G M Coclite

C Donadello

T N T Nguyen

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

Keywords: 2020 Mathematics Subject Classification. 35Q92, 35M33 population dynamics, predator-prey systems, parabolic-hyperbolic equations, nonlocal conservation laws, nonlocal boundary value problem

. We establish existence of solutions by applying the vanishing viscosity method, and we prove stability by a doubling of variables type argument.

Introduction oduction

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 [START_REF] Coclite | A PDE model for the spatial dynamics of a voles population structured in age[END_REF]. To this end we couple the latter equation to the hyperbolic equation for predators proposed in [START_REF] Colombo | Hyperbolic predators vs. parabolic prey[END_REF] 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 odel_new odel_new

(1.1)                        ∂ t u + div x (uν(φ)) = (b(φ) -β)u, (t, x) ∈ (0, T ) × R 2 ,
∂ t ρ + ∂ a ρ + div x (ρχ 1 (a)v(x)Y θ (φ -R)) = µ∆ x ρ -d(t, a, x)ρ -p(a, u)ρ, (t, a, x) ∈ (0, T ) 2 × R 2 , ρ(t, 0, x) = A (φ) ∞ 0 ρ(t, a, x)χ 3 (a) da ω(t, x), (t, x) ∈ (0, T ) × R 2 , ρ(0, a, x) = ρ 0 (a, x), (a, x) ∈ (0, T ) × R 2 , u(0, x) = u 0 (x),

x ∈ R 2 ,
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 a at (t, x). More precisely, the relation between φ and ρ is given by eq:defnu eq:defnu

(1.2) φ(t, x) = ∞ 0 ρ(t, a, x)χ 2 (a) da,
where χ 2 (a) is an approximation of the indicator function of the interval (σ, T ), where T 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 function b(φ) represents the reproduction rate of predators depending on preys' availability, while β > 0, the predators' mortality rate, is assumed to be constant. As in [START_REF] Colombo | Hyperbolic predators vs. parabolic prey[END_REF] the flux of u is driven to the direction of higher preys' concentration by a nonlinear, nonlocal velocity ν of the form

ν(φ) = κ ∇(φ * η) 1 + ∇(φ * η) 2
, where κ > 0 is the maximal speed of predators and η is a positive smooth mollifier with R 2 η dx = 1 so that the convolution (φ(t) * η)(x) represents an average of the density of preys in a neighborhood of x at time t.

The equation for the preys, introduced in [START_REF] Coclite | A PDE model for the spatial dynamics of a voles population structured in age[END_REF], is related to classical models for the dynamics of a population structured in age, see [START_REF] Colombo | Stability and optimization in structured population models on graphs[END_REF][START_REF] Murray | Mathematical Biology I. An Introduction[END_REF][START_REF] Webb | Population Models Structured by Age, Size, and Spatial Position[END_REF], but the choice of the coefficients and boundary conditions at a = 0 takes into account the data collections and ecological considerations in [START_REF] Berthier | Dispersal, landscape and travelling waves in cyclic vole populations[END_REF][START_REF] Delattre | Le campagnol terrestre: prévention et contrôle des populations[END_REF][START_REF] Giraudoux | Population dynamics of fosserial water vole (arvicola terrestris scherman) : a land use and a landscape perspective[END_REF][START_REF] Halliez | Historical agricultural changes and the expansion of a water vole population in an alpine valley[END_REF][START_REF] Saucy | Juvenile dispersal in the vole arvicola terrestris during rainy nights: A preliminary report[END_REF]. We recall here the essential assumptions on the form of the coefficients.

We introduce constants 0 < A 1 < A 2 so that a vole is young (baby) if its age a is in (0, A 1 ), juvenile if its age is in (A 1 , A 2 ) 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, if ξ ≥ 0, 0, if ξ ≤ -1, Y (ξ) ≥ 0, Y θ (ξ) = Y ξ θ .
From Y θ we costruct the approximations of the indicator functions of the intervals (σ, T ), (A 1 , A 2 ), and (A 1 , T )

χ 1 (a) = Y θ (a -A 1 )Y θ (A 2 -a), χ 2 (a) = Y θ (a -σ)Y θ (T -a), χ 3 (a) = Y θ (a -A 1 )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 is u, while d = d(t, a, x) stands for all other casualties (sickness, starvation, generic predation, etc).

The second-order term µ∆ x ρ represents short range spatial dynamics related to foraging activities.

Everywhere in the following θ 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 [START_REF] Giraudoux | Weather influences m. arvalis reproduction but not population dynamics in a 17-year time series[END_REF] and references therein. The function A(φ) describes the influence of the total density of voles on natality. Examples of non constant A are functions of the form function function

(1.3) A(φ) = αφ γ (β + φ) γ ,
for different choices of α, β and γ.

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

∈ C ∞ (R) ∩ W 1,∞ (R), b(•) ≥ 0, ass:b ass:b (1.4) v ∈ C ∞ (R 2 ) ∩ L 2 (R 2 ) ∩ L ∞ (R 2 ), div x (v) ∈ L 1 (R 2 ) ∩ W 2,∞ (R 2 ), v > 0, ass:v ass:v (1.5) d ∈ C ∞ ([0, ∞) × [0, ∞) × R 2 ) ∩ W 2,∞ ((0, ∞) × (0, ∞) × R 2 ), 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, |A (ξ)ξ|, |A (ξ)ξ| ≤ C 0 , ass:A ass:A (1.8) ω ∈ C ∞ ([0, ∞) × R 2 ) ∩ W 1,∞ ((0, ∞) × R 2 ), ω(•, •) ≥ 0,
ass:w ass:w

(1.9)

ρ 0 ∈ L 1 ((0, ∞) × R 2 ) ∩ L ∞ ((0, ∞) × R 2 ), ρ 0 ≥ 0, s:init.1 s:init.1 (1.10) sup x∈R 2 ρ 0 (•, x) L 1 (0,∞) , sup a>0 ρ 0 (a, •) L 1 (R 2 ) , R 2 T V (ρ 0 (•, x))dx ≤ C 0 , s:init.2 s:init.2
(1.11)

u 0 ∈ L 1 (R 2 ) ∩ BV (R 2 ) ∩ L ∞ (R 2 ), u 0 ≥ 0, s:init.3 s:init.3
(1.12)

for some positive constants

d * , d * , p * , p * , C 0 .
For what the velocity of predators, ν, is concerned, we assume that the mollifier η satisfies sumption sumption

(1.13) ∇ x η ∈ (C 2 ∩ W 2,2 ∩ W 1,∞ )(R 2 , R 2 ).
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.

solution 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 ; L 1 ((0, ∞) × R 2 )) ∩ L ∞ ((0, T ) × (0, ∞) × R 2 ) ∩ L 2 ((0, T ) × (0, ∞); H 2 (R 2 )). def:2 (D.2) u ∈ L 1 ((0, T ) × R 2 ) ∩ BV ((0, T ) × R 2 ). def:3 (D.3) For almost every (t, x) ∈ (0, T ) × R 2 , ρ(t, •, x) ∈ BV (0, ∞) and ρ(t, 0 + , x) = A (φ) ∞ 0 ρ(t, a, x)χ 3 (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 (R 4 ) ∞ 0 ∞ 0 R 2 (ρ∂ t ξ + ρ∂ a ξ + ρχ 1 (a)v • ∇ x ξY θ (φ -R) + µρ∆ x ξ -dρξ -pρξ) dx da dt + ∞ 0 ∞ 0 R 2 A (φ) ρ(t, a, x)χ 3 (a)ω(t, x)ξ(t, 0, x)dx da dt + ∞ 0 R 2 ρ 0 (a, x)ξ(0, a, x)dx da = 0, ∞ 0 ∞ 0 R 2 (u∂ t ξ + uν(φ) • ∇ x ξ + (b(φ) -β)uξ) dx da dt + ∞ 0 R 2 u 0 (x)ξ(0, a, x)dx da = 0. solution Definition 1.2.
We say that a weak solution (u, ρ) is an entropy solution of (1.1) if for any nonnegative test function ξ ∈ C ∞ (R 4 ) with compact support and for any constant c ∈ R there hold ) 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

∞ 0 ∞ 0 R 2 (|ρ -c| (∂ t ξ + ∂ a ξ) -div x (|ρ -c|χ 1 vY θ (φ -R)) ξ +µ∆ x |ρ -c|ξ -sign (ρ -c) (d + p)ρξ) dx da dt + ∞ 0 R 2 |ρ(t, 0 + , x) -c|ξ(t, 0, x)dx dt + ∞ 0 R 2 |ρ 0 (a, x) -c|ξ(0, a, x) dx da ≥ ∞ 0 ∞ 0 R 2 c sign (ρ -c) χ 1 (a)div x (v(x)Y θ (φ -R)) ξ dx da dt ropy_rho ropy_rho (1.14) and ∞ 0 ∞ 0 R 2 (|u -c|∂ t ξ + |u -c|ν(φ) • ∇ x ξ + sign (u -c) (b(φ) -β)uξ) dx da dt + ∞ 0 R 2 |u 0 (x) -c|ξ(0, a, x) dx da ≥ ∞ 0 ∞ 0 R 2 c sign (u -c) div x (ν(φ)) ξ dx da dt.
u 1 (t, •) -u 2 (t, •) L 1 (R 2 ) + ρ 1 (t, •, •) -ρ 2 (t, •, •) L 1 ((0,∞)×R 2 ) ≤ Ce Ce Ct ρ 1,0 -ρ 2,0 L 1 ((0,∞)×R 2 ) + Ce Ce Ct u 1,0 -u 2,0 L 1 ((R 2 )
.

eq:stab eq:stab

(1.16)

The fixed point argument used in [START_REF] Colombo | Hyperbolic predators vs. parabolic prey[END_REF] 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 [START_REF] Panov | Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux[END_REF], 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.

Existence exist

The existence argument is based on the compactness analysis of a sequence of solutions to approximating 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 odel_eps odel_eps

(2.1)

                             ∂ t u ε + div x (u ε ν(φ ε )) = (b(φ ε ) -β)u ε + ε∆ x u ε , (t, x) ∈ (0, T ) × R 2 , ∂ t ρ ε + ∂ a ρ ε + div x (ρ ε χ 1 (a)v(x)Y θ (φ ε -R)) = µ∆ x ρ ε -d(t, a, x)ρ ε -p(a, u ε )ρ ε + ε∂ a (χ(a)∂ a ρ ε ), (t, a, x) ∈ (0, T ) 2 × R 2 , ρ ε (t, 0, x) = A (|φ ε |) ∞ 0 |ρ ε (t, a, x)|χ 3 (a)da ω(t, x), (t, x) ∈ (0, T ) × R 2 , ρ ε (0, a, x) = ρ 0,ε (a, x), (a, x) ∈ (0, T ) 2 × R 2 , u ε (0, x) = u 0,ε (x), x ∈ R 2 ,
where

φ ε (t, x) = ∞ 0 ρ ε (t, a, x)χ 2 (a)da, def:pe def:pe (2.2) χ(a) ∈ C ∞ ([0, +∞), [0, 1]) satisfies χ(0) = 0, and {(ρ 0,ε , u 0,ε )} ε is a family of approximations of the initial condition (ρ 0 , u 0 ) such that it-v-eps it-v-eps (2.3)
ρ 0,ε ∈ C ∞ ((0, ∞) × R 2 ), u 0,ε ∈ C ∞ (R 2 ), ε > 0, ρ 0,ε → ρ 0 , a.e. and in L p ((0, ∞) × R 2 ), 1 ≤ p < ∞, as ε → 0, u 0,ε → u 0 , a.e. and in L p (R 2 ), 1 ≤ p < ∞, as ε → 0, ρ 0,ε ≥ 0, ρ 0,ε L 1 ((0,∞)×R 2 ) ≤ C, ε ≥ 0, sup x∈R 2 ρ 0,ε (•, x) L 1 (R) , sup a≥0 ρ 0,ε (a, •) L 1 (R 2 ) , ∂ a ρ 0,ε L 1 ((0,∞)×R 2 ) ≤ C, ε ≥ 0, u 0,ε L 1 (R 2 ) , ∇ x u 0,ε L 1 (R 2 ) , ε ∆ x u 0,ε L 1 (R 2 ) ≤ 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 notation C for all the positive constants independent on ε appearing 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 [START_REF] Colombo | Hyperbolic predators vs. parabolic prey[END_REF]Lemma 4.1]).

lem:eta Lemma 2.1. Let η be such that sumption sumption

(2.4) ∇ x η ∈ (C 2 ∩ W 2,2 ∩ W 1,∞ )(R 2 , R 2 ).
Then the map ν :

L 1 (R 2 ; R) → C ∞ (R) ∩ W 1,∞ (R 2 ; R 2 ) satisfies ν(φ) L ∞ (R 2 ;R 2 ) ≤ K φ L 1 (R 2 ,R) ,
ass:nu1 ass:nu1

(2.5)

div x (ν(φ)) L 2 (R 2 ,R) ≤ K φ L 1 (R 2 ;R) ,
ass:nu2 ass:nu2

(2.6)

∇ x ν(φ) L ∞ (R 2 ,R 2×2 ) ≤ K φ L 1 (R 2 ;R) ,
ass:nu3 ass:nu3

(2.7)

ν(φ 1 ) -ν(φ 2 ) L ∞ (R 2 ,R 2 ) ≤ K φ 1 -φ 2 L 1 (R 2 ,R) ,
ass:nu4 ass:nu4

(2.8)

∇ x (div x (ν(φ))) L 2 (R 2 ,R 2 ) ≤ K 1 + K φ L 1 (R 2 ,R) φ L 1 (R 2 ,R) ,
ass:nu5 ass:nu5

(2.9)

div x (ν(φ 1 ) -ν(φ 2 )) L 2 (R 2 ,R) ≤ K 1 + K φ 2 L 1 (R 2 ,R) φ 1 -φ 2 L 1 (R 2 ,R) ,
ass:nu6 ass:nu6

(2.10)

where K is a positive constant.

Lemma 2.2 (Nonnegativity of ρ ε , φ ε , u ε ). We have that lem:pos lem:pos

(2.11) ρ ε ≥ 0, φ ε ≥ 0, u ε ≥ 0.
Proof. Consider the function

x → η(x ) = -x1 (-∞,0) (x ).
and observe that

η (x ) = -1 (-∞,0) (x ), η(x ) = x η (x ).
From (2.1) we obtain

d dt ∞ 0 R 2 η(ρ ε )dx da = ∞ 0 R 2 η (ρ ε )∂ t ρ ε dx da = - ∞ 0 R 2 η (ρ ε )∂ a ρ ε dx da - ∞ 0 R 2 div x (ρ ε χ 1 vY θ ) η (ρ ε )dx da + µ ∞ 0 R 2 η (ρ ε )∆ x ρ ε dx da - ∞ 0 R 2 η (ρ ε )(d + p)ρ ε dx da + ε ∞ 0 R 2 η (ρ ε )∂ a (χ(a)∂ a ρ ε )ρ ε dx da = R 2 η(ρ ε (t, 0, x)) =0 dx + ∞ 0 R 2 ρ ε χ 1 (v • ∇ x ρ ε )Y θ η (ρ ε )dx da =0 -µ ∞ 0 R 2 η (ρ ε ) (∇ x ρ ε ) 2 dx da ≤0 - ∞ 0 R 2 (d + p)η(ρ ε )dx da ≤0 + ε ∞ 0 R 2 ∂ a η (ρ ε )χ(a)∂ a ρ ε dx da -ε ∞ 0 R 2 η (ρ ε )χ(a)(∂ a ρ ε ) 2 ≤0 dx da ≤ -ε R 2 η (ρ ε (t, 0, x)) =0 χ(0) =0 ∂ a ρ ε (t, 0, x)dx = 0.
Thus, integrating on (0, t) we obtain η(ρ ε (t, a, x)) = 0 which implies that ρ ε ≥ 0, and then φ ε ≥ 0.

From (2.1) we obtain

d dt R 2 η(u ε )dx = R 2 η (u ε )∂ t u ε dx = - R 2 η (u ε )div x (u ε ν(φ ε )) dx =0 + R 2 η (u ε ) (b(φ) -β) ≤c u ε dx + ε R 2 η (u ε )∆ x u ε dx ≤0 ≤ c R 2 η(u ε )dx.
Integrating on (0, t) and applying Gronwall Lemma we obtain η(u ε (t, x)) = 0, so that u ε ≥ 0.

_eps_new Remark 2.1. We can remove the absolute value in the boundary condition for ρ ε in (2.1).

of rho

Lemma 2.3 (L 1 estimate on ρ ε ). For all t ≥ 0, we have that

ρ ε (t, •, •) L 1 ((0,∞)×R 2 ) ≤ e Ct C.
eq:re1 eq:re1

(2.12)

Proof. Due to the nonnegativity of ρ ε and the boundedness of d, p we have

d dt ∞ 0 R 2 |ρ ε |dx da = d dt ∞ 0 R 2 ρ ε dx da = - ∞ 0 R 2 ∂ a ρ ε dx da - ∞ 0 R 2 div x (ρ ε χ 1 vY θ ) dx da + µ ∞ 0 R 2 ∆ x ρ ε dx da =0 - ∞ 0 R 2 ρ ε (d + p) ≤c dx da + ε ∞ 0 R 2 ∂ a (χ(a)∂ a ρ ε ) dx da =0 ≤ R 2 ρ ε (t, 0, x)dx + c ∞ 0 R 2 ρ ε dx da = R 2 A (φ ε ) ∞ 0 ρ ε (t, a, x)χ 3 (a)da ω(t, x)dx + c ∞ 0 R 2 ρ ε dx da ≤c ∞ 0 R 2 ρ ε dx da.
Integrating on (0, t) we get (2.12) thanks to the Gronwall Lemma and the assumptions in (2.3) .

m:L2 rho Lemma 2.4 (L 2 estimate on ρ ε ). For any t ≥ 0, we have

ρ ε (t, •, •) L 2 ((0,∞)×R 2 ) , ∇ x ρ ε L 2 ((0,t)×(0,∞)×R 2 ) , √ ε χ∂ a ρ ε L 2 ((0,t)×(0,∞)×R 2 ) ≤ e Ct C.
eq:re2 eq:re2

(2.13)

Proof. We recall the equation of ρ ε

∂ t ρ ε + ∂ a ρ ε + div x (ρ ε χ 1 (a)v(x)Y θ (φ ε -R)) =µ∆ x ρ ε -d(t, a, x)ρ ε -p(a, u ε )ρ ε + ε∂ a (χ(a)∂ a ρ ε ).
eq:re eq:re

(2.14)
We multiply (2.14) by ρ ε and have

d dt ∞ 0 R 2 ρ 2 ε 2 dx da = ∞ 0 R 2 ρ ε ∂ t ρ ε dx da = - ∞ 0 R 2 ρ ε ∂ a ρ ε dx da - ∞ 0 R 2 ρ ε div x (ρ ε χ 1 vY θ ) dx da + µ ∞ 0 R 2 ρ ε ∆ x ρ ε dx da - ∞ 0 R 2 ρ 2 ε (d + p) ≤c dx da + ε ∞ 0 R 2 ρ ε ∂ a (χ(a)∂ a ρ ε ) dx da ≤ R 2 ρ ε (t, 0, x) 2 2 dx - µ 2 ∞ 0 R 2 |∇ x ρ ε | 2 dx da + c ∞ 0 R 2 (ρ ε χ 1 vY θ ) 2 dx da + c ∞ 0 R 2 ρ 2 ε dx da -ε ∞ 0 R 2 χ(a) (∂ a ρ ε ) 2 dx da ≤ 1 2 R 2 A 2 (φ ε ) ∞ 0 ρ ε (t, a, x)χ 3 (a)da 2 ω 2 (t, x)dx + c ∞ 0 R 2 ρ 2 ε dx da - µ 2 ∞ 0 R 2 |∇ x ρ ε | 2 dx da -ε ∞ 0 R 2 χ(a) (∂ a ρ ε ) 2 dx da ≤c ∞ 0 R 2 ρ 2 ε dx da - µ 2 ∞ 0 R 2 |∇ x ρ ε | 2 dx da -ε ∞ 0 R 2 χ(a) (∂ a ρ ε ) 2 dx da.
Integrating over (0, t) and using the Gronwall Lemma we gain

ρ ε (t, •, •) 2 L 2 ((0,∞)×R 2 ) + e ct µ t 0 ∞ 0 R 2 e -cs |∇ x ρ ε | 2 dx da ds + e ct 2ε t 0 ∞ 0 R 2 e -cs χ(a) (∂ a ρ ε ) 2 dx da ds ≤ e ct ρ 0,ε (•, •) 2 L 2 ((0,∞)×R 2 ) .
We obtain (2.13) by using the fact that

χ∂ a ρ ε 2 L 2 ((0,t)×(0,∞)×R 2 ) ≤ χ L ∞ (0,∞) t 0 ∞ 0 R 2 χ(a)(∂ a ρ ε ) 2 dx da ds ≤ t 0 ∞ 0 R 2 χ(a)(∂ a ρ ε ) 2 dx da ds.
We consider the class of functions def:psi def:psi

(2.15) ψ ε (t, x) = ∞ 0 ρ ε (t, a, x)ξ(a)da, for ξ ∈ C ∞ c ((0, ∞)) such that q:suppxi q:suppxi (2.16) supp (ξ) ⊂ (0, T ).
In particular any of the functions χ i , i = 1, 2, 3, can play the role of ξ, 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

ψ ε (t, •) L 1 (R 2 ) ≤ e Ct C, :nonloc1 :nonloc1
(2.17)

ψ ε (t, •) L 2 (R 2 ) , ∇ x ψ ε L 2 ((0,t)×R 2 ) ≤ e Ct C.
:nonloc2 :nonloc2

(2.18) imate re Lemma 2.5 (H 2 estimate on ρ ε ). For any t ≥ 0, we have

∇ x ρ ε (t, •, •) L 2 ((0,∞)×R 2 ) , D 2 x ρ ε L 2 ((0,∞)×(0,t)×R 2 ) , ε χ(a)(∇ x ∂ a ρ ε ) 2 L 1 ((0,∞)×(0,t)×R 2 ) ≤ e Ct C,
eq:re3 eq:re3

(2.19)

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

Y θ (φ ε -R)φ ε , Y θ (φ ε -R)φ ε ≤ C. con:Y con:Y (2.20) We multiply (2.14) by -∆ x ρ ε then d dt ∞ 0 R 2 |∇ x ρ ε | 2 2 dx da = ∞ 0 R 2 ∇ x ρ ε • ∂ t ∇ x ρ ε dx da = - ∞ 0 R 2 ∆ x ρ ε ∂ t ρ ε dx da = -µ ∞ 0 R 2 |D 2 x ρ ε | 2 dx da + ∞ 0 R 2 ∆ x ρ ε div x (ρ ε χ 1 vY θ ) dx da + ∞ 0 R 2 ∆ x ρ ε ∂ a ρ ε dx da + ∞ 0 R 2 ρ ε (-d -p) ≤c ∆ x ρ ε dx da -ε ∞ 0 R 2 ∂ a (χ(a)∂ a ρ ε )∆ x ρ ε dx da ≤ - µ 2 ∞ 0 R 2 |D 2 x ρ ε | 2 dx da + ∞ 0 R 2 (div x (ρ ε χ 1 vY θ )) 2 dx da + R 2 |∇ x ρ ε (t, 0, x)| 2 2 dx + c ∞ 0 R 2 ρ 2 ε dx da -ε ∞ 0 R 2 χ(a)|∇ x ∂ a ρ ε | 2 dx da ≤ - µ 2 ∞ 0 R 2 |D 2 x ρ ε | 2 dx da -ε ∞ 0 R 2 χ(a)|∇ x ∂ a ρ ε | 2 dx da + c ∞ 0 R 2 ρ 2 ε dx da + c R 2 ∞ 0 ∇ x ρ ε .vχ 1 Y θ da 2 dx + c R 2 ∞ 0 ρ ε χ 1 daY θ 2 ≤c, since (2.20) (v.∇ x φ ε ) 2 dx + c R 2 ∞ 0 ρ ε χ 1 Y θ div x (v) da 2 dx + c R 2 |A (φ ε )| 2 ∞ 0 ρ ε (t, a, x)χ 3 (a)da 2 ≤c, since (1.8) |∇ x φ ε | 2 ω 2 (t, x)dx + c R 2 A (φ ε ) ∞ 0 |∇ x ρ ε (t, a, x)|χ 3 (a)da ω(t, x) 2 dx + c R 2 A (φ ε ) ∞ 0 ρ ε (t, a, x)χ 3 (a)da|∇ x ω(t, x)| 2 dx ≤ - µ 2 ∞ 0 R 2 |D 2 x ρ ε | 2 dx da -ε ∞ 0 R 2 χ(a)|∇ x ∂ a ρ ε | 2 dx da + c ∞ 0 R 2 ρ 2 ε dx da + c ∞ 0 R 2 |∇ x ρ ε | 2 dx da + c R 2 |∇ x φ ε | 2 dx.
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

fty of u fty of u (2.21) u ε (t, •) L ∞ (R 2 ) ≤ Ce Ct .
Proof. Let C be a positive constant that will be fixed later. We define

u ε (t, x) = e -Ct u ε (t, x),
and we consider the associated problem

   ∂ t u ε + div x (u ε ν(φ ε )) = (b(φ ε ) -β -C)u ε + ε∆ x u ε , u ε (0, x) = u 0,ε (x).
We claim that for any given T > 0 there exist a sufficiently large constant k > 0 and a suitable C such that u ε (t, x) ≤ k for any t ≤ T and x ∈ R 2 , provided u 0,ε (x) ≤ k for all x ∈ R 2 .

Consider the function

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

η (x ) = 1 (k,∞) (x ), x η (x ) = η(x ) + kη (x ).
We have

d dt R 2 η(u ε )dx = R 2 η (u ε )∂ t u ε dx = - R 2 η (u ε )div x (u ε ν(φ ε )) dx + R 2 η (u ε )(b(φ ε ) -β -C)u ε dx + ε R 2 η (u ε )∆ x u ε dx ≤0 ≤ - R 2 η (u ε )div x ((u ε -k)ν(φ ε )) dx =0 -k R 2 η (u ε )div x (ν(φ ε )) dx + R 2 (η(u ε ) + kη (u ε ))(b(φ ε ) -β -C) dx ≤ R 2 η(u ε )(b(φ ε ) -β -C) dx -k R 2 η (u ε )(C + β + div x (ν(φ ε )) -b(φ ε )) dx.
From the inequality div x (ν(φ ε )) L ∞ ≤ 2 ∇ν(φ ε ) 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 R 2 η(u ε )dx ≤ 0.
Integrating over (0, t) we obtain

0 ≤ R 2 η(u ε (t, x)) dx ≤ R 2 η(u 0,ε (x)) dx = 0, which means u ε ≤ k. The inequality (2.21) follows.
te for u Lemma 2.7. For all t ≥ 0, the following estimates on u ε hold

u ε (t, •) L 1 (R 2 ) ≤ e Ct C,
eq:u1 eq:u1

(2.22)

u ε (t, •) L 2 (R 2 ) , ∇ x u ε L 2 ((0,t)×R 2 ) ≤ e Ct C,
eq:u2 eq:u2

(2.23)

∇ x u ε (t, •) L 2 (R 2 ) , ∆ x u ε L 2 ((0,t)×R 2 ) ≤ e Ct C.
eq:u3 eq:u3

(2.24)

Proof. We recall the equation of u ε eq:ue eq:ue

(2.25) ∂ t u ε + div x (u ε ν(φ ε )) = (b(φ ε ) -β)u ε + ε∆ x u ε .
(2.22). Using the nonnegativity of u ε we have

d dt R 2 |u ε |dx = d dt R 2 u ε dx = ε R 2 ∆ x u ε dx - R 2 div x (u ε ν(φ ε )) dx =0 + R 2 (b(φ ε ) -β) ≤c u ε dx ≤ c R 2 u ε dx.
Then, applying the Gronwall Lemma we gain

u ε (t, •) L 1 (R 2 ) ≤ u 0,ε (•) L 1 (R 2 ) e ct .
(2.23). We multiply (2.25) by u ε to obtain

d dt R 2 u 2 ε 2 dx = R 2 u ε ∂ t u ε dx = - R 2 div x (u ε ν(φ ε )) u ε dx + R 2 (b(φ ε ) -β)u 2 ε dx + ε R 2 ∆ x u ε u ε dx ≤ R 2 u ε ν(φ ε ).∇u ε dx + c R 2 u 2 ε dx -ε R 2 |∇ x u ε | 2 dx ≤ R 2 ∇ x u 2 ε 2 .ν(φ ε )dx + c R 2 u 2 ε dx -ε R 2 |∇ x u ε | 2 dx ≤ R 2 u 2 ε 2 div x (ν(φ ε )) dx + c R 2 u 2 ε dx -ε R 2 |∇ x u ε | 2 dx ≤ c T R 2 |u ε |dx + c R 2 u 2 ε dx -ε R 2 |∇ x u ε | 2 dx.
Integrating over (0, t) and then using the Gronwall Lemma we get

u ε (t, •) 2 L 2 (R 2 ) + 2ε e ct t 0 e -cs ∇ x u ε (s, •) 2 L 2 (R 2 ) ds ≤ e ct u ε (0, •) 2 L 2 (R 2 ) + ce ct .
(2.24). We multiply (2.25) by -∆ x u ε to obtain

d dt R 2 |∇ x u ε | 2 2 dx = R 2 ∇ x u ε • ∂ t ∇ x u ε dx = - R 2 ∆ x u ε ∂ t u ε dx = R 2 ∆ x u ε div x (u ε ν(φ ε )) dx - R 2 ∆ x u ε (b(φ ε ) -β)u ε dx -ε R 2 |∆ x u ε | 2 dx = - R 2 ∇ x u ε ∇ x (∇ x u ε .ν(φ) + u ε div x (ν(φ))) dx + R 2 (b(φ) -β)|∇ x u ε | 2 dx + R 2 b (φ)∇ x φ.∇ x u ε u ε dx -ε R 2 |∆ x u ε | 2 dx ≤ -ε R 2 |∆ x u ε | 2 dx + c R 2 |∇ x u ε | 2 dx + c T R 2 |∇ x ψ ε | 2 dx + R 2 ∇ x |∇ x u ε | 2 2 .ν(φ)dx + R 2 |∇ x u| 2 |∇ x ν(φ)| ≤c T dx + R 2 |∇ x u| 2 |div x (ν(φ)) | ≤c T dx + R 2 |∇ x u ε |u ε |∇ x div x (ν(φ)) |dx ≤ -ε R 2 |∆ x u ε | 2 dx + c T R 2 |∇ x u ε | 2 dx + c T R 2 |∇ x ψ ε | 2 dx + c T R 2 |∇ x div x (ν(φ)) | 2 dx.
Integrating over (0, t) and using the estimates in (2.9) and (2.18) we gain

∇ x u ε (t, •) L 2 (R 2 ) + ε ∆ x u ε (•, •) L 2 ((0,t)×R 2 ) ≤ ce ct . phi_e 3
Lemma 2.8. For every t ≥ 0, the following estimates on ψ ε hold (2.26)

∇ x ψ ε (t, •) L 2 (R 2 ) , D 2 x ψ ε L 2 ((0,t)×R 2 ) ≤ e Ct C,
D 2 x ψ ε (t, •) L 2 (R 2 ) , D 3 x ψ ε L 2 ((0,t)×R 2 ) ≤ e Ct C, onloc2.2 onloc2.2
(2.27)

ψ ε (t, •) L ∞ (R 2 ) ≤ e Ct C, onloc2.3 onloc2.3
(2.28)

D 3 x ψ ε (t, •) L 2 (R 2 ) , D 4 x ψ ε L 2 ((0,t)×R 2 ) ≤ e Ct C, onloc2.4 onloc2.4
(2.29)

∂ t ψ ε (t, •) L 1 (R 2 ) , ∂ t ψ ε (t, •) L 2 (R 2 ) , ∇ x ψ ε (t, •) L ∞ (R 2 ) ≤ e Ct C.
onloc2.5 onloc2.5

(2.30)

Proof. We multiply by ξ(a) the equation for ρ ε in system (2.1) then, integrating with respect to a, we get

∂ t ψ ε -µ∆ x ψ ε + div x ∞ 0 ρ ε χ 1 ξda vY θ = - ∞ 0 (d + p(a, u ε ))ρ ε ξda + ∞ 0 ρ ε (1 + εχ )ξ + εχξ da.
eq:psi eq:psi

(2.31)

(2.26). We multiply (2.31) by -∆ x ψ ε d dt R 2 |∇ x ψ ε | 2 2 dx = R 2 ∇ x ψ ε • ∂ t ∇ x ψ ε dx = - R 2 ∆ x ψ ε ∂ t ψ ε dx = -µ R 2 |D 2 x ψ ε | 2 dx + R 2 ∆ x ψ ε div x ∞ 0 ρ ε χ 1 ξda vY θ dx + R 2 ∞ 0 ρ ε ξ (d + p) ≤c ∆ x ψ ε da dx - R 2 ∞ 0 ρ ε (1 + εχ )ξ + εχξ ≤c ∆ x ψ ε da dx ≤ - µ 2 R 2 |D 2 x ψ ε | 2 dx + c R 2 div x ∞ 0 ρ ε χ 1 ξda vY θ 2 dx + c R 2 ψ 2 ε dx + c ∞ 0 R 2 ρ 2 ε dx da ≤ - µ 2 R 2 |D 2 x ψ ε | 2 dx + c R 2 ∇ x ∞ 0 ρ ε χ 1 ξda 2 |v| 2 Y 2 θ ≤c dx + c R 2 ∞ 0 ρ ε χ 1 ξda 2 |div x (v) | 2 Y 2 θ ≤c dx + c R 2 ∞ 0 ρ ε χ 1 ξda 2 (Y θ ) 2 |v| 2 ≤c (see (2.16)) |∇ x φ ε | 2 dx + c R 2 ψ 2 ε dx + c ∞ 0 R 2 ρ 2 ε dx da.
Since the functions

φ ε , (t, x) → ∞ 0 ρ ε χ 1 ξda
have the same structure as ψ ε , using the Gronwall Lemma and estimates (2.13), (2.18) we get (2.26).

(2.27). We multiply (2.31) by D 4

x ψ ε to have

d dt R 2 |∆ x ψ ε | 2 2 dx = R 2 ∆ x ψ ε • ∂ t ∆ x ψ ε dx = R 2 D 4 x ψ ε ∂ t ψ ε dx =µ R 2 ∆ x ψ ε D 4 x ψ ε dx - R 2 D 4 x ψ ε div x ∞ 0 ρ ε χ 1 ξda vY θ dx + R 2 ∞ 0 ρ ε ξ(-d -p(a, u ε ))D 4 x ψ ε da dx + R 2 ∞ 0 ρ ε (1 + εχ )ξ + εχξ D 4 x ψ ε da dx = -µ R 2 |D 3 x ψ ε | 2 dx + R 2 D 3 x ψ ε • ∇ x div x ∞ 0 ρ ε χ 1 ξda vY θ dx - R 2 ∇ x ∞ 0 ρ ε ξ (-d -p(a, u ε )) da • D 3 x ψ ε dx + R 2 ∞ 0 ∇ x ρ ε (1 + εχ )ξ + εχξ ≤c .D 3 x ψ ε da dx ≤ - µ 2 R 2 |D 3 x ψ ε | 2 dx + c R 2 ∇ x div x ∞ 0 ρ ε χ 1 ξda vY θ 2 dx + c R 2 ∇ x ∞ 0 ρ ε ξ (-d -p(a, u ε )) da 2 dx + c ∞ 0 R 2 |∇ x ρ ε | 2 dx da ≤ - µ 2 R 2 |D 3 x ψ ε | 2 dx + c R 2 D 2 x ∞ 0 ρ ε χ 1 ξda 2 |v| 2 Y 2 θ ≤c dx + c R 2 ∞ 0 ρ ε χ 1 ξda 2 |D 2 x v| 2 Y 2 θ ≤c dx + c R 2 ∞ 0 ρ ε χ 1 ξda 2 |v| 2 (Y θ ) 2 ≤c, see (2.16) |∇ x φ ε | 4 dx + c R 2 ∞ 0 ρ ε χ 1 ξda 2 |v| 2 (Y θ ) 2 ≤c, see (2.16) |D 2 x φ ε | 2 dx + c R 2 ∇ x ∞ 0 ρ ε χ 1 ξda 2 |∇ x v| 2 Y 2 θ ≤c dx + c R 2 ∞ 0 ρ ε χ 1 ξda 2 |∇ x v| 2 (Y θ ) 2 ≤c, see (2.16) |∇ x φ ε | 2 dx + c R 2 ∇ x ∞ 0 ρ ε χ 1 ξda 2 |v| 2 (Y θ ) 2 ≤c |∇ x φ ε | 2 dx + c R 2    ∞ 0 ρ ε ξ |∇ x d| ≤c da    2 dx + c R 2 ∞ 0 ∇ x ρ ε ξ (d + p(a, u ε )) ≤c da 2 dx + c R 2 ∞ 0 ρ ε ∂ u p(a, u ε ) ≤c ∇ x u ε ξda 2 dx + c ∞ 0 R 2 |∇ x ρ ε | 2 dx da ≤ - µ 2 R 2 |D 3 x ψ ε | 2 dx + c R 2 D 2 x ∞ 0 ρ ε χ 1 ξda 2 dx + c R 2 ∞ 0 ρ ε χ 1 ξda 2 dx + c R 2 ∇ x ∞ 0 ρ ε χ 1 ξda 2 dx + c R 2 |∇ x φ ε | 2 dx + c R 2 |∇ x φ ε | 4 dx + c R 2 |D 2 x φ ε | 2 dx + c R 2 ∇ x ∞ 0 ρ ε χ 1 ξda 4 dx + c R 2 ψ 2 ε dx+ + c R 2 |∇ x ψ ε | 2 dx + c R 2 ψ 4 ε dx + c R 2 |∇ x u ε | 4 dx + c ∞ 0 R 2 |∇ x ρ ε | 2 dx da,
Integrating over (0, t) we get

D 2 x ψ ε (t, •) 2 L 2 (R 2 ) + µ D 3 x ψ ε 2 L 2 ((0,t)×R 2 ) ≤ D 2 x ψ ε (0, •) 2 L 2 (R 2 ) + c t 0 ∞ 0 ρ ε χ 1 ξda (s) 2 H 2 (R 2 ) ds + c t 0 φ ε (s, •) 2 H 2 (R 2 ) ds + c t 0 ∇ x ∞ 0 ρ ε χ 1 ξda (s) 4 L 4 (R 2 ) ds + c t 0 ∇ x φ ε (s, •) 4 L 4 (R 2 ) ds + c t 0 ψ ε (s, •) 2 L 2 (R 2 ) ds + c t 0 ∇ x ρ ε (s, •, •) 2 L 2 ((0,∞)×R 2 ) ds + c t 0 ∇ x ψ ε (s, •) 2 L 2 (R 2 ) ds + c t 0 R 2 ψ ε (s, •) 4 L 4 (R 2 ) ds + c t 0 ∇ x u ε (s, •) 4 L 4 (R 2 ) ds + c t 0 ∇ x ρ ε (s, •, •) 2 L 2 ((0,∞)×R 2 ) ds.
Using (2.13), (2.18), (2.26), (2.24) and the embedding H 1 (R 2 ) ⊂ L 4 (R 2 ) we gain (2.27).

(2.28). Directly follows from (2.18), (2.26), (2.27) and the embedding

H 2 (R 2 ) ⊂ L ∞ (R 2 ).
(2.29). We multiply (2.31) by

-D 6 x ψ ε d dt R 2 |D 3 x ψ ε | 2 2 dx = R 2 D 3 x ψ ε • ∂ t D 3 x ψ ε dx = - R 2 D 6 x ψ ε ∂ t ψ ε dx = -µ R 2 ∆ x ψ ε D 6 x ψ ε dx + R 2 D 6 x ψ ε div x ∞ 0 ρ ε χ 1 ξda vY θ dx - R 2 ∞ 0 ρ ε ξ (-d -p(a, u ε )) D 6 x ψ ε da dx - R 2 ∞ 0 ρ ε (1 + εχ )ξ + εχξ ≤c D 6 x ψ ε da dx = -µ R 2 |D 4 x ψ ε | 2 dx + R 2 D 4 x ψ ε ∆ x div x ∞ 0 ρ ε χ 1 ξda vY θ dx - R 2 ∆ x ∞ 0 ρ ε ξ (-d -p(a, u ε )) da D 4 x ψ ε dx - R 2 ∆ x ∞ 0 ρ ε (1 + εχ )ξ + εχξ da D 4 x ψ ε dx ≤ - µ 2 R 2 |D 4 x ψ ε | 2 dx + c R 2 ∆ x div x ∞ 0 ρ ε χ 1 ξda vY θ 2 dx + c R 2 ∆ x ∞ 0 ρ ε dξda 2 dx + c R 2 ∆ x ∞ 0 ρ ε p(a, u ε )ξda 2 dx + c R 2 ∞ 0 |∆ x ρ ε | 2 da dx ≤ - µ 2 R 2 |D 4 x ψ ε | 2 dx + c R 2 D 3 x ∞ 0 ρ ε χ 1 ξda 2 |v| 2 Y 2 θ ≤c dx + c R 2 D 2 x ∞ 0 ρ ε χ 1 ξda 2 |∇ x v| 2 Y 2 θ ≤c dx + c R 2 ∇ x ∞ 0 ρ ε χ 1 ξda 2 |D 2 x v| 2 Y 2 θ ≤c dx + c R 2 ∞ 0 ρ ε χ 1 ξda 2 |D 3 x v| 2 Y 2 θ ≤c dx + c R 2 D 2 x ∞ 0 ρ ε χ 1 ξda 2 |v| 2 (Y θ ) 2 ≤c |∇ x φ ε | 2 dx + c R 2 ∞ 0 ρ ε χ 1 ξda 2 |D 2 x v| 2 (Y θ ) 2 ≤c |∇ x φ ε | 2 dx + c R 2 ∇ x ∞ 0 ρ ε χ 1 ξda 2 |∇ x v| 2 (Y θ ) 2 ≤c |∇ x φ ε | 2 dx + c R 2 ∇ x ∞ 0 ρ ε χ 1 ξda 2 |v| 2 (Y θ ) 2 ≤c |∇ x φ ε | 4 dx + c R 2 ∇ x ∞ 0 ρ ε χ 1 ξda 2 |v| 2 (Y θ ) 2 ≤c |D 2 x φ ε | 2 dx + c R 2 ∞ 0 ρ ε χ 1 ξda 2 |∇ x v| 2 (Y θ ) 2 ≤c |∇ x φ ε | 4 dx + c R 2 ∞ 0 ρ ε χ 1 ξda 2 |∇ x v| 2 (Y θ ) 2 ≤c |D 2 x φ ε | 2 dx + c R 2 ∞ 0 ρ ε χ 1 ξda 2 |v| 2 (Y θ ) 2 ≤c |∇ x φ ε | 6 dx + c R 2 ∞ 0 ρ ε χ 1 ξda 2 |v| 2 (Y θ ) 2 ≤c |D 2 x φ ε | 2 |∇ x φ ε | 2 dx + c R 2 ∞ 0 ρ ε χ 1 ξda 2 |v| 2 (Y θ ) 2 ≤c |D 3 x φ ε | 2 dx + c R 2 |∇ x d| 2 ≤c •|∇ x ψ ε | 2 dx + c R 2 d 2 ≤c |∆ x ψ ε | 2 dx + c R 2 |∆ x d| 2 ≤c |ψ ε | 2 dx + c R 2 ∞ 0 ∇ x ρ ε • ∇ x u ε ξ ∂ u p(a, u ε ) ≤c da 2 dx + c R 2 ∞ 0 ∆ x ρ ε ξ p(a, u ε )da 2 dx + c R 2 ∞ 0 ρ ε ξ ≤c T ∂ 2 uu p(a, u ε ) ≤c |∇ x u ε | 2 da 2 dx + c R 2 ∞ 0 ρ ε ξ ≤c T ∂ u p(a, u ε ) ≤c ∆ x u ε da 2 dx + c R 2 ∞ 0 |∆ x ρ ε | 2 da dx ≤ - µ 2 R 2 |D 4 x ψ ε | 2 dx + c R 2 D 3 x ∞ 0 ρ ε χ 1 ξda 2 dx + c R 2 D 2 x ∞ 0 ρ ε χ 1 ξda 2 dx + c R 2 ∇ x ∞ 0 ρ ε χ 1 ξda 2 dx + c R 2 ∞ 0 ρ ε χ 1 ξda 2 dx + c R 2 D 2 x ∞ 0 ρ ε χ 1 ξda 4 dx + c R 2 ∇ x ∞ 0 ρ ε χ 1 ξda 4 dx + c R 2 |D 3 x φ ε | 2 dx + c R 2 |D 2 x φ ε | 2 dx + c R 2 |D 2 x φ ε | 4 dx + c R 2 |∇ x φ ε | 2 dx + c R 2 |∇ x φ ε | 4 dx + c R 2 |∇ x φ ε | 6 dx + c R 2 |∇ x φ ε | 8 dx + c R 2 ψ 2 ε dx + c R 2 |∇ x ψ ε | 2 dx + c R 2 |D 2 x ψ ε | 2 dx + c R 2 |∇ x ψ ε | 4 dx + c T R 2 |∇ x u ε | 4 dx + c T R 2 |∆ x u ε | 2 dx + c R 2 ∞ 0 |∆ x ρ ε | 2 da dx.
We integrate over (0, t) and using the estimates (2.19), (2.18), (2.26), (2.27), (2.24), together with the embedding H 1 (R 2 ) ⊂ L p (R 2 ) for very 1 ≤ p < ∞ give us

D 3 x ψ ε (t, •) 2 L 2 (R 2 ) + µ D 4 x ψ ε 2 L 2 ((0,t)×R 2 ) ≤ ce ct .
(

(R 2 ) ⊂ L ∞ (R 2 ).
n:nonloc Lemma 2.9 (Estimates on the trace of ρ ε at a = 0). Define ρ 0 ε (t, x) = ρ ε (t, 0, x). For every t ≥ 0 the following estimates hold

ρ 0 ε (t, •) L 1 (R 2 ) , D 2 x ρ 0 ε L 1 ((0,t)×R 2 ) , ∂ t ρ 0 ε L 1 ((0,t)×R 2 ) ≤ e Ct C, ρ 0 ε (t, •) L 2 (R 2 ) , ∇ x ρ 0 ε (t, •) L 2 (R 2 ) ≤ e Ct C, ∇ x ρ 0 ε L 2 ((0,t)×R 2 ) , D 2 x ρ 0 ε L 2 ((0,t)×R 2 ) , ∂ t ρ 0 ε L 2 ((0,t)×R 2 ) ≤ e Ct C, ρ 0 ε (t, •) L ∞ (R 2 ) ≤ e Ct C.
Proof. Directly deduce from the definition of ρ ε at a = 0, using Lemma 2.8, the estimates in (2.17),

(2.18), and the assumptions on A(φ) in (1.8).

linftyro Lemma 2.10 (L ∞ estimate on ρ ε ). We have that linftyro linftyro

(2.32) ρ ε (t, •, •) L ∞ ((0,∞)×R 2 ) ≤ Ce Ct ,
for every t ≥ 0.

Proof. Let M be a positive constant that will be fixed later. Define

ρ ε (t, a, x) = e -M t ρ ε (t, a, x). ρ ε satisfies the problem deleps-o deleps-o (2.33)          ∂ t ρ ε + M ρ ε + ∂ a ρ ε + div x (ρ ε χ 1 (a)vY θ (φ ε -R)) = µ∆ x ρ ε + ε∂ a (χ(a)∂ a ρ ε ) -(d + p)ρ ε , ρ ε (t, 0, x) = A (φ ε ) ∞ 0 ρ ε (t, a, x)χ 3 (a)da ω(t, x), ρ ε (0, a, x) = ρ 0,ε (a, x).
Moreover, for any given T > 0 there exists a sufficiently large constant B > 0 such that for any t ≤ T and x ∈ R 2 (2.3), (2.11), and (2.9) imply eq:ore eq:ore

(2.34) 0 ≤ ρ ε (t, 0, x), ρ 0,ε (a, x) ≤ B.
Consider

η(ξ) = (ξ -B)1 (B,∞) (ξ),
and observe that

η (ξ) = 1 (B,∞) (ξ), ξη (ξ) = η(ξ) + Bη (ξ).
From the equation (2.33) we have

d dt ∞ 0 R 2 η(ρ ε )dx da = ∞ 0 R 2 η (ρ ε )∂ t ρ ε dx da = -M ∞ 0 R 2 η (ρ ε )ρ ε dx da - ∞ 0 R 2 η (ρ ε )∂ a ρ ε dx da - ∞ 0 R 2 div x (ρ ε χ 1 vY θ ) η (ρ ε )dx da + µ ∞ 0 R 2 η (ρ ε )∆ x ρ ε dx da - ∞ 0 R 2 η (ρ ε )(d + p)ρ ε dx da ≤0 +ε ∞ 0 R 2 η (ρ ε )∂ a (χ(a)∂ a ρ ε )dx da = -M ∞ 0 R 2 η(ρ ε )dx da ≤0 -M B ∞ 0 R 2 η (ρ ε )dx da + R 2 η(ρ ε (t, 0, x))dx - ∞ 0 R 2 div x ((ρ ε -B)χ 1 vY θ ) η (ρ ε )dx da -B ∞ 0 R 2 div x (χ 1 vY θ ) η (ρ ε )dx da -µ ∞ 0 R 2 η (ρ ε )χ(a) (∇ x ρ ε ) 2 dx da ≤0 -ε R 2 η (ρ ε (t, 0, x))∂ a ρ ε (t, 0, x)dx -ε ∞ 0 R 2 η (ρ ε )(∂ a ρ ε ) 2 dadx ≤0 ≤ R 2 η(ρ ε (t, 0, x))dx + ∞ 0 R 2 (ρ ε -B)χ 1 (v • ∇ x ρ ε )Y θ η (ρ ε )dx da =0 -B ∞ 0 R 2 (M + div x (χ 1 vY θ ))η (ρ ε )dx da -ε R 2 η (ρ ε (t, 0, x))∂ a ρ ε (t, 0, x)dx.
Thanks to (2.9), (2.34), and (2.30) we have for M large enough

η(ρ ε (t, 0, x)) = 0, η (ρ ε (t, 0, x)) = 0, M + div x (χ 1 vY θ ) ≥ 0, which imply d dt ∞ 0 R 2 η(ρ ε )dx da ≤ 0.
Thus, we integrate over (0, t) we obtain 0

≤ ∞ 0 R 2 η(ρ ε (t, a, x))dx da ≤ ∞ 0 R 2 η(ρ 0,ε (a, x))dx da = 0, and
η(ρ ε (t, a, x)) = 0.
As a consequence we have that ρ ε ≤ B.

m:l1ro_a

Lemma 2.11 (L 1 estimate on ∂ a ρ ε ). We have that

∂ a ρ ε (t, •, •) L 1 ((0,∞)×R 2 ) ≤ Ce Ct
for every t ≥ 0 and a suitable constant C independent on ε.

Proof. Differentiating the equation in (2.14) with respect to a we get

∂ 2 ta ρ ε + ∂ 2 aa ρ ε + div x (∂ a ρ ε χ 1 vY θ ) + div x ρ ε χ 1 vY θ = µ∆ x ∂ a ρ ε + ε∂ 2 aa (χ(a)∂ a ρ ε ) -(∂ a d + ∂ a p(a, u ε ))ρ ε -(d + p)∂ a ρ ε .
eq:rea eq:rea

(2.35) Then d dt ∞ 0 R 2 |∂ a ρ ε |dx da = ∞ 0 R 2 ∂ 2 ta ρ ε sign (∂ a ρ ε ) dx da = ∞ 0 R 2 (ε∂ 2 aa (χ(a)∂ a ρ ε ) -∂ 2 aa ρ ε )sign (∂ a ρ ε ) dx da +µ ∞ 0 R 2 ∆ x ∂ a ρ ε sign (∂ a ρ ε ) dx da ≤0 - ∞ 0 R 2 div x (∂ a ρ ε χ 1 vY θ ) sign (∂ a ρ ε ) dx da =0 - ∞ 0 R 2 ∇ x ρ ε • vχ 1 Y θ sign (∂ a ρ ε ) dx da - ∞ 0 R 2 ρ ε χ 1 div x (v) Y θ sign (∂ a ρ ε ) dx da - ∞ 0 R 2 ρ ε χ 1 v • ∇ x φ ε Y θ sign (∂ a ρ ε ) dx da - ∞ 0 R 2 (∂ a d + ∂ a (p(a, u ε ))) ≤c ρ ε sign (∂ a ρ ε ) dx da - ∞ 0 R 2 (d + p)|∂ a ρ ε |dx da ≤0 ≤ R 2 |ε∂ a (χ(0)∂ a ρ ε (t, 0, x)) -∂ a ρ ε (t, 0, x)|dx + c ∇ x ρ ε (t, •, •) L 2 ((0,∞)×R 2 ) v L 2 (R 2 ) + c ρ ε (t, •, •) L 2 ((0,∞)×R 2 ) div x (v) L 2 (R 2 ) + c ρ ε (t, •, •) L 2 ((0,∞)×R 2 ) ∇ x φ ε (t, •) L 2 (R 2 ) + c ρ ε (t, •, •) L 1 ((0,∞)×R 2 ) .
Since

ε∂ a (χ(0)∂ a ρ ε (t, 0, x)) -∂ a ρ ε (t, 0, x) = ∂ t ρ 0 ε -µ∆ x ρ 0 ε + dρ 0 ε ,

BV in x

Lemma 2.12 (BV estimate w.r.t. x on u ε ). For every t ≥ 0 the following estimate holds

∇ x u ε (t, •) L 1 (R 2 ) ≤ Ce Ct .
:BV in x :BV in x

(2.36)

Proof. From the equation of u ε in (2.25) we have the equation

∂ t ∇ x u ε + ∇ x (∇u ε • ν(φ ε ) + u ε div x (ν(φ ε ))) = (b(φ ε ) -β)∇ x u ε + b (φ ε )∇ x φ ε u ε + ε∇ x ∆ x u ε .
We define

sign (∇ x u ε ) = (sign (∂ x 1 (u ε )) , sign (∂ x 2 (u ε ))),
where x = (x 1 , x 2 ). Then, using the L ∞ bounds on ∇ x φ ε and ν(φ ε ) in (2.30), (2.5), we get

d dt R 2 |∇ x u ε |dx = R 2 sign(∇ x u ε )∂ t ∇ x u ε dx = - R 2 ∇ x (∇ x u ε • ν(φ ε )) sign(∇ x u ε )dx =0 - R 2 ∇ x (u ε div x (ν(φ ε ))) sign(∇ x u ε )dx + ε R 2 ∇ x ∆ x u ε sign(∇ x u ε )dx ≤0 + R 2 b (φ ε )∇ x φ ε ≤c T u ε sign(∇ x u ε )dx + R 2 (b(φ ε ) -β) ≤c |∇ x u ε |dx ≤ R 2 |∇ x u ε ||div x (ν(φ ε )) | dx - R 2 u ε ∇ x (div x (ν(φ ε ))) sign (∇ x u ε ) dx + c T R 2 u ε dx + c R 2 |∇ x u ε |dx ≤c R 2 |∇ x u ε |dx + c T R 2 |u ε |dx + c R 2 u 2 ε dx + c R 2 |∇ x u ε | 2 dx + R 2 |div x (ν(φ ε )) | 2 dx + R 2 |∇ x (div x (ν(φ ε )))| 2 dx.
We obtain the desired inequality (2.36) integrating over (0, t), using the Gronwall Lemma and the estimates in (2.6) and (2.9) and Lemma 2.7.

BV in t

Lemma 2.13 (BV estimate w.r.t. t on u ε ). For every t ≥ 0, the following estimate holds

∂ t u ε (t, •) L 1 (R 2 ) ≤ Ce Ct .
:BV in t :BV in t

(2.37)

Proof. From the definition of ν(φ) in (1.2) we compute

∂ t ν(φ ε ) = κ ∂ t φ ε * ∇ x η 1 + φ ε * ∇ x η 2 3/2 ,
then from (2.30) we have

∂ t ν(φ ε ) L ∞ (R 2 ;R 2 ) ≤ κ ∇ x η L ∞ (R 2 ;R 2 ) ∂ t φ ε L 1 (R 2 ,R) ≤ Ce Ct .
Similarly, explicit computations give us eq:divnu eq:divnu

(2.38) div x (ν(φ ε )) = κ φ ε * ∆ x η 1 + φ ε * ∇ x η 2 3/2 ,
and

∂ t div x (ν(φ ε )) = κ ∂ t φ ε * ∆ x η 1 + φ ε * ∇ x η 2 3/2 -3κ(φ ε * ∆ x η) φ ε * ∇ x η 1 + φ ε * ∇ x η 2 ∂ t φ ε * ∇ x η 1 + φ ε * ∇ x η 2 2 .
Therefore, from (2.17), (2.30), we obtain

∂ t div x (ν(φ ε )) L 2 (R 2 ;R) ≤ κ ∂ t φ ε * ∆ x η L 2 (R 2 ;R) + 3κ φ ε * ∆ x η L ∞ (R 2 ;R) ∂ t φ ε * ∇ x η L 2 (R 2 ;R 2 ) ≤ κ ∂ t φ ε L 1 (R 2 ;R) ∆ x η L 2 (R 2 ;R) + 3 ∆ x η L ∞ (R 2 ;R) ∇ x η L 2 (R 2 ;R 2 ) φ ε L 1 (R 2 ;R) ≤ Ce Ct .
The estimates above allow us to prove (2.37). From the equation (2.25) for u ε we have

∂ 2 tt u ε + div x (∂ t u ε ν(φ ε )) + div x (u ε ∂ t ν(φ ε )) = (b(φ ε ) -β)∂ t u ε + b (φ ε )∂ t φ ε u ε + ε∆ x ∂ t u ε , then we consider d dt R 2 |∂ t u ε | dx = - R 2 div x (∂ t u ε ν(φ ε )) sign (∂ t u ε ) dx =0 +ε R 2 ∆ x ∂ t u ε sign (∂ t u ε ) dx ≤0 - R 2 (∇u ε ∂ t ν(φ ε ) ≤c T +u ε ∂ t div x (ν(φ ε )))sign (∂ t u ε ) dx + R 2 (b(φ ε ) -β) ≤c |∂ t u ε |dx + R 2 b (φ ε )∂ t φ ε u ε sign (∂ t u ε ) dx ≤c T R 2 |∇ x u ε |dx + c R 2 u 2 ε dx + R 2 |∂ t div x (ν(φ ε )) | 2 dx + c T R 2 |∂ t φ ε |dx + c R 2 |∂ t u ε |dx.
Integrating on (0, t) we obtain (2.37) thanks to (2.23), (2.30) and (2.36) and the Gronwall Lemma.

We are now ready to prove the compactness of the families {ρ ε } ε and {u ε } ε , and the first part of Theorem 1.1, establishing the existence of entropy solutions for the system (1.1).

em:exist Lemma 2.14 (Strong compactness of {ρ ε } ε and {u ε } ε ). There exists a couple of functions (u, ρ) and a sequence {ε k } k∈N ∈ (0, ∞), ε k → 0, such that, for every T > 0

ρ ε k → ρ, a.e. in (0, T ) × (0, ∞) × R 2 and in L p loc ((0, ∞) × (0, ∞) × R 2 ), 1 ≤ p < ∞, ρ(•, •, •) ≥ 0, ρ(t, •, x) ∈ BV (0, ∞), for a.e. (t, x) ∈ (0, ∞) × R 2 , ρ ∈ L ∞ (0, T ; L 1 ((0, ∞) × R 2 )) ∩ L ∞ ((0, T ) × (0, ∞) × R 2 ) ∩ L 2 ((0, T ) × (0, ∞); H 2 (R 2 )), ence rho ence rho (2.39) and u ε k h → u a.e. in (0, T ) × R 2 and in L p loc ((0, ∞) × R 2 ), 1 ≤ p < ∞, u(•, •) ≥ 0, u ∈ L ∞ ((0, T ) × R 2 ) ∩ BV ((0, T ) × R 2 ).
rgence u rgence u

(2.40)

Proof. Rewrite equation of ρ ε in (2.14)

∂ t ρ ε + div (a,x) ρ 2 ε /2 ρ 2 ε /2 = (ρ ε -1)∂ a ρ ε + ρ ε div x (ρ ε ) -div x (ρ ε χ 1 (a)vY θ (φ ε -R)) + µ∆ x ρ ε + ε∂ a (χ(a)∂ a ρ ε ) -(d + p)ρ ε . q:convex q:convex (2.41) Let η ∈ C 2 (R 2 ) be a convex entropy with flux Q ∈ C 2 (R; R 2 ) such that Q (ρ) = ρη (ρ) ρη (ρ) .
Multiplying both sides of (2.41) by η (ρ ε ) we get

∂ t η(ρ ε ) + div (a,x) (Q(ρ ε )) = η (ρ ε ) (ρ ε -1)∂ a ρ ε + ρ ε div x (ρ ε ) -div x (ρ ε χ 1 (a)vY θ (φ ε -R)) + µ∆ x ρ ε -(d + p)ρ ε L 1,ε + ∂ a (εη (ρ ε )χ(a)∂ a ρ ε ) L 2,ε -εη (ρ ε )χ(a)(∂ a ρ ε ) 2 L 3,ε
.

eq:entr eq:entr

(2.42)

For every K ⊂⊂ (0, ∞) × (0, ∞) × R 2 , thanks to Lemmas 2.3, 2.4, 2.5, 2.8, 2.10, 2.11 and the estimates in (2.17), (2.18), we have

L 1,ε L 1 (K) ≤ η (ρ ε ) L ∞ (K) ( ρ ε L ∞ (K) + 1) ∂ a ρ ε L 1 (K) + ρ ε L 2 (K) ∇ x ρ ε L 2 (K) + c ρ ε L 2 (K) ∇ x φ ε L 2 (K) + c φ ε L 2 (K) ρ ε L 2 (K) + c φ ε L 2 (K) ∇ x ρ ε L 2 (K) + c ∆ x ρ ε L 2 (K) + d L ∞ (K) ρ ε L 1 (K) ≤ c, εη (ρ ε )χ(a)∂ a ρ ε L 2 (K) ≤ε η (ρ ε ) L ∞ (K) χ(a)∂ a ρ ε L 2 (K) ≤ c √ ε → 0, L 3,ε L 1 (K) ≤ε η (ρ ε ) L ∞ (K) χ(a)∂ a ρ 2 ε L 1 (K) ≤ c.
Applying the Murat Lemma in [START_REF] Murat | L'injection du cône positif de H -1 dans W -1, q est compacte pour tout q < 2[END_REF] to get

{∂ t η(ρ ε ) + div (a,x) (Q(ρ ε ))} ε is compact in H -1 loc ((0, ∞) × (0, ∞) × R 2 ),
this implies the strong compactness of {ρ ε } ε .

Thanks to Lemmas 2.21, 2.13, 2.12,

{u ε } ε is bounded in L ∞ ((0, T ) × R 2 ) ∩ BV ((0, T ) × R 2 ) so that
Helly's Theorem applies.

Lemma 2.15. The couple of functions (u, ρ) introduced in Lemma 2.14 is an entropy solution of (1.1)

in the sense of Definition 1.2.

Proof. It is clear that the couple (u, ρ) is a weak solution of (1.1) in the sense of Definition 1.1 thanks to the strong convergence results in Lemma 2.14. In particular, the fact that ρ is a weak solution comes directly from [START_REF] Panov | Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux[END_REF]Th. 5].

We obtain (1.14) as in [2, Lemma 2.10], then we only have to verify that (1.15) holds. Let ξ ∈ C ∞ (R 4 ) be a nonnegative text function with compact support and c ∈ R be a constant. Multiplying the equation of u ε in (2.1) by sign(u ε -c) we have

∂ t |u ε -c| + div x (|u ε -c|ν(φ ε )) + c sign(u ε -c)div x (ν(φ ε )) ≤ sign(u ε -c)(b(φ ε ) -β)u ε + ε∆ x |u ε -c|. Then, ∞ 0 ∞ 0 R 2 (|u ε -c|∂ t ξ + |u ε -c|ν(φ ε ) • ∇ x ξ + sign(u ε -c)(b(φ ε ) -β)u ε ξ +ε|u ε -c|∆ x ξ -c sign(u ε -c)div x (ν(φ ε )) ξ) dx da dt + ∞ 0 R 2
|u ε,0 (x) -c|ξ(0, a, x)dx da ≥ 0.

By taking the limit for ε → 0, we get (1.15).

Uniqueness and stability tability

In this section we establish the inequality (1.16), which concludes the proof of Theorem 1.1. To this end we introduce the following preliminary lemma.

doubling Lemma 3.1. Let (u 1 , ρ 1 ) and (u 2 , ρ 2 ) be two entropy solutions of (1.1) obtained from the initial data (u 1,0 , ρ 1,0 ) and (u 2,0 , ρ 2,0 ) respectively. For every nonnegative test function

ξ ∈ C ∞ c (R 4 ) the following inequalities hold ∞ 0 ∞ 0 R 2 |ρ 1 -ρ 2 | (∂ t ξ + ∂ a ξ) -div x (|ρ 1 -ρ 2 |χ 1 v(Y θ (φ 1 -R) + Y θ (φ 2 -R))) ξ + µ∆ x |ρ 1 -ρ 2 |ξ -|ρ 1 -ρ 2 |dξ -|ρ 1 -ρ 2 |p(a, u 1 )ξ -sign (ρ 1 -ρ 2 ) (p(a, u 1 ) -p(a, u 2 ))ρ 2 ξ dx da dt + ∞ 0 R 2 |ρ 1 (t, 0 + , x) -ρ 2 (t, 0 + , x)|ξ(t, 0, x)dx dt + ∞ 0 R 2 |ρ 1,0 (a, x) -ρ 2,0 (a, x)|ξ(0, a, x) dx da ≥ ∞ 0 ∞ 0 R 2 sign (ρ 1 -ρ 2 ) χ 1 (ρ 2 div x (vY θ (φ 1 -R)) -ρ 1 div x (v(x)Y θ (φ 2 -R))) ξ dx da dt,
for rho for rho (3.1) where

φ i (t, x) := ∞ 0 ρ i (t, a, x)χ 2 (a)da, for i = 1, 2,
and

∞ 0 ∞ 0 R 2 |u 1 -u 2 |∂ t ξ -div x (|u 1 -u 2 |(ν(φ 1 ) + ν(φ 2 ))) ξ + |u 1 -u 2 |(b(φ 1 ) -β)ξ + sign (u 1 -u 2 ) (b(φ 1 ) -b(φ 2 ))u 2 ξ dx da dt + ∞ 0 R 2 |u 1,0 (x) -u 2,0 (x)|ξ(0, a, x) dx da ≥ ∞ 0 ∞ 0 R 2 sign (u 1 -u 2 ) (u 2 div x (ν(φ 1 )) -u 1 div x (ν(φ 2 ))) ξ dx da dt.
ng for u ng for u

(3.2)
Proof. We double the variables and write

ρ 1 = ρ 1 (t, a, x), ρ 2 = ρ 2 (s, b, y), φ 1 = φ 1 (t, x), φ 2 = φ 2 (s, y) u 1 = u 1 (t, x) u 2 = u 2 (s, y),
where x = (x 1 , x 2 ) and y = (y 1 , y 2 ).

Consider the test function

Ξ n (t, s, a, b, x, y) = ξ t + s 2 , a + b 2 , x + y 2 λ n s -t 2 λ n b -a 2 λ n y 1 -x 1 2 λ n y 2 -x 2 2 ,
where

λ n (u) = nλ(nu), λ ∈ C ∞ (R), λ ≥ 0, λ L 1 (R) = 1, supp (λ) ⊂ [-1, 1].
To prove inequality (3.1) we follow the doubling of variables argument appearing in [2, Lemma 3.1],

and use the regularity of p and the L ∞ bounds on u 1 , u 2 . Then we have only to verify (3.2). We write (1.15) for u 1 (t, x) using u 2 (s, y) as a constant and integrate over (s, y)

(|u 1 -u 2 |∂ t Ξ n -div x (|u 1 -u 2 |ν(φ 1 )) Ξ n + sign (u 1 -u 2 ) (b(φ 1 ) -β)u 1 Ξ n ) dx
and we write (1.15) for u 2 (s, y) using u 1 (t, x) as a constant and integrate over (t, x) As n → ∞ we get (3.2).

(|u 1 -u 2 |∂ s Ξ n -div y (|u 1 -u 2 |ν(φ 2 )) Ξ n -sign (u 1 -u 2 ) (b(φ 2 ) -β)u 2 Ξ n ) dx
(|u 1 -u 2 |(∂ t Ξ n + ∂ s Ξ n ) -div x (|u 1 -u 2 |ν(φ 1 )) Ξ n -div y (|u 1 -u 2 |ν(φ 2 )) Ξ n +|u 1 -u 2 |(b(φ 1 ) -β)Ξ n + sign (u 1 -u 2 ) (b(φ 1 ) -b(φ 2 ))u 2 Ξ n ) dx
Lemma 3.2. For every t ≥ 0, the following inequality holds

φ 1 (t, •) -φ 2 (t, •) 2 L 2 (R 2 ) + µe Ct t 0 R 2 e -Cs |∇ x (φ 1 -φ 2 )| 2 dx ds ≤ Ce Ct ρ 1,0 -ρ 2,0 2 L 1 ((0,∞)×R 2 ) +Ce Ct t 0 ∞ 0 R 2 e -Cs (ρ 1 -ρ 2 ) 2 dx da ds + Ce Ct t 0 R 2
e -Cs (u 1 -u 2 ) 2 dx ds.

q:stab.6 q:stab.6

(3.5)

In particular, we have that ine:new ine:new

(3.6) t 0 R 2 e -Cs |∇ x (φ 1 -φ 2 )| 2 dxds 1/2 ≤ Ce Ct ρ 1,0 -ρ 2,0 L 1 ((0,∞)×R 2 ) + Ce Ct t 0 ρ 1 (s, •, •) -ρ 2 (s, •, •) 2 L 1 ((0,∞)×R 2 ) ds 1/2 + Ce Ct t 0 u 1 (s, •) -u 2 (s, •) 2 L 1 (R 2 ) ds 1/2 . Proof. Since φ ε k satisfies ∂ t φ ε k -µ∆ x φ ε k +div x ∞ 0 ρ ε k χ 1 da vY θ (φ ε k -R) = ∞ 0 ρ ε k χ 2 (-d -p(a, u ε k )) da + ∞ 0 ρ ε k (1 + εχ )χ 2 + εχχ 2 da,
as k → ∞ we get the equation of φ. Then, subtracting the equation for φ 2 from the equation for φ 1 we obtain

∂ t (φ 1 -φ 2 ) -µ∆ x (φ 1 -φ 2 ) + div x ∞ 0 (ρ 1 -ρ 2 )χ 1 da vY θ (φ 1 -R) + div x ∞ 0 ρ 2 χ 1 da v(Y θ (φ 1 -R) -Y θ (φ 2 -R)) = ∞ 0 (ρ 1 -ρ 2 ) χ 2 -dχ 2 -p(a, u 1 )χ 2 da - ∞ 0 ρ 2 (p(a, u 1 ) -p(a, u 2
)) χ 2 da.

q:stab.5 q:stab.5

Then d dt R 2 (φ 1 -φ 2 ) 2 2 dx + µ R 2 |∇ x (φ 1 -φ 2 )| 2 dx = R 2 (φ 1 -φ 2 ) ∞ 0 (ρ 1 -ρ 2 ) χ 2 -dχ 2 -p(a, u 1 )χ 2 da dx - R 2 (φ 1 -φ 2 ) ∞ 0 ρ 2 (p(a, u 1 ) -p(a, u 2 )) ≤ c|u 1 -u 2 | χ 2 da dx - R 2 (φ 1 -φ 2 )div x ∞ 0 (ρ 1 -ρ 2 )χ 1 da vY θ (φ 1 -R) dx - R 2 (φ 1 -φ 2 )div x ∞ 0 ρ 2 χ 1 da v(Y θ (φ 1 -R) -Y θ (φ 2 -R)) dx ≤ c R 2 (φ 1 -φ 2 ) 2 dx + ce ct R 2 (u 1 -u 2 ) 2 dx + ce ct ∞ 0 R 2 (ρ 1 -ρ 2 ) 2 dx da + µ 2 R 2 |∇ x (φ 1 -φ 2 )| 2 dx. (3.7)
Using Gronwall Lemma we get

φ 1 (t, •) -φ 2 (t, •) 2 L 2 (R 2 ) + µe ct t 0 R 2 e -cs |∇ x (φ 1 -φ 2 )| 2 dx ds ≤ e ct φ 1,0 -φ 2,0 2 L 2 (R 2 ) + ce ct t 0 ∞ 0 R 2 e -cs (ρ 1 -ρ 2 ) 2 dx da ds + ce ct t 0 R 2 e -cs (u 1 -u 2 ) 2 dx ds.
Finally, we obtain (3.5) and (3.6) from the definition of φ 1 and φ 2 .

We are now ready to complete the proof of Theorem 1.1.

Proof. Our goal is to prove the inequality (1.16). We rewrite (3.1) and (3.2) as

∞ 0 ∞ 0 R 2 |ρ 1 -ρ 2 | (∂ t ξ n + ∂ a ξ n ) -|ρ 1 -ρ 2 |χ 1 (Y θ (φ 1 -R) + Y θ (φ 2 -R))v • ∇ x ξ n +µ|ρ 1 -ρ 2 |∆ x ξ n -|ρ 1 -ρ 2 |dξ n -|ρ 1 -ρ 2 |p(a, u 1 )ξ n -sign (ρ 1 -ρ 2 ) (p(a, u 1 ) -p(a, u 2 ))ρ 2 ξ n dx da dt + ∞ 0 R 2 |ρ 1 (t, 0 + , x) -ρ 2 (t, 0 + , x)|ξ n (t, 0, x)dx dt + ∞ 0 R 2 |ρ 1,0 (a, x) -ρ 2,0 (a, x)|ξ n (0, a, x) dx da ≥ ∞ 0 ∞ 0 R 2 sign (ρ 1 -ρ 2 ) χ 1 (ρ 2 div x (vY θ (φ 1 -R)) -ρ 1 div x (v(x)Y θ (φ 2 -R))) ξ n dx da dt, (3.8)
and

∞ 0 ∞ 0 R 2 |u 1 -u 2 |∂ t ξ n + |u 1 -u 2 |(ν(φ 1 ) + ν(φ 2 )) • ∇ x ξ n + |u 1 -u 2 |(b(φ 1 ) -β)ξ n + sign (u 1 -u 2 ) (b(φ 1 ) -b(φ 2 ))u 2 ξ n dx da dt + ∞ 0 R 2 |u 1,0 (x) -u 2,0 (x)|ξ n (0, a, x) dx da ≥ ∞ 0 ∞ 0 R 2 sign (u 1 -u 2 ) (u 2 div x (ν(φ 1 )) -u 1 div x (ν(φ 2 ))) ξ n dx da dt, (3.9)
where {ξ n } n is a sequence of nonnegative test functions approximating the characteristic function of the strip (-∞, t) × R × R 2 . Sending n → ∞, we have that Using (3.6), we have

u 1 (t, •) -u 2 (t, •) L 1 (R 2 ) ≤ u 1,0 -u 2,0 L 1 (R 2 ) - t 0 R 2 sign (u 1 -u 2 ) (u 2 div x (ν(φ 1 )) -u 1 div x (ν(φ 2 ))) dx ds + t 0 R 2 |u 1 -u 2 | b(φ 1 ) ≤c +sign (u 1 -u 2 ) (b(φ 1 ) -b(φ 2 )) ≤c|φ 1 -φ 2 | u 2 dx ds ≤ u 1,0 -u 2,0 L 1 (R 2 ) + t 0 R 2 |u 1 -u 2 |div x (ν(φ 1 )) dx ds - t 0 R 2 sign (u 1 -u 2 ) u 1 div x (ν(φ 1 ) -ν(φ 2 )) see (2.10) dx ds + c t 0 R 2 |u 1 -u 2 | dx ds + ce ct t 0 R 2 |φ 1 -φ 2 | dx ds ≤ u 1,0 -u 2,0 L 1 (R 2 ) + ce ct t 0 R 2 |u 1 -u 2 | dx ds + ce ct t 0 R 2 |div x (ν(φ 1 ) -ν(φ 2 )) | 2 dx 1/2 ds + c t 0 R 2 |u 1 -u 2 | dx ds + ce ct t 0 R 2 |φ 1 -φ 2 | dx ds ≤ u 1,0 -u 2,0 L 1 (R 2 ) + c(1 + e ct ) t 0 u 1 (s, •) -u 2 (s, •) L 1 (R 2 ) ds + ce ct t 0 ρ 1 (s, •, •) -ρ 2 (s, •, •) L 1 ((0,∞)×R 2 ) ds, and
ρ 1 (t, •, •) -ρ 2 (t, •, •) L 1 ((0,∞)×R 2 ) ≤ ρ 1,0 -ρ 2,0 L 1 ((0,∞)×R 2 ) + t 0 R 2 |ρ 1 (s, 0 + , x) -ρ 2 (s, 0 + , x)|dx ds - t 0 ∞ 0 R 2 sign (ρ 1 -ρ 2 )
u 1 (t, •) -u 2 (t, •) 2 L 1 (R 2 ) + ρ 1 (t, •, •) -ρ 2 (t, •, •) 2 L 1 ((0,∞)×R 2 )
≤ u For completeness we sketch the proof of the other estimates.

(2.6). directly comes from the definition of ν in (1.2) and the expression of div x (ν(φ)) in (2.38).

(2.7). We compute ∇ x ν(φ) by using the fact that ∇ x (f ν)

= f ∇ x ν + ν ⊗ ∇ x f ∇ x ν(φ) = κ 1 1 + φ * ∇ x η 2 1/2 ∇ x (φ * ∇ x η) + κ(φ * ∇ x η) ⊗ ∇ x 1 1 + φ * ∇ x η 2 1/2 = κ φ * ∇ 2 x η 1 + φ * ∇ x η 2 1/2 -κ(φ * ∇ x η) ⊗ (φ * ∇ 2 x η)(φ * ∇ x η) 1 + φ * ∇ x η 2 3/2 ,
then (notice that to shorten the notations we write L ∞ for L ∞ (R 2 ; R 2×2 )

∇ x ν(φ) L ∞ ≤ κ φ * ∇ 2 x η L ∞ 1 + φ * ∇ x η 2 1/2 + κ φ * ∇ x η 1 + φ * ∇ x η 2 1/2 ⊗ φ * ∇ 2 x η 1 + φ * ∇ x η 2 1/2 φ * ∇ x η 1 + φ * ∇ x η 2 1/2 L ∞ ≤ κ φ * ∇ 2 x η L ∞ + κ φ * ∇ x η 1 + φ * ∇ x η 2 1/2 L ∞ φ * ∇ 2 x η L ∞ φ * ∇ x η 1 + φ * ∇ x η 2 1/2 L ∞ ≤ 2κ φ * ∇ 2 x η L ∞ ≤ 2κ ∇ 2 x η L ∞ φ L 1 (R 2 ;R) .
(2.9). We compute gradient of (2.38)

∇ x div x (ν(φ)) = κ φ * ∇∆ x η 1 + φ * ∇ x η 2 3/2 -3κ(φ * ∆ x η) φ * ∇ 2 x η 1 + φ * ∇ x η 2 2 φ * ∇ x η 1 + φ * ∇ x η 2 1/2 , then ∇ x div x (ν(φ)) L 2 (R 2 ;R 2 ) ≤ κ φ * ∇∆ x η L 2 (R 2 ;R 2 ) + 3κ φ * ∆ x η L 2 (R 2 ;R 2 ) φ * ∇ 2 x η L ∞ (R 2 ;R 2×2 ) ≤ κ φ L 1 (R 2 ;R) ∇∆ x η L 2 (R 2 ;R 2 ) + 3 ∆ x η L 2 (R 2 ;R) ∇ 2 x η L ∞ (R 2 ;R 2×2 ) φ L 1 (R 2 ;R) .
(2.10). We have that ≤ κ (φ 1 -φ 2 ) * ∆ x η L 2 (R 2 ;R) + 48 25

√ 5 κ φ 2 * ∆ x η L ∞ (R 2 ;R) (φ 1 -φ 2 ) * ∇ x η L 2 (R 2 ;R 2 ) ≤ κ φ 1 -φ 2 L 1 (R 2 ,R) ∆ x η L 2 (R 2 ;R) + 48 25 √ 5 ∇ x η L 2 (R 2 ,R 2 ) ∆ x η L ∞ (R 2 ;R) φ 2 L 1 (R 2 ,R) .
(Giuseppe Maria Coclite)

Department of Mechanics, Mathematics and Management, Polytechnic University of Bari, Via E. Orabona 4, 70125

Bari, Italy Email address: giuseppemaria.coclite@poliba.it (Carlotta Donadello and Thi Nhu Thao Nguyen)

Laboratoire de Mathématiques CNRS UMR6623, Université de Bourgogne Franche-Comté, 16 route de Gray, 25030

Besanc ¸on Cedex, France.

Email address: carlotta.donadello@univ-fcomte.fr, thi nhu thao.nguyen@univ-fcomte.fr

umptions 1 . 2 .

 12 Assumptions. The coefficients b, v, d, p, A, ω and the initial data ρ 0 , u 0 of system (1.1) satisfy the following conditions: b

Theorem 1 . 1 .

 11 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 (u 1 , ρ 1 ) and (u 2 , ρ 2 ) are the two entropy solutions of (1.1) having initial data (u 1,0 , ρ 1,0

onloc2.1 onloc2. 1

 1

2 sign (ρ 1 - 2 |A(φ 1 2 sign (ρ 1 - 2 sign (ρ 1 - 2 A 2 |A (φ 1 ) 2 |ρ 1 - 2 sign (ρ 1 - 2 sign (ρ 1 - 0 u 1 0 ρ 1 R 2 |A (φ 1 )ce ct t 0 R 2 |Y θ (φ 1 -ce ct t 0 R 2 |∇ 2 + ce ct t 0 u 1 0 ρ 1 R 2 2 + ce ct t 0 u 1

 21212121221212121010121212201012201 χ 1 (a) (ρ 2 div x (vY θ (φ 1 -R)) -ρ 1 div x (vY θ (φ 2 -R))) dx da ds ρ 2 ) (p(a, u 1 ) -p(a, u 2 ))ρ 2 dx da ds ≤ ρ 1,0 -ρ 2,0 L 1 ((0,∞)×R 2 ) )ρ 1 -A(φ 2 )ρ 2 |χ 3 ω dx ds ρ 2 ) χ 1 (a) (ρ 2 div x (vY θ (φ 1 -R)) -ρ 1 div x (vY θ (φ 2 -R))) dx da ds ρ 2 ) (p(a, u 1 ) -p(a, u 2 ))ρ 2 dx da ds ≤ ρ 1,0 -ρ 2,0 L 1 ((0,∞)×R 2 ) (φ 1 ) |ρ 1 -ρ 2 | χ 3 ωdx da ds + -A (φ 2 )| ρ 2 χ 3 ωdx da ds + ρ 2 |χ 1 (a)div x (vY θ (φ 1 -R)) dx da ds ρ 2 ) χ 1 (a)ρ 1 div x (v) (Y θ (φ 1 -R) -Y θ (φ 2 -R)) dx da ds ρ 2 ) χ 1 (a)ρ 1 v • ∇ x (Y θ (φ 1 -R) -Y θ (φ 2 -R)) dx da ds + ce ct t (s, •) -u 2 (s, •) L 1 (R 2 ) ds ≤ ρ 1,0 -ρ 2,0 L 1 ((0,∞)×R 2 ) + c t (s, •, •) -ρ 2 (s, •, •) L 1 ((0,∞)×R 2 ) ds + ce ct t 0 -A (φ 2 )| dx ds + R) -Y θ (φ 2 -R)| dx ds + x (Y θ (φ 1 -R) -Y θ (φ 2 -R))| 2 dx ds 1/(s, •) -u 2 (s, •) L 1 (R 2 ) ds ≤ ρ 1,0 -ρ 2,0 L 1 ((0,∞)×R 2 ) + c(1 + e ct ) t (s, •, •) -ρ 2 (s, •, •) L 1 ((0,∞)×R 2 ) ds + ce ct t 0 |∇ x (φ 1 -φ 2 )| 2 dx ds 1/(s, •) -u 2 (s, •) L 1 (R 2 ) ds.

div x (ν(φ 1 ) 1 + 1 1 + φ 2 * ∇ x η 2 3

 11123 -ν(φ 2 )) = κ (φ 1 -φ 2 ) * ∆ x η 1 + φ 1 * ∇ x η 2 3/2 + κ(φ 2 * ∆ x η) φ 1 * ∇ x η 2 3/2 -Using the inequality |(1 + x 2 ) -3/2 -(1 + y 2 ) -3/2 | ≤ 48 25 √ 5 |x -y|, we obtain div x (ν(φ 1 ) -ν(φ 2 )) L 2 (R 2 ;R)

  .30). It follows from (2.17), (2.18), (2.26), (2.27), (2.29), (2.31) and the embedding H 2

  dy da db dt ds+ |u 1,0 (x) -u 2 |Ξ n (0, s, a, b, x, y)dx dy da db ds ≥ sign (u 1 -u 2 ) u 2 div x (ν(φ 1)) Ξ n dx dy da db dt ds,

	ntropy.1 ntropy.1

  -u 2 ) u 1 div x (ν(φ 2 )) Ξ n dx dy da db dt ds.

	+ sign (u 1 ntropy.2 ntropy.2 |u 1 -u 2,0 (y)|Ξ n (t, 0, a, b, x, y)dx dy da db dt ≥ -(3.4)	dy da db dt ds
	Summing (3.3) and (3.4) we have

  1,0 -u 2,0 2 L 1 (R 2 ) + c(1 + e ct ) (s, •) -u 2 (s, •) 2 L 1 (R 2 ) ds + c(1 + e ct ) ρ 1,0 -ρ 2,0 2 L 1 ((0,∞)×R 2 ) + c(1 + e ct ) (s, •, •) -ρ 2 (s, •, •) 2 L 1 ((0,∞)×R 2 ) ds.Finally, we use the Gronwall Lemma to obtain the result (1.16).4. Proof of Lemma 2.1 This lemma is similar to [4, Lemma 4.1], in particular, (2.5) and (2.8) already appeared there.

	t
	0	u 1 t
		ρ 1
		0

proof

Proof.

Acknowledgments

The second and the third author acknowledge the support of the Région Bourgogne Franche-Comté, projet 2017-2020 "Analyse mathématique et simulation numérique d'EDP issus de problèmes de contrôle et du trafic routier". The first author is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). He has been partially supported by the Research Project of National Relevance "Multiscale Innovative Materials and Structures" granted by the Italian Ministry of Education, University and Research (MIUR Prin 2017, project code 2017J4EAYB and the Italian Ministry of Education, University and Research under the Programme Department of Excellence Legge 232/2016 (Grant No. CUP -D94I18000260001).


D Chourrout
email: daniel.chourrout@sars.uib.no

F Delsuc

P Chourrout
email: chourrout@sars.uib.no

R B Edvardsen

F Rentzsch

E Renfer

M F Jensen

B Zhu

P De Jong

R E Steele

& U Technau

MINIMAL PROTOHOX CLUSTER INFERRED FROM BILATERIAN AND CNIDARIAN HOX COMPLEMENTS

Recent molecular phylogenies support that the non-bilaterian Cnidaria is the sister group to bilaterians [START_REF] Collins | Evaluating multiple alternative hypotheses for the origin of Bilateria: an analysis of 18S molecular evidence[END_REF][START_REF] Philippe | Multigene analysis of bilaterian animals corroborate the monophyly of Ecdysozoa, Lophotrochozoa and Protostomia[END_REF] and are therefore informative to reconstruct the early history of bilaterian homeobox gene complements. Earlier searches for cnidarian homeobox genes have revealed the presence of anterior-like Hox genes as well as Gsx, so that the ProtoHox cluster must have been duplicated prior to the cnidarian-bilaterian split. Earlier reports (6-13) have proposed that posterior Hox genes but not central genes or Hox3 are also present in cnidarians. We used publicly available high coverage genome shotgun sequence to identify the complete set of homeobox genes of two distantly related cnidarians, the freshwater polyp Hydra magnipapillata (Hydrozoa) and the sea anemone Nematostella vectensis (Anthozoa).

Nematostella is particularly informative as it is considered to represent the basal group within the Cnidaria (14,15).

Eighteen Nematostella candidate homeodomains were allocated to the Antennapedia class / Hox subclass and named based upon relationship with vertebrate and fly sequences. Nine of them were clearly not Hox genes, and included the ParaHox gene Gsx, four distinct Mox genes, Mnx, Gbx, Rough, and Evx. Seven were considered as candidate Hox genes, including five genes already known (13) which we renamed to avoid confusions with bilaterian Hox paralogs: HoxA (formerly anthox6), HoxB (novel), HoxC (formerly anthox7), HoxDa and HoxDb (two distinct copies with identical homeobox of a gene known as anthox8), HoxE (formerly anthox1a) and HoxF (formerly anthox1). Finally, two novel genes showed higher divergence from Antp/Hox genes of other species (see below). In contrast, a search in the Hydra genome revealed only eight Antp/Hox genes, including one Mox gene, one Gsx gene and six candidate Hox genes.

To clarify the relationships between bilaterian and cnidarian Hox/ParaHox genes, we carried out phylogenetic analyses (Methods in Supporting Online Material, SOM), including evaluation of.alternative hypotheses that receive some support in the data but would remain hindered with traditional tree building methods. Most reliable trees were obtained with large datasets of homeodomain proteins from putative slow-evolving bilaterian species such as Nereis virens, Cupiennius salei, Ptychodera flava and Branchiostoma floridae. First, using only bilaterian sequences, we could recognise the major groups of Hox and ParaHox genes (Fig. 1a and SOM) and confirm that ParaHox genes Gsx and Xlox cluster with the anterior genes Hox1/2 and Hox3, respectively. Surprisingly, the group of ParaHox Cdx genes appeared divergent from all three groups of bilaterian Hox genes, and showed only slight phylogenetic affinity (BP = 38) with posterior Hox genes (Fig. 1a andSOM). The existence of a precursor for Hox posterior genes and Cdx in the ProtoHox cluster is therefore questionable.

Neighbour-net analyses clearly showed that within cnidarians Nematostella is the slowest evolving species, and the Nematostella dataset was consequently used for extensive sequence comparisons. HoxA, HoxB, HoxC, HoxDa/Db and one divergent gene (Hox-related) clustered with the anterior genes (Hox1-3) of bilaterians (Fig. 1b andSOM). The precise relationships within the anterior Hox homeodomains could not be statistically resolved despite weak preferential affinities of HoxA/HoxB with Hox1/2, and of HoxC/HoxDa/HoxDb with Hox3 in several analyses (not shown). By comparison, HoxE and HoxF which have been previously tentatively classified as posterior genes [START_REF] Finnerty | Origins of bilateral symmetry: Hox and Dpp expression in a sea anemone[END_REF] did not preferentially group with anterior, central or posterior genes (Fig. 1b and SOM). Instead, they formed an independent and strongly supported branch. Finally, one of the two divergent genes showed conflicting affinities in network analyses with Cdx and Xlox (Fig. 1b) and is therefore most likely a ParaHox gene. This gene named Xlox/Cdx is not a second Gsx gene, according to likelihood-based statistical tests (SOM). When Hox genes from other cnidarians were included, the same picture appeared with cnidarian-specific non-anterior Hox genes, although their long branches tended to be attracted towards the fast evolving posterior genes (SOM). Overall, cnidarians possess (i) bona fide anterior Hox genes, (ii) several non-anterior Hox genes not clearly related to central and posterior genes of bilaterians, and (2) Gsx and a second candidate ParaHox gene related to Xlox and Cdx.

Using BAC library screening and sequence reconstruction from shotgun data, we established linkages between distinct homeobox genes in the Nematostella genome. The results (Fig. 2a and SOM) showed a main cluster of five genes (HoxC/HoxDa/HoxDb/Evx/HoxA) in a tandem array over approximately 50 kilobases. Linkage between the non-Hox gene Evx and a Hox gene has been shown earlier in another anthozoan cnidarian [START_REF] Miller | Homeobox genes and the zootype[END_REF]. We extended the cluster sequence and reached two other homeobox genes approximately 200 kilobases downstream: one is Mnx, a gene also present in the neighbourhood of chordate Hox clusters [START_REF] Castro | Chromosomal mapping of ANTP class homeobox genes in Amphioxus: piecing together ancestral genomes[END_REF], and the other is the ortholog of the fly gene Rough which is on the chromosome arm 3R as the ANT-C and BX-C complexes. Another linkage could be established between the anterior gene Hoxrelated and the non-anterior gene HoxE, whereas HoxB and HoxF appeared to be isolated in two distinct genome locations. In sum, linkages in the Nematostella genome provide support for the existence of an ancient cnidarian Hox cluster associating Evx, several anterior Hox genes and at least one non-anterior Hox gene. In the Nematostella lineage, the cluster was expanded through gene duplications. Our analyses indeed robustly support that two precursor genes have been amplified into HoxC/HoxDa/HoxDb and into HoxE/HoxF subsets, respectively. A common origin for HoxA and HoxB is also suggested by the analyses, though with weaker support. The cluster was also affected by a rearrangement placing Evx between anterior Hox genes, and by at least two splits. Most interestingly, our linkage studies placed the candidate Xlox/Cdx gene immediately downstream of Gsx, in tandem orientation. Thus, this association can represent a ParaHox cluster. By comparison, genomic sequence reconstruction of the Hydra Hox gene complement provided evidence for strong degeneration of the Hox cluster in Hydra: only two non-anterior genes which most likely result from a recent duplication (HmHoxc1 and HmHoxc3) could be linked (Fig. 2c), and Gsx was the only ParaHox gene detected.

As lineage-specific duplications in the Hox cluster are rare within bilaterians, we wanted to know whether or not those found in Nematostella are limited to the Hox genes. For this we examined potential linkages between all other homeobox genes. Including the nine Hox-like genes mentioned above, we found a total of 139 homeobox genes, a surprisingly high number compared to other invertebrates (Table 1 and SOM). Phylogenetic analyses allowed placing at least 87 of those into 58 known groups of homeobox genes, out of 76 groups known for bilaterians. They also suggested that 42 unclassified genes have arisen through recent amplifications of maximally ten genes. By comparison, Hydra has a considerably smaller number of homeobox genes and gene groups. Since all Hydra homeobox gene groups have representatives in the Nematostella genome, we assume that the homeobox complement has been dramatically reduced in the Hydra lineage (Table 1). Comparisons between the extended homeobox sequences of Nematostella allowed the detection of an additional 13 physical clusters (Fig. 2b andSOM). Four of these clusters may have a more ancient origin, either because they have been identified in the genomes of bilaterians or because they associate distantly related genes. The other nine clusters are undoubtedly the result of recent tandem duplications. We also did not detect obvious synteny conservation between the environments of well related but unlinked homeobox genes, as might be expected after whole genome duplication (not shown). Hence, the gene duplications observed among the Nematostella Hox genes represent a general phenomenon for the homeobox gene complement.

From our data, the most parsimonious scenario for the evolution of Hox/ParaHox clusters is as follows (Fig. 3): two ProtoHox genes (P1/2 and P3) gave rise to the Hox cluster consisting of two anterior Hox genes Hox1/2 and Hox3, and to the ParaHox cluster with Gsx and Xlox. Subsequently, internal duplications expanded the Hox cluster, first by adding a precursor of the non-anterior genes. This gene may have appeared before the cnidarian-bilaterian split or independently in both lineages. Similarly, Cdx was added in the ParaHox cluster, either in the bilaterian lineage, or before the cnidarian-bilaterian split. After this split, the cnidarian and bilaterian Hox clusters were further expanded through lineage-specific duplications. In this scenario, cnidarians never had bona fide central and posterior Hox genes, and perhaps no Cdx gene either. Alternative scenarios can be proposed but they appear to be less parsimonious.

For example, the precursor of non-anterior genes could have already existed in the ProtoHox cluster but had then to evolve into multiple directions leading to central/posterior Hox genes of bilaterians, HoxE/F of cnidarians, and Cdx. Or there were even more such precursors in the ProtoHox cluster and some had to be lost in one lineage and/or in one of the two clusters.

Another main angle to interpret the existing data is that cnidarians are in fact simplified bilaterians, with a reduced mesoderm as one of the possible regressions [START_REF] Finnerty | Origins of bilateral symmetry: Hox and Dpp expression in a sea anemone[END_REF]18). A loss of central and posterior genes, or their rapid divergence into current non-anterior Hox genes of cnidarians could have accompanied the process of simplification, as would have the Hox cluster breakdowns too. While this cannot be ruled out completely, extensive EST (19) and genome surveys in Nematostella identified a surprising complexity of genes, including many genes lost in bilaterian lineages. This and the considerable degree of gene sequence conservation of the Nematostella lineage argues against a highly reduced and derived organism. In addition, more complex Hox gene complements in more basal metazoan phyla such as Placozoa, which would be indicative of gene loss in Cnidaria, could not be found (20).

We therefore argue, that the common ancestor of cnidarians and bilaterians contained rather simple primordial Hox and ParaHox clusters that had very distinct fates in Cnidaria and Bilateria. The ProtoHox cluster itself may have consisted of only two anterior genes.

METHODS:

Homeodomains were searched into each cnidarian genome data using TBLASTN at low stringency (10 -8) with representative query sequences from mouse and the other cnidarian and classified into known classes, subclasses and groups. Alignments of shotgun datasets with expressed sequence tags had indicated that the probability of a given gene to be represented exceeds 98%. Phylogenetic analyses were based on Maximum likelihood, Bayesian and distance-based network analyses of homeodomain proteins. Phylogenetic network analyses allowed to visualise conflicting signals and areas of uncertainty in the data, which appear with homeodomains due to reduced alignment lengths. Genomic sequences were extended using the in-house developed software MàD (for Marche à Droite) which performs walks and jumps using shotgun sequences and their physical links. Linkages between homeobox genes were established through BAC library screening, confirmed using PCR on positive BAC clones, with MàD reconstruction of long sequence contigs. All technical details are provided in Supporting online Information.

Sequences have been deposited in Genbank with accession numbers DQ500742-DQ500879

with a synthetic presentation in Supporting Online Material.

Supporting Online Material accompanies the paper on www.nature.com/nature.

(18) Seipel, K. & Schmid, V. Evolution of striated muscle: jellyfish and the origin of triploblasty. Dev. Biol. 282, 14-26 (2005). Bayesian posterior probabilities (PP) for the major groups have been mapped on the network.

FIGURE AND TABLE LEGENDS:

A dash (-) indicates that the corresponding split was not observed in the Bayesian analysis.

Bootstrap proportions (BP) were inferred using the BioNJ algorithm for the same dataset and

Table 1 | Classification of homeobox genes detected in the genomes of Nematostella

vectensis and Hydra magnipapillata, based on phylogenetic analyses. In Nematostella, a significant subset of genes could be allocated to known classes and superfamilies but diverged from known homeobox gene groups, while three genes could not be allocated to known classes (see also SOM). The Hydra genome shows a substantial reduction of the homeobox gene complement, and all identified gene groups are also present in Nematostella.

Bilaterian animals have a

  Hox gene cluster essential for patterning the main body axis, and a ParaHox gene cluster. Comparison of Hox/ParaHox genes has led to postulate that both clusters originated from the duplication of an ancient cluster named ProtoHox, which contained up to four genes with at least the precursors of anterior and posterior Hox/ParaHox genes (1-3). However, the way genes diversified within the ProtoHox, Hox and ParaHox clusters remains unclear as no systematic study of nonbilaterian animals exists. Our characterisation of the full Hox/ParaHox gene complements and genomic organisation in two cnidarian species (Nematostella vectensis and Hydra magnipapillata) suggests a ProtoHox cluster simpler than originally thought: (i) both species possess bilaterian-like anterior Hox genes, but their non-anterior genes do not appear as counterparts of either bilaterian central or posterior genes, (ii) two clustered ParaHox genes, Gsx and a gene related to Xlox and Cdx, are found in Nematostella vectensis, (iii) we do not find clear phylogenetic support for a common origin of bilaterian Cdx and posterior genes, which therefore may have appeared after the ProtoHox cluster duplication. Consequently, the ProtoHox cluster may have consisted of only two anterior genes. Non-anterior genes can have appeared independently in the Hox and ParaHox clusters, possibly after the separation of bilaterians and cnidarians.

( 19 )

 19 Technau, U. et al. Maintenance of ancestral complexity and non-metazoan genes in two basal cnidarians. Trends Genet. 21, 633-639 (2005). (20) Monteiro, A.S., Schierwater, B., Dellaporta, S.L. & Holland, P.W.H. A low diversity of ANTP class homeobox genes in Placozoa. Evol. Dev. 8, 174-182 (2006).

Figure 1 |

 1 Figure 1 | Phylogenetic analyses of bilaterian and Nematostella Hox genes based on their homeodomain. More detailed figures including all sequence origins are available in SOM.

JTT+G 8

 8 ML distance estimates. Scale bar represents estimated number of changes per site. Note the position of the Cdx group which is distinct from the posterior Hox group. (b) Neighbour-Net network of 92 Hox and ParaHox genes from bilaterian and Nematostella based on JTT+G 8 ML distance estimates. Note the position of non-anterior Nematostella genes (Nv_HoxE and Nv_HoxF). The major groups of bilaterian Hox and ParaHox proteins remain unaffected by the addition of the Nematostella dataset. Bayesian Posterior Probabilities (PP) and Bootstrap proportions (BP) were calculated as for the bilaterian dataset.

Figure 2 |

 2 Figure 2 | All physical linkages detected between cnidarian homeobox genes. (a). Linkages associating Nematostella ANTP/Hox-like genes were validated using BAC library screening with individual Hox-like genes, before an in silico genome walk using the local assembler of shotgun traces allowed to reconstruct the cluster DNA sequences. The position of Hox and Hox-related genes including the ParaHox gene Gsx and their respective orientations are represented by red arrows, and other linked homeobox genes by blue arrows. Putative intervening non-homeobox genes are represented by circles, filled with green when expression is confirmed by EST alignments; sequence similarity with known genes is detected by BLASTX (SOM). (b). Linkages associating other Nematostella homeobox genes were validated with in silico genome walk using the local assembler of shotgun traces. (a).Linkages associating Hydra ANTP/Hox-like genes including Gsx were validated with in silico genome walk. Genes found in their environments have been annotated using BLASTX and alignments with Hydra ESTs (SOM).

Figure 3 |

 3 Figure 3 | Parsimonious consensus scenario for the evolution of Hox/ParaHox clusters of

Acknowledgements:

We acknowledge Dan Rokhsar for his efforts in heading the Nematostella Genome project at DoE-JGI and the J. Craig Venter Institute and the National Human Genome Research Institute for the Hydra Genome Project. The initial EST screen in Nematostella was funded by the DFG (Te311-3/1). The construction of the Nematostella BAC library was funded by the National Science Foundation (NSF IBN-0208335).

Authors' contribution

The project was conceived and the manuscript was written by D. Chourrout and U. Technau.