Tiến-Tài Nguyễn

LINEAR AND NONLINEAR ANALYSIS OF THE VISCOUS RAYLEIGH-TAYLOR SYSTEM WITH NAVIER-SLIP BOUNDARY CONDITIONS

Keywords: 2010 Mathematics Subject Classification. 34B07, 47A10, 47B07, 76D05, 76E30 Navier-Stokes equations, linear growth rate, spectral analysis, Rayleigh-Taylor instability

In this paper, we are interested in the linear and the nonlinear Rayleigh-Taylor instability for the gravity-driven incompressible Navier-Stokes equations with Navier-slip boundary conditions around a smooth increasing density profile ρ 0 px 2 q in a slab domain 2πLT ˆp´1, 1q (L ą 0, T is the usual 1D torus). The linear instability study of the viscous Rayleigh-Taylor model amounts to the study of the following ordinary differential equation on the finite interval p´1, 1q,

with the boundary conditions

where λ ą 0 is the growth rate in time, g ą 0 is the gravity constant, k is the wave number and two Navier-slip coefficients ξ ˘are nonnegative constants.

For each k P L ´1Zzt0u, we define a threshold of viscosity coefficient µcpk, Ξq for the linear instability. So that, in the k-supercritical regime, i.e. µ ą µcpk, Ξq, we describe a spectral analysis adapting the operator method initiated by Lafitte-Nguyễn [13] and then prove that there are infinite nontrivial solutions pλn, φnq ně1 of (0.1)-(0.2) with λn Ñ 0 as n Ñ 8 and φn P H 4 pp´1, 1qq. Based on the existence of infinitely many normal modes of the linearized problem, we construct a wide class of initial data to the nonlinear equations, extending the previous framework of Guo-Strauss [5] and of Grenier [7], to prove the nonlinear Rayleigh-Taylor instability in a high regime of viscosity coefficient, namely µ ą 3 sup kPL ´1Zzt0u µcpk, Ξq.

The Rayleigh-Taylor (RT) instability, studied first by Lord Rayleigh in [17] and then Taylor [18] is well known as a gravity-driven instability in two semi-infinite inviscid and incompressible fluids when the heavy one is on top of the light one. It has attracted much attention due to both its physical and mathematical importance. Two applications worth mentioning are implosion of inertial confinement fusion capsules [START_REF] Lindl | Inertial Confinement Fusion[END_REF] and core-collapse of supernovae [START_REF] Remington | A review of astrophysics experiments on intense lasers[END_REF]. For a detailed physical comprehension of the RT instability, we refer to three survey papers [START_REF] Kull | Theory of the Rayleigh-Taylor instability[END_REF]19,20]. Mathematically speaking, for the inviscid and incompressible regime with a smooth density profile, the classical RT instability was investigated by Lafitte [12], by Guo and Hwang [START_REF] Guo | On the dynamical Rayleigh-Taylor instability[END_REF] and by Helffer and Lafitte [9].

Concerning the viscous RT instability, one of the first studies can be found in the book of Chandrasekhar [1, Chap. X]. He considered two uniform viscous fluid separated by a horizontal boundary and generalized the classical result of Rayleigh and Taylor. We refer the readers to mathematical viscous RT studies for two compressible channel flows by Guo and Tice [START_REF] Guo | Linear Rayleigh-Taylor instability for viscous, compressible fluids[END_REF], for incompressible fluid in the whole space R 3 by Jiang et. al [START_REF] Jiang | Nonlinear instability for nonhomogeneous incompressible viscous fluids[END_REF] and Lafitte and Nguyễn [START_REF] Lafitte | Spectral analysis of the incompressible viscous Rayleigh-Taylor system[END_REF], respectively.

In this paper, we are concerned with the viscous RT of the gravity-driven incompressible Navier-Stokes equations in a 2D slab domain Ω " 2πLT ˆp´1, 1q with L ą 0 and T is the 1D-torus, that read as $ ' & ' % B t ρ `divpρ uq " 0, B t pρ uq `divpρ u b uq `∇P " µ∆ u ´ρ g, div u " 0, (1.1) where t ě 0, x " px 1 , x 2 q P 2πLT ˆp´1, 1q. The unknowns ρ :" ρpt, xq, u :" upt, xq and P :" P pt, xq denote the density, the velocity and the pressure of the fluid, respectively, while µ is the viscosity coefficient and g :" g e 2 , g ą 0 being the gravity constant. Let Σ ˘" 2πLT ˆt˘1u, the Navier-slip boundary conditions proposed by Navier (see [START_REF] Navier | Sur les lois de l'équilibre et du mouvement des corps élastiques[END_REF]) are given on Σ ˘as follows u ¨ n " 0 on Σ `Y Σ ´, pµp∇ u `∇ u T q ¨ nq τ " ξp xq u on Σ `Y Σ ´.

(1.2)

Here, n is the outward normal vector of the boundary, pµp∇ u `∇ u T q ¨ nq τ is the tangential component of µp∇ u `∇ u T q ¨ n and ξp xq is a scalar function describing the slip effect on the boundary, only taking nonnegative constant values ξ ˘on Σ ˘, respectively.

Let ρ 0 and P 0 be two C 1 -functions on x 2 such that P 1 0 " ´gρ 0 with 1 " d{dx 2 . Then, the laminar flow pρ 0 px 2 q, 0, P 0 px 2 qq is a steady-state solution of (1.1). Of interest of this paper is to study the nonlinear instability of the above laminar flow to Eq. (1.1)-(1.2) that satisfies ρ 0 P C 1 pr´1, 1sq, ρ 1 0 ą 0 on r´1, 1s, ρ 0 p˘1q " ρ ˘P p0, `8q, (

i.e. to study the nonlinear Rayleigh-Taylor instability problem.

Linearizing (1.1) in the vicinity of pρ 0 px 2 q, 0, P 0 px 2 qq and then seeking a normal mode at a horizontal spatial frequency k P L ´1Zzt0u of the form e λpkqt U p xq " e λpkqt pcospkx 1 qωpx 2 q, sinpkx 1 qθpx 2 q, cospkx 1 qφpx 2 q, cospkx 1 qqpx 2 qq T , the linear RT instability amounts to the investigation of the parameter λpkq P C (Reλ ą 0) such that there exists a nontrivial solution φ P H 4 pp´1, 1qq of the following ordinary differential equation for the second component of velocity ´λ2 pρ 0 k 2 φ ´pρ 0 φ 1 q 1 q " λµpφ p4q ´2k 2 φ 2 `k4 φq ´gk 2 ρ 1 0 φ, (1.4) with the boundary conditions $ ' & ' % φp´1q " φp1q " 0, µφ 2 p1q " ξ `φ1 p1q, µφ 2 p´1q " ´ξ´φ 1 p´1q.

(1.5) Note that the embedding H 4 pp´1, 1qq ãÑ C 3 pp´1, 1qq allows us to write (1.5). In this case, such a λ is called a growth rate of the instability or a characteristic value of the linearized problem (see Eq. (2.5) below) as in [1, Sect. 92-93, Chap. X]). We will present the derivation of the physical model in Section 2.

As the density profile is increasing, we first show that λ is always real in Lemma 2.1. Since our goal is to study the instability, we are left to look for λ ą 0. Hence, for the linear instability, we continue the spectral analysis of Helffer and Lafitte [9], Lafitte and Nguyễn [START_REF] Lafitte | Spectral analysis of the incompressible viscous Rayleigh-Taylor system[END_REF] for Eq. (1.4)-(1.5).

For any horizontal spatial frequency k P L ´1Zzt0u, we then define a k-supercritical regime of the viscosity coefficient µ ą µ c pk, Ξq (see µ c pk, Ξq in Proposition 3.1), we prove that there exists an infinite sequence of characteristic values pλ n pk, µqq ně1 , decreasing towards 0 as n Ñ 8. This is stated in Theorem 2.1.

The second goal, described in Section 5 is to obtain a nonlinear instability result on more general initial data using the linear result of Theorem 2.1 (see (2.15)) and working in the regime µ ą 3 sup kPL ´1Zzt0u µ c pk, Ξq. The classical way of proving the nonlinear instability is to estimate the difference between the solution to the nonlinear problem and the normal mode solution to the linearized problem. In order to show that, in the spirit of and Grenier [7], only the maximal normal mode e λ1t U 1 p xq was taken to derive a solution of the nonlinear equation whose initial datum is δ U 1 p xq with 0 ă δ ! 1. Our nonlinear result, Theorem 2.2, generalizes the previous results of Guo-Strauss and of Grenier, by showing that a wide class of initial data (related to a linear combination of normal modes) to the nonlinear problem departing from the equilibrium gives rise to the nonlinear instability.

This paper is organized as follows. In Section 2, we present the governing equations and state the main results. Section 3 is devoted to some materials for the linear study. Then, in Section 4, we prove the linear instability, i.e. Theorem 2.1. Section 5 is to prove the nonlinear instability, i.e. Theorem 2.2.

Main results

2.1. The governing equations. Let us recall the steady state pρ 0 px 2 q, 0, P 0 px 2 qq of (1.1), with ρ 0 satisfies (1.3) and P 1 0 " ´gρ 0 . We now derive the linearization of Eq. (1.1) around the equilibrium state pρ 0 px 2 q, 0, P 0 px 2 qq. The perturbations σ " ρ ´ρ0 , u " u ´ 0, p " P ´P0 thus satisfy

$ ' & ' % B t σ ` u ¨∇pρ 0 `σq " 0, pρ 0 `σqB t u `pρ 0 `σq u ¨∇ u `∇p " µ∆ u ´σ g, div v " 0. (2.1)
Note that pµp∇ u `∇ u T q ¨ nq τ " n ˆpµp∇ u `∇ u T q ¨ nq ˆ n and that n " p0, ˘1q T . Hence, the boundary conditions are

$ ' & ' % u 2 " 0, on Σ ˘, µB x2 u 1 " ξ `u1 on Σ `, µB x2 u 1 " ´ξ´u1 on Σ ´.
(2.

2)

The linearized equations are

$ ' & ' % B t σ `ρ1 0 u 2 " 0, ρ 0 B t u `∇p " µ∆ u ´σ g, div u " 0, (2.3)
and the corresponding boundary conditions remaining (2.2).

The linear RT instability problem is to seek a normal mode of the form

$ ' ' ' & ' ' ' %
σpt, xq " e λt cospkx 1 qωpx 2 q, u 1 pt, xq " e λt sinpkx 1 qθpx 2 q, u 2 pt, xq " e λt cospkx 1 qφpx 2 q, qpt, xq " e λt cospkx 1 qqpx 2 q.

(2.4)

where k P L ´1Zzt0u, λ P Czt0u and Reλ ě 0. It follows from (2.3) that

$ ' ' ' & ' ' ' % λω `ρ1
0 φ " 0, λρ 0 θ ´kq `µpk 2 θ ´θ2 q " 0, λρ 0 φ `q1 `µpk 2 φ ´φ2 q " ´gω, kθ `φ1 " 0 (2.5) and from (2.2) that φp˘1q " 0, µθ 1 p1q " ξ `θp1q, µθ 1 p´1q " ´ξ´θ p´1q.

(2.6)

We obtain

ω " ´ρ1 0 λ φ, θ " ´1 k φ 1 , q " ´1 k 2 pλρ 0 φ 1 `µpk 2 φ 1 ´φ3 qq. (2.7)
Then, we substitute q, ω into (2.5) 3 to get a fourth-order ordinary differential equation (1.4). We have the boundary conditions (1.5) deduced from (2.2), which are obtained by assuming the solution to be in C 2 pr´1, 1sq.

Main results

. Before stating our main results, we present our material for the linearized equations.

When the density profile ρ 0 is increasing, we show that all characteristic values λ are real. Let L 0 be the characteristic length, such that L ´1 0 " } Proof of Lemma 2.1 is given in Section 3.1. In view of Lemma 2.1, we look for functions φ being real and we only consider the vector space of real-valued functions in what follows in the linear analysis.

We now study the linearized problem, i.e. (1.4)-(1.5). Of importance is to construct a continuous and coercive bilinear form B k,λ,µ as λ ě 0 and k P Rzt0u (i.e. we do not restrict λ P p0, b g L0 q and k P L ´1Zzt0u at this step) on the functional space H2 pp´1, 1qq :" tϕ P H 2 pp´1, 1qq, ϕp˘1q " 0u, so that the finding of a solution φ P H 4 pp´1, 1qq of Eq. (1.4)-(1.5) on p´1, 1q is equivalent to finding a weak solution φ P H2 pp´1, 1qq to the variational problem

λB k,λ,µ pφ, θq " gk 2 ż 1 ´1 ρ 1 0 φθdx 2 for all θ P H2 pp´1, 1qq, (2.8)
and thus improving the regularity of that weak solution φ.

The desired bilinear form B k,λ,µ is

B k,λ,µ pϑ, q :"λ ż 1 ´1 ρ 0 pk 2 ϑ `ϑ1 1 qdx 2 `µ ż 1 ´1pϑ 2 2 `2k 2 ϑ 1 1 `k4 ϑ qdx 2 ´ξ´ϑ 1 p´1q 1 p´1q ´ξ`ϑ 1 p1q 1 p1q, (2.9)
For all λ ě 0 and k P Rzt0u, we will place ourselves in a k-supercritical regime of the viscosity coefficient µ ą µ c pk, Ξq with Ξ " pξ ´, ξ `q (see the precise formula µ c pk, Ξq in Proposition 3.1) such that

B k,0,µ is coercive if and only if µ ą µ c pk, Ξq, (2.10)
it yields that B k,λ,µ is coercive for all λ ě 0 and µ ą µ c pk, Ξq. As B k,λ,µ is a coercive form on H 2 pp´1, 1qq, we have that a B k,λ,µ p¨, ¨q is a norm on H 2 pp´1, 1qq. Let p H2 pp´1, 1qqq 1 be the dual space of H2 pp´1, 1qq associated with the norm a B k,λ,µ p¨, ¨q. In view of Riesz's representation theorem, we thus obtain an abstract operator Y k,λ,µ from H2 pp´1, 1qq to p H2 pp´1, 1qqq 1 q such that B k,λ,µ pϑ, q " xY k,λ,µ ϑ, y for all ϑ, P H2 pp´1, 1qq.

(

Y k,λ,µ ϑ " λpρ 0 k 2 ϑ ´pρ 0 ϑ 1 q 1 q `µpϑ p4q ´2k 2 ϑ 2 `k4 ϑq in D 1 pp´1, 1qq.
Owing to a bootstrap argument, we further get that a weak solution φ of (2.11) actually belongs to H 4 pp´1, 1qq and φ satisfies the boundary conditions (1.5). Hence, we are able to define the inverse operator of Y ´1 k,λ,µ of Y k,λ,µ (see Proposition 3.4), from L 2 pp´1, 1qq to a subspace of H 4 pp´1, 1qq requiring all elements satisfy (1.5). Composing the above operator Y ´1 k,λ,µ with the continuous injection from H 4 pp´1, 1qq to L 2 pp´1, 1qq, we obtain that Y ´1 k,λ,µ is a compact and self-adjoint operator from L 2 pp´1, 1qq to itself.

Denoting by M the operator of multiplication by a ρ 1 0 in L 2 pp´1, 1qq. Note from (2.12) that, we thus find pλ, vq such that

λ gk 2 v " MY ´1 k,λ,µ Mv.
Once it is proven that the operator MY ´1 k,λ,µ M is compact and self-adjoint from L 2 pp´1, 1qq to itself, then the discrete spectrum of MY ´1 k,λ,µ M is an infinite sequence of eigenvalues (denoted by tγ n pk, λ, µqu ně1 ).

Let v n,k,λ,µ be an eigenfunction of the operator MY ´1 k,λ,µ M associated with the eigenvalue γ n pk, λ, µq and let φ n,k,λ,µ " Y ´1 k,λ,µ Mv n,k,λ,µ , we have

γ n pk, λ, µqY k,λ,µ φ n,k,λ,µ " M 2 φ n,k,λ,µ " ρ 1 0 φ n,k,λ,µ . (2.13)
From (2.13), it can be seen that, for each n, we have to solve the equation

γ n pk, λ, µq " λ gk 2 .
(2.14)

We will show that Eq. (2.14) has a unique root λ n pk, µq P R `because of the decrease of γ n in λ (see Lemma 4.1), which is an extension of Kato's perturbation theory of the spectrum of operators [START_REF] Kato | Perturbation theory for linear operators[END_REF]. In addition, when λ n is a characteristic value, we have λ n ď b g L0 for all n ě 1.

This yields that for any horizontal spatial frequency k P L ´1Zzt0u, there exists a sequence of characteristic values pλ n pk, µqq ně1 , that is uniformly bounded and we further obtain that λ n decreases towards 0 as n Ñ 8. For each λ n , we have that φ n,k,λn,µ " Y ´1 k,λn,µ v n,k,λn,µ is a solution in H 4 pp´1, 1qq of (1.4)-(1.5) associated with λ " λ n .

We sum up the above arguments in our first theorem.

Theorem 2.1. Let k P L ´1Zzt0u be fixed and let ρ 0 satisfy that (1.3), i.e. ρ 0 P C 1 pr´1, 1sq, ρ 0 p˘1q " ρ ˘P p0, 8q, ρ 1 0 ą 0 everywhere on r´1, 1s. For all µ ą µ c pk, Ξq, there exists an infinite sequence pλ n , φ n q ně1 with λ n ą 0 decreasing towards 0 and φ n P H 4 pp´1, 1qq, φ n non trivial, satisfying (1.4)-(1.5).

Once Eq. (1.4)-(1.5) is solved, we go back to the linearized equations (2.3). For a fixed k P L ´1Zzt0u, we obtain a sequence of solutions to the linearized equations (2.3) as indicated in Proposition 4.2, which are pe λj pk,µqt U j pk, µ, xqq jě1 , with U j pk, µ, xq " pσ j , u j , p j q T pk, µ, xq.

Let us choose a k 0 P L ´1Zzt0u and fix µ ą 3µ c pΞq. In view of getting infinitely many characteristic values of the linearized problem, we introduce a linear combination of normal modes

U M pt, xq " M ÿ j"1
c j e λj pk0,µqt U j pk 0 , µ, xq (2.15) to construct an approximate solution to the nonlinear problem (1.1)-(1.2), with constants c j being chosen such that at least one of c j p1 ď j ď N q is non-zero (2.16) and let j m :" mintj : 1 ď j ď N, c j ‰ 0u,

1 2 |c jm |} u jm } L 2 pΩq ą ÿ jějm`1 |c j |} u j } L 2 pΩq . (2.17) Eq. (2.1)-(2.
2) is supplemented with the initial data δ U M p0, xq (0 ă δ ! 1) has a unique local strong solution pσ δ , u δ q with an associated pressure q δ on r0, T max q (see Proposition 5.1). We define the differences

σ d " σ δ ´δσ M , u d " u δ ´δ u M , q d " q δ ´δq M ,
which satisfy Eq. (5.7), along with the initial condition (5.8) and the boundary conditions (5.9). For t small enough, we estimate the bound in time of }pσ

}pσ d , u d qptq} 2 L 2 pΩq ď Cδ 3 ´N ÿ j"jm |c j |e λj t `maxp0, M ´N q ´max N `1ďjďM |c j | ¯e 2 3 ν0Λt ¯3.
The nonlinear result follows.

Theorem 2.2. Let µ c pΞq be defined as in (2.18) and µ ą 3µ c pΞq. Let ρ 0 satisfies (1.3), i.e.

ρ 0 P C 1 pr´1, 1sq, ρ 0 p˘1q " ρ ˘P p0, 8q, ρ 1 0 ą 0 everywhere on r´1, 1s. Let M P N ‹ , there exist a positive constant m 0 and two positive constants δ 0 and 0 sufficiently small such that for any δ P p0, δ 0 q, the nonlinear equations (2.1) with boundary conditions (2.2) and the initial data 2.17) has a unique local strong solution pσ δ , u δ q with an associated pressure q δ such that } u δ pT δ q} L 2 pΩq ě m 0 0 , (2.22)

δ M ÿ j"1 c j U j p xq satisfying (2.16)-(
where T δ P p0, T max q is given by δ ř M j"jm |c j |e λj T δ " 0 .

Preliminaries

The first aim is to prove Lemma 3.1, showing that all characteristic values λ are real for any increasing density profile ρ 0 . In the second part, we find the exact formula of the k-critical viscosity coefficient µ c pk, Ξq (see (2.10) above) for all k ą 0. The last goal is to study the bilinear form B k,λ,µ in Section 3.3 to prepare for our linear study. Proof. Let φ P H 4 pp´1, 1qq satisfy (1.4)-(1.5). Multiplying by φ on both sides of (1.4) and using the integration by parts, we get that

´ż 1 ´1pρ 0 φ 1 q 1 φdx 2 " ´ρ0 φ 1 φ ˇˇ1 ´1 `ż 1 ´1 ρ 0 |φ 1 | 2 dx 2 that ´ż 1 ´1 φ 2 φdx 2 " ´φ1 φ ˇˇ1 ´1 `ż 1 ´1 |φ 1 | 2 dx 2
and that ż 1

´1 φ p4q φdx 2 " φ 3 φ ˇˇ1 ´1 ´φ2 φ 1 ˇˇ1 ´1 `ż 1 ´1 |φ 2 | 2 dx 2 , we obtain that λ ´µ ż 1 ´1p|φ 2 | 2 `2k 2 |φ 1 | 2 `k4 |φ| 2 qdx 2 ´ξ´| φ 1 p´1q| 2 ´ξ`| φ 1 p1q| 2 λ2 ż 1 ´1pk 2 ρ 0 |φ| 2 `ρ0 |φ 1 | 2 qdx 2 " gk 2 ż 1 ´1 ρ 1 0 |φ| 2 dx 2 . (3.1)
Suppose that λ " λ 1 `iλ 2 , then one deduces from (3.1) that

λ 1 ´µ ż 1 ´1p|φ 2 | 2 `2k 2 |φ 1 | 2 `k4 |φ| 2 qdx 2 ´ξ´| φ 1 p´1q| 2 ´ξ`| φ 1 p1q| pλ 2 1 ´λ2 2 q ż 1 ´1pk 2 ρ 0 |φ| 2 `ρ0 |φ 1 | 2 qdx 2 " gk 2 ż 1 ´1 ρ 1 0 |φ| 2 dx 2 (3.2)
and that

λ 2 ´µ ż 1 ´1p|φ 2 | 2 `2k 2 |φ 1 | 2 `k4 |φ| 2 qdx 2 ´ξ´| φ 1 p´1q| 2 ´ξ`| φ 1 p1q| " ´2λ 1 λ 2 ż 1 ´1pk 2 ρ 0 |φ| 2 `ρ0 |φ 1 | 2 qdx 2 . (3.3) If λ 2 ‰ 0, Eq. (3.3) leads us to ´2λ 1 ż 1 ´1pk 2 ρ 0 |φ| 2 `ρ0 |φ 1 | 2 qdx 2 " µ ż 1 ´1p|φ 2 | 2 `2k 2 |φ 1 | 2 `k4 |φ| qdx 2 ´ξ´| φ 1 p´1q| 2 ´ξ`| φ 1 p1q| 2 ,
which yields

´pλ 2 1 ´λ2 2 q ż 1 ´1pk 2 ρ 0 |φ| 2 `ρ0 |φ 1 | 2 qdx 2 " ´2λ 2 1 ż 1 ´1pk 2 ρ 0 |φ| 2 `ρ0 |φ 1 | 2 qdx 2 ´gk 2 ż 1 ´1 ρ 1 0 |φ| 2 dx 2 .
Equivalently,

pλ 2 1 `λ2 2 q ż 1 ´1pk 2 ρ 0 |φ| 2 `ρ0 |φ 1 | 2 qdx 2 " ´gk 2 ż 1 ´1 ρ 1 0 |φ| 2 dx 2 . (3.4) That implies pλ 2 1 `λ2 2 qk 2 ρ ´ż 1 ´1 |φ| 2 dx 2 ď ´gk 2 ż 1 ´1 ρ 1 0 |φ| 2 dx 2 .
The positivity of ρ 1 0 yields a contradiction, then λ is real. Due to (3.2) again, we further get that

λ 2 ż 1 ´1 ρ 0 pk 2 |φ| 2 `|φ 1 | 2 qdx 2 ď gk 2 ż 1 ´1 ρ 1 0 |φ| 2 dx 2 .
It tells us that λ is bounded by b g L0 . This finishes the proof of Lemma 3.1.

Note again that, thanks to Lemma 2.1, in what follows in this section, we only use real-valued functions for the linear analysis.

3.2.

The threshold of viscosity coefficient. We obtain the precise formula of the critical viscosity coefficient µ c pk, Ξq for all k P Rzt0u. Note that µ c pk, Ξq " µ c p´k, Ξq for all k P Rzt0u, it suffices to find µ c pk, Ξq for k P R `.

Proposition 3.1. The following results hold.

(1) For all k P R `, we have

µ c pk, Ξq " max φP H2 pp´1,1qq ξ ´pφ 1 p´1qq 2 `ξ`p φ 1 p1qq 2 ş 1 ´1ppφ 2 q 2 `2k 2 pφ 1 q 2 `k4 φ 2 qdx 2 .
(3.5)

Moreover,

µ c pk, Ξq " 1 4k sinh 2 p2kq ¨psinhp2kq coshp2kq ´2kqpξ ``ξ ´q `˜psinhp2kq ´2k coshp2kqq 2 pξ ``ξ ´q2 `sinh 2 p2kqpsinh 2 p2kq ´4k 2 qpξ `´ξ ´q2 ¸1 2 ‹ ‹ ' . (3.6) (2) µ c pk, Ξq is a decreasing function in k P R `and lim kÑ0 µ c pk, Ξq " sup kPRzt0u µ c pk, Ξq ": µ s c pΞq. (3.7)
We have the asymptotic expansion of µ c pk, Ξq as k Ñ 0

`, µ c pk, Ξq " 1 3 ´ξ``ξ´`b ξ 2 `´ξ `ξ´`ξ 2 ´2 15 ´4pξ ``ξ ´q `4ξ 2 `´ξ `ξ´`4 ξ 2 bξ 2 `´ξ `ξ´`ξ 2 ´¯k 2 `Opk 3 q. ( 3.8)

That implies

µ s c pΞq "

1 3 ´ξ``ξ´`b ξ 2 `´ξ `ξ´`ξ 2 ´¯.
(3.9)

As k " 1, we obtain the limit

µ c pk, Ξq ď b 2pξ 2 ``ξ 2 ´q k Ñ 0. (3.10)
(3) We have

µ s c pΞq " max φP H2 pp´1,1qq ξ ´pφ 1 p´1qq 2 `ξ`p φ 1 p1qq 2 ş 1 ´1pφ 2 q 2 dx 2 " 1 3 ´ξ``ξ´`b ξ 2 `´ξ `ξ´`ξ 2 ´¯. (3.11)
The proof of Proposition 3.1 is postponed to Appendix A.

Remark 3.1. We see that µ s c pΞq in [3, Proposition 2.2] should be revised and we redo the computation the critical viscosity coefficient. The authors in [START_REF] Ding | Stability analysis for the incompressible Navier-Stokes equations with Navier boundary conditions[END_REF] consider 2πLT ˆp0, 1q instead of 2πLT ˆp´1, 1q and constant values k 0,1 instead of ξ ˘. The formula of the critical viscosity defined as in [3, (1.29)] is

µ c :" sup φPY Zpφq,
where

Y " tφ P H 1 0 pp0, 1qq X H 2 pp0, 1qq, 1 2
ż 1 0 |φ 2 | 2 dx " 1u, Zpφq " k 0 2 |φ 1 p0q| 2 `k1 2 |φ 1 p1q| 2 .
The authors in [3, Proposition 2.2] claim that µ c " k 6 if k 0 " k 1 " k ą 0. However, in that case, let us take φ " 1 ? 2 px 2 ´xq P Y, then we have a contradiction that

Z ´1 ? 2 px 2 ´xq ¯" k 2 ą k 6 .

3.3.

A bilinear form and a self-adjoint invertible operator. In what follows in this section we have λ ě 0 and k P R `being fixed. Let us recall the definition of B k,λ,µ from (2.9),

B k,λ,µ pϑ, q :" λ ż 1 ´1 ρ 0 pk 2 ϑ `ϑ1 1 qdx 2 `µ ż 1 ´1pϑ 2 2 `2k 2 ϑ 1 1 `k4 ϑ qdx 2 ´ξ´ϑ 1 p´1q 1 p´1q ´ξ`ϑ 1 p1q 1 p1q. Lemma 3.2.
We have the followings.

' For all µ ą 0, B k,λ,µ is a continuous bilinear form on H2 pp´1, 1qq. ' For all µ ą µ c pk, Ξq, the bilinear form B k,λ,µ is coercive.

Proof of Lemma 3.2. Clearly, B k,λ,µ is a bilinear form on H2 pp´1, 1qq. We then establish the boundedness of B k,λ,µ . The integral terms of B k,λ,µ are bounded by

C ‹ pλ `1q}ϑ} H2 pp´1,1qq } } H2 pp´1,1qq , (3.12)
where C ‹ is generic constant depending on physical parameters. Meanwhile, it follows from the general Sobolev inequality that

pϑ 1 p´1qq 2 `pϑ 1 p1qq 2 ď C ‹ }ϑ 1 } 2 H 1 pp´1,1qq . Consequently, we get |B k,λ,µ pϑ, q| ď C ‹ p1 `λq}ϑ} H2 pp´1,1qq } } H2 pp´1,1qq , (3.13)
i.e. B k,λ,µ is bounded.

We show the coercivity of B k,λ,µ . We have that

B k,λ,µ pϑ, ϑq " λ ż 1 ´1 ρ 0 pk 2 ϑ 2 `pϑ 1 q 2 qdx 2 `µ ż 1 ´1ppϑ 2 q 2 `2k 2 pϑ 1 q 2 `k4 ϑ 2 qdx 2 ´ξ´p ϑ 1 p´1qq 2 ´ξ`p ϑ 1 p1qq 2 .
As λ ě 0 and µ ą µ c pk, Ξq, we have

B k,λ,µ pϑ, ϑq ě λ ż 1 ´1 ρ 0 pk 2 ϑ 2 `pϑ 1 q 2 qdx 2 `pµ ´µc pk, Ξqq ż 1 ´1ppϑ 2 q 2 `2k 2 pϑ 1 q 2 `k4 ϑ 2 qdx 2 ě pµ ´µc pk, Ξqq ż 1 ´1ppϑ 2 q 2 `2k 2 pϑ 1 q 2 `k4 ϑ 2 qdx 2 .
(3.14)

It then follows from (3.13) and (3.14) that B k,λ,µ is a continuous and coercive bilinear form on H2 pp´1, 1qq.

With the above property of B k,λ,µ , we then establish: Proposition 3.2. Let µ ą µ c pk, Ξq and p H2 pp´1, 1qqq 1 be the dual space of H2 pp´1, 1qq, associated with the norm a B k,λ,µ p¨, ¨q. There is a unique operator Y k,λ,µ P LpH 2 pp´1, 1qq, p H2 pp´1, 1qqq 1 q, which is also bijective, such that

B k,λ,µ pϑ, q " xY k,λ,µ ϑ, y (3.15)
for all ϑ, P H2 pp´1, 1qq.

Proof. It follows from Riesz's representation theorem that there exists a unique operator Y k,λ,µ P Lp H2 pp´1, 1qq, p H2 pp´1, 1qqq 1 q such that B k,λ,µ pϑ, q " xY k,λ,µ ϑ, y for all P H2 pp´1, 1qq. Proof of Proposition 3.2 is complete.

Proposition 3.3. We have the following results.

(1) For all ϑ P H2 pp´1, 1qq,

Y k,λ,µ ϑ " λpk 2 ρ 0 ϑ ´pρ 0 ϑ 1 q 1 q `µpϑ p4q ´2k 2 ϑ 2 `k4 ϑq in D 1 pp´1, 1qq. (2) Let f P L 2 pp´1, 1qq be given, there exists a unique solution ϑ P H2 pp´1, 1qq of Y k,λ,µ ϑ " f in p H2 pp´1, 1qqq 1 .
(3.16) Moreover, we have that ϑ P H 4 pp´1, 1qq satisfies the boundary conditions (1.5).

Proof. It follows from Proposition 3.2 that there is a unique ϑ P H2 pp´1, 1qq such that

λ ż 1 ´1 ρ 0 pk 2 ϑ `ϑ1 1 qdx 2 `µ ż 1 ´1pϑ 2 2 `2k 2 ϑ 1 1 `k4 ϑ qdx 2 " xY k,λ,µ ϑ, y (3.17)
for all P C 8 0 pp´1, 1qq. We respectively define pϑ 2 q 1 and pϑ 2 q 2 in the distributional sense as the first and second derivative of ϑ 2 which is in L 2 pp´1, 1qq. Hence, Eq. (3.17) is equivalent to

λ ż 1 ´1 ρ 0 pk 2 ϑ `ϑ1 1 qdx 2 `µxpϑ 2 q 2 , y `µ ż 1 ´1p2k 2 ϑ 1 1 `k4 ϑ qdx 2 " xY k,λ,µ ϑ, y (3.18) for all P C 8 0 pp´1, 1qq. We deduce from (3.18) that λ ż 1 ´1pk 2 ρ 0 ϑ ´pρ 0 ϑ 1 q 1 q dx 2 `µxpϑ 2 q 2 ´2k 2 ϑ 2 `k4 ϑ, y " xY k,λ,µ ϑ, y (3.19)
for all P C 8 0 pp´1, 1qq. The resulting equation implies that

µppϑ 2 q 2 ´2k 2 ϑ 2 `k4 ϑq `λpk 2 ρ 0 ϑ ´pρ 0 ϑ 1 q 1 q " Y k,λ,µ ϑ in D 1 pp´1, 1qq. (3.20)
The first assertion holds.

Under the assumption f P L 2 pp´1, 1qq, we improve the regularity of the weak solution ϑ P H2 pp´1, 1qq of (3.20). Indeed, we rewrite (3.20) as µxpϑ 2 q 2 , y " ż 1 ´1pY k,λ,µ ϑ `2µk 2 ϑ 2 ´µk 4 ϑ ´λk 2 ρ 0 ϑ `λpρ 0 ϑ 1 q 1 q dx 2 for all P C 8 0 pp´1, 1qq. Since pf `2µk 2 ϑ 2 ´µk 4 ϑ ´λk 2 ρ 0 ϑ `λpρ 0 ϑ 1 q 1 q belongs to L 2 pp´1, 1qq, it then follows from (3.19) that pϑ 2 q 2 P L 2 pp´1, 1qq. Let χ P C 8 0 pp´1, 1qq satisfy ş 1 ´1 χpyqdy " 1. Using the distribution theory, we define Σ P D 1 pp´1, 1qq such that xΣ, θy " xpϑ 2 q 2 , ζ θ y (3.21) for all θ P C 8 0 pp´1, 1qq, where

ζ θ px 2 q " ż x2 ´1 ´θpyq ´χpyq ż 1
´1 θpsqds ¯dy for all ´1 ă x 2 ă 1. We obtain xΣ 1 , θy " ´xΣ, θ 1 y " ´xpϑ 2 q 2 , ζ θ 1 y.

Note that

xpϑ 2 q 2 , ζ θ 1 y " xpϑ 2 q 2 , θpx 2 q ´ż x2 ´1 χpyq ż 1
´1 θ 1 psqdsdyy " xpϑ 2 q 2 , θy, this yields xΣ 1 , θy " ´xpϑ 2 q 2 , θy. Hence, we have that pϑ 2 q 1 `Σ " constant. In view of pϑ 2 q 2 P L 2 pp´1, 1qq and (3.21), we know that pϑ 2 q 1 P L 2 pp´1, 1qq. Since ϑ P H2 pp´1, 1qq and pϑ 2 q 1 , pϑ 2 q 2 P L 2 pp´1, 1qq, it tells us that ϑ belongs to H 4 pp´1, 1qq and we can take their traces of derivatives of ϑ up to order 3.

By performing (3.19), we then show that ϑ satisfies (1.5). Indeed, for all P H2 pp´1, 1qq, we perform the integration by parts to obtain from (3.19) that

λ ż 1 ´1 ρ 0 pk 2 ϑ `ϑ1 1 qdx 2 `µ ż 1 ´1pϑ 2 2 `2k 2 ϑ 1 1 `k4 ϑ qdx 2 ´λρ 0 ϑ 1 ˇˇ1 ´1 `µ´ϑ 3 ˇˇ1 ´1 ´ϑ2 1 ˇˇ1 ´1 ´2k 2 ϑ 1 ˇˇ1 ´1¯" ż 1 ´1pY k,λ,µ ϑq dx 2 .
It then follows from the definition of the bilinear form B k,λ,µ that

λρ 0 ϑ 1 ˇˇ1 ´1 ´µ´ϑ 3 ˇˇ1 ´1 ´ϑ2 1 ˇˇ1 ´1 ´2k 2 ϑ 1 ˇˇ1 ´1¯" ξ ´ϑ1 p´1q 1 p´1q `ξ`ϑ 1 p1q 1 p1q, (3.22
) for all P H2 pp´1, 1qq. By collecting all terms corresponding to 1 p˘1q in (3.22), we deduce that µϑ 2 p˘1q " ˘ξ˘ϑ 1 p˘1q.

This yields that ϑ satisfies (1.5). The proof of Proposition 3.3 is complete.

We obtain more information on the inverse operator Y ´1 k,λ,µ .

Proposition 3.4. The operator Y ´1 k,λ,µ : L 2 pp´1, 1qq Ñ L 2 pp´1, 1qq is compact and self-adjoint.

Proof. It follows from Proposition 3.3 that Y k,λ,µ , being supplemented with (1.5), admits an inverse operator Y ´1 k,λ,µ from L 2 pp´1, 1qq to a subspace of H 4 pp´1, 1qq requiring all elements satisfy (1.5), which is symmetric due to Proposition 3.2. We compose Y ´1 k,λ,µ with the continuous injection from H 4 pp´1, 1qq to L 2 pp´1, 1qq. Notice that the embedding H p pp´1, 1qq ãÑ H q pp´1, 1qq for p ą q ě 0 is compact. Therefore, the operator Y ´1 k,λ,µ is compact and self-adjoint from L 2 pp´1, 1qq to L 2 pp´1, 1qq.

Linear instability

4.1.

A sequence of characteristic values. We continue considering λ ě 0 and k P L ´1Zzt0u being fixed. We study the operator S k,λ,µ :" MY ´1 k,λ,µ M, where M is the operator of multiplication by a ρ 1 0 . Proposition 4.1. Under the hypothesis (1.3), the operator S k,λ,µ : L 2 pp´1, 1qq Ñ L 2 pp´1, 1qq is compact and self-adjoint.

Proof. Due to the assumption on ρ 0 (1.3), the operator S k,λ,µ is well-defined and bounded from L 2 pp´1, 1qq to itself. Y ´1 k,λ,µ is compact, so is S k,λ,µ . Moreover, because both the inverse Y ´1 k,λ,µ and M are self-adjoint, the self-adjointness of S k,λ,µ follows.

As a result of the spectral theory of compact and self-adjoint operators, the point spectrum of S k,λ,µ is discrete, i.e. is a sequence tγ n pk, λ, µqu ně1 of eigenvalues of S k,λ,µ , associated with normalized orthogonal eigenfunctions t n u ně1 in L 2 pp´1, 1qq. That means

S k,λ,µ n " MY ´1 k,λ,µ M n " γ n pk, λ, µq n . So that φ n " Y ´1
k,λ,µ M n belongs to H 4 pp´1, 1qq and satisfies (1.5). One thus has

γ n pk, λ, µqY k,λ,µ φ n " ρ 1 0 φ n (4.1)
and φ n satisfies (1.5). Eq. (4.1) also tells us that γ n pk, λ, µq ą 0 for all n. Indeed, we obtain γ n pk, λ, µq

ż 1 ´1pY k,λ,µ φ n qφ n dx 2 " ż 1 ´1 ρ 1 0 φ 2 n dx 2 .
That implies

γ n pk, λ, µqB k,λ,µ pφ n , φ n q " ż 1 ´1 ρ 1 0 φ 2 n dx 2 . (4.2)
Since B k,λ,µ pφ n , φ n q ą 0 and ρ 1 0 ą 0 on p´1, 1q, we know that γ n pk, λ, µq is positive for all n ě 1. Hence, by reordering and using the spectral theory of compact and self-adjoint operators again, we have that tγ n pk, λ, µqu ně1 is a positive sequence decreasing towards 0 as n Ñ 8.

For each n, we have that φ n is a solution of (1.4)-(1.5) if and only if there are positive λ n such that (2.14) holds. To solve (2.14), we use the two following lemmas. Lemma 4.1. For each n, ' γ n pk, λ, µq and φ n are differentiable in λ. ' γ n pk, λ, µq is decreasing in λ.

Proof. The proof of Lemma 4.1(1) is the same as [13, Lemma 3.3], we omit the details here. We now prove that γ n pk, λ, µq is decreasing in λ.

Let z n " dφn dλ , it follows from (4.1) that

k 2 ρ 0 φ n ´pρ 0 φ 1 n q 1 `Yk,λ,µ z n " 1 γ n pk, λ, µq ρ 1 0 z n `d dλ ´1 γ n pk, λ, µq ¯ρ1 0 φ n (4.3) on p´1, 1q. At x 2 " ˘1, we have $ ' & ' % z n p´1q " z n p1q " 0, µz 2 n p1q " ξ `z1 n p1q, µz 2 n p´1q " ´ξ´z 1 n p´1q. (4.4)
Multiplying by φ n on both sides of (4.3), we obtain that

ż 1 ´1pk 2 ρ 0 φ n ´pρ 0 φ 1 n q 1 qφ n dx 2 `ż 1 ´1pY k,λ,µ z n qφ n dx 2 " 1 γ n pk, λ, µq ż 1 ´1 ρ 1 0 z n φ n dx 2 `d dλ ´1 γ n pk, λ, µq ¯ż 1 ´1 ρ 1 0 φ 2 n dx 2 .
(4.5)

Note that z n enjoys (4.4), then

ż 1 ´1pY k,λ,µ z n qφ n dx 2 " ż 1 ´1pY k,λ,µ φ n qz n dx 2 " 1 γ n pk, λ, µq ż 1 ´1 ρ 1 0 z n φ n dx 2 .
That implies

d dλ ´1 γ n pk, λ, µq ¯ż 1 ´1 ρ 1 0 φ 2 n dx 2 " ż 1 ´1pk 2 ρ 0 φ n ´pρ 0 φ 1 n q 1 qφ n dx 2 . (4.6)
Using the integration by parts, we obtain from (4.6) that

d dλ ´1 γ n pk, λ, µq ¯ż 1 ´1 ρ 1 0 φ 2 n dx 2 " ż 1 ´1 ρ 0 pk 2 φ 2 n `pφ 1 n q 2 qdx 2 ą 0.
Consequently, γ n pk, λ, µq is decreasing in λ ą 0.

4.2.

Proof of Theorem 2.1 and normal modes of the linearized equations.

In view of Lemma 4.1, we are able to prove Theorem 2.1.

Proof of Theorem 2.1. For each n, there is only one solution λ n of (2.14). Indeed, using (4.2), we know that

1 γ n pk, λ, µq ż 1 ´1 ρ 1 0 φ 2 n dx 2 " ż 1 ´1pY k,λ,µ φ n qφ n dx 2 " B k,λ,µ pφ n , φ n q. Hence, it follows from (3.14) that 1 γ n pk, λ, µq ż 1 ´1 ρ 1 0 φ 2 n dx 2 ě λ ż 1 ´1 ρ 0 pk 2 φ 2 n `pφ 1 n q 2 qdx 2 `pµ ´µc pk, Ξqq ż 1 ´1ppφ 2 n q 2 `2k 2 pφ 1 n q 2 `k4 φ 2 n qdx 2 ě λk 2 ż 1 ´1 ρ 0 φ 2 n dx 2 `pµ ´µc pk, Ξqqk 4 ż 1 ´1 φ 2 n dx 2 .
That implies 1 L 0 γ n pk, λ, µq ě λk 2 `pµ ´µc pk, Ξqqk 4 ρ `. Consequently, for all n ě 1, λ γ n pk, λ, µq ą gk 2 for λ large. We now prove that pλ n q ně1 decreases towards 0 as n Ñ 8. If λ m ă λ m`1 for some m ě 1, we have γ m pk, λ m , µq ą γ m pk, λ m`1 , µq.

Meanwhile, we also have

γ m pk, λ m`1 , µq ą γ m`1 pk, λ m`1 , µq. That implies λ m gk 2 " γ m pk, λ m , µq ą γ m`1 pk, λ m`1 , µq " λ m`1 gk 2 .
That contradiction tells us that pλ n q ně1 is a decreasing sequence. Suppose that lim nÑ8 λ n " d 0 ą 0.

Note that for all n, γ n pk, λ n , µq " λn gk 2 , then

γ n pk, d 0 , µq ě γ n pk, λ n , µq " λ n gk 2 .
Let n Ñ 8, we get that 0 ě d 0 , a contradiction. Hence λ n decreases towards 0 as n Ñ 8. The proof of Theorem 2.1 is complete.

We derive the following property for the largest characteristic value λ 1 found in Theorem 2.1. Lemma 4.2. Let us recall the bilinear form B k,λ,µ on H 2 pp´1, 1qq (2.9) and pλ 1 , φ 1 q from Theorem 2.1. We have that

1 gk 2 " max φPH 2 pp´1,1qq ş 1 ´1 ρ 1 0 φ 2 dx 2 λ 1 B k,λ1,µ pφ, φq , (4.9)
and the variational problem (4.9) is attained by the function φ 1 .

Proof. For all λ ą 0, we solve the variational problem βpk, λ, µq " max ´ż 1

´1 ρ 1 0 φ 2 dx 2 ˇˇφ P H2 pp´1, 1qq, λB k,λ,µ pφ, φq " 1

¯.

Let us define the Lagrangian functional

L B pφ, βq " ż 1 ´1 ρ 1 0 φ 2 dx 2 ´βpλB k,λ,µ pφ, φq ´1q.
Thanks to the Lagrange multiplier theorem, the extrema of the quotient

ş 1 ´1 ρ 1 0 φ 2 dx 2 λB k,λ,µ pφ, φq
are necessarily the stationary points pβ ‹ , φ ‹ q of L B , which satisfy

λB k,λ,µ pφ ‹ , φ ‹ q " 1 (4.10) and ż 1 ´1 ρ 1 0 φ ‹ θdx 2 ´β‹ λB k,λ,µ pφ ‹ , θq " 0, (4.11)
for all θ P H2 pp´1, 1qq. Restricting θ P C 8 0 pp´1, 1qq and following the line of the proof of Proposition 3.3, one deduces from (4.11) that φ ‹ has to satisfy Two inequalities (4.13) and (4.14) tell us that βpk, λ, µq " λ ´1γ 1 pk, λ, µq for all λ ą 0. We thus obtain βpk, λ 1 , µq " 1 gk 2 and the variational problem (4.9) is attained by the function φ 1 . Proof of Lemma 4.2 is complete.

β ‹ λY k,λ,µ φ ‹ " ρ 1 0 φ ‹ (4.
We now solve the linearized equations (2.3) to prepare for our nonlinear part. Proposition 4.2. For each k P L ´1Zzt0uzt0u and for all µ ą µ c pk, Ξq, there exists an infinite sequence of solutions pn ě 1q e λnpk,µqt U n pk, µ, xq " e λnpk,µqt pσ n , u n , p n q T pk, µ, xq " e λnpk,µqt ¨cospkx 1 qω n pk, µ, x 2 q sinpkx 1 qθ n pk, µ, x 2 q cospkx 1 qφ n pk, µ, x 2 q

cospkx 1 qq n pk, µ, x 2 q ‹ ‹ ‹ ‹ ‹ ‹ ' to the linearized equation (2.3)-(2.
2), such that σ n P H 2 pΩq, u n P pH 3 pΩqq 2 and p n P H 1 pΩq.

Proof. For each solution λ n P p0, b g L0 q of (2.14), we have that φ n " Y ´1 k,λn,µ M n is a solution of (1.4)-(1.5) in H 4 pp´1, 1qq as λ " λ n . We now find a solution to the system (2.5) as λ " λ n . First, we obtain θ n " ´φ1 n k and ω n " ´ρ1 0 φn λn . Due to (2.7), we get

q n " ´1 k 2 pλ n ρ 0 φ 1 n `µpk 2 φ 1 n ´φ3 n qq P H 1 pp´1, 1qq.
With a solution pω n , θ n , φ n , q n q of (2.5), we then conclude that e λnpk,µqt pσ n , u n,1 , u n,2 , p n q T pk, µ, xq " e λnpk,µqt ¨cospkx 1 qω n pk, µ, x 2 q sinpkx 1 qθ n pk, µ, x 2 q cospkx 1 qφ n pk, µ, x 2 q

cospkx 1 qq n pk, µ, x 2 q ‹ ‹ ‹ ‹ ‹ ‹ '
is a solution to the linearized equations (2.3)-(2.2).

Nonlinear instability

5.1. The local existence. The first important things are the local existence of strong solutions to the nonlinear equations and a priori energy estimates to those solutions. We restate Proposition 4.1 of [2].

Proposition 5.1. Suppose that the steady state satisfies (1.3). For any given initial data pσ 0 , u 0 q P pH 1 pΩq X L 8 pΩqq ˆpH 2 pΩqq 2 satisfying div u 0 " 0, and also being compatible with the boundary conditions (1.2), the nonlinear equations (2.1) has a local strong solution pσ, u, ∇qq P Cpr0, T max q, H 1 pΩq ˆpH 2 pΩqq 2 ˆpL 2 pΩqqq 2 .

(5.1)

Let Eptq :" b }σptq} 2 H 1 pΩq `} uptq} 2
H 2 pΩq and δ 0 ą 0 be sufficiently small, we further get that if sup 0ďsďt Eptq ď δ 0 , there holds

E 2 ptq `}p∇q, B t uq} 2 L 2 pΩq `ż t 0 p}B t upsq} 2 H 1 pΩq `} upsq} 2 H 2 pΩq qds ď C 0 ´E2 p0q `ż t 0 }pσ, uqpsq} 2 L 2 pΩq ds ¯.
(5.2)

Thanks to Proposition 4.2, we will formulate a sequence of approximate solutions e λnpk,µq U n pk, µ, xq to the nonlinear equations (2.1)-(2.2), which are solutions to the linearized equations (2.3)-(2.2). Let us fix a k " k 0 P L ´1Zzt0u such that (2.21) holds and µ ą 3µ c pΞq. We recall (2.15),

pσ M , u M , q M qpt, xq :" M ÿ j"1
c j e λj pk,µqt U j pk, µ, xq.

Keeping in mind that min r´1,1s ρ 0 ą 0, then due to the embedding from H 2 pΩq to L 8 pΩq, there exists a constant δ 0 ą 0 such that

δ 0 } ÿ jě1 σ j p0, xq} L 8 pΩq ą 1 2 min r´1,1s
ρ 0 px 2 q.

(5.3)

Hence, for δ ď δ 0 , 1 2 min r´1,1s ρ 0 px 2 q ď min Ω pρ 0 px 2 q `δσ M p0, xqq.

By virtue of Proposition 5.1, the nonlinear equations (2.1)-(2.2) with the initial data δpσ M , u M qp0q admits a local solution pσ δ , u δ q P C 0 pr0, T max q, H 1 pΩq ˆpH 2 pΩqq 2 q with an associated pressure q δ P C 0 pr0, T max q, L 2 pΩqq. Furthermore, we have 1 2 min r´1,1s ρ 0 px 2 q ď inf Ω pρ 0 px 2 q `σδ pt, xqq for all t P r0, T max q.

In what follows, the constants C i pi ě 1q are universal ones depending only on physical parameters, M and c j pj ě 1q.

Let F M ptq " ř M j"jm |c j |e λj t and 0 ă 0 ! 1 be fixed later (5.42). There exists a unique T δ such that δF M pT δ q " 0 . Let

C 1 " b }σ M p0q} 2 H 1 pΩq `} u M p0q} 2 H 2 pΩq , C 2 " b }σ M p0q} 2 L 2 pΩq `} u M p0q} 2 L 2 pΩq .
(5.4)

In view of (5.6), we then obtain that for s P t0, 1, 2u and τ P t0, 1u,

}B τ t u d ptq} H s pΩq ď δ}B τ t u M ptq} H s pΩq `}B τ t u δ ptq} H s pΩq ď C 8 δF M ptq.
To prove (5.13), we use (5.7) 1 and (5.6) again,

}σ d ptq} H 1 pΩq `}B t σ d ptq} L 2 pΩq ď }σ δ ptq} H 1 pΩq `δ}σ M ptq} H 1 pΩq `C9 }u d 2 ptq} L 2 pΩq `} u δ ptq} L 2 pΩq }∇σ δ } L 2 pΩq ď C 10 δF M ptq.
Lemma 5.1 is proven.

Lemma 5.2. There holds

}B t u d p0q} 2 L 2 pΩq ď C 11 δ 3 .
(5.14)

Proof. From (5.7) 2,3 and the boundary conditions (5.9), we use the integration by parts to obtain that ż

Ω ρ 0 |B t u d | 2 d x " ż Ω µ∆ u d ¨Bt u d d x ´żΩ pσ δ B t u δ `pρ 0 `σδ q u δ ¨∇ u δ q ¨Bt u d d x ´żΩ gσ d B t u d 2 d x.
Thanks to Lemma 5.1, one has

´żΩ pσ δ B t u δ `pρ 0 `σδ q u δ ¨∇ u δ q ¨Bt u d d x ď C 12 δ 3 F 3 M ptq.
(5.15)

That implies

}B t u d ptq} 2 L 2 pΩq ď C 13 ´p} u d ptq} H 2 pΩq `}σ d ptq} L 2 pΩq q}B t u d ptq} L 2 pΩq `δ3 F 3 M ptq ¯.
Using Young's inequality, we further get

}B t u d ptq} 2 L 2 pΩq ď 1 2 }B t u d ptq} 2 L 2 pΩq `C14 p} u d ptq} 2 H 2 pΩq `}σ d ptq} 2 L 2 pΩq q `C13 δ 3 F 3 M ptq.
That implies

}B t u d ptq} 2 L 2 pΩq ď C 15 ´} u d ptq} 2 H 2 pΩq `}σ d ptq} 2 L 2 pΩq `δ3 F 3 M ptq ¯.
(5.16)

Letting t Ñ 0 in (5.16), we complete the proof Lemma 5.2.

Lemma 5.3. Let X :" t w P pH 2 pΩqq 2 , w satisfies (1.2) and div w " 0u.

There holds for all w P H 2 ‹ pΩq, ż

Ω gρ 1 0 |w 2 | 2 d x `Λ ż p2πLTq 2 pξ `|w 1 px 1 , 1q| 2 `ξ´| w 1 px 1 , ´1q| 2 qdx 1 ď Λ 2 ż Ω ρ 0 | w| 2 d x `Λµ ż Ω |∇ w| 2 d x.
(5.17)

The proof of Lemma 5.3 is due to the definition of Λ (2.20) and Lemma 4.2, that is similar to [2, Lemma 5.1], hence we omit the details here.

Lemma 5.4. There holds for all w P Xzt 0u,

sup wPX ş 2πLT pξ `|w 1 px 1 , 1q| 2 `ξ´| w 1 px 1 , ´1q| 2 qdx 1 }∇ w} 2 L 2 pΩq
ď µ c pΞq.

(5.18)

Proof. Let us fix a horizontal frequency k P L ´1Z and introduce the horizontal Fourier transform f pk, x 2 q " ż 2πLT f p xqe ´ikx1 dx 1 .

For w P X, we write ŵ1 pk, x 2 q " ´iθpk, x 2 q, ŵ2 pk, x 2 q " φpk, x 2 q.

Then, kθ`φ 1 " 0 and pθ, φq enjoys (2.6). Following Fubini's and Parseval's theorem, one thus deduces

ż 2πLT pξ `|w 1 px 1 , 1q| 2 `ξ´| w 1 px 1 , ´1q| 2 qdx 1 " 1 2πL ÿ kPL ´1Z pξ `p|θpk, 1q| 2 `ξ´| θpk, ´1q| 2 q (5.19) and }∇ w} 2 L 2 pΩq " 1 2πL ÿ kPL ´1 Z ż 1 ´1pk 2 p|θ| 2 `|φ| 2 q `|θ 1 | 2 `|φ 1 | 2 qpk, x 2 qdx 2 .
(5.20)

We may reduce to estimate (5.18) when θ and φ are real-valued and continue the estimate to the real and imaginary part of θ and φ. For any k P L ´1Zzt0u, we have from kθ `φ1 " 0 that ξ `pθpk, 1qq 2 `ξ´p θpk, ´1qq 2 " 1 k 2 pξ `ppφ 1 pk, 1qq 2 `ξ´p φ 1 pk, ´1qq 2 q (5.21) and that

ż 1 ´1 ´k2 pθ 2 `φ2 q `pθ 1 q 2 `pφ 1 q 2 ¯pk, x 2 qdx 2 " 1 k 2 ż 1 ´1pk
4 φ 2 `2k 2 pφ 1 q 2 `pφ 2 q 2 qpk, x 2 qdx 2 .

(5.22)

Owing to (5.19), (5.21) and the definition of µ c pk, Ξq (see (3.5)), we get

ż 2πLT pξ `|w 1 px 1 , 1q| 2 `ξ´| w 1 px 1 , ´1q| 2 qdx 1 ď 1 2πL ¨lim sup kÑ0 1 k 2 pξ `pφ 1 pk, 1qq 2 `ξ´p φ 1 pk, ´1qq 2 q `ÿ kPL ´1Zzt0u 1 k 2 pξ `pφ 1 pk, 1qq 2 `ξ´p φ 1 pk, ´1qq 2 q ‹ ‹ ‹ ' ď 1 2πL ¨lim sup kÑ0 µ c pk, Ξq k 2 ż 1 ´1pk 4 φ 2 `2k 2 φ 12 `φ22 qpk, x 2 qdx 2 `ÿ kPL ´1Zzt0u µ c pk, Ξq k 2 ż 1 ´1pk 4 φ 2 `2k 2 φ 12 `φ22 qpk, x 2 qdx 2 ‹ ‹ ‹ ‹ ' .
Thanks to Proposition 3.1, we obtain

ż 2πLT pξ `|w 1 px 1 , 1q| 2 `ξ´| w 1 px 1 , ´1q| 2 qdx 1 ď µ c pΞq 2πL ¨lim sup kÑ0 1 k 2 ż 1 ´1pk 4 φ 2 `2k 2 pφ 1 q 2 `pφ 2 q 2 qpk, x 2 qdx 2 `ÿ kPL ´1Zzt0u 1 k 2 ż 1 ´1pk 4 φ 2 `2k 2 pφ 1 q 2 `pφ 2 q 2 qpk, x 2 qdx 2 ‹ ‹ ‹ ‹ ' .
(5.23)

Combining (5.20), (5.22) and (5.23), it gives

ż 2πLT pξ `|w 1 px 1 , 1q| 2 `ξ´| w 1 px 1 , ´1q| 2 qdx 1 ď µ c pΞq}∇ w} 2 L 2 pΩq .
Lemma 5.4 is proven.

We now prove Proposition 5.2.

Proof of Proposition 5.2. We rewrite (5.7) 2 as

pρ 0 `σδ qB t u d ´µ∆ u d `∇q d " f δ ´gσ d e 2 ,
where f δ " ´σδ B t u M ´pρ 0 `σδ q u δ ¨∇ u δ . Differentiate the resulting equation with respect to t and then multiply by B t u d , we obtain after integration that ż

Ω B t σ δ |B t u d | 2 d x `żΩ pρ 0 `σδ qB 2 t u d ¨Bt u d d x " ż Ω µ∆B t u d ¨Bt u d d x ´żΩ ∇B t q d ¨Bt u d d x `żΩ pB t f δ ´gB t σ d e 2 q ¨Bt u d d x.
Since divB t u d " 0, we use the integration by parts to further obtain ż

Ω B t σ δ ptq|B t u d ptq| 2 d x `żΩ pρ 0 `σδ ptqqB 2 t u d ptq ¨Bt u d ptqd x " ż Ω pB t f δ ptq ´gB t σ d ptq e 2 q ¨Bt u d ptqd x ´µ ż Ω |∇B t u d ptq| 2 d x `ż2πLT pξ `|B t u d 1 pt, x 1 , 1q| 2 `ξ´| B t u d 1 pt, x 1 , ´1q| 2 qdx 1 .
That means, 1 2

d dt ż Ω pρ 0 `σδ ptqq|B t u d ptq| 2 d x " ´1 2
ż Ω B t σ δ ptq|B t u d ptq| 2 d x `żΩ pB t f δ ptq ´gB t σ d ptq e 2 q ¨Bt u d ptqd x ´µ ż Ω |∇B t u d ptq| 2 d x `ż2πLT pξ `|B t u d 1 pt, x 1 , 1q| 2 `ξ´| B t u d 1 pt, x 1 , ´1q| 2 qdx 1 .
Using (5.7) 1 , we then get

d dt ż Ω ´pρ 0 `σδ ptqq|B t u d ptq| 2 ´gρ 1 0 |u d 2 ptq| 2 ¯d x `2µ ż Ω |∇B t u d ptq| 2 d x ´2 ż 2πLT pξ `|B t u d 1 pt, x 1 , 1q| 2 `ξ´| B t u d 1 pt, x 1 , ´1q| 2 qdx 1 " ´żΩ B t σ δ ptq|B t u d ptq| 2 d x `2 ż Ω pB t f δ ptq `g u δ ptq ¨∇σ δ ptq e 2 q ¨Bt u d ptqd x.
Integrating in time variable, we get

} b ρ 0 `σδ ptqB t u d ptq} 2 L 2 pΩq `2µ ż t 0 }∇B t u d psq} 2 L 2 pΩq ds ´2 ż t 0 ż 2πLT pξ `|u d 1 ps, x 1 , 1q| 2 `ξ´| u d 1 ps, x 1 , ´1q| 2 qdx 1 ds " ż Ω gρ 1 0 |u d 2 ptq| 2 d x `´ż Ω pρ 0 `σδ ptqq|B t u d ptq| 2 d x ¯ˇˇt "0 `ż t 0 ż Ω p2B t f δ psq `2g u δ psq ¨∇σ δ psq e 2 ´Bt σ δ psqB t u d psqq ¨Bt u d psqds.
(5.24)

We continue using (5.12), (5.13) and (5.14) to estimate each term of the r.h.s of (5.24). This yields

} b ρ 0 `σδ ptqB t u d ptq} 2 L 2 pΩq `2µ ż t 0 }B t u d psq} 2 L 2 pΩq ds ´2 ż 2πLT pξ `|B t u d 1 pt, x 1 , 1q| 2 `ξ´| B t u d 1 pt, x 1 , ´1q| 2 qdx 1 ď ż Ω gρ 1 0 |u d 2 ptq| 2 d x `C16 δ 3 F 3 M ptq.
(5.25)

Due to (5.17), we further get that

} b ρ 0 `σδ ptqB t u d ptq} 2 L 2 pΩq `2µ ż t 0 }∇B t u d psq} 2 L 2 pΩq ds ´2 ż t 0 ż 2πLT pξ `|B t u d 1 ps, x 1 , 1q| 2 `ξ´| B t u d 1 ps, x 1 , ´1q| 2 qdx 1 ds ď Λ 2 ż Ω ρ 0 | u d ptq| 2 d x `Λµ ż Ω |∇ u d ptq| 2 d x ´Λ ż 2πLT pξ `|u d 1 pt, x 1 , 1q| 2 `ξ´| u d 1 pt, x 1 , ´1q| 2 qdx 1 `C16 δ 3 F 3 M ptq ď Λ 2 ż Ω pρ 0 `σδ ptqq| u d ptq| 2 d x `Λµ ż Ω |∇ u d ptq| 2 d x ´Λ ż 2πLT pξ `|u d 1 pt, x 1 , 1q| 2 `ξ´| u d 1 pt, x 1 , ´1q| 2 qdx 1 `C17 δ 3 F 3 M ptq. (5.26)
On the other hand, we have

d dt } b ρ 0 `σδ ptq u d ptq} 2 L 2 pΩq " 2 ż Ω pρ 0 `σδ ptqq u d ptq ¨Bt u d ptqd x `żΩ B t σ δ ptq| u d ptq| 2 d x.
Let us recall 0 from (2.19) and ν 0 " 3` 0 2` 0 P p1, 3 2 q. We fix two positive constants m 1,2 such that

m 1 " ν 0 `bν 2 0 ´1 (5.27)
and that

m 2 " µpm 2 1 ´m1 `1q´µ c pΞqpm 2 1 `1q`bpµpm 2 1 ´m1 `1q ´µc pΞqpm 2 1 `1qq 2 ´µ2 m 2 1 .
(5.28) With m 1 ą 0 from (5.27), we use Young's inequality to observe

2 ż Ω pρ 0 `σδ ptqq u d ptq ¨Bt u d ptqd x ď 1 Λm 1 } b ρ 0 `σδ ptqB t u d ptq} 2 L 2 pΩq `Λm 1 } b ρ 0 `σδ ptq u d ptq} 2 L 2 pΩq .
That will imply

d dt } b ρ 0 `σδ ptq u d ptq} 2 L 2 pΩq ď 1 Λm 1 } b ρ 0 `σδ ptqB t u d ptq} 2 L 2 pΩq `Λm 1 } b ρ 0 `σδ ptq u d ptq} 2 L 2 pΩq `C18 δ 3 F 3 M ptq. (5.29)
With m 2 ą 0 defined as in (5.28), we obtain from (5.26) and (5.29) that

d dt } b ρ 0 `σδ ptq u d ptq} 2 L 2 pΩq `m2 }∇ u d ptq} 2 L 2 pΩq ď ´m1 `1 m 1 ¯Λ} b ρ 0 `σδ ptq u d ptq} 2 L 2 pΩq `´µ m 1 `m2 ¯}∇ u d } 2 L 2 pΩq `2 Λm 1 ż t 0 ż 2πLT pξ `|B t u d 1 ps, x 1 , 1q| 2 `ξ´| B t u d 1 ps, x 1 , ´1q| 2 qdx 1 ds ´2µ Λm 1 ż t 0 }∇B t u d psq} 2 L 2 pΩq ds `C19 δ 3 F 3 M ptq.
Together with (5.18), we deduce

d dt } b ρ 0 `σδ ptq u d ptq} 2 L 2 pΩq `m2 }∇ u d ptq} 2 L 2 pΩq ď ´m1 `1 m 1 ¯Λ} b ρ 0 `σδ ptq u d ptq} 2 L 2 pΩq `´µ m 1 `m2 ¯}∇ u d ptq} 2 L 2 pΩq ´2pµ ´µc pΞqq Λm 1 ż t 0 }∇B t u d psq} 2 L 2 pΩq ds `C19 δ 3 F 3 M ptq.
(5.30)

We use Young's inequality to get that Because of (2.21), we have λ j ą 2 3 ν 0 Λ for j m ď j ď N and λ j ă 2 3 ν 0 Λ for j ě N `1. It yields that for j m ď j ď N , ż t 0 e p3λj ´2ν0Λqs ds " 1 3λ j ´2ν 0 Λ pe p3λj ´2ν0Λqt ´1q ď 1 3λ j ´2ν 0 Λ e p3λj ´2ν0Λqt

´µ m 1 `m2 ¯}∇ u d ptq} 2 L 2 pΩq " 2 ´µ m 1 `m2 ¯ż t 0 ż Ω ∇ u d psq
(5.36) and that for j ě N `1, ż t 0 e p3λj ´2ν0Λqs ds " 1 3λ j ´2ν 0 Λ pe p3λj ´2ν0Λqt ´1q ď 1 2ν 0 Λ ´3λ j .

(5.37)

In view of (5.36) and (5.37), we obtain from (5.35

) that if M ď N , } u d ptq} 2 L 2 pΩq ď C 20 δ 3 ´M ÿ j"jm |c j | 3λ j ´2ν 0 Λ e 3λj t ānd if M ě N `1, } u d ptq} 2 L 2 pΩq ď C 20 δ 3 ´M ÿ j"jm |c j | 3λ j ´2ν 0 Λ e 3λj t `M ÿ j"N `1 |c j | 2ν 0 Λ ´3λ j e 2ν0Λt ¯.
That means

} u d ptq} 2 L 2 pΩq ď C 21 δ 3 ´N ÿ j"jm |c j |e λj t `maxp0, M ´N q ´max N `1ďjďM |c j | ¯e 2 3 ν0Λt ¯3.
(5.38)

To show the bound of }σ d ptq} L 2 pΩq , we use Cauchy-Schwarz's inequality to deduce from (5.7) 1 that

d dt }σ d ptq} L 2 pΩq ď }σ d ptq} L 2 pΩq ď pmax ρ 1 0 q}u d 2 ptq} L 2 pΩq `} u δ ptq} L 2 pΩq }σ δ ptq} H 1 pΩq .
Using (5.6), we obtain

d dt }σ d ptq} L 2 pΩq ď C 22 p}u d 2 ptq} L 2 pΩq `δ2 F 2 M ptqq.
Note that σ d p0q " 0, integrating the resulting inequality in time, we have

}σ d ptq} L 2 pΩq ď C 22 ż t 0 p}u d 2 psq} L 2 pΩq `δ2 F 2 M psqqds.
Together with (5.38), we have

}σ d ptq} 2 L 2 pΩq ď C 23 δ 3 ´N ÿ j"jm |c j |e λj t `maxp0, M ´N q ´max N `1ďjďM |c j | ¯e 2 3 ν0Λt ¯3.
(5.39) The inequality (5.11) follows from (5.38) and (5.39). Proof of Proposition 5.2 is complete.

´1 2 c 2 jm e pλj m `λjm`1qt } u jm } 2 L 2 pΩq .
This yields

} u M ptq} 2 L 2 pΩq ě c 2 jm ´e2λj m t ´1 2 e pλj m `λjm`1qt ´1 4 e pλj m`1 `λjm`2qt ¯} u jm } 2 L 2 pΩq `M ÿ j"jm`1 c 2 j e 2λj t } u j } 2 L 2 pΩq .
Notice that for all t ě 0, e 2λj m t ´1 2 e pλj m `λjm`1qt ´1 4 e pλj m`1 `λjm`2qt ě 1 4 e 2λj m t .

Hence, we have } u M ptq} L 2 pΩq ě C 24 F M ptq, (5.41) for all t ď minpT δ , T ‹ , T ‹‹ q.

Let cpM q " max N `1ďjďM

|c j | |c jm | ě 0.
We recall the definition of T ‹ and T ‹‹ from (5.5) and the fact that T δ satisfies uniquely δF M pT δ q " 0 , provided that 0 is taken to be (5.42)

0 ă min ´C2 δ 0 C 3 , C 2
We then prove that T δ ď mintT ‹ , T ‹‹ u.

(5.43) In fact, if T ‹ ă T δ , we have from (5.6) that

Eppσ δ , u δ qpT ‹ qq ď C 3 δF M pT ‹ q ď C 3 δF M pT δ q " C 3 0 ă C 2 δ 0 .
And if T ‹‹ ă T δ , we have by (5.11) and the definition of C 2 (5.4) that

}pσ δ , u δ qpT δ q} L 2 pΩq ď δ}pσ M , u M qpT δ q} L 2 pΩq `}pσ d , u d qpT δ q} L 2 pΩq ď C 2 δF M pT δ q `aC 4 δ 3 2 ´N ÿ j"1 |c j |e λj T δ `maxp0, M ´N q ´max N `1ďjďM |c j | ¯e 2 3 ν0ΛT δ ¯3 2 .
(5.44)

Notice from (2.21) that for N `1 ď j ď M ,

|c j |δe 2 3 ν0ΛT δ ă |c j | |c jm | pδ|c jm |e λ1T δ q ă |c j | |c jm | δF M pT δ q " |c j | |c jm | 0 .
Then, it follows from (5.44) that

}pσ δ , u δ qpT δ q} L 2 pΩq ď C 2 δF M pT δ q `aC 4 p1 `McpM qq 3 2 δ 3 2 F 3 2 M pT δ q ď C 2 0 `aC 4 p1 `McpM qq 3 2 3 2
0 . Using (5.42) again, we deduce }pσ δ , u δ qpT δ q} L 2 pΩq ă 2C 2 0 " 2C 2 δF M pT δ q.

which also contradicts the definition of T ‹‹ . Once we have (5.43), we then get from (5.11) and (5.41) that

} u δ pT δ q} L 2 pΩq ě δ} u M pT δ q} L 2 pΩq ´} u d pT δ q} L 2 pΩq ě C 24 δF M pT δ q ´aC 4 δ 3 2 ´N ÿ j"1 |c j |e λj T δ `maxp0, M ´N q ´max N `1ďjďM |c j | ¯e 2 3 ν0ΛT δ ¯3 2 .
Therefore,

} u δ pT δ q} L 2 pΩq ě C 24 0 ´aC 4 p1 `McpM qq 3 2 3 2 0 ě C 24 0 2 ą 0.
(5.45)

The inequality (2.22) is proven by taking δ 0 satisfying (5.3), 0 satisfying (5.42) and m 0 " 1 2 C 24 . This ends the proof of Theorem 2.2.

is bounded because of the embedding H 2 pp´1, 1qq ãÑ C 1 pp´1, 1qq. To prove (3.6), let us consider the Lagrangian functional

L k pφ, βq " β ´ż 1 ´1ppφ
2 q 2 `2k 2 pφ 1 q 2 `k4 φ 2 qdx 2 ´1q ´pξ ´pφ 1 p´1qq 2 `ξ`p φ 1 p1qq 2 q (A.1) for any φ P H2 pp´1, 1qq and β ‰ 0. Using Lagrange multiplier theorem, the extrema of the quotient ξ ´pφ 1 p´1qq 2 `ξ`p φ 1 p1qq 2 ş 1 ´1ppφ 2 q 2 `2k 2 pφ 1 q 2 `k4 φ 2 qdx 2 are necessarily the stationary points of pφ k , β k q of L k , which satisfy

ż 1 ´1ppφ 2 k q 2 `2k 2 pφ 1 k q 2 `k4 φ 2 k qdx 2 " 1, (A.2)
and

β k ż 1 ´1pφ 2 k ω 2 `2k 2 φ 1 k ω 1 `k4 φ k ωqdx 2 " ξ ´φ1 k p´1qω 1 p´1q `ξ`φ 1 k p1qω 1 p1q (A.3)
for all ω P H2 pp´1, 1qq. We obtain from (A.3) after taking integration by parts that

φ p4q k ´2k 2 φ 2 k `k4 φ k " 0, and $ ' ' ' & ' ' ' % β k pφ 3 k p1q `2k 2 φ 1 k p1qqωp1q " 0, pβ k φ 2 k p1q ´ξ`φ 1 k p1qqω 1 p1q " 0, β k pφ 3 k p´1q `2k 2 φ 1 k p´1qqωp´1q " 0, pβ k φ 2 k p´1q `ξ´φ 1 k p´1qqω 1 p´1q " 0,
for all ω P H2 pp´1, 1qq. This yields

# β k φ 2 k p1q ´ξ`φ 1 k p1q " 0, β k φ 2 k p´1q `ξ´φ 1 k p´1q " 0. (A.4)
Hence, φ k is of the form φ k px 2 q " pAx 2 `Bq sinhpkx 2 q `pCx 2 `Dq coshpkx 2 q, with A, B, C, D are four constants such that A 2 `B2 `C2 `D2 ą 0. Since φ k P H2 pp´1, 1qq, we get # pA `Bq sinh k `pC `Dq cosh k " 0, p´A `Bq sinhp´kq `p´C `Dq coshp´kq " 0.

It yields

C " ´B tanh k and D " ´A tanh k. (A.5)

We then compute φ 1 k px 2 q " pA `kD `kCx 2 q sinhpkx 2 q `pC `kB `kAx 2 q coshpkx 2 q and φ 2 k px 2 q " p2kC `k2 B `k2 Ax 2 q sinhpkx 2 q `p2kA `k2 D `k2 Cx 2 q coshpkx 2 q.

That implies (3.10). The proof of the second assertion of Proposition 3.1 is complete.

Appendix C. Proof of Proposition 3.1(3)

In this appendix, we prove Proposition 3.1(3). We first show that µ s c pΞq " sup φP H2 pp´1,1qq

ξ ´pφ 1 p´1qq 2 `ξ`p φ 1 p1qq 2 ş 1 ´1pφ 2 q 2 dx 2 (C.1)
Indeed, let μc pΞq :" sup φP H2 pp´1,1qq

ξ ´pφ 1 p´1qq 2 `ξ`p φ 1 p1qq 2 ş 1 ´1pφ 2 q 2 dx 2 and then prove that µ s c pΞq " μc pΞq. Clearly, we have µ c pk, Ξq ď μc pΞq for all k P Rzt0u. It yields µ s c pΞq ď μc pΞq. It suffices to show that μc pΞq ě µ s c pΞq. For any ε ą 0, from the definition of µ c pΞq, we fix a function φ ε P H2 pp´1, 1qq such that

ξ ´pφ 1 ε p´1qq 2 `ξ`p φ 1 ε p1qq 2 ş 1 ´1pφ 2 ε q 2 dx 2 ě μc pΞq ´ε.
Let k ‰ 0 be small enough, we then obtain

ξ ´pφ 1 ε p´1qq 2 `ξ`p φ 1 ε p1qq 2 ş 1 ´1ppφ 2 ε q 2 `2k 2 pφ 1 ε q 2 `k4 φ 2 ε qdx 2 ą ξ ´pφ 1 ε p´1qq 2 `ξ`p φ 1 ε p1qq 2 ş 1 ´1pφ 2 ε q 2 dx 2 ´ε.
That implies µ c pk, Ξq ą μc pΞq ´2ε. We deduce that μc pΞq " sup kPRzt0u µ c pk, Ξq, i.e. (C.1).

Then, we show that max φP H2 pp´1,1qq

ξ ´pφ 1 p´1qq 2 `ξ`p φ 1 p1qq 2 ş 1 ´1pφ 2 q 2 dx 2 " 1 3 ´ξ``ξ´`b ξ 2 `´ξ `ξ´`ξ 2 ´¯. (C.2)
Let us consider the Lagrangian functional

L 0 pφ, βq " β ´ż 1 ´1pφ 2 q 2 dx 2 ´1¯´ξ ´pφ 1 p´1qq 2 ´ξ`p φ 1 p1qq 2 . (C.3)
for any φ P H2 pp´1, 1qq and β ‰ 0. Owing to Lagrange multiplier theorem, the extrema of the quotient ξ ´pφ 1 p´1qq 2 `ξ`p φ 1 p1qq 2 ş 1 ´1pφ 2 q 2 dx 2 are necessarily the stationary points pφ 0 , β 0 q of L 0 , which satisfy

ż 1 ´1pφ 2 0 q 2 dx 2 " 1, (C.4) and β 0 ż 1 ´1 φ 2 0 ω 2 dx 2 ´pξ ´φ1 0 p´1qω 1 p´1q `ξ`φ 1 0 p1qω 1 p1qq " 0 (C.5)
for all ω P H2 pp´1, 1qq. We obtain from (C.5) after taking integration by parts that φ p4q 0 " 0 on p´1, 1q. and # φ 2 0 p1q " ξ `φ1 0 p1q, φ 2 0 p´1q " ´ξ´φ

1 0 p´1q. (C.6)
Hence, φ 0 is of the form φ 0 px 2 q " px 2 2 ´1qpAx 2 `Bq. Substituting this form of φ 0 into (C.6), we have that # β 0 p3A `Bq " ξ `pA `Bq, β 0 p3A ´Bq " ξ ´pA ´Bq.

Hence, # Ap3β 0 ´ξ`q `Bpβ 0 ´ξ`q " 0, Ap3β 0 ´ξ´q ´Bpβ 0 ´ξ´q " 0.

(C.7) System (C.7) admits a nontrivial solution pA, Bq if and only if p3β 0 ´ξ`q pβ 0 ´ξ´q `p3β 0 ´ξ´q pβ 0 ´ξ`q " 0.

It yields 3β 2 0 ´2pξ ``ξ ´qβ 0 `ξ´ξ`" 0.

(C.8) The discriminant of (C.8) is ∆ 0,ξ " pξ ``ξ ´q2 ´3ξ ´ξ`" ξ 2 We take the higher value β 0,`. As β 0 " β 0,`, we have from (C.7) 2 that Ap3β 0,`´ξ´q " Bpβ 0,`´ξ´q .

It is obvious that 3β 0,`´ξ´" ξ ``b ξ 2

`´ξ `ξ´`ξ 2 ´ą 0. Then we have A " B β 0,`´ξ3 β 0,`´ξá nd φ 0 px 2 q " Bz 0 px 2 q, with z 0 px 2 q " ´β0,`´ξ3 β 0,`´ξ´x 2 `1¯p

x 2 2 ´1q.

We continue using (C.4) to find a non-zero B. This yields

B 2 ż 1 ´1pz 2 2 px 2 qq 2 dx 2 " 1.
That is equivalent to ´.

8B 2 ´3
That means, we observe max φP H2 pp´1,1qq

ξ ´pφ 1 p´1qq 2 `ξ`p φ 1 p1qq 2 ş 1 ´1pφ 2 q 2 dx 2 That variational problem is attained by the function φ 0 px 2 q " ˘1 2 ? 2 3β 0,`´ξb In [2], the authors Ding, Zi and Li construct an approximate solution generated by the maximal normal mode, pσ a , u a , q a qpt, xq " δe λ1pkqt U 1 p xq with k being fixed such that 2Λ 3 ă λ 1 pkq ă Λ. Applying Proposition 5.1, the nonlinear equations (2.1)-(2.2) with the initial data pσ δ , u δ , q δ qp0q " pσ a , u a , q a qp0q. admits a strong solution pσ δ , u δ q P C 0 pr0, T max q, H 1 ˆH2 q with an associated pressure q δ P C 0 pr0, T max q, L 2 q. Let T δ such that δe λ1T δ " 0 ! 1. We define T ‹ :" sup ! t P p0, T max q|Epσ δ ptq, u δ ptqq ď Cδ 0 u ą 0, T ‹‹ :" suptt P p0, T max q|}pσ δ , u δ qptq} L 2 pΩq ď Cδe λ1t ) ą 0.

Then for all t ď mintT δ , T ‹ , T ‹‹ u, we have E 2 pσ δ ptq, u δ ptqq `}B t u δ ptq} 2 In [2, Proposition 5.2], they claim that the difference functions pσ d , u d , q d q " pσ δ , u δ , q δ q ´pσ a , u a , q a q enjoy }pσ d , u d q} 2 L 2 pΩq ď Cδ 3 e 3λ1t (D.1) for all µ ą 0. We believe that (D.1) needs to be corrected, not for all µ ą 0. Precisely, we are in doubt about inequality (137) in that paper, that is for all t ď mintT δ , T ‹ , T ‹‹ u, The inequality (D.1) is followed by applying Gronwall's inequality to (D.4).

We shall explain the arguments of (D.2) in [2]. First, we still have

}

3. 2 . 11 4. Linear instability 13 4. 1 . A sequence of characteristic values 13 4. 2 . 15 5

 21113113215 The threshold of viscosity coefficient 10 3.3. A bilinear form and a self-adjoint invertible operator Proof of Theorem 2.1 and normal modes of the linearized equations

ρ 1 0

 1 ρ0 } L 8 pp´1,1qq , we further obtain the uniform upper bound b g L0 of λ. Lemma 2.1. For any k P L ´1Zzt0u, ' all characteristic values λ are always real, ' all characteristic values λ satisfy that λ ď b g L0 .

3. 1 .Lemma 3 . 1 .

 131 The positivity of characteristic values λ. For any k P L ´1Zzt0u, ' all characteristic values λ are always real, ' all characteristic values λ satisfy that λ ď b g L0 .

Meanwhile, for all n ě 1 and λ ď 1 8 )

 18 In view of (4.7), (4.8) and Lemma 4.1, we obtain only one solution λ n of (2.14) and pλ n , φ n q satisfies (1.4)-(1.5). That means for all n, λ n is a characteristic value, hence it is bounded by b g L0 .

2 2C 4 2 244C 4

 2424 p1 `McpM qq 3 , C p1 `McpM qq 3 ¯.

2 `1¯p x 2 2

 22 ´3 0β 0,`ξ´`7 ξ 2 ´´β 0,`´ξ3 β 0,`´ξ´x ´1q. Combining (C.1) and (C.2), we obtain Proposition 3.1(3). Appendix D. Comments on the paper of Ding, Zi and Li

L 2 pΩq `ż t 0 }∇B t u δ pτ q} 2 L 2

 022 pΩq dτ ď Cδ 2 e 2λ1t .

} b ρ 0 `σδ ptqB t u d ptq} 2 L 2 pΩq `Λµ}∇ u d ptq} 2 L 2 pΩq`µ ż t 0 }∇B t u d psq} 2 L 2 pΩq ds ď Λ ´ż t 0 } b ρ 0 } b ρ 0 `σδ ptq u d ptq} 2 L 2 ptqq u d ptq ¨Bt u d ptqd x `żΩ B t σ δ ptq| u d ptq| 2 d x ď 1 Λ } b ρ 0 `σδ ptqB t u d ptq} 2 L 2 2 L 2 pΩq ď Λ ż t 0 ´}b ρ 0

 2222020022122200 `σδ psq u d psq} 2 L 2 pΩq `Λµ}∇ u d psq} 2 L 2 pΩq Λ} b ρ 0 `σδ ptq u d ptq} 2 L 2 pΩq `Cδ 3 e 3λ1t . (D.2) Due to (D.2) and the following inequality d dt pΩq `Λ} b ρ 0 `σδ ptq u d ptq} 2 L 2 pΩq `Cδ 3 e 3λ1t , (D.3) it is claimed in [2, (138)] that d dt } b ρ 0 `σδ ptq u d ptq} 2 L 2 pΩq `´} b ρ 0 `σδ ptqB t u d ptq} 2 L 2 pΩq `Λµ}∇ u d ptq} `σδ psqB t u d psq} 2 L 2 pΩq `Λµ}∇ u d psq} 2 L 2 pΩq ¯ds `Λ} b ρ 0 `σδ ptq u d ptq} 2 L 2 pΩq `Cδ 3 e 3λ1t . (D.4)

  ¨∇B t u d psqd xds Λm 2 . M ptq ď M 2 max jmďjďM |c j | 2 F M p3tq, we then have from (5.34) that } u d ptq} 2 L 2 pΩq ď C 20 δ 3 e 2ν0Λt

	Since F 3
							M ÿ	ż t	|c j |e p3λj ´2ν0Λqs ds.	(5.35)
							j"jm	0
							ď	2pµ ´µc pΞqq Λm 1	ż t 0	}∇B t u d psq} 2 L 2 pΩq ds	(5.31)
							¯2 2pµ ´µc pΞqq `Λm 1 ´µ m1 `m2	ż t 0	}∇ u d psq} 2 L 2 pΩq ds.
	Combining (5.30) and (5.31) gives us
			d dt	b } ρ 0 `σδ ptq u d ptq} 2 L 2 pΩq `m2 }∇ u d ptq} 2 L 2 pΩq
			ď ´m1	`1 m 1	b ¯Λ} ρ 0 `σδ ptq u d ptq} 2 L 2 pΩq	(5.32)
						¯2 2pµ ´µc pΞqq `Λm 1 ´µ m1 `m2	ż t 0	}∇ u d psq} 2 L 2 pΩq ds `C19 δ 3 F 3 M ptq.
	It follows from (5.27) and (5.28) that
	¯2 2pµ ´µc pΞqq Λm 1 ´µ m1 `m2 ¯m2 " 2ν 0 Therefore, (5.32) becomes " Λ ´m1 `1 m 1
		d dt	b }	ρ 0 `σδ ptq u d ptq} 2 L 2 pΩq `m2 }∇ u d ptq} 2 L 2 pΩq
		ď 2ν 0 Λ ´}b ρ 0 `σδ ptq u d ptq} 2 L 2 pΩq `m2	ż t 0	}∇ u d psq} 2 L 2 pΩq ds ¯`C 19 δ 3 F 3 M ptq.
							(5.33)
	Recalling that u d p0q " 0, thus, applying Gronwall's inequality to (5.33), one obtains
	b						ż t	ż t
	}	ρ 0 `σδ ptq u d ptq} 2 L 2 pΩq `m2	0	}∇ u d psq} 2 L 2 pΩq ds ď C 19 δ 3 e 2ν0Λt	0	e ´2ν0Λs F 3 M psqds.
							(5.34)

  b ρ 0 `σδ ptqB t u d ptq} 2 L 2 pΩq `2µ ps, x 1 , 1q| 2 `ξ´| u d 1 ps, x 1 , ´1q| 2 qdx 1 ds p2B t f δ psq `2g u δ psq ¨∇σ δ psqe 2 ´Bt σ δ psqB t u d psqq ¨Bt u d psqds. ps, x 1 , 1q| 2 `ξ´| u d 1 ps, x 1 , ´1q| 2 qdx 1 ds |u d 2 ptq| 2 d x `Cδ 3 e 3λ1t . |B t u d 1 ps, x 1 , 1q| 2 `ξ´| B t u d 1 ps, x 1 , ´1q| 2 qdx 1 ds , 1q| 2 `ξ´| u d 1 pt, x 1 , ´1q| 2 qdx 1 `Cδ 3 e 3λ1t . pt, x 1 , 1q| 2 `ξ´| u d 1 pt, x 1 , ´1q| 2 qdx 1 ps, x 1 , 1q| 2 `ξ´| u d 1 ps, x 1 , ´1q| 2 qdx 1 ds

						ż t
						0	}∇B t u d psq} 2 L 2 pΩq ds
	ż 2πLT 1 " ´2 ż t 0 pξ `|u d Ω gρ 1 0 |u d 2 ptq| 2 d x `´ż Ω ż	pρ 0	`σδ ptqq|B t u d ptq| 2 d x	"0 ¯ˇˇt	(D.5)
		`ż t	ż
		0	Ω
	We estimate
		b			ż t
		}	ρ 0 `σδ ptqB t u d ptq} 2 L 2 pΩq `2µ	0	}∇B t u d psq} 2 L 2 pΩq ds
		ż 2πLT 1 ď ´2 ż t 0 pξ `|u d ż gρ 1 0 (D.6)
			Ω
	That implies
	b				ż t
	}	ρ 0 `σδ ptqB t u d ptq} 2 L 2 pΩq `2µ	0	}∇B t u d psq} 2 L 2 pΩq ds
	ż 2πLT pξ `ď Λ 2 ´2 ż t 0 ż			(D.7)
	By using the inequality
		Λµ}∇ u d } 2 L 2 pΩq ď Λ 2 µ	ż t 0	}∇ u d psq} 2 L 2 pΩq ds	`µ ż t 0	}∇B t u d psq} 2 L 2 pΩq ds
	and the identity
		ż
	Λ 1 " Λ 2 2πLT pξ `|u d ż t ż 0 2πLT 1 `ż t pξ `|u d ż
		0

Ω pρ 0 `σδ ptqq| u d ptq| 2 d x `Λµ ż Ω |∇ u d ptq| 2 d x ´Λ ż 2πLT pξ `|u d 1 pt, x 1 2πLT pξ `|B t u d 1 ps, x 1 , 1q| 2 `ξ´| B t u d 1 ps, x 1 , ´1q| 2 qdx 1 ds ´ż t 0 ż 2πLT pξ `|Λu d 1 ´Bt u d 1 | 2 ps, x 1 , 1q `ξ´| Λu d 1 ´Bt u d 1 | 2 ps, x 1 , ´1qqdx 1 ds,

Email address: tientai.nguyen@math.univ-paris13.fr

Acknowledgments

The author is deeply grateful to Prof. Jean-Marc Delort and Prof. Olivier Lafitte for their fruitful discussions on this paper. Thanks also go to Prof. Jeffrey Rauch for his advice on this study. The author wishes to thank Assoc. Prof. Quốc-Anh Ngô for his encouragement. The author would also thank the hospitality of Centre International de Rencontre Mathématique and Université de Montréal during the visit where parts of this work were accomplished. This work is supported by a grant from Région Île-de-France.

We define

T ‹ :" sup ! t P p0, T max q|Epσ δ ptq, u δ ptqq ď C 1 δ 0 u ą 0, T ‹‹ :" suptt P p0, T max q|}pσ δ , u δ qptq} L 2 pΩq ď 2C 2 δF M ptq ) ą 0.

(5.5)

Note that Epσ δ p0q, u δ p0qq " C 1 δ ă C 1 δ 0 and because of (5.1), we then have T ‹ ą 0.

Similarly, we have T ‹‹ ą 0. Then for all t ď mintT δ , T ‹ , T ‹‹ u, it follows from the a priori energy estimate (5.2) that E 2 pσ δ ptq, u δ ptqq `}B t u δ ptq} 2

(5.6) 5.2. The difference functions. Let us recall σ d " σ δ ´δσ M , u d " u δ ´δ u M , q d " q δ ´δq M .

Hence, pσ d , u d , q d q satisfies $ ' & ' %

´ u δ ¨∇σ δ , ρ 0 B t u d ´µ∆ u d `∇q d " ´σδ B t u δ ´pρ 0 `σδ q u δ ¨∇ u δ ´gσ d e 2 , div u d " 0, (5.7) along with the initial condition, pσ d , u d qp0q " 0

(5.8)

and the boundary conditions,

(5.9)

The compatibility conditions read as

(5.10)

We now establish the error estimate for }pσ d , u d q} L 2 pΩq .

Proposition 5.2. For all t ď minpT δ , T ‹ , T ‹‹ q, there holds

(5.11)

The proof of Proposition 5.2 relies on Lemmas 5.

(5.40)

It can be seen that

By Cauchy-Schwarz's inequality, we obtain

This yields

Due to the assumption (2.17), we deduce that

Appendix A. The precise value of µ c pk, Ξq

In this appendix, we prove Proposition 3.1(1). The equality (3.5) can be seen immediately from the definition of B k,0,µ .

Note that the quotient

Substituting these formulas into (A.4), we have

β k ´p2kC `k2 pB ´Aqq sinhp´kq `p2kA `k2 pD ´Cqq coshp´kq " ´ξ´´p A `kpD ´Cqq sinhp´kq `pC `kpB ´Aqq coshp´kqq.

Thanks to (A.5), that reduces to

Then, pA, Bq is a solution of the following system

(A.7) System (A.7) admits a nontrivial solution if and only

We rewrite Eq. (A.8) as a quadratic equation of β k , that is

The discriminant is

´4k 2 pcosh 2 p2kq ´1qpsinh 2 p2kq ´4k 2 qξ `ξ"

`k2 sinh 2 p2kqpsinh 2 p2kq ´4k 2 qpξ `´ξ ´q2 .

Because tanhp2kq ă 2k for all k ą 0 and ξ 2 ``ξ 2 ´ą 0, we have ∆ k,ξ is always positive. Hence, we have that (A.9) has two roots

4k 2 sinh 2 p2kq .

We take the higher value β k,`ą 0 and then solve the system (A.7) as β k " β k,`.

If ξ ´ě ξ `, we have

Then, we obtain from (A.7) 1 that

So that pA, B, C, Dq " Ap1, a k,Ξ , ´ak,Ξ tanh k, ´tanh kq with A ‰ 0 and φ k px 2 q " Az k px 2 q, with z k px 2 q " px 2 `ak,Ξ q sinhpkx 2 q ´tanh kpa k,Ξ x 2 `1q coshpkx 2 q.

We find A from (A.2), such that

If 0 ă ξ ´ă ξ `, that will imply 2kpcoshp2kq ´1qβ k,`´ξ´p sinhp2kq ´2kq ą 0.

We further get from (A.7) 2 that B " ´A 2kp1 `coshp2kqqβ k,`´ξ´p 2k `sinhp2kqq 2kpcoshp2kq ´1qβ k,`´ξ´p sinhp2kq ´2kq ": ´Ab k,Ξ .

So that, we have pA, B, C, Dq " Ap1, ´bk,Ξ , b k,Ξ tanh k, ´tanh kq with A ‰ 0 and φ k px 2 q " Aw k px 2 q, with w k px 2 q " px 2 ´bk,Ξ q sinhpkx 2 q `pb k,Ξ x 2 ´1q tanh k coshpkx 2 q.

We still find A from (A.2),

We have just shown that µ c pk, Ξq " max φP H2 pp´1,1qq

`sinh 2 p2kqpsinh 2 p2kq ´4k 2 qpξ `´ξ ´q2

That variational problem is attained by the function φ k px 2 q " Az k px 2 q, where A satisfies (A.11) or φ k px 2 q " Aw k px 2 q, where A satisfies (A.12). The equality (3.6) is shown and the first part of Proposition 3.1 then follows.

Appendix B. Asymptotic behavior of µ c pk, Ξq in low/high regime of wave number

Let us prove Proposition 3.1 (2). Clearly, we have that µ c pk, Ξq is a decreasing function in k ą 0. It yields (3.7).

We first consider k Ñ 0. Let us recall the Taylor's expansion of sinhp2kq and coshp2kq. We have sinhp2kq " 2k `4 3 k 3 `4 15 k 5 `Opk 6 q, and coshp2kq " 1 `2k 2 `2 3 k 4 `Opk 5 q.

We deduce that sinhp2kq coshp2kq ´2k 4k sinh 2 p2kq "

and that sinh 2 p2kqpsinh 2 p2kq ´4k 2 q 16k 2 sinh 4 p2kq "

We deduce that

That will imply (3.9), i.e.

µ s c pΞq "

Furthermore, we have that

it is obtained from (D.7) that (see (134) in [2]) We are not clear about the way in [2] to remove all integral terms over 2πLT in the r.h.s of (D.9) to get (D.2) for all µ ą 0, especially the following term


Paul Cousin

Aude Maignan

Organic Structures Emerging From Bio-Inspired Graph-Rewriting Automata

Keywords: cellular automata, graph-rewriting automata, dynamical systems, artificial life, complexity

Graph-Rewriting Automata (GRA) are an extension of Cellular Automata to a dynamic structure using local graph-rewriting rules. This work introduces linear algebra based tools that allow for a practical investigation of their behavior in deeply extended time scales. A natural subset of GRA is explored in different ways thereby demonstrating the benefits of this method. Some elements of the subset were discovered to create chaotic patterns of growth and others to generate organic-looking graph structures. These phenomena suggest a strong relevance of GRA in the modeling natural complex systems. The approach presented here can be easily adapted to a wide range of GRA beyond the chosen subset.

Introduction

The idea of extending the study of discrete dynamical systems like cellular automata to systems with evolving typologies has already been approached in several ways. DEM-Systems [START_REF] Edwards | A class of discrete dynamical systems with properties of both cellular automata and l-systems[END_REF][START_REF]Dem-systems: a new type of adaptive system[END_REF][START_REF]Complex self-reproducing systems[END_REF], Structurally Dynamic Cellular Automata [START_REF] Ilachinski | Structurally dynamic cellular automata[END_REF][START_REF] Alonso-Sanz | A structurally dynamic cellular automaton with memory in the hexagonal tessellation[END_REF][START_REF] Alonso-Sanz | A structurally dynamic cellular automaton with memory[END_REF], Generative Network Automata [START_REF] Sayama | Generative network automata: A generalized framework for modeling complex dynamical systems with autonomously varying topologies[END_REF][START_REF] Sayama | Generative network automata: A generalized framework for modeling adaptive network dynamics using graph rewritings[END_REF][START_REF] Schmidt | Designing and evaluating algorithms for automated discovery of adaptive network models based on generative network automata[END_REF] and Graph-Rewriting Automata [START_REF] Tomita | Self-description for construction and computation on graph-rewriting automata[END_REF][START_REF]Asynchronous graph-rewriting automata and simulation of synchronous execution[END_REF][START_REF] Tomita | Graph-rewriting automata as a natural extension of cellular automata[END_REF] are related concepts sharing this goal. These systems can produce a range of new behaviors compared to their fixed topology counterparts, however their implementation is much less straightforward.

In this paper, new linear algebra based tools will be presented, then applied to study a natural subset of GRA. Links to the Mathematica source code, a more recent GPU-accelerated Python implementation, and more information can be found at paulcousin.github.io/graph-rewriting-automata.

1

A GRA consists of an initial graph G0 defined at time step t = 0 and a rule to iteratively evolve it to any discrete time step t > 0. Gt is the graph obtained at time t. The studied subset of GRA will be defined by the following restrictions:

1. 3-regular, undirected and finite graphs, 2. binary labeling of vertices v with two possible states s:

• s(v) = 1, called "alive" and colored purple v ,

• s(v) = 0, called "dead" and colored orange v , 3. rules local to a vertex and its adjacent vertices, deterministic and applied simultaneously on the entire graph, 4. only one type of topology altering operation called division.

Divisions will be operated as follows:



In this model, vertices can, for example, be seen as cells that divide under specific local conditions (internal and environmental). An initially simple graph can thus grow into a large organism with a complex self-organized structure.

→ → → → → . . . G0 G1 G2 G3 G4
Figure 1: Example of evolution starting from a simple initial graph.

Graphs and Rules

A graph G is fully described by an adjacency matrix A and a state vector S: The neighborhood N (v) of a vertex v is the set of vertices which are adjacent to it. The state s of a vertex being either dead or alive and there being from 0 to 3 alive vertices in its neighborhood, there are only 8 possible configurations c, which can be ordered in this way:

Gt = { At , St } G 0 ≡ A 0 =     0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0     S 0 =     1 0 0 0    
c(v) = 4 × s(v) + i∈N (v) s(i) (1)
For instance, if v is a dead vertex surrounded by 3 dead vertices, c(v) will be equal to 0 (see Figure 3 for all the other configurations). The space of possible rules applicable in the subset of GRA defined in Section 1 is finite. Every rule must specify, for each configuration, whether the vertex will be alive or dead at t + 1 and whether it will have undergone a division, leading to 4 possible final states. Thus, there are 4 8 = 65, 536 possible rules. Each rule can be described by two functions:

0 ≡ 1 ≡ 2 ≡ 3 ≡ 4 ≡ 5 ≡ 6 ≡ 7 ≡
R : [[0, 7]] → { 0 ≡ v , 1 ≡ v } R(c(vt)) = s(vt+1) (2)
R ′ : [[0, 7]] → { 0 ≡ , 1 ≡ } R ′ (c(vt)) = 1, if c leads to a division at t+1 0, otherwise (3)
Every rule can thus be labeled by a unique rule number n.

n = 7 i=0 2 i R(i) + 2 i+8 R ′ (i) (4)
This labeling system, inspired by the Wolfram code [START_REF] Wolfram | A New Kind of Science[END_REF], is such that a rule number in its binary form displays the behavior of the rule. Starting from the right, the 8 first digits indicate the future state for each configuration as they have been ordered previously. The 8 following digits show when a division occurs.

       

Implementation

Implementing GRA as described below must leverage a sparse array format to be efficient.

The application of a rule to a graph will come in several steps which are strictly equivalent to the rule being applied all at once. o being the order of the graph and @ being the operator applying a function to every element of a vector, the first step consists of computing a configuration vector C.

C =    c(v1) . . . c(vo)    = 4 × S + A • S (5)
S can then be updated as follows.

S = R @ C (6)
A division vector D is computed similarly.

D = R ′ @ C (7)
Divisions can now be performed one by one as a combination of simple operations on matrices. The first 1 in D signals the vertex to divide. For the state vector, suffice to triple the line corresponding to this vertex. For the adjacency matrix, both the line and the column must be tripled. Ones then have to be spread across these lines and columns and the intersection must be filled with a sub-matrix containing zeros on the diagonal and ones everywhere else. Finally, the first 1 in the division vector is turned into a 0 and then tripled. This process must be repeated until D is a null vector.

    1 0 0 0     →         1 0 0 0 0 0         (a) State vector     0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0     →         0 1 0 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 0         (b) Adjacency matrix     0 1 1 1     →     0 0 1 1     →         0 0 0 0 1 1         (c) Division vector

Exploration

An exhaustive exploration will be performed on the set of rules with exactly one division. This set being color symmetric, if it is the case as well for G0, considering the half of it were the division is applied to a dead cell will give the full picture. This makes a total of 1024 rules to be explored.

n ∈ {i + 2 j | i ∈ [[0, 255]], j ∈ [[8, 11]]} (8)
To investigate the behavior of these rules, a simple graph G0 depicted in Figure 6 and containing all 8 configurations will serve as a starting point. This ensures that G1 ̸ = G0. All the results and plots in this paper were computed in Mathematica 13.0 [START_REF] Inc | Mathematica, Version 13.0[END_REF], on a MacBook Pro 2020 with an Intel Core i5 Quad-Core 2.0GHz and 16GB of memory. Graphs are displayed by the default graph plotting function in Mathematica and can be better seen by enlarging this pdf.

Growth

One natural way to classify rules is by observing the pattern of growth they produce. Table 1 shows a classification obtained using several methods, mainly the least squares method.

Halted growth

A total of 422 rules lead to a halt when reaching a cycle. These cycles have periods of 1, 2, 3, 6, or 8.

Linear growth

73 rules produce a strictly linear growth, with the same number of vertices gained at each time step. 19 rules create a repeating pattern of growth that averages to linearity. The largest period spanned 42 time steps and was produced by rule 2183. This comes from the two main structures of the graph: a loop growing in 7 steps {2, 0, 0, 2, 0, 0, 0} and a braid growing in 6 steps {4, 0, 0, 4, 2, 0}. More surprising though are the 3 rules giving rise to a chaotic linear growth: 2222, 2238 and 2239. These rules produce ladder-like loops in which states change according to an Elementary Cellular Automaton rule [START_REF] Weisstein | Elementary cellular automaton[END_REF]. For the sake of concision, only rule 2222 will be presented. This rule, although producing a rather simple graph structure, is interesting by the rhythm of its growth. At any time step, it will either grow of 0, 2, 4 or 6 new vertices which averages to a linear growth. It can be noted that there are no increase of 6 vertices at once for t ∈ [[1217, 20608]]. Model fit: o = 72.8095 + 0.630684

× t Adjusted R 2 : 0.999992
What is remarkable here is the fact that this growth does not follow a repeating pattern but, instead, seems to have chaotic properties. To illuminate the peculiar aspect of this phenomenon, Figure 10 shows the distribution of the lengths of time intervals without growth in a log-linear plot. Rule 2182 was found to be unique in the set regarding its growth. It produces a particular structure that grows almost quadratically, the exponent being slightly above 2. Model fit: o = 0.0620341 × t 2.01874 Adjusted R 2 : 0.999999

Exponential growth

Rule 256 produces a perfect exponential growth from t = 1. Among the 373 other exponential rules, some display simple fractal structures, others are more complex.

Unclassified

132 rules have not allowed for a confident classification. Suspecting interesting behaviors to be responsible for these less trivial growth patterns, all of them were observed individually and a selection of rules producing remarkable graph structures is presented in Section 6.

As Figure 11 already hinted at, some rules create organic-looking graph structures.

Figure 14 shows a selection of graphs with this property, labeled with the number of the rule that produced them. These structures are particularly evocative. Some even seem familiar, resembling macroscopic algae or related photosynthetic eukaryotes.

Conclusion

In this paper, a new approach to Graph-Rewriting Automata based on linear algebra was presented. When leveraging a sparse array format, it can be used to implement GRA much more efficiently than with previous methods. To demonstrate the benefits of this approach, a natural subset of GRA was explored only using a MacBook Pro 2020.

A focus was first put on the evolution of the number of vertices over time, leading to a classification of rules by growth pattern. Several noteworthy phenomena were discovered through this approach. It also led to a category of unclassified growths in which interesting behaviors were suspected. The graphs of this category were thus visually inspected on a individual basis and some remarkable structures were discovered.

In future work, the ideas presented here could easily be adapted to cover a variety of other types of GRA. Here are a few examples of possible trajectories:

• using continuous valued or non-binary discrete states,

• working with d-regular or non-regular graphs,

• reaching an extended neighborhood with powers of A,

• working with directed graphs using A • S and A T • S,

• including other topology altering operations.

Other tools to analyse graphs directly from their adjacency matrix and state vector could also be developed to gain new insights into the behaviors and properties of GRA.

Figure 2 :

 2 Figure 2: Example of the graph G 0 from Figure 1.

Figure 3 :

 3 Figure 3: Eight possible configurations.

Figure 4 :

 4 Figure 4: Rule of the example in Figure 1 (n = 765 = 10, 1111, 1101 2 ).

Figure 5 :

 5 Figure 5: Division operated on the second vertex of the graph from Figure 2.

Figure 6 :

 6 Figure 6: G 0 : Initial graph with all 8 configurations.

Figure 7 :

 7 Figure 7: Rule 2183 at t = 600 along with a close-up of the braid.

Figure 8 :

 8 Figure 8: Rule 2222 at t = 200.

Figure 9 :

 9 Figure 9: Rule 2222: linear growth.

Figure 10 :

 10 Figure 10: Rule 2222: length distribution of time intervals without growth computed for 0 ≤ t ≤ 200, 000.

Figure 11 :

 11 Figure 11: Rule 2182.

Figure 12 :

 12 Figure 12: Rule 2182: quasi-quadratic growth.

Figure 13 :

 13 Figure 13: Examples of fractal structures found in exponential rules.

Figure 14 :

 14 Figure 14: Some organic-looking graph structures.

Table 1 :

 1 Number Of Rules By Growth Pattern.

	Halted		Linear		Quadratic Exponential Unclassified
		strict periodic chaotic
	422	73	19	3	1	374	132

Table 2 :

 2 Period distribution in rules leading to cycles.

	Period	1	2	3 6 8
	Number of rules 310 102 2 6 2

Acknowledgment

This work was partially supported by the Université Grenoble Alpes, through an Excellence Internship which took place between June 7 th and July 15 th 2022.