alpha_{\eta}}{N}\sum\limits_{j = 1}^N K^{\prime}\big(X_{\eta}^i(t)-X_{\rho}^j(t)\big)d\xi, \qquad i = 1,\cdots N, \end{cases} \end{equation} | (63) |
where the densities are reconstructed as
\begin{equation} \begin{cases} \rho^i(t) = \frac{1}{N ( X_l^{i+1}(t) - X_l^i(t) )}, \qquad i = 1,\cdots, N-1, \\ \rho^0(t) = 0, \\ \rho^N(t) = 0, \end{cases} \end{equation} | (64) |
and the same for the
\int_{X_{\rho,0}^i}^{X_{\rho,0}^{i+1}} \rho(t = 0) dX = \frac{1}{N-1}, \qquad i = 1,\cdots,N-1. |
The second method we use is the finite volume method which introduced in [12] and extended to systems in [13], that consists in a
\begin{equation*} \widetilde{\rho}_{i}(t) : = \frac{1}{\Delta x}\int_{U_{i}}\rho(x,t)dx,\qquad \widetilde{\eta}_{i}(t) : = \frac{1}{\Delta x}\int_{U_{i}}\eta(x,t)dx, \end{equation*} |
the averages of the solutions
\begin{equation} \begin{cases} \frac{d\widetilde{\rho}_{i}(t)}{dt} = -\frac{F_{i+\frac{1}{2}}^{\rho}(t) - F_{i-\frac{1}{2}}^{\rho}(t) }{\Delta x}, \\ \frac{d\widetilde{\eta}_{i}(t)}{dt} = -\frac{F_{i+\frac{1}{2}}^{\eta}(t) - F_{i-\frac{1}{2}}^{\eta}(t) }{\Delta x}, \end{cases} \end{equation} | (65) |
where the numerical flux
\begin{equation} F_{i+\frac{1}{2}}^{\rho} = \max (\vartheta_{\rho}^{i+1},0)\Big[ \widetilde{\rho}_{i} + \frac{\Delta x}{2}(\rho_{x})_{i} \Big] + \min (\vartheta_{\rho}^{i+1},0)\Big[ \widetilde{\rho}_{i} - \frac{\Delta x}{2}(\rho_{x})^{i} \Big], \end{equation} | (66) |
where
\begin{equation} \begin{aligned} \vartheta_{\rho}^{i+1} = &- \frac{\theta}{\Delta x} \big( \widetilde{\rho}_{i+1} - \widetilde{\rho}_{i} \big) - \sum\limits_{j}\widetilde{\rho}_{j}\big( S_{\rho}(x_{i+1} - x_{j}) - S_{\rho}(x_{i} - x_{j}) \big) \\ &- \alpha_{\rho} \sum\limits_{j}\widetilde{\eta}_{j}\big( K(x_{i+1} - x_{j}) - K(x_{i} - x_{j}) \big), \end{aligned} \end{equation} | (67) |
and
\begin{equation} (\rho_{x})^{i} = {\rm{minmod}} \Bigg( 2\frac{\widetilde{\rho}_{i+1} - \widetilde{\rho}_{i}}{\Delta x},\; \frac{\widetilde{\rho}_{i+1} - \widetilde{\rho}_{i-1}}{2\Delta x},\; 2\frac{\widetilde{\rho}_{i} - \widetilde{\rho}_{i-1}}{\Delta x} \Bigg). \end{equation} | (68) |
The minmod limiter in (68) has the following definition
\begin{equation} {\rm{minmod}}(a_{1}, a_{2},\; \dots) : = \begin{cases} \min (a_{1}, a_{2},\; \dots), \quad {\rm{if}} \; a_{i} > 0\quad \forall i \\ \max (a_{1}, a_{2},\; \dots), \quad {\rm{if}} \; a_{i} < 0\quad \forall i \\ 0, \qquad\qquad\qquad{\rm{otherwise.}} \end{cases} \end{equation} | (69) |
We have the same as the above expresions for
In all the simulations below, we will fix the kernels as a normalised Gaussian potentials
S_{\rho}(x) = S_{\eta}(x) = K(x) = \frac{1}{\sqrt{\pi}} e^{-x^2}, |
that are under the assumptions on the kernels (A1), (A2) and (A3). This choice helps us in better understanding the variation in the behavior of the solutions w.r.t. the change in the initial data and the parameter
This figure shows how from the initial densities
● in the first row steady states are plotted at the level of density, on the l.h.s. we compare the two methods illustrated above, while on the r.h.s. we show the evolution by the finite volume method,
● in the second row we plot the particles paths for both species obtained with the particles method,
● in the last row we show the pseudo inverse functions corresponding to the steady state densities.
The last example we present shows an interesting traveling waves-type evolution.
The first example is devoted to validating existence of mixed steady state and separated steady state. By choosing the initial data
(70) |
and fixing
(71) |
with the same
We then, test two cases where we validate the existence of steady states of more bumps. Let us fix
(72) |
as initial data and we take
(73) |
and
Finally, in this example, we detect existence of traveling waves. Indeed, by choosing initial data as
(74) |
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