Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation

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Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation

Grégory Faye

To cite this version:

Grégory Faye. Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn

equation. Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical

Sciences, 2016, 36 (5), pp.2473 - 2496. �10.3934/dcds.2016.36.2473�. �hal-01682322�

Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation

Gr´ egory Faye

^∗1

1CAMS - Ecole des Hautes Etudes en Sciences Sociales, 190-198 avenue de France, 75013, Paris, France

September 16, 2015

Abstract

We prove the multidimensional stability of planar traveling waves for scalar nonlocal Allen-Cahn equations using semigroup estimates. We show that if the traveling wave is spectrally stable in one space dimension, then it is stable inn-space dimension,n≥2, with perturbations of the traveling wave decaying liket^−(n−1)/4as t→+∞inH^k(Rⁿ) fork≥n+1

2

.

Key words: Nonlocal equation; Traveling wave; Nonlinear stability.

AMS subject classifications: 35K57, 34K20 and 47D06.

1 Introduction

We consider the scalar nonlocal Allen-Cahn equation

∂

_t

u(x, t) = −u(x, t) +

Z

Rⁿ

K(x − y)u(y, t)dy + f (u(x, t)) := −u(x, t) + K ∗

_x

u(x, t) + f (u(x, t)) (1.1) where u ∈

R

, (x, t) ∈

Rⁿ

×

R⁺

and f is a smooth function of bistable type with three zeros, 0, 1 and a ∈ (0, 1). A prototypical example for f is the cubic nonlinearity of form f

_cu

(u) := u(1 − u)(u − a). Here K ∈ L

¹

(

R

) is a nonnegative function with

R

Rⁿ

K(x)dx = 1 and that is even with respect to each variable. A traveling wave ϕ(ξ) is a smooth function of the variable ξ = e · x − ct, for e ∈

Sⁿ⁻¹

and some c ∈

R

, which is a solution of (1.1) satisfying the limits lim

ξ→−∞

ϕ(ξ) = 1 and lim

ξ→+∞

ϕ(ξ) = 0. Without loss of generality, we suppose that e = (1, 0, . . . , 0). In the moving frame x = (ξ, z) ∈

R

×

Rⁿ⁻¹

, equation (1.1) can be written as

∂

_t

u(x, t) − c∂

_ξ

u(x, t) = −u(x, t) +

Z

Rⁿ

K(x − y)u(y, t)dy + f (u(x, t)) (1.2)

∗Corresponding Author, gfaye@ehess.fr

such that the traveling wave ϕ(ξ) is a stationary solution of (1.2). If we define K

₀

:

R

→

R

as K

₀

(ξ) =

Z

Rⁿ⁻¹

K(ξ, z)dz (1.3)

then (ϕ, c) satisfies

− c ϕ(ξ) = ˙ −ϕ(ξ) +

Z

R

K

₀

(ξ − ζ)ϕ(ζ )dζ + f(ϕ(ξ)), lim

ξ→−∞

ϕ(ξ) = 1 and lim

ξ→+∞

ϕ(ξ) = 0, (1.4) where ˙ stands for

_dξ^d

and ϕ is decreasing.

Main assumptions. Throughout the paper, we will assume the following hypotheses for f and K which ensure the existence and uniqueness (modulo translation) of a solution (ϕ, c) to (1.4), see [3].

Hypothesis (H1) We suppose that the nonlinearity f satisfies the following properties:

(i) f ∈ C

^∞

(

R

);

(ii) f (u) = 0 precisely when u ∈ {0, a, 1};

(iii) f

⁰

(0) < 0, f

⁰

(1) < 0 and f

⁰

(a) > 0.

Note that we only need f ∈ C

²

(

R

) to obtain the existence result of [3] and here we require more regularity to obtain uniform bounds on the nonlinear terms in our stability analysis.

Hypothesis (H2) We suppose that the kernel K satisfies the following properties:

(i) K ≥ 0, is even with respect to each variable;

(ii) K ∈ W

^1,1

(

Rⁿ

);

(iii)

R

Rⁿ

K(x)dx = 1,

R

Rⁿ

kxkK(x)dx < ∞ and

R

Rⁿ

kxk

²

K(x)dx < ∞;

(iv) K(k) = 1

b

− d

0

kkk

²

+ o(kkk

²

) as k → 0 with d

0

> 0.

Here, W

^k,p

(R

ⁿ

) denotes the Sobolev space with its usual norm and we use the notation H

^k

(R

ⁿ

) :=

W

^k,2

(

Rⁿ

). The symbol K

b

denotes the Fourier transform of K defined as K(k) =

b

Z

Rⁿ

K(x)e

^−ik·x

dx, k ∈

Rⁿ

.

The first assumption is natural from a modeling point of view while the second and third assumptions are required to ensure the existence of traveling wave solution ϕ to equation (1.4). The third and forth assumptions also imply that

∀j ∈

J

1, n

K Z

Rⁿ

x

_j

K(x)dx = 0 and d

₀

= 1 2n

Z

Rⁿ

kxk

²

K(x)dx > 0.

Furthermore, as −1 + K(k)

b

∼ −d

₀

kkk

²

for k → 0, in the long wavelength limit, the linear operator u 7→ −u+K ∗ u approaches the Laplacian d

₀

∆

_Rⁿ

and we recover the classical Allen-Cahn equation. Remark that in the short wavelength limit we have −1 + K(k)

b

∼ −1 for kkk → ∞ such that u 7→ −u + K ∗ u is a bounded operator which is a very different feature from the Laplacian. Note that with Hypothesis (H2) for the kernel K we recover all the hypotheses of [3] for K

₀

.

In this paper, we are concerned with determining the stability of the traveling wave ϕ. We are thus let to study the spectral properties of the linear operator L

L : H

¹

(

Rⁿ

) −→ L

²

(

Rⁿ

)

u 7−→ −u + K ∗

_x

u + c

_dξ^d

+ f

⁰

(ϕ)u. (1.5) It is natural to assume that the wave ϕ is linearly stable in one space dimension to get stability in higher in space dimensions. In fact, it is consequence of Hypotheses (H1) and (H2) on f and K. First, define the linear operator L

₀

associated to equation (1.4)

L

₀

: H

¹

(

R

) −→ L

²

(

R

)

u 7−→ −u + K

₀

∗

_ξ

u + c

_dξ^d

+ f

⁰

(ϕ)u. (1.6) and its adjoint operator L

^∗₀

L

^∗₀

: D(L

^∗₀

) ⊂ L

²

(

R

) −→ L

²

(

R

)

u 7−→ −u + K

₀

∗

_ξ

u − c

_dξ^d

+ f

⁰

(ϕ)u. (1.7) Lemma 1.1 ([2,

6]).

Suppose that Hypotheses (H1) and (H2) are satisfied, then

(i) 0 is an algebraic simple eigenvalue of L

₀

with negative eigenfunction ϕ

⁰

; (ii) there exists γ

₀

> 0 such that σ

_ess

(L

₀

) ⊂ {λ | |<(λ)| < −γ

₀

};

(iii) there exists a unique negative solution ψ ∈ H

¹

(

R

) which solves L

^∗₀

ψ = 0 with

R

R

ϕ

⁰

(ξ)ψ(ξ)dξ = 1.

Since the eigenvalue zero is isolated, there exists a spectral projection operator, P , onto the null space of L

₀

given by

P u = 1 2πi

Z

Γ

(L

₀

− λ)

⁻¹

udλ, (1.8)

where Γ is a simple closed curve in the complex plane enclosing the zero eigenvalue. If h·, ·i denotes the scalar product on L

²

(R) then we can write P as

P u(ξ, z) = hψ, ui(z)ϕ

⁰

(ξ) :=

Z

R

ψ(ξ)u(ξ, z)dξ

ϕ

⁰

(ξ). (1.9)

We define the operator Q as Qu := u − Pu.

Main result. We can now state our main result. The perturbation of the wave will be written as u(x, t) := ϕ(ξ − ρ(z, t)) + v(ξ − ρ(z, t), z, t) (1.10) where ρ :

Rⁿ⁻¹

→

R

∈ H

^k

(

Rⁿ⁻¹

) and v :

Rⁿ

→

R

∈ H

^k

(

Rⁿ

) is in the range of the operator L

₀

that is P v = 0. And we set

E

₀

:= kv

₀

k

_W1,1(Rⁿ)

+ kv

₀

k

_Hk(Rⁿ)

+ kρ

₀

k

_W1,1(Rⁿ⁻¹)

+ kρ

₀

k

_Hk+1(Rⁿ⁻¹)

.

Theorem 1. Let n ≥ 2 and k ≥

_n+1

2

. Suppose that Hypotheses (H1) and (H2) are satisfied. There exists C > 0 such that if E

₀

is sufficiently small, then the traveling wave solution ϕ of equation (1.2) is stable in the sense that the perturbation (ρ, v) given in (1.10) satisfies the decay estimates for all t ≥ 0

kv(t)k

_Hk(Rⁿ)

≤ C(1 + t)

^{−n−14 −1}

E

₀

, (1.11a) kρ(t)k

_Hk(Rⁿ⁻¹)

≤ C(1 + t)

⁻ⁿ⁻¹⁴

E

₀

, (1.11b) k∇

_z

· ρ(t)k

_Hk(Rⁿ⁻¹)

≤ C(1 + t)

⁻ⁿ⁺¹⁴

E

₀

, (1.11c) where ∇

_z

= (∂

_x2

, · · · , ∂

_xn

).

Note that Theorem

1

is well known in the case of local diffusion, namely when the nonlocal term −u+K∗

_x

u in equation (1.1) is replaced by the standard Laplacian ∆ =

Pn

i=1

∂

_x²_i

on

Rⁿ

. Xin [18] was the first to prove these results in dimension n ≥ 4 in the local case. His results were then extended to the remaining dimensions n = 2, 3 in [14] and generalized to systems of bistable reaction-diffusion equations by Kapitula [13]. Our strategy of proof will be similar to as [13,

18] where semigroup estimates for the associated

linearized operator are used to prove the multidimensional stability of the traveling wave ϕ. It is important to remark that in dimension n ≥ 4, these semigroup estimates are sufficient to prove Theorem

1. For the

remaining dimensions n = 2, 3, the proof essentially relies on the decomposition of the perturbation as written in (1.10) which basically allows one to split the problem into two parts. One part controls the drift of the perturbations along the translates of the wave and another part which controls the remaining part of the perturbations and will decay faster in time. Although our proof will follow the strategy developed in [13,

18], we still have to deal with the nonlocal nature of our equations. In our case, we use point-wise

Green’s functions estimates to obtain sharp decay estimates of the linear part of our linearized operator.

These types of estimates are reminiscent of the ones obtained by Hoffman and coworkers [11] in the study of multi-dimensional stability of planar traveling of lattice differential equations, which are discrete version of equation (1.1). In the nonlocal setting, using super- and sub- solution technique, Chen [5] has been able to prove the uniform multidimensional stability of the traveling wave ϕ of equation (1.1). As a direct consequence, our Theorem

1

generalizes Chen’s result.

An application. This present work was initially motivated by the study of Bates and Chen [1] where they prove a multidimensional stability result for a slightly different multidimensional nonlocal Allen-Cahn equation. Their idea was to consider a generalization of the Laplacian in n-dimension for which, each component ∂

_x²_i

of ∆ is approximated by the convolution operator −u + J ∗

_xi

u. They obtain an equation of form

∂

_t

u =

n

X

i=1

(−u + J ∗

_xi

u) + f (u), (1.12)

with

J ∗

_xi

u(x) :=

Z

R

J (y)u(x

1

, · · · , x

i

− y, · · · , x

n

)dy.

The kernel J satisfies the following Hypothesis.

Hypothesis (H3) We suppose that the kernel J satisfies the following properties:

(i) J ≥ 0, is even;

(ii) J ∈ W

η^1,1

(

R

) for η > 0.

Here W

η^1,1

(

R

) denotes the exponentially weighted function space defined as W

_η^1,1

(

R

) :=

n

u ∈ W

^1,1

(

R

) | e

^η|·|

u ∈ L

¹

(

R

) and e

^η|·|

∂

x

u ∈ L

¹

(

R

)

o

.

A direct consequence of Hypothesis (H3) is that J

b

(k) = 1 − d

₀

k

²

+ o(k

²

) as k → 0 for d

₀

> 0. In this setting, the traveling wave ϕ is solution of (1.4) with K

₀

= J and the linearized operator L

₀

has the same expression as in equation (1.6) and thus Lemma

1.1

is also verified provided that f satisfies Hypothesis (H1).

To study the stability of the traveling wave ϕ, we work with the same decomposition

u(x, t) = ϕ(ξ − ρ(z, t)) + v(ξ − ρ(z, t), x, t), t ≥ 0, (1.13) with ρ :

Rⁿ⁻¹

→

R

∈ H

^k

(

Rⁿ⁻¹

) and v :

Rⁿ

→

R

∈ H

^k

(

Rⁿ

) is in the range of the operator L

₀

that is P v = 0. We also set

E

e₀

:= kv

₀

k

_L1(Rⁿ)

+ kv

₀

k

_Hk(Rⁿ)

+ kρ

₀

k

_W1,1(Rⁿ⁻¹)

+ kρ

₀

k

_Hk+1(Rⁿ⁻¹)

. As a bi-product of our proof, we obtain the following result.

Theorem 2. Let n ≥ 2 and k ≥

_n+1

2

. Suppose that Hypotheses (H1) and (H3) are satisfied. There exists C > 0 such that if E

e₀

is sufficiently small, then the traveling wave solution ϕ of equation (1.12) is stable in the sense that the perturbation (ρ, v) given in (1.13) satisfies the decay estimates for all t ≥ 0

kv(t)k

_Hk(Rⁿ)

≤ C(1 + t)

⁻ⁿ⁺¹²

E

e₀

, (1.14a) kρ(t)k

_Hk(Rⁿ⁻¹)

≤ C(1 + t)

⁻ⁿ⁻¹⁴

E

e₀

, (1.14b) k∇

_z

· ρ(t)k

_Hk(Rⁿ⁻¹)

≤ C(1 + t)

⁻ⁿ⁺¹⁴

E

e₀

. (1.14c)

Note that Bates and Chen [1] only proved Theorem

2

in dimension n ≥ 4 and thus our result generalizes their analysis to the remaining dimensions n = 2, 3. Compared to Theorem

1, we obtain a sharper decay

of v component of the perturbation. This is a consequence of the fact that the projection P commutes with each linear operator −u + J ∗

_xi

u for i = 2 · · · n.

Outline. The paper is organized in three parts. In section

2, we prove Theorem 1

in dimension n ≥ 4.

In the following section we study the semigroup associated to the linear operator L and derive estimates crucial for our nonlinear stability analysis. Finally, in section

4, we prove Theorem 1

for the remaining dimensions n = 2, 3 and Theorem

2

is proved in section

5.

2 Stability in high dimension

In this section, we give a simple proof of Theorem

1

in the high-dimensional case n ≥ 4, following ideas that

have been developed for the multidimensional local Allen-Cahn equations [13,

18] and then, generalized to

reaction-diffusion systems for example [10]. The main ingredient of the proof is an estimate (see (2.4)) for the linearized evolution operator L which will be established in the following section

3.

We consider a solution u(x, t) = ϕ(ξ) + v(x, t) of (1.1) which satisfies the equation

∂

t

v(x, t) = Lv(x, t) + H(v(x, t)), (2.1)

where

H(v) := f (ϕ + v) − f (ϕ) − f

⁰

(ϕ)v. (2.2) The Cauchy problem associated to equation (2.1) with initial condition v

0

∈ H

^k

(R

ⁿ

) ∩ L

¹

(R

ⁿ

), with k ≥ n+1 and n ≥ 4 is locally well-posed in H

^k

(

Rⁿ

). This is equivalent to say that for any v

₀

∈ H

^k

(

Rⁿ

) ∩L

¹

(

Rⁿ

) there exists a time T > 0 such that (2.1) as a unique mild solution in H

^k

(

Rⁿ

) defined on [0, T ] satisfying v(0) = v

0

. The integral formulation of (2.1) is given by

v(t) = S

_L

(t)v

0

+

Z t

0

S

_L

(t − s)H(v(s))ds, (2.3)

where S

_L

is the semigroup associated to the linear operator L. Anticipating the estimates derived in the following sections (see (3.21)), there exist positive constants C and θ such that

kS

L

(t)vk

_Hk(Rⁿ)

≤ C

(1 + t)

⁻ⁿ⁻¹⁴

kvk

_L1(Rⁿ)

+ e

^−θt

kvk

_Hk(Rⁿ)

. (2.4)

The nonlinear contribution H(v) is at least quadratic in v close to the origin. As a consequence, we can find a positive nondecreasing function κ :

R+

→

R+

such that, for all t ∈ [0, T ],

|H(v)| ≤ κ(R)|v|

²

, for |v| ≤ R.

Let T

∗

> 0 be the maximal time of existence of a solution v ∈ H

^k

(

Rⁿ

) with initial condition v

0

∈ H

^k

(

Rⁿ

) ∩ L

¹

(

Rⁿ

). For t ∈ [0, T ∗) we define

Φ(t) = sup

0≤s≤t

(1 + s)

ⁿ⁻¹⁴

kv(s)k

_Hk(Rⁿ)

. Using estimate (2.4) directly into the integral formulation (2.3) yields

kv(t)k

_Hk(Rⁿ)

≤ kS

_L

(t)v

₀

k

_Hk(Rⁿ)

+

Z t

0

kS

_L

(t − s)H(v(s))k

_Hk(Rⁿ)

ds

.

(1 + t)

⁻ⁿ⁻¹⁴

kv

₀

k

_L1(Rⁿ)

+ e

^−θt

kv

₀

k

_Hk(Rⁿ)

+ κ(Φ(t))

Z t

0

(1 + t − s)

⁻ⁿ⁻¹⁴

kv(s)k

²_Hk(Rⁿ)

ds + κ(Φ(t))

Z t

0

e

^−θ(t−s)

kv(s)k

²_Hk(Rⁿ)

ds.

Here, and throughout the paper we use the notation A

.

B whenever A ≤ κB for κ > 0 a constant independent of time t. From Lemma

A.1, there exist constants

C

1

> 0 and C

2

> 0 so that

Z t

0

(1 + t − s)

⁻ⁿ⁻¹⁴

(1 + s)

⁻ⁿ⁻¹²

ds ≤ C

₁

(1 + t)

⁻ⁿ⁻¹⁴

,

Z t

0

e

^−θ(t−s)

(1 + s)

⁻ⁿ⁻¹²

ds ≤ C

2

(1 + t)

⁻ⁿ⁻¹⁴

.

Note that the first inequality is a consequence of our careful choice of n. Indeed, this inequality is only true for

ⁿ⁻¹₄

>

¹₂

(n ≥ 4). Then, for all t ∈ [0, T

∗

) we have

Φ(t) ≤ C

₀

kv

₀

k

_L1(Rⁿ)

+ kv

₀

k

_Hk(Rⁿ)

+ C

e₀

κ(Φ(t))Φ(t)

²

,

for some positive constants C

₀

and C

e₀

. Suppose that the initial condition v

₀

is small enough so that 2C

₀

kv

₀

k

_L1(Rⁿ)

+ kv

₀

k

_Hk(Rⁿ)

< 1 and 4C

₀

C

e₀

κ(1)

kv

₀

k

_L1(Rⁿ)

+ kv

₀

k

_Hk(Rⁿ)

< 1, then

Φ(t) ≤ 2C

0

kv

₀

k

_L1(Rⁿ)

+ kv

₀

k

_Hk(Rⁿ)

< 1,

for all t ∈ [0, T

∗

). This implies that the maximal time of existence is T

∗

= +∞ and the solution v of (2.1) satisfies:

sup

t≥0

(1 + t)

ⁿ⁻¹⁴

kv(t)k

_Hk(Rⁿ)

≤ 2C

₀

kv

₀

k

_L1(Rⁿ)

+ kv

₀

k

_Hk(Rⁿ)

.

3 Linear estimates

We first start this section by deriving the nonlinear problem that we will be solving in the next section and then derive estimates of the corresponding linear parts.

3.1 Setup of the problem

We represent each solution u(x, t) of (1.2) through the decomposition

u(x, t) := ϕ(ξ − ρ(z, t)) + v(ξ − ρ(z, t), z, t) (3.1) where ρ :

Rⁿ⁻¹

→

R

∈ H

¹

(

Rⁿ⁻¹

) and v :

Rⁿ

→

R

∈ H

¹

(

Rⁿ

). Note that the perturbation v can be written as

v(x, t) = u(ξ + ρ(z, t), z, t) − ϕ(ξ),

and v is in range of L

₀

such that P v = 0. Note that such a decomposition (1.10) is always possible. Indeed, suppose that w ∈ H

^k

(

Rⁿ

) is given and small enough. In order to use the decomposition (1.10), we need to find a unique pair (ρ(w), v(w)) ∈ H

^k

(R

ⁿ⁻¹

) × H

^k

(R

ⁿ

) with P v = 0 that satisfies

ϕ + w = T

_ρ

· (ϕ + v) ,

where T

_ρ

· ϕ(ξ) = ϕ(ξ −ρ) and T

_ρ

·v(ξ, z, t) = v(ξ −ρ, z, t). Reproducing the standard argument of Kapitula in [13], we use Taylor’s theorem to write

ϕ − T

_−ρ

· ϕ = −ρ

Z 1

0

T

_sρ

· ϕ

⁰

ds, so that we obtain the equivalent equation

T

_−ρ

· w = v − ρ

Z ₁

0

T

_sρ

· ϕ

⁰

ds.

Taking the inner product with ψ the eigenfunction of the adjoint operator L

^∗₀

associated to the zero eigenvalue yields

hT

_−ρ

· w, ψi = −ρ

Z 1

0

T

_sρ

· ϕ

⁰

ds, ψ

.

Here, we have used the fact that we look for solution v so such that P v = 0. We can then define the functional F : H

^k

(

Rⁿ

) × H

^k

(

Rⁿ⁻¹

) → H

^k

(

Rⁿ⁻¹

) by

F (w, ρ) = hT

_−ρ

· w, ψi + ρ

Z 1

0

T

_sρ

· ϕ

⁰

ds, ψ

,

with F(0, 0) = 0 and Fr´ echet derivative D

_ρ

F (0, 0) = id where id is the identity operator. By the implicit function theorem with have on a neighborhood of (0, 0) the existence of ρ(w) such that F (w, ρ(w)) = 0.

We then apply the projection Q to the equation T

_−ρ

· w = v − ρ

R1

0

T

_sρ

· ϕ

⁰

ds and obtain v = QT

_−ρ(w)

· w + Q

ρ(w)

Z 1 0

T

_sρ(w)

· ϕ

⁰

ds

,

which clearly admits a solution v(w) with P v(w) = 0. As a conclusion, all sufficiently small perturbation can be written as in equation (1.10).

We can now substitute the Ansatz (3.1) into (1.2) to get the evolution equation:

−∂

_t

ρ ϕ ˙

_ρ

− cϕ

_ρ

+ ∂

_t

v

_ρ

− c∂

_ξ

v

_ρ

− ∂

_t

ρ∂

_ξ

v

_ρ

= −ϕ

_ρ

− v

_ρ

+ K ∗

_x

ϕ

_ρ

+ K ∗

_x

v

_ρ

+ f (ϕ

_ρ

+ v

_ρ

), with ϕ

ρ

= ϕ(· − ρ) and v

ρ

= v(· − ρ, ·, ·). As ϕ is solution of (1.4), we obtain:

(∂

t

− L) v = (∂

t

− L) (ρ ϕ) + ˙ H(v) + N (ρ, v) + R(ρ, v) (3.2) where the nonlinear term H has been defined in (2.2), R(ρ, v) := ∂

t

ρ∂

ξ

v and the remaining term N is split into two different parts

N (ρ, v) := N

₁

(ρ) + N

₂

(ρ, v), (3.3)

where

N

₁

(ρ)(x, t) :=

Z

Rⁿ

K(x − x

⁰

)ϕ ξ

⁰

+ ρ(z, t) − ρ(z

⁰

, t)

dξ

⁰

dz

⁰

−

Z

Rⁿ

K(x − x

⁰

)ϕ(ξ

⁰

)dξ

⁰

dz

⁰

−

Z

Rⁿ

K(x − x

⁰

) ˙ ϕ(ξ

⁰

) ρ(z, t) − ρ(z

⁰

, t)

dξ

⁰

dz

⁰

, (3.4)

and

N

₂

(ρ, v)(x, t) :=

Z

Rⁿ

K(x − x

⁰

)v ξ

⁰

+ ρ(z, t) − ρ(z

⁰

, t), z

⁰

, t dξ

⁰

dz

⁰

−

Z

Rⁿ

K(x − x

⁰

)v(ξ

⁰

, z

⁰

, t)dξ

⁰

dz

⁰

. (3.5) One can check that the third term of N

₁

(ϕ, ρ) is actually L(ρ ϕ) as ˙

L(ρ ϕ) = ˙ −ρ ϕ ˙ + K ∗

_x

(ρ ϕ) + ˙ cρ ϕ ¨ + f

⁰

(ϕ)ρ ϕ ˙

= K ∗

_x

(ρ ϕ) ˙ − ρK

₀

∗ ϕ ˙

= −

Z

Rⁿ

K(x − x

⁰

) ˙ ϕ(ξ

⁰

) ρ(z, t) − ρ(z

⁰

, t)

dξ

⁰

dz

⁰

.

Finally, if we denote S

_L

(t) the semigroup generated by the linear operator L, applying Duhamel’s formula to (3.2), we obtain

v(t) = S

_L

(t)v

₀

+ ρ(t) ˙ ϕ − S

_L

(t)(ρ

₀

ϕ) + ˙

Z t

0

S

_L

(t − s) (H(v(s)) + N (ρ(s), v(s)) + R(ρ(s), v(s))) ds.

As v is in the range of L

₀

, we must have v(t) = QS

_L

(t)v

0

− QS

_L

(t)(ρ

0

ϕ) + ˙

Z t

0

QS

_L

(t − s) (H(v(s)) + N (ρ(s), v(s)) + R(ρ(s), v(s))) ds, (3.6a) ρ(t) = hS

_L

(t)(ρ

0

ϕ ˙ − v

0

), ψi −

Z t

0

hS

_L

(t − s) (H(v(s)) + N (ρ(s), v(s)) + R(ρ(s), v(s))) , ψids. (3.6b) In the following sections, we will derive estimates on S

L

(t).

3.2 Study of S

_L

(t)

We consider the initial value problem

∂

_t

u = Lu, u( · , 0) = u

₀

∈ H

^k

(

Rⁿ

), (3.7) which has the solution u(ξ, z, t) = S

_L

(t)u

0

(ξ, z). From its definition, L can be written as

L = L

₀

+ A, where the operator A is defined as

Au := −K

₀

∗

_ξ

u + K ∗

_x

u, u ∈ L

²

(

Rⁿ

). (3.8) Let ˆ u represent the Fourier transform of u in z:

ˆ

u(ξ,

e

k, t) =

Z

Rⁿ⁻¹

u(ξ, z, t)e

^−iz·ek

dz so that (3.7) is transformed into

∂

t

u(ξ, ˆ

e

k, t) = L

₀

u(ξ, ˆ

e

k, t) + B(

e

k)ˆ u(ξ,

e

k, t), where

B(

e

k)ˆ u(ξ, k) :=

e

− K

b_n−1

(0

n−1

) ∗

_ξ

u(ξ, ˆ

e

k) + K

b_n−1

( k)

e

∗

_ξ

u(ξ, ˆ

e

k), with

K

b_n−1

(ξ,

e

k) :=

Z

Rⁿ⁻¹

K(x)e

^−iz·ek

dz.

We have used the fact that K

b_n−1

(ξ, 0

n−1

) = K

₀

by definition. We readily note that for each k

e

∈

Rⁿ⁻¹

, the operator

L(e k) : H

¹

(

R

) −→ L

²

(

R

)

u 7−→ L

₀

u + B(e k)u

defines a C

⁰

semigroup with L(0

n−1

) = L

₀

.

Lemma 3.1. The family of operators L(

e

k) satisfies the following properties.

(i) Near

e

k = 0

n−1

, the only eigenvalue λ is a smooth function of

e

k and the expression of λ(e k) reads:

λ(

e

k) = −Ake kk

²

+ o

ke kk

³

, (3.9)

with A := d

0

hK

₀

∗

_ξ

ϕ

⁰

, ψi.

(ii) σ L(e k)

⊂ {<(λ) < 0}, for k

e

6= 0

n−1

.

Proof. For the first property (i), we apply perturbation theory to the linear operator L(e k) for

e

k near zero. To this end, we define

F :

Rⁿ⁻¹

× C × H

_⊥¹

(

R

) −→ L

²

(

R

) (

e

k, λ, w) 7−→

L(

e

k) − λ

( ˙ ϕ + w), where H

_⊥¹

(R) =

u ∈ H

¹

(R) | hu, ϕi ˙ = 0 . Applying the implicit function theorem, we see that there exist a small neighborhood of the origin and smooth functions λ(

e

k) and w( k) such that

e

F(

e

k, λ(

e

k), w(

e

k)) = 0 on that neighborhood. We denote q(e k) = ˙ ϕ + w(e k). Similarly for the adjoint operator L

^∗

(e k), we have a smooth continuation of ψ given by q

^∗

(

e

k) so that

hq( k), q

e ^∗

( k)i

e

= 1.

Differentiating F(e k, λ(e k), w(e k)) = 0 with respect to

e

k

_j

, for any j, we find

∂

ekj

λ(0) = ∂

ekj

hB(e k)ϕ

⁰

, ψi

ek=0n−1

= 0.

Indeed, if ` ∈

R

, we have that

B(e

\

k)u(`) =

Z

R

B(e k)u(ξ)e

^−ξ`

dξ

=

− K(`,

b

0

n−1

) + K(`,

b

k)

e

ˆ u(`)

∼ −d

₀

ke kk

²

u(`) ˆ as ke kk → 0. Similarly, we find that for any j and l,

∂

²

ekjekl

λ(0) = 0.

Finally, for any j, we have that

∂

²

ekjekj

λ(0) = −2d

₀

K

₀

∗

_ξ

ϕ

⁰

, ψ , which gives the desired expansion.

For the second property (ii), we have just seen that λ(e k) 6= 0 for small values of

e

k. For any u ∈ H

¹

(

R

) we have that

hL(

e

k)u, ui = −hu, ui + h K

b_n−1

( k)

e

∗

_ξ

u, ui + hf

⁰

(ϕ)u, ui

≤ −1 + K

b_n−1

(0,

e

k) + sup

ϕ∈[0,1]

f

⁰

(ϕ)

!

hu, ui.

As K

b_n−1

(0, k)

e

→ 0 as ke kk → ∞, there exist M > 0 and c

_M

> 0 so that for all ke kk ≥ M ,

−1 + K

b_n−1

(0,

e

k) + sup

ϕ∈[0,1]

f

⁰

(ϕ) < −c

_M

.

This implies that <(λ) < −c

_M

< 0 for all λ ∈ σ(L(e k)) with ke kk ≥ M . For the region in-between, compactness and local robustness of the spectrum ensure that σ

L( k)

e

⊂ {<(λ) < 0}.

Based on Lemma

3.1, there exists

> 0, so that λ(e k) is a simple eigenvalue of L(e k) in ke kk ≤ 2. As a consequence, there exists a smooth spectral projection operator, P ( k), given by

e

P (e k)u = 1 2πi

Z

Γ

L(e k) − λ

−1

udλ, (3.10)

where Γ is a simple closed curve in the complex plane enclosing the zero eigenvalue. More conveniently, we write P (e k) as

P ( k)u(ξ) =

e Z

R

q

^∗

( k, ξ)u(ξ)dξ

e

q( k, ξ) :=

e

hq

^∗

(

e

k), uiq( k),

e

ke kk ≤ 2. (3.11) Following some ideas developed in [12,

17] for viscous conservation laws, we introduce a smooth cutoff

function χ(

e

k) that is identically one for ke kk ≤ and identically zero for ke kk ≥ 2. We can then split the solution operator S

_L

(t)u

₀

into a low-frequency part

S

_L^I

(t)u

₀

(ξ, z) := 1 (2π)

ⁿ⁻¹

Z

Rⁿ⁻¹

e

^iek·z

e

^L(ek)th

χ(e k)P (e k)ˆ u

₀

(ξ, k)

e i

de k (3.12)

and the associated high-frequency part S

_L^II

(t)u

0

(ξ, z) := 1

(2π)

ⁿ⁻¹ Z

Rⁿ⁻¹

e

^iek·z

e

^L(ek)th

id − χ( k)P

e

(

e

k) ˆ

u

0

(ξ,

e

k)

i

d

e

k, (3.13) where id denotes the identity. One can easily check that we have S

_L

(t) = S

_L^I

(t) + S

_L^II

(t).

3.2.1 Low-frequency bounds

We introduce the Green kernel associated with S

_L^I

(t) as

G

^I

(x, t; x

⁰

) := S

_L^I

(t)δ

_x⁰

(x), (3.14) where x = (ξ, z) and x

⁰

= (ξ

⁰

, z

⁰

).

Proposition 3.1. The Green kernel G

^I

satisfies G

^I

(x, t; x

⁰

) = 1

(2π)

ⁿ⁻¹ Z

Rⁿ⁻¹

e

^{ik·(z−ze 0)}

χ(

e

k)e

^λ(ek)t

q( k, ξ)q

e ^∗

(

e

k, ξ

⁰

)d k.

e

(3.15) Proof. First of all, through a direct computation, we have

δ

c_x⁰

(ξ,

e

k) =

Z

Rⁿ⁻¹

e

^−iek·z

δ

_(ξ⁰_,z⁰₎

(ξ, z)dz = e

^−iek·z0

δ

_ξ⁰

(ξ).

Using the properties of the spectral projection P (e k), we further have P (e k) δ

c_x⁰

(ξ,

e

k) = q

^∗

(e k, ξ

⁰

)q(e k, ξ).

Finally, noticing that e

^L(k)te

q(

e

k, ξ) = e

^λ(ek)t

q( k, ξ), we obtain the desired formula.

e

Proposition 3.2. The low-frequency Green function G

^I

(x, t; x

⁰

) of (3.14) can be decomposed as G

^I

(x, t; x

⁰

) :=

˙

ϕ(ξ)Ψ(z − z

⁰

, t; ξ

⁰

) + G

e^I

(x, t; x

⁰

), for which the following estimates hold:

sup

ξ⁰

kΨ(·, t; ξ

⁰

)k

_L2(Rⁿ⁻¹) .

(1 + t)

⁻ⁿ⁻¹⁴

, (3.16a) sup

ξ⁰

kΨ(·, t)k

_Hk(Rⁿ⁻¹) .

(1 + t)

^{−n−14 −|α|2}

, (3.16b) sup

x⁰

k G

e^I

(·, t; x

⁰

)k

_L2(Rⁿ) .

(1 + t)

^{−n−14 −1}

, (3.16c) sup

x⁰

k G

e^I

(·, t; x

⁰

)k

_Hk(Rⁿ) .

(1 + t)

^{−n−14 −|α|+22}

. (3.16d) for α ∈

Zⁿ⁻¹+

with |α| ≤ k. Moreover, P G

e^I

(x, t; x

⁰

) = 0.

Proof. The idea of the proof is based on the remark that for ke kk ≤ 2, the smooth eigenfunctions q(

e

k, ξ ) and q

^∗

( k, ξ) have an expansion of the form

e

q(e k, ξ) = ˙ ϕ(ξ) + O ke kk

²

, q

^∗

(

e

k, ξ

⁰

) = ψ(ξ

⁰

) + O

ke kk

²

.

This leads us to introduce an auxiliary function Ψ(z, t) of the form

e

Ψ(z, t) :=

e

1

(2π)

ⁿ⁻¹ Z

Rⁿ⁻¹

e

^iek·z

χ(e k)e

^λ(k)te

de k, so that we formally have

G

^I

(x, t; x

⁰

) − ϕ(ξ)ψ(ξ ˙

⁰

) Ψ(z

e

− z

⁰

, t) = 1 (2π)

ⁿ⁻¹

Z

Rⁿ⁻¹

e

^{ik·(z−ze 0)}

χ(e k)e

^λ(ek)t

O ke kk

²

de k. (3.17) On the one hand, a simple Fourier transform computation shows that

1 (2π)

ⁿ⁻¹

Z

Rⁿ⁻¹

e

^{iek·(z−z0)}

e

^−Akekk2t

d k

e

= (4πAt)

⁻ⁿ⁻¹²

exp

− kz − z

⁰

k

²

4At

,

where A = d

₀

hK

₀

∗

_ξ

ϕ

⁰

, ψi, which directly gives us bounds for Ψ(·, t) that are similar to the standard

e

diffusive bounds satisfied for the heat equations:

k Ψ(·, t)k

e _L2(Rⁿ⁻¹) .

(1 + t)

⁻ⁿ⁻¹⁴

, (3.18a) k Ψ(·, t)k

e _Hk(Rⁿ⁻¹) .

(1 + t)

^{−n−14 −|α|2}

, (3.18b) for α ∈

Zⁿ⁻¹₊

with |α| ≤ k. On the other hand, because of the presence of terms of the form ke kk

²

e

^−Akekk2t

in the rest term of equation (3.17), the decay rate is improved by factor (1 + t)

⁻¹

so that we have the following estimates for G

e^I

:= G

^I

− ϕψ ˙ Ψ.

e

sup

x⁰

k G

e^I

(·, t; x

⁰

)k

_L2(Rⁿ) .

(1 + t)

^{−n−14 −1}

, (3.19a) sup

x⁰

k G

e^I

(·, t; x

⁰

)k

_Hk(Rⁿ) .

(1 + t)

^{−n−14 −|α|2 −1}

, (3.19b)

for α ∈

Zⁿ⁻¹+

with |α| ≤ k. Now, we can define Ψ(z − z

⁰

, t; ξ

⁰

) as Ψ(z − z

⁰

, t; ξ

⁰

) := 1

(2π)

ⁿ⁻¹ Z

Rⁿ⁻¹

e

^iek·z

χ(e k)e

^λ(k)te

q

^∗

(e k, ξ

⁰

)hq(e k, ·), ψie k (3.20) and set G

e^I

(x, t; x

⁰

) := G

^I

(x, t; x

⁰

) − ϕ(ξ)Ψ(z ˙ − z

⁰

, t; ξ

⁰

). And all the estimates (3.16) are readily obtained from (3.18) and (3.19).

Proposition 3.3. The linear operator S

_L^I

(t) satisfies the decay estimate

S_L^I

(t)u

H^k(Rⁿ) .

(1 + t)

⁻ⁿ⁻¹⁴

kuk

_L1(Rⁿ)

. Furthermore, we have

QS_L^I

(t)u

_Hk(

Rⁿ) .

(1 + t)

^{−n−14 −1}

kuk

_L1(Rⁿ)

.

Proof. The proof of the proposition easily follows from the estimates (3.16) by first noticing that S

_L^I

(t)u(x) =

Z

Rⁿ

G

^I

(x, t; x

⁰

)u(x

⁰

)dx

⁰

, x ∈

Rⁿ

, and

Z

Rⁿ

S_L^I

(t)u(x)

2

dx ≤

sup

x⁰

kG

^I

(·, t; x

⁰

)k

_L2(Rⁿ)

2Z

Rⁿ

|u(x

⁰

)|dx

⁰ 2

.

The estimates in H

^k

(

Rⁿ

) are obtained via similar computations. Finally, we recall the decomposition of G

^I

(x, t; x

⁰

) implies that

QG

^I

(x, t; x

⁰

) = G

e^I

(x, t; x

⁰

), which completes the proof of the proposition.

3.2.2 High-frequency bounds

We now study the high-frequency bounds associated to S

_L^II

(t). By definition, we have S

_L^II

(t)u(ξ, z) := 1

(2π)

ⁿ⁻¹ Z

Rⁿ⁻¹

e

^ik·ze

e

^L(ek)th

id − χ(e k)P (e k) ˆ u(ξ,

e

k)

i

de k, where P (e k) is such that

Z

R

e

^L(ek)th

id − χ(

e

k)P ( k)

e

ˆ

u(ξ,

e

k)

i

2

dξ ≤ e

^−2θt

u(·,

e

k)

2 L²(R)

,

where θ > 0 is a positive constant which depends only on . Then, using Parseval’s inequality, one obtains

S_L^II

(t)u

L²(Rⁿ) .

e

^−θt

kuk

_L2(Rⁿ)

, from which one can deduce H

^k

(

Rⁿ

)-estimates.

Proposition 3.4. The linear operator S

_L^II

(t) satisfies the decay estimate

S_L^II

(t)u

H^k(Rⁿ) .

e

^−θt

kuk

_Hk(Rⁿ)

.

As a conclusion, combining Proposition

3.3

and

3.4, we arrive at the linear estimate for semigroup

S

_L

(t) of L. For u ∈ H

^k

(

Rⁿ

), we have

kS

L

(t)uk

_Hk(Rⁿ) .

(1 + t)

⁻ⁿ⁻¹⁴

kuk

_L1(Rⁿ)

+ e

^−θt

kuk

_Hk(Rⁿ)

. (3.21) As a consequence of our analysis, we also have

kQS

L

(t)uk

_Hk(Rⁿ) .

(1 + t)

^{−n−14 −1}

kuk

_L1(Rⁿ)

+ e

^−θt

kuk

_Hk(Rⁿ)

, (3.22) and

k∇

_z

· S

_L

(t)uk

_Hk(Rⁿ) .

(1 + t)

^{−n−14 −12}

kuk

_L1(Rⁿ)

+ t

⁻¹²

e

^−θt

kuk

_Hk(Rⁿ)

. (3.23)

4 Nonlinear stability in dimension 2 and 3

In this section, we prove Theorem

1

for the remaining dimensions 2 and 3.

4.1 Some nonlinear estimates

We first give estimates on the nonlinear terms that appear in our system (3.6). More precisely, we will prove the following lemma.

Lemma 4.1. Let k ≥

_n+1

2

. There exists a δ > 0 such that for any v ∈ H

^k

(

Rⁿ

) and ρ ∈ H

^k+1

(

Rⁿ⁻¹

) with kvk

_Hk(Rⁿ)

≤ δ, kρk

_Hk(Rⁿ⁻¹)

≤ δ and k∇

_z

· ρk

_Hk(Rⁿ⁻¹)

≤ δ we have

kH(v)k

_L1(Rⁿ)

, kH(v)k

_Hk(Rⁿ)

≤ C kvk

²_Hk(Rⁿ)

, (4.1a) kN

₁

(ρ)k

_L1(Rⁿ)

, kN

₁

(ρ)k

_Hk(Rⁿ)

≤ C k∇

_z

· ρk

²_Hk(Rⁿ⁻¹)

, (4.1b) kN

₂

(ρ, v)k

_L1(Rⁿ)

, kN

₂

(ρ, v)k

_Hk(Rⁿ)

≤ C kρk

_Hk(Rⁿ⁻¹)

kvk

_Hk(Rⁿ)

, (4.1c)

kR(ρ, v)k

_L1(Rⁿ)

, kR(ρ, v)k

_Hk(Rⁿ)

≤ C

kvk

²_Hk(Rⁿ)

+ kρk

_Hk(Rⁿ⁻¹)

kvk

_Hk(Rⁿ)

+ k∇

_z

· ρk

²_Hk(Rⁿ⁻¹)

. (4.1d)

Proof. Throughout the proof we will use that from Sobolev embedding we have kuvk

_Hk(Rⁿ)

≤ C kuk

_Hk(Rⁿ)

kvk

_Hk(Rⁿ)

and kuk

_L∞(Rⁿ)

≤ C kuk

_Hk(Rⁿ)

.

Note that in order to obtain the nonlinear estimates (4.1), we will only use the above Sobolev embedding and Taylor’s theorem together with the fact that both f and ϕ are smooth with the a priori bounds on v and ρ. As the proofs of each estimate are almost similar, we will present only the key points.

•

For H(v). We use Taylor’s formula to write

H(v) = f (ϕ + v) − f (ϕ) − f

⁰

(ϕ)v = v

² Z 1

0

(1 − s)f

⁰⁰

(ϕ + sv)ds. (4.2) As ϕ and v are both bounded, we have

kH(v)k

_L1(Rⁿ)

≤ C kvk

²_Hk(Rⁿ)

and kH(v)k

_L2(Rⁿ)

≤ C kvk

²_Hk(Rⁿ)

.