AN HYPERBOLIC-PARABOLIC PREDATOR-PREY MODEL INVOLVING A VOLE POPULATION STRUCTURED IN AGE

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AN HYPERBOLIC-PARABOLIC PREDATOR-PREY MODEL INVOLVING A VOLE POPULATION

STRUCTURED IN AGE

Giuseppe Maria Coclite, Carlotta Donadello, Thi Nhu Thao Nguyen

To cite this version:

Giuseppe Maria Coclite, Carlotta Donadello, Thi Nhu Thao Nguyen. AN HYPERBOLIC- PARABOLIC PREDATOR-PREY MODEL INVOLVING A VOLE POPULATION STRUCTURED IN AGE. 2021. �hal-02936776v2�

AN HYPERBOLIC-PARABOLIC PREDATOR-PREY MODEL INVOLVING A VOLE POPULATION STRUCTURED IN AGE

G. M. COCLITE, C. DONADELLO, AND T. N. T. NGUYEN

Abstract. We prove existence and stability of entropy solutions for a predator-prey system consisting of an hyperbolic equation for predators and a parabolic-hyperbolic equation for preys. The preys’ equation, which represents the evolution of a population of voles as in [2], depends on time,t, age,a, and on a 2-dimensional space variable x, and it is supplemented by a nonlocal boundary condition ata= 0. The drift term in the predators’ equation depends nonlocally on the density of preys and the two equations are also coupled via classical source terms of Lotka-Volterra type, as in [4]. We establish existence of solutions by applying the vanishing viscosity method, and we prove stability by a doubling of variables type argument.

1. Introduction

sec:introduction

1.1. The model and the assumptions. Our goal in this paper is to investigate the wellposedness of a predator-prey model extending the model for a vole population structured in age we introduced in [2]. To this end we couple the latter equation to the hyperbolic equation for predators proposed in [4] in which the drift depends nonlocally on the density of preys, so that the predators tend to move toward the regions in which preys are more abundant. The system we consider writes as follows

eq:model_new

eq:model_new (1.1)



























∂_tu+ div_x(uν(φ)) = (b(φ)−β)u, (t, x)∈(0, T)×R²,

∂tρ+∂aρ+ divx(ρχ1(a)v(x)Yθ(φ−R)) =µ∆xρ−d(t, a, x)ρ−p(a, u)ρ, (t, a, x)∈(0, T)²×R², ρ(t,0, x) =A(φ)

Z ∞ 0

ρ(t, a, x)χ₃(a) da

ω(t, x), (t, x)∈(0, T)×R²,

ρ(0, a, x) =ρ₀(a, x), (a, x)∈(0, T)×R²,

u(0, x) =u₀(x), x∈R²,

where u = u(t, x) and φ = φ(t, x) represent the respective density of predators and preys at (t, x).

Since the prey population is also structured on age its dynamics is better described by ρ=ρ(t, a, x), which is the density of preys of age aat (t, x). More precisely, the relation between φ and ρ is given by

eq:defnu

eq:defnu (1.2) φ(t, x) =

Z ∞ 0

ρ(t, a, x)χ2(a) da,

whereχ2(a) is an approximation of the indicator function of the interval (σ, T), whereT is the target time of our observation and 0 < σ 1. The parameter σ does not play a role in the modeling, but allows to avoid technical difficulties in our analysis. In the first equation, the functionb(φ) represents the reproduction rate of predators depending on preys’ availability, while β >0, the predators’ mor- tality rate, is assumed to be constant. As in [4] the flux ofuis driven to the direction of higher preys’

Date: February 19, 2021.

2020Mathematics Subject Classification. 35Q92, 35M33.

Key words and phrases. population dynamics, predator-prey systems, parabolic-hyperbolic equations, nonlocal conservation laws, nonlocal boundary value problem.

1

concentration by a nonlinear, nonlocal velocity ν of the form ν(φ) =κ ∇(φ∗η)

q

1 +k∇(φ∗η)k² ,

whereκ >0 is the maximal speed of predators and η is a positive smooth mollifier with R

R²ηdx= 1 so that the convolution (φ(t)∗η)(x) represents an average of the density of preys in a neighborhood of xat time t.

The equation for the preys, introduced in [2], is related to classical models for the dynamics of a population structured in age, see [3, 10, 13], but the choice of the coefficients and boundary conditions at a= 0 takes into account the data collections and ecological considerations in [1, 5, 6, 8, 12]. We recall here the essential assumptions on the form of the coefficients.

We introduce constants 0< A₁ < A₂so that a vole is young (baby) if its ageais in (0, A₁), juvenile if its age is in (A₁, A₂) and adult otherwise. The three age classes differ as babies do not reproduce, adults’ mortality rate is lower and juveniles exhibit a significant spatial dynamic during dispersals.

Dispersal is a characteristic phenomenon of vole populations, correlated to overcrowding. Whenever the density of voles φ rises above a threshold value R >0, representing a fraction of the capacity of the environment, the juvenile individuals leave their original colony and disperse over relatively large distances (0.5 to 5 km) with velocity v(x). We fix θ > 0 and we consider an approximation of the Heaviside function, Y_θ, defined as

Y(ξ) =







1, ifξ ≥0, 0, ifξ ≤ −1,

Y⁰(ξ)≥0, Y_θ(ξ) =Y ξ

θ

.

From Y_θ we costruct the approximations of the indicator functions of the intervals (σ, T), (A1, A2), and (A₁, T)

χ₁(a) =Y_θ(a−A₁)Y_θ(A₂−a), χ₂(a) =Y_θ(a−σ)Y_θ(T −a), χ₃(a) =Y_θ(a−A₁)Y_θ(T −a).

The mortality rate of voles splits into two terms: p = p(a, u) represents the mortality due to the presence of the specific predator whose density isu, whiled=d(t, a, x) stands for all other casualties (sickness, starvation, generic predation, etc).

The second-order termµ∆_xρrepresents short range spatial dynamics related to foraging activities.

Everywhere in the followingθ and µ >0 are fixed.

In the boundary condition at a = 0, the function ω = ω(t, x) is the reproduction rate of voles depending on time and position. Both these parameters are significant here, as the beginning and the end of the reproduction season are strongly connected to the average temperature over one week, see [7] and references therein. The function A(φ) describes the influence of the total density of voles on natality. Examples of non constant Aare functions of the form

def.Hallee.function

def.Hallee.function (1.3) A(φ) = αφ^γ

(β+φ)^γ, for different choices ofα,β andγ.

The fourth and fifth equations are the respective initial conditions att= 0 for voles and predators.

assumptions

1.2. Assumptions. The coefficientsb,v,d,p,A, ω and the initial data ρ0, u0 of system (1.1) satisfy the following conditions:

b∈C^∞(R)∩W^1,∞(R), b(·)≥0,

ass:b

ass:b (1.4)

v∈C^∞(R²)∩L²(R²)∩L^∞(R²), div_x(v)∈L¹(R²)∩W^2,∞(R²), v>0,

ass:v

ass:v (1.5)

d∈C^∞([0,∞)×[0,∞)×R²)∩W^2,∞((0,∞)×(0,∞)×R²), 0< d∗ ≤d(·,·,·)≤d^∗,

ass:d

ass:d (1.6)

p∈C^∞([0,∞)×R)∩W^2,∞((0,∞)×R), 0< p∗ ≤p(·,·)≤p^∗,

ass:p

ass:p (1.7)

A ∈C^∞(R)∩L^∞(R), A(·)≥0, A(0) = 0, |A⁰(ξ)ξ|,|A⁰⁰(ξ)ξ| ≤C0,

ass:A

ass:A (1.8)

ω ∈C^∞([0,∞)×R²)∩W^1,∞((0,∞)×R²), ω(·,·)≥0,

ass:w

ass:w (1.9)

ρ₀∈L¹((0,∞)×R²)∩L^∞((0,∞)×R²), ρ₀ ≥0,

ass:init.1

ass:init.1 (1.10)

sup

x∈R²

kρ₀(·, x)k_L1(0,∞),sup

a>0

kρ₀(a,·)k_L1(R²), Z

R²

T V(ρ₀(·, x))dx≤C₀,

ass:init.2

ass:init.2 (1.11)

u₀∈L¹(R²)∩BV(R²)∩L^∞(R²), u₀≥0,

ass:init.3

ass:init.3 (1.12)

for some positive constantsd∗,d^∗,p∗,p^∗,C0.

For what the velocity of predators,ν, is concerned, we assume that the mollifierη satisfies

eq:eta_assumption

eq:eta_assumption (1.13) ∇_xη ∈(C²∩W^2,2∩W^1,∞)(R²,R²).

1.3. Main result. Our main result is the wellposedness of entropy weak solutions for system (1.1), stated in Theorem 1.1. We adopt the following definitions of weak solution and entropy solution.

def:weaksolution Definition 1.1. We say that the pair(u, ρ)is a weak solution of (1.1)if the following holds for every T >0.

def:1 (D.1) ρ≥0, ρ∈L^∞(0, T;L¹((0,∞)×R²))∩L^∞((0, T)×(0,∞)×R²)∩L²((0, T)×(0,∞);H²(R²)).

def:2 (D.2) u∈L¹((0, T)×R²)∩BV((0, T)×R²).

def:3 (D.3) For almost every (t, x)∈(0, T)×R², ρ(t,·, x)∈BV(0,∞) and ρ(t,0⁺, x) =A(φ)

Z ∞ 0

ρ(t, a, x)χ₃(a)da

ω(t, x),

where ρ(t,0⁺, x) is the trace of ρ(t,·, x) at a= 0.

def:4 (D.4) For every test function ξ ∈C_c^∞(R⁴) Z ∞

0

Z ∞ 0

Z

R²

(ρ∂tξ+ρ∂aξ+ρχ1(a)v· ∇_xξYθ(φ−R) +µρ∆xξ−dρξ−pρξ)dxdadt +

Z ∞ 0

Z ∞ 0

Z

R²

A(φ)ρ(t, a, x)χ3(a)ω(t, x)ξ(t,0, x)dxdadt +

Z ∞ 0

Z

R²

ρ₀(a, x)ξ(0, a, x)dxda= 0, Z ∞

0

Z ∞ 0

Z

R²

(u∂tξ+uν(φ)· ∇_xξ+ (b(φ)−β)uξ)dxdadt+ Z ∞

0

Z

R²

u0(x)ξ(0, a, x)dxda= 0.

def:entropyweaksolution Definition 1.2. We say that a weak solution (u, ρ) is an entropy solution of (1.1) if for any non- negative test function ξ∈C^∞(R⁴) with compact support and for any constant c∈Rthere hold

Z ∞ 0

Z ∞ 0

Z

R²

(|ρ−c|(∂tξ+∂aξ)−divx(|ρ−c|χ₁vYθ(φ−R))ξ +µ∆x|ρ−c|ξ−sign (ρ−c) (d+p)ρξ) dxdadt +

Z ∞ 0

Z

R²

|ρ(t,0⁺, x)−c|ξ(t,0, x)dxdt +

Z ∞ 0

Z

R²

|ρ₀(a, x)−c|ξ(0, a, x)dxda

≥ Z ∞

0

Z ∞ 0

Z

R²

csign (ρ−c)χ₁(a)div_x(v(x)Y_θ(φ−R))ξdxdadt

eq:entropy_rho

eq:entropy_rho (1.14)

and

Z ∞ 0

Z ∞ 0

Z

R²

(|u−c|∂_tξ+|u−c|ν(φ)· ∇_xξ+ sign (u−c) (b(φ)−β)uξ)dxdadt +

Z ∞ 0

Z

R²

|u₀(x)−c|ξ(0, a, x)dxda≥ Z ∞

0

Z ∞ 0

Z

R²

csign (u−c) div_x(ν(φ))ξdxdadt.

eq:entropy_u

eq:entropy_u (1.15)

thr:main Theorem 1.1. Assume (1.4)-(2.4), then the initial boundary value problem (1.1) admits a unique entropy solution (u, ρ) in the sense of Definition 1.2. Moreover, if (u₁, ρ₁) and (u₂, ρ₂) are the two entropy solutions of (1.1) having initial data (u_1,0, ρ_1,0) and (u_2,0, ρ_2,0), then there exists a positive constant C such that the following estimate holds for almost every t≥0

ku₁(t,·)−u₂(t,·)k_L1(R²)+kρ₁(t,·,·)−ρ₂(t,·,·)k_L1((0,∞)×R²)

≤ Ce^CeCtkρ_1,0−ρ_2,0k_L1((0,∞)×

R²)+Ce^CeCtku_1,0−u_2,0k_L1((

R²).

eq:stab

eq:stab (1.16)

The fixed point argument used in [4] to prove existence and stability for a predator-prey system does not apply to our system in a straightforward way because it requires extremely fine information on the coefficients appearing in the a priori estimates for both predators’ and preys’ equations. This is not easy to obtain in our case, as the equation we use for preys comes from a specific population model and its analytical study is rather technical.

In Section 2 we introduce a sequence of parabolic approximations of problem (1.1), for which we prove suitable a priori estimates. Then we apply the compensated compactness lemma by Panov, see [11], to show the strong compactness of the sequence for voles, while the strong convergence of the sequence of predators is ensured by Helly’s theorem. Lemma 2.14 establishes the existence of an entropy solution in the sense of Definition 1.2. The uniqueness and stability of such solutions are proved in Section 3 using a doubling of variables type argument.

2. Existence

exist

The existence argument is based on the compactness analysis of a sequence of solutions to approx- imating problems defined as follows. For any given ε >0, we let (ρ_ε =ρ_ε(t, a, x), u_ε =u_ε(t, x)) be a

solution of the problem

eq:model_eps

eq:model_eps (2.1)































∂tuε+ divx(uεν(φε)) = (b(φε)−β)uε+ε∆xuε, (t, x)∈(0, T)×R²,

∂_tρ_ε+∂_aρ_ε+ div_x(ρ_εχ₁(a)v(x)Y_θ(φ_ε−R))

=µ∆_xρ_ε−d(t, a, x)ρ_ε−p(a, u_ε)ρ_ε+ε∂_a(χ(a)∂_aρ_ε), (t, a, x)∈(0, T)²×R², ρ_ε(t,0, x) =A(|φ_ε|)

Z ∞ 0

|ρ_ε(t, a, x)|χ₃(a)da

ω(t, x), (t, x)∈(0, T)×R², ρε(0, a, x) =ρ0,ε(a, x), (a, x)∈(0, T)²×R²,

u_ε(0, x) =u_0,ε(x), x∈R²,

where

φε(t, x) = Z ∞

0

ρε(t, a, x)χ2(a)da,

def:pe

def:pe (2.2)

χ(a) ∈ C^∞([0,+∞),[0,1]) satisfies χ(0) = 0, and {(ρ_0,ε, u0,ε)}_ε is a family of approximations of the initial condition (ρ0, u0) such that

eq:init-v-eps

eq:init-v-eps (2.3)

ρ_0,ε∈C^∞((0,∞)×R²), u_0,ε ∈C^∞(R²), ε >0, ρ0,ε→ρ0, a.e. and in L^p((0,∞)×R²),1≤p <∞, asε→0,

u_0,ε→u₀, a.e. and in L^p(R²),1≤p <∞,as ε→0,

ρ0,ε≥0,kρ_0,εk_L1((0,∞)×R²)≤C, ε≥0,

sup

x∈R²

kρ_0,ε(·, x)k_L1(R),sup

a≥0

kρ_0,ε(a,·)k_L1(R²),k∂_aρ0,εk_L1((0,∞)×R²)≤C, ε≥0, ku_0,εk_L1(R²),k∇_xu_0,εk_L1(R²), εk∆_xu_0,εk_L1(R²) ≤C, u_0,ε≥0, ε >0.

2.1. A priori estimates. In this section we establish the a priori estimates on (u_ε, ρ_ε) which are necessary to pass to the limit as ε→0 in (2.1).

From now on we use the notationC for all the positive constants independent onεappearing in the text or in the statements of our results, while in proofs we write c to indicate any positive constant non depending on ε, and c_T for quantities of the form ce^ct, t ∈ (0, T). The proof of the following preliminary Lemma is postposed to Section 4 (see also [4, Lemma 4.1]).

lem:eta Lemma 2.1. Let η be such that

eq:eta_assumption

eq:eta_assumption (2.4) ∇_xη ∈(C²∩W^2,2∩W^1,∞)(R²,R²).

Then the map ν :L¹(R²;R)7→C^∞(R)∩W^1,∞(R²;R²) satisfies kν(φ)k_L∞(R²;R²)≤Kkφk_L1(R²,R),

ass:nu1

ass:nu1 (2.5)

kdiv_x(ν(φ))k_L2(R²,R)≤Kkφk_L1(R²;R),

ass:nu2

ass:nu2 (2.6)

k∇_xν(φ)k_L∞(R²,R^2×2)≤Kkφk_L1(R²;R),

ass:nu3

ass:nu3 (2.7)

kν(φ₁)−ν(φ2)k_L∞(R²,R²) ≤Kkφ₁−φ2k_L1(R²,R),

ass:nu4

ass:nu4 (2.8)

k∇_x(divx(ν(φ)))k_L2(R²,R²)≤K

1 +Kkφk_L1(R²,R)

kφk_L1(R²,R),

ass:nu5

ass:nu5 (2.9)

kdiv_x(ν(φ1)−ν(φ2))k_L2(R²,R) ≤K

1 +Kkφ₂k_L1(R²,R)

kφ₁−φ2k_L1(R²,R),

ass:nu6

ass:nu6 (2.10)

where K is a positive constant.

Lemma 2.2 (Nonnegativity of ρε, φε, uε). We have that

lem:pos

lem:pos (2.11) ρ_ε≥0, φ_ε≥0, u_ε≥0.

Proof. Consider the function

x 7→η(x) =−x1(−∞,0)(x).

and observe that

η⁰(x) =−1(−∞,0)(x), η(x) =xη⁰(x).

From (2.1) we obtain d dt

Z ∞ 0

Z

R²

η(ρε)dxda= Z ∞

0

Z

R²

η⁰(ρε)∂tρεdxda

=− Z ∞

0

Z

R²

η⁰(ρ_ε)∂_aρ_εdxda− Z ∞

0

Z

R²

div_x(ρ_εχ₁vY_θ)η⁰(ρ_ε)dxda +µ

Z ∞ 0

Z

R²

η⁰(ρε)∆xρεdxda− Z ∞

0

Z

R²

η⁰(ρε)(d+p)ρεdxda +ε

Z ∞ 0

Z

R²

η⁰(ρε)∂a(χ(a)∂aρε)ρεdxda

= Z

R²

η(ρ_ε(t,0, x))

| {z }

=0

dx+ Z ∞

0

Z

R²

ρ_εχ₁(v· ∇_xρ_ε)Y_θη⁰⁰(ρ_ε)dxda

| {z }

=0

−µ Z ∞

0

Z

R²

η⁰⁰(ρε) (∇_xρε)²dxda

| {z }

≤0

− Z ∞

0

Z

R²

(d+p)η(ρε)dxda

| {z }

≤0

+ε Z ∞

0

Z

R²

∂_a η⁰(ρ_ε)χ(a)∂_aρ_ε

dxda−ε Z ∞

0

Z

R²

η⁰⁰(ρ_ε)χ(a)(∂_aρ_ε)²

| {z }

≤0

dxda

≤ −ε Z

R²

η⁰(ρ_ε(t,0, x))

| {z }

=0

χ(0)

| {z }

=0

∂_aρ_ε(t,0, x)dx= 0.

Thus, integrating on (0, t) we obtain η(ρε(t, a, x)) = 0 which implies thatρε≥0, and then φε ≥0.

From (2.1) we obtain d

dt Z

R²

η(uε)dx= Z

R²

η⁰(uε)∂tuεdx

=− Z

R²

η⁰(uε)divx(uεν(φε)) dx

| {z }

=0

+ Z

R²

η⁰(uε) (b(φ)−β)

| {z }

≤c

uεdx+ε Z

R²

η⁰(uε)∆xuεdx

| {z }

≤0

≤c Z

R²

η(uε)dx.

Integrating on (0, t) and applying Gronwall Lemma we obtainη(uε(t, x)) = 0, so thatuε≥0.

eq:model_eps_new Remark 2.1. We can remove the absolute value in the boundary condition for ρε in (2.1).

L1 of rho Lemma 2.3 (L¹ estimate on ρ_ε). For all t≥0, we have that kρ_ε(t,·,·)k_L1((0,∞)×R²) ≤e^CtC.

eq:re1

eq:re1 (2.12)

Proof. Due to the nonnegativity ofρε and the boundedness of d,pwe have d

dt Z ∞

0

Z

R²

|ρ_ε|dxda= d dt

Z ∞ 0

Z

R²

ρ_εdxda

=− Z ∞

0

Z

R²

∂aρεdxda− Z ∞

0

Z

R²

divx(ρεχ1vYθ) dxda+µ Z ∞

0

Z

R²

∆xρεdxda

| {z }

=0

− Z ∞

0

Z

R²

ρε(d+p)

| {z }

≤c

dxda+ε Z ∞

0

Z

R²

∂a(χ(a)∂aρε) dxda

| {z }

=0

≤ Z

R²

ρ_ε(t,0, x)dx+c Z ∞

0

Z

R²

ρ_εdxda

= Z

R²

A(φε) Z ∞

0

ρε(t, a, x)χ3(a)da

ω(t, x)dx+c Z ∞

0

Z

R²

ρεdxda

≤c Z ∞

0

Z

R²

ρεdxda.

Integrating on (0, t) we get (2.12) thanks to the Gronwall Lemma and the assumptions in (2.3) .

Lem:L2 rho Lemma 2.4 (L² estimate on ρ_ε). For any t≥0, we have kρ_ε(t,·,·)k_L2((0,∞)×R²),k∇_xρ_εk_L2((0,t)×(0,∞)×R²),√

εkχ∂_aρ_εk_L2((0,t)×(0,∞)×R²)≤e^CtC.

eq:re2

eq:re2 (2.13)

Proof. We recall the equation ofρε

∂_tρ_ε+∂_aρ_ε+ div_x(ρ_εχ₁(a)v(x)Y_θ(φ_ε−R))

=µ∆xρε−d(t, a, x)ρε−p(a, uε)ρε+ε∂a(χ(a)∂aρε).

eq:re

eq:re (2.14)

We multiply (2.14) by ρε and have d

dt Z ∞

0

Z

R²

ρ²_ε

2 dxda= Z ∞

0

Z

R²

ρ_ε∂_tρ_εdxda

=− Z ∞

0

Z

R²

ρε∂aρεdxda− Z ∞

0

Z

R²

ρεdivx(ρεχ1vY_θ) dxda+µ Z ∞

0

Z

R²

ρε∆xρεdxda

− Z ∞

0

Z

R²

ρ²_ε(d+p)

| {z }

≤c

dxda+ε Z ∞

0

Z

R²

ρ_ε∂_a(χ(a)∂_aρ_ε) dxda

≤ Z

R²

ρ_ε(t,0, x)²

2 dx− µ 2

Z ∞ 0

Z

R²

|∇_xρ_ε|²dxda+c Z ∞

0

Z

R²

(ρ_εχ₁vY_θ)²dxda +c

Z ∞ 0

Z

R²

ρ²_εdxda−ε Z ∞

0

Z

R²

χ(a) (∂_aρ_ε)²dxda

≤1 2

Z

R²

A²(φε) Z ∞

0

ρε(t, a, x)χ3(a)da 2

ω²(t, x)dx+c Z ∞

0

Z

R²

ρ²_εdxda

− µ 2

Z ∞ 0

Z

R²

|∇_xρε|²dxda−ε Z ∞

0

Z

R²

χ(a) (∂aρε)²dxda

≤c Z ∞

0

Z

R²

ρ²_εdxda−µ 2

Z ∞ 0

Z

R²

|∇_xρ_ε|²dxda−ε Z ∞

0

Z

R²

χ(a) (∂_aρ_ε)²dxda.

Integrating over (0, t) and using the Gronwall Lemma we gain kρ_ε(t,·,·)k²_L2((0,∞)×R²)+e^ctµ

Z t 0

Z ∞ 0

Z

R²

e^−cs|∇_xρε|²dxdads

+e^ct2ε Z t

0

Z ∞ 0

Z

R²

e^−csχ(a) (∂aρε)²dxdads≤e^ctkρ_0,ε(·,·)k²_L2((0,∞)×R²). We obtain (2.13) by using the fact that

kχ∂_aρ_εk²_L2((0,t)×(0,∞)×R²)≤ kχk_L∞(0,∞)

Z t 0

Z ∞ 0

Z

R²

χ(a)(∂_aρ_ε)²dxdads

≤ Z t

0

Z ∞ 0

Z

R²

χ(a)(∂aρε)²dxdads.

We consider the class of functions

def:psi

def:psi (2.15) ψ_ε(t, x) =

Z ∞ 0

ρ_ε(t, a, x)ξ(a)da, forξ ∈C_c^∞((0,∞)) such that

eq:suppxi

eq:suppxi (2.16) supp (ξ)⊂(0, T).

In particular any of the functionsχ_i,i= 1,2,3, can play the role ofξ, so that the estimates obtained for ψ_ε apply to φ_ε. Thanks to the definition of ψ_ε in (2.15) and the results in (2.12), (2.13), the following inequalities hold for every t≥0

kψ_ε(t,·)k_L1(R²) ≤e^CtC,

eq:nonloc1

eq:nonloc1 (2.17)

kψ_ε(t,·)k_L2(R²),k∇_xψ_εk_L2((0,t)×R²) ≤e^CtC.

eq:nonloc2

eq:nonloc2 (2.18)

H^2 estimate re Lemma 2.5 (H² estimate on ρε). For any t≥0, we have k∇_xρ_ε(t,·,·)k_L2((0,∞)×R²),

D²_xρ_ε

L²((0,∞)×(0,t)×R²), ε

χ(a)(∇_x∂_aρ_ε)²

L¹((0,∞)×(0,t)×R²)≤e^CtC,

eq:re3

eq:re3 (2.19)

Proof. In the proofs, we will use the following remark

Y_θ⁰(φε−R)φε, Y_θ⁰⁰(φε−R)φε ≤C.

con:Y

con:Y (2.20)

We multiply (2.14) by −∆_xρε then d

dt Z ∞

0

Z

R²

|∇_xρε|²

2 dxda= Z ∞

0

Z

R²

∇_xρε·∂t∇_xρεdxda=− Z ∞

0

Z

R²

∆xρε∂tρεdxda

=−µ Z ∞

0

Z

R²

|D_x²ρ_ε|²dxda+ Z ∞

0

Z

R²

∆_xρ_εdiv_x(ρ_εχ₁vY_θ) dxda+ Z ∞

0

Z

R²

∆_xρ_ε∂_aρ_εdxda +

Z ∞ 0

Z

R²

ρε(−d−p)

| {z }

≤c

∆xρεdxda−ε Z ∞

0

Z

R²

∂a(χ(a)∂aρε)∆xρεdxda

≤ −µ 2

Z ∞ 0

Z

R²

|D²_xρε|²dxda+ Z ∞

0

Z

R²

(divx(ρεχ1vYθ))²dxda+ Z

R²

|∇_xρ_ε(t,0, x)|²

2 dx

+c Z ∞

0

Z

R²

ρ²_εdxda−ε Z ∞

0

Z

R²

χ(a)|∇_x∂aρε|²dxda

≤ −µ 2

Z ∞ 0

Z

R²

|D²_xρ_ε|²dxda−ε Z ∞

0

Z

R²

χ(a)|∇_x∂_aρ_ε|²dxda+c Z ∞

0

Z

R²

ρ²_εdxda +c

Z

R²

Z ∞ 0

∇_xρε.vχ1Yθda 2

dx+c Z

R²

Z ∞ 0

ρεχ1daY_θ⁰ 2

| {z }

≤c,since (2.20)

(v.∇_xφε)²dx

+c Z

R²

Z ∞ 0

ρεχ1Y_θdivx(v) da 2

dx

+c Z

R²

|A⁰(φ_ε)|² Z ∞

0

ρ_ε(t, a, x)χ₃(a)da 2

| {z }

≤c,since (1.8)

|∇_xφ_ε|²ω²(t, x)dx

+c Z

R²

A(φε)

Z ∞ 0

|∇_xρε(t, a, x)|χ₃(a)da

ω(t, x) 2

dx +c

Z

R²

A(φε)

Z ∞ 0

ρε(t, a, x)χ3(a)da|∇_xω(t, x)|

2

dx

≤ −µ 2

Z ∞ 0

Z

R²

|D²_xρε|²dxda−ε Z ∞

0

Z

R²

χ(a)|∇_x∂aρε|²dxda+c Z ∞

0

Z

R²

ρ²_εdxda +c

Z ∞ 0

Z

R²

|∇_xρ_ε|²dxda+c Z

R²

|∇_xφ_ε|²dx.

Integrating over (0, t), using the Gronwall Lemma, and estimates (2.13), (2.18) we gain (2.19).

Lemma 2.6 ( L^∞ estimate on u_ε). For every t≥0, we have

L_infty of u

L_infty of u (2.21) ku_ε(t,·)k_L∞(R²) ≤Ce^Ct.

Proof. LetC be a positive constant that will be fixed later. We define uε(t, x) =e^−Ctuε(t, x),

and we consider the associated problem







∂tuε+ divx(uεν(φε)) = (b(φε)−β− C)u_ε+ε∆xuε, u_ε(0, x) =u_0,ε(x).

We claim that for any givenT >0 there exist a sufficiently large constantk >0 and a suitableCsuch thatu_ε(t, x)≤kfor any t≤T and x∈R², providedu_0,ε(x)≤k for allx∈R².

Consider the function

x 7→η(x) = (x−k)1(k,∞)(x), and observe that

η⁰(x) =1(k,∞)(x), xη⁰(x) =η(x) +kη⁰(x).

We have d dt

Z

R²

η(uε)dx= Z

R²

η⁰(uε)∂tuεdx

=− Z

R²

η⁰(uε)divx(uεν(φε)) dx+ Z

R²

η⁰(uε)(b(φε)−β− C)u_εdx+ε Z

R²

η⁰(uε)∆xuεdx

| {z }

≤0

≤ − Z

R²

η⁰(uε)divx((uε−k)ν(φε)) dx

| {z }

=0

−k Z

R²

η⁰(uε)divx(ν(φε)) dx

+ Z

R²

(η(uε) +kη⁰(uε))(b(φε)−β− C) dx

≤ Z

R²

η(uε)(b(φε)−β− C) dx−k Z

R²

η⁰(uε)(C+β+ divx(ν(φε))−b(φε)) dx.

From the inequalitykdiv_x(ν(φε))k_L∞ ≤2k∇ν(φ_ε)k_L∞ and the estimates in (2.7) and (2.17), it follows that for C large enough

C+β−b(φ_ε)≥0, C+β+ div_x(ν(φ_ε))−b(φ_ε)≥0, thus,

d dt

Z

R²

η(u_ε)dx≤0.

Integrating over (0, t) we obtain 0≤

Z

R²

η(uε(t, x)) dx≤ Z

R²

η(u0,ε(x)) dx= 0,

which means uε≤k. The inequality (2.21) follows.

estimate for u Lemma 2.7. For allt≥0, the following estimates on u_ε hold

ku_ε(t,·)k_L1(R²)≤e^CtC,

eq:u1

eq:u1 (2.22)

ku_ε(t,·)k_L2(R²),k∇_xu_εk_L2((0,t)×R²)≤e^CtC,

eq:u2

eq:u2 (2.23)

k∇_xu_ε(t,·)k_L2(R²),k∆_xu_εk_L2((0,t)×R²)≤e^CtC.

eq:u3

eq:u3 (2.24)

Proof. We recall the equation ofuε eq:ue

eq:ue (2.25) ∂_tu_ε+ div_x(u_εν(φ_ε)) = (b(φ_ε)−β)u_ε+ε∆_xu_ε. (2.22). Using the nonnegativity ofuε we have

d dt

Z

R²

|u_ε|dx= d dt

Z

R²

uεdx=ε Z

R²

∆xuεdx− Z

R²

divx(uεν(φε)) dx

| {z }

=0

+ Z

R²

(b(φε)−β)

| {z }

≤c

uεdx≤c Z

R²

uεdx.

Then, applying the Gronwall Lemma we gain

ku_ε(t,·)k_L1(R²)≤ ku_0,ε(·)k_L1(R²)e^ct. (2.23). We multiply (2.25) byu_ε to obtain

d dt

Z

R²

u²_ε 2 dx=

Z

R²

uε∂tuεdx

=− Z

R²

divx(uεν(φε))uεdx+ Z

R²

(b(φε)−β)u²_εdx+ε Z

R²

∆xuεuεdx

≤ Z

R²

uεν(φε).∇u_εdx+c Z

R²

u²_εdx−ε Z

R²

|∇_xuε|²dx

≤ Z

R²

∇_x u²_ε

2

.ν(φ_ε)dx+c Z

R²

u²_εdx−ε Z

R²

|∇_xu_ε|²dx

≤ Z

R²

u²_ε

2 div_x(ν(φ_ε)) dx+c Z

R²

u²_εdx−ε Z

R²

|∇_xu_ε|²dx

≤c_T Z

R²

|u_ε|dx+c Z

R²

u²_εdx−ε Z

R²

|∇_xu_ε|²dx.

Integrating over (0, t) and then using the Gronwall Lemma we get ku_ε(t,·)k²_L2(R²)+ 2ε e^ct

Z t 0

e^−csk∇_xuε(s,·)k²_L2(R²)ds≤e^ctku_ε(0,·)k²_L2(R²)+ce^ct.

(2.24). We multiply (2.25) by−∆_xuε to obtain d

dt Z

R²

|∇_xuε|² 2 dx=

Z

R²

∇_xuε·∂t∇_xuεdx=− Z

R²

∆xuε∂tuεdx

= Z

R²

∆xuεdivx(uεν(φε)) dx− Z

R²

∆xuε(b(φε)−β)uεdx−ε Z

R²

|∆_xuε|²dx

=− Z

R²

∇_xuε∇_x(∇_xuε.ν(φ) +uεdivx(ν(φ))) dx+ Z

R²

(b(φ)−β)|∇_xuε|²dx +

Z

R²

b⁰(φ)∇_xφ.∇_xuεuεdx−ε Z

R²

|∆_xuε|²dx

≤ −ε Z

R²

|∆_xu_ε|²dx+c Z

R²

|∇_xu_ε|²dx+c_T Z

R²

|∇_xψ_ε|²dx

+ Z

R²

∇_x

|∇_xuε|² 2

.ν(φ)dx+ Z

R²

|∇_xu|²|∇_xν(φ)|

| {z }

≤c_T

dx+ Z

R²

|∇_xu|²|div_x(ν(φ))|

| {z }

≤c_T

dx

+ Z

R²

|∇_xu_ε|u_ε|∇_xdiv_x(ν(φ))|dx

≤ −ε Z

R²

|∆_xu_ε|²dx+c_T Z

R²

|∇_xu_ε|²dx+c_T Z

R²

|∇_xψ_ε|²dx+c_T Z

R²

|∇_xdiv_x(ν(φ))|²dx.

Integrating over (0, t) and using the estimates in (2.9) and (2.18) we gain k∇_xuε(t,·)k_L2(R²)+εk∆_xuε(·,·)k_L2((0,t)×R²)≤ce^ct.

estimate for phi_e 3 Lemma 2.8. For everyt≥0, the following estimates on ψ_ε hold k∇_xψε(t,·)k_L2(R²),

D_x²ψε

L²((0,t)×R²) ≤e^CtC,

eq:nonloc2.1

eq:nonloc2.1 (2.26)

D²_xψ_ε(t,·)

L²(R²), D_x³ψ_ε

L²((0,t)×R²) ≤e^CtC,

eq:nonloc2.2

eq:nonloc2.2 (2.27)

kψ_ε(t,·)k_L∞(R²) ≤e^CtC,

eq:nonloc2.3

eq:nonloc2.3 (2.28)

D³_xψε(t,·)

L²(R²), D_x⁴ψε

L²((0,t)×R²) ≤e^CtC,

eq:nonloc2.4

eq:nonloc2.4 (2.29)

k∂_tψ_ε(t,·)k_L1(R²),k∂_tψ_ε(t,·)k_L2(R²),k∇_xψ_ε(t,·)k_L∞(R²) ≤e^CtC.

eq:nonloc2.5

eq:nonloc2.5 (2.30)

Proof. We multiply byξ(a) the equation forρε in system (2.1) then, integrating with respect toa, we get

∂_tψ_ε−µ∆_xψ_ε+ div_x

Z ∞ 0

ρ_εχ₁ξda

vY_θ

=− Z ∞

0

(d+p(a, u_ε))ρ_εξda+ Z ∞

0

ρ_ε (1 +εχ⁰)ξ⁰+εχξ⁰⁰ da.

eq:psi

eq:psi (2.31)

(2.26). We multiply (2.31) by−∆_xψε

d dt

Z

R²

|∇_xψ_ε|² 2 dx=

Z

R²

∇_xψ_ε·∂_t∇_xψ_εdx=− Z

R²

∆_xψ_ε∂_tψ_εdx

=−µ Z

R²

|D_x²ψ_ε|²dx+ Z

R²

∆_xψ_εdiv_x

Z ∞ 0

ρ_εχ₁ξda

vY_θ

dx+ Z

R²

Z ∞ 0

ρ_εξ(d+p)

| {z }

≤c

∆_xψ_εdadx

− Z

R²

Z ∞ 0

ρ_ε (1 +εχ⁰)ξ⁰+εχξ⁰⁰

| {z }

≤c

∆_xψ_εdadx

≤ −µ 2

Z

R²

|D_x²ψε|²dx+c Z

R²

divx

Z ∞ 0

ρεχ1ξda

vY_θ 2

dx+c Z

R²

ψ_ε²dx+c Z ∞

0

Z

R²

ρ²_εdxda