STABILITY ESTIMATE FOR THE SEMI-DISCRETE LINEARIZED BENJAMIN-BONA-MAHONY EQUATION

https://doi.org/10.1051/cocv/2021087 www.esaim-cocv.org

STABILITY ESTIMATE FOR THE SEMI-DISCRETE LINEARIZED BENJAMIN-BONA-MAHONY EQUATION

Rodrigo Lecaros

, Jaime H. Ortega

and Ariel P´ erez

Abstract. In this work we study the semi-discrete linearized Benjamin-Bona-Mahony equation (BBM) which is a model for propagation of one-dimensional, unidirectional, small amplitude long waves in non-linear dispersive media. In particular, we derive a stability estimate which yields a unique continuation property. The proof is based on a Carleman estimate for a finite difference approximation of Laplace operator with boundary observation in which the large parameter is connected to the mesh size.

Mathematics Subject Classification.35B60, 35L05, 35Q35, 35R45, 65M06.

Received November 26, 2020. Accepted August 29, 2021.

Dedicated to Enrique Zuazua on the occasion of his 60th birthday

1. Introduction and results

In this work, we are interested in a linearized version of the Benjamin-Bona-Mahony equation (BBM)

ut+ux+uux−uxxt= 0, (1.1)

proposed by T. Benjaminet al.[2] as a model for propagation of one-dimensional, unidirectional, small amplitude long waves in non-linear dispersive media.

In the last years, several authors have widely studied dispersive equations in the context of controllability and inverse problems. Nevertheless, the BBM equation presents several particularities due to the structure of the operator. In particular, the infinitesimal generator of the semigroup is given by (I−∂_x²)⁻¹∂x, which is a

∗This work is dedicated to Prof. Enrique Zuazua. Dear Enrique, thanks for these years of friendship and for your valuable contributions to the study of control and partial differential equations, and also, for the formation of several generations of researchers in Latin America.

∗∗A. P´erez was founded by the National Agency for Research and Development (ANID)/Scholarship Program/ Doctorado Nacional Chile/2017 – 21170495. R. Lecaros was partially supported by FONDECYT(Chile) Grant 11180874. J.H. Ortega was partially supported by Centro de Modelamiento Matem´atico (AFB170001) and FONDECYT(Chile) Grant 1201125.

Keywords and phrases:Benjamin-Bona-Mahony equation, unique continuation property, Carleman estimate discrete Carleman inequalities, dispersive equations, water wave equation, finite difference method, semi-discrete equations.

1 Departamento de Matem´atica, Universidad T´ecnica Federico Santa Mar´ıa, Valparaiso, Chile.

2 Centro de Modelamiento Matem´atico, IRL 2807 CNRS-UChile, Universidad de Chile, Santiago, Chile.

3 Departamento de Ingenier´ıa Matem´atica, Universidad de Chile, Santiago, Chile.

***Corresponding author:[email protected]

c

The authors. Published by EDP Sciences, SMAI 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

compact operator, in opposition with the most common situation in PDE where the generator is an unbounded operator among others.

We note some interesting results about the unique continuation property (UCP) for BBM for the continuous case. We refer to the reader to those works and their references for a more detailed discussion: L. Rosier and B.-Y. Zhang in [14] developed a UCP for (1.1) on a periodic domain. Moreover, in [6] P. L. da Silva and I. L.

Freire give an alternative proof using geometrical arguments for the periodic case, and for the case when (1.1) is solved in R. In [16], X. Zhang and E. Zuazua considered a linearized BBM equation with space-dependent potential

ut−uxxt= [α(x)u]_x+β(x)u, (x, t)∈(0,1)×(0, T). (1.2) In that work, the authors established that the only solution of (1.2), such thatu(0, t) =u(1, t) = 0, is the trivial one u≡0 provided that bothαandβ do not vanish on some open subset of (0,1). Furthermore, ifα(x) =−1 and β(x) = 0 in (1.2), S. Micu proved in [12] a UCP with the additional boundary conditionu_x(1, t) = 0, and study controllabily results. On the other hand, in [15], M. Yamamoto established a UCP for BBM-like equation with time and space dependent potential

∂tu(x, t)−∂_x²∂tu(x, t) =p(x, t)∂xu(x, t) +q(x, t)u(x, t), (x, t)∈(0,1)×(0, T), (1.3) where p∈L^∞((0, T)×(0,1)) and q∈L^∞(0, T;L²(0,1)). It was shown that the solution of (1.3) shall vanish in (0,1)×(0, T), provided u(1, t) =∂_xu(1, t) = 0 for allt∈(0, T) andu(x,0) = 0 forx∈(0,1). The main tool to prove this result is a Carleman estimate for the Laplacian operator. Through a more refined version of this Carleman estimate, a stability estimate can be formulated for equation (1.3) (see Sect.5).

We note that the UCP’s have been well studied in several continuous partial differential equations, but it is possible to see that the corresponding discrete case does not holds. For instance, ifuis a harmonic function in a domain Ω andu=∂nu= 0 on Γ⊂∂Ω thenu≡0 in Ω, it does not generally hold its discrete formulation. We refer to the counterexample due to O. Kavian, presented by E. Zuazua in [18]. However, for (1.4) it is possible to obtain a quantitative UCP under restriction over the mesh size. This result shall be discussed in more detail in Section1.2.

In this paper, we are interested whether a unique continuation property, as in the work of Yamamoto [15], still holds for a semi-discrete approximation in space of (1.3). In this sense, for N ∈Ngiven, we set the space discretization parameter h:= 1/(N + 1). We consider the pairs (x_i, t) with t ∈(0, T), T >0, and x_i =ih, for i= 1, . . . , N. Thus, the space semi-discrete approximation of equation (1.3) by using the centered finite difference method with respect to the space variable is given by

∂tui(t)−∂tui+1(t)−2∂tui(t) +∂tui−1(t)

h² =pi(t)ui+1(t)−ui−1(t)

2h +qi(t)ui(t), (1.4) fori∈ {1,2, . . . , N}andt∈(0, T), whereu_i(t) stands for u(x_i, t).

1.1. Notation

We introduce the notation of meshes and operators that shall be used throughout this paper and necessary to state our results. We consider the following regular partition of the interval [0,1] as

Mh:={xi|x_i:=ih, i= 0,1, . . . , N+ 1},

forN ∈Nand h:= 1/(N+ 1). For any sets of pointsWh⊂ Mh, we define the following dual meshesW_h⁰ and W_h^∗ as

W_h⁰ :=τ+(Wh)∩τ−(Wh), W_h^∗:=τ+(Wh)∪τ−(Wh), (1.5)

Figure 1.Primal meshes, discretization of space variable.

Figure 2.Dual meshes, discretization of space variable.

where

τ_±(W_h) :=

x_h±h

2 |x_h∈ W_h

.

We shall denoteWh=W_h^∗∗:= (W_h^∗)^∗ and ˚Wh=W_h⁰⁰:= (W_h⁰)⁰.

We note that if W˚h=Wh, then for two consecutive pointsx_i, x_i+1∈ Wh we havex_i+1−x_i=h. Thus, any subset Wh⊂ Mh that verifiesW˚h =Wh is called regular mesh. Finally, we define the boundary of a regular mesh Wh as ∂Wh:=Wh\ Wh. These sets defined above can be seen in Figures1 and2.

We introduce, using (1.5), the semi-discrete sets. Let us consider T >0, we define Q_h:=Wh×(0, T). We also define the dual semi-discrete sets byQ⁰_h:=W_h⁰ ×(0, T) andQ^∗_h:=W_h^∗×(0, T). Similarly, the semi-discrete boundary is given by ∂Q_h=∂W_h×(0, T).

We define the average operatorA_h and the difference operatorD_h by A_h(u_h)(x_h, t) := τ₊u_h(x_h, t) +τ₋u_h(x_h, t)

2 ,

Dh(uh)(xh, t) := τ+uh(xh, t)−τ₋uh(xh, t)

h ,

where τ_±u_h(x_h, t) :=u_h x_h±^h₂, t .

We denote byC(Q_h) the set of real-valued functions defined inQ_h, and byL²_h(Q_h) the set C(Q_h) endowed with the norm

kuhk²_L2 h(Q_h):=

Z T 0

kuk²_L2

h(Wh)dt, where kuhk²_L2

h(Wh) is induced by the inner product huh, vhi_Wh:=

Z

Wh

uhvh:=h X

x_h∈W_h

uh(xh)vh(xh).

Foruh∈C(Qh), we define itsL^∞_h (Qh)-norm as kuhk_L∞

h(Qh):= max

(xh,t)∈Qh

{|uh(xh, t)|}.

To introduce the boundary conditions, we define the outward normal for (xh, t)∈∂Qh as

n_h(x_h, t) :=







1 (τ₋(x_h), t)∈Q^∗_h and (τ₊(x_h), t)∈/Q^∗_h,

−1 (τ₋(xh), t)∈/Q^∗_h and (τ+(xh), t)∈Q^∗_h, 0 otherwise.

We indicate by ∂Q⁺ (resp. ∂Q⁻) the set of points such that n_h(x_h, t) = 1 (resp. n_h(x_h, t) =−1). We also introduce the trace operator foru_h∈C(Q^∗_h) as

∀(xh, t)∈∂Q_h, t_r(u_h) :=







τ−uh(xh, t) nh(xh, t) = 1, τ+uh(xh, t) nh(xh, t) =−1,

0 nh(xh, t) = 0.

Let us finally introduce the discrete integration on the boundary foruh∈C(∂Wh) as Z

∂Wh

uh:= X

x_h∈∂Wh

uh(xh).

1.2. Discrete Carleman estimates with boundary observation

For the discrete Carleman estimate, we consider the weight function of the form r(x, t) =e^sϕ(x,t)fors≥1, withϕ(x, t) =e^λψ(x,t)whereψis a continuous function whose domain of definition Ω is contained in an enlarged smooth open and connected neighborhood ˜Ω. We also assume ψ∈C^k

Ω˜

with k large enough such that it satisfies the following property

∂xψ(x, t)>0, (x, t)∈Ω˜ ×(0, T). (1.6) The assumption of the higher-order derivatives is needed to obtain the estimates on the weight function presented in Section4, in contrast to the continuous case. We shall use the same notation for the sample of the continuous function on the discrete or semi-discrete sets. We now state a uniform Carleman estimate for the operatorD_h²with boundary observation. Although for the Carleman estimate just the condition (1.6) is needed, to achieve the UCP property for the system (1.4) we also consider the following assumption

∂tψ(x, t)<0, (x, t)∈Ω˜ ×(0, T). (1.7) It is not difficult to find a function that verifies conditions (1.6) and (1.7), for instance the following function, ψ(x, t) := (x−x₀)²−t², x₀<0. (1.8) Theorem 1.1. (Discrete Carleman estimate)

Letψbe a function verifying (1.6)andT >0. For the parameterλ0≥1sufficiently large, there exists0(λ0)≥1, h₀>0,ε₀>0andC=C(ε₀, s₀, λ₀)independent ofh >0 such that

C

e^sϕD_h²vh

2

L²_h(Q_h)+s Z

∂Q⁺_h

ϕ∂xψtr(e^2sϕ)tr(|Dhvh|²) +s³ Z

∂Q⁺_h

(ϕ∂xψ)³tr(e^2sϕ)tr(Ah(|vh|²))

!

≥s³ke^sϕvhk²_L2

h(Q_h)+ske^sϕDhvhk²_L2

h(Q^∗_h), (1.9) for all h∈(0, h0),s∈(s0, ε0/h)andvh defined in Qh:= ˚Mh×(0, T).

Several recent works have been concerned with discrete and semi-discrete Carleman estimates for second- order differential operators. The hyperbolic case has been developed for the one-dimensional case by L. Baudouin and S. Ervedoza [1], to study the stability of an inverse problem to recover a potential term in a semi-discrete wave equation. The elliptic case has been developed by F. Boyer et al.in [3], for the one-dimensional case, to establish a relaxed observability estimate for the associated semi-discrete parabolic equation. In [7], S. Ervedoza and F. de Gournay study the Laplacian operator in arbitrary dimension to prove the stability for the discrete Calderon problem, with limiting Carleman weight function. Semi-discrete Carleman estimates for parabolic operators have been established by F. Boyer and J. Le Rousseau in [4] for multidimensional Cartesian grids.

Moreover, in [13], T. N. T. Nguyen studied in the one-dimensional setting a semi-discrete parabolic operator with discontinuous diffusion coefficient; both of them obtain relaxed controllability results for their respective systems.

In the aforementioned works, the discretization was based on a finite difference scheme. Also, the Carleman parameter cannot be arbitrarily large, which is related to the discretization step size, in contrast to the continuous setting. Let us finally mentioned that recently in [10], a fully discrete Carleman estimates for parabolic operator have been obtained by V. Hern´andez-Santamar´ıa and P. Gonz´alez Casanova, where the spatial and time discrete step-size parameters are connected to the Carleman parameter.

We recall that T. Carleman introduced an estimate, known as the Carleman estimate, to prove a UCP for second-order elliptic partial differential equations in [5] when coefficients fail to be analytic. Nowadays, it has become an efficient tool to prove UCP, the study of controllability, observability, and stabilization for PDEs.

We refer to the work of X. Fuet al.[8], and references therein, where the authors present a unified approach to Carleman estimates for second-order PDEs and their applications to control theory and inverse problems.

One of the main difficulties in the development of discrete or semi-discrete Carleman estimates is to compute multiple discrete operators such asD_handA_h on the Carleman weight functions. For this reason, we establish Theorem 4.9 (see Sect.4) to reduce some tedious computation, and it represents an extension of the results presented by F. Boyer et al.in [3] related to discrete estimate on the weight function.

1.3. Semi-discrete setting and main results

With the notation we have introduced we can rewrite the semi-discretization (1.4) as

∂tuh−D_h²∂tuh=phDhAhuh+qhuh inQh:= ˚Mh×(0, T), (1.10) where ph, qh ∈L^∞_h (Qh). In (1.10) uh(t) provides an approximation of u(xh, t), u being the solution of the continuous equation (1.3).∂tuh stands for the first order differentiation with respect totand the operatorsDh

and D²_h are the classical central finite-difference approximation of the space derivatives. We assume that there exists a constantM >0 independent of hsuch that max{kphk_L∞

h(Q_h),kqhk_L∞

h(Q_h)} ≤M. Theorem 1.2. (Stability for space semi-discrete BBM equation)

Letψbe a function verifying(1.6)and (1.7), andT >0. Forλ0≥1sufficiently large, there exists0(λ0, M)≥1, h0>0 depending on M, ε0>0 and a constant C >0 independent of h >0 such that the following estimate

holds

s³ke^sϕu_hk²_L2

h(Qh)+s³ke^sϕ∂_tu_hk²_L2

h(Qh)+ske^sϕD_h∂_tu_hk²_L2 h(Q^∗_h)

≤Cs Z

∂Q⁺_h

ϕ∂xψtr(e^2sϕ)tr(|Dh∂tuh|²) +Cs³

Z

∂Q⁺_h

(ϕ∂_xψ)³t_r(e^2sϕ)t_r(A_h(|∂_tu_h|²)),

(1.11)

for0< h≤h0 ands∈(s0, ε0/h)anduh(xh,0) = 0in M˚h.

As a consequence of Theorem1.2, we have the following unique continuation property for semi-discrete BBM equation (1.10).

Corollary 1.3. (UCP for a semi-discrete BBM equation)

There exists h0>0 depending onM such that ifuh= 0 on{1} ×(0, T),Dhuh= 0 on {1−h/2} ×(0, T) and uh(·,0) = 0in M˚h; then uh(xh, t) = 0 inQh for allh∈(0, h0).

The methodology of the proof of Theorem1.2is similar in spirit to [15], where it have been obtained a UCP for equation (1.3). However, it cannot be followed straightly from the proof of the continuous case since the parameter s cannot be arbitrarily large. As we mentioned above, this parameter is related to the mesh size.

Thus, a semi-discrete version of the Carleman estimate used in [15] is not enough. For this reason, we develop a more refined Carleman estimate, (5.2), and its semi-discrete counterpart, see Theorem 1.1.

The rest of this article is organized as follows. First, in Section 2, we introduce and prove some discrete calculus formulas for uniform meshes. Then, we establish a stability estimate for the semi-discrete numerical approximation based on a uniform spatial discretization of equation (1.3) in Section 3. The last three sections are devoted to the proof of the discrete Carleman estimate for a finite-difference approximation of Laplacian operator with boundary observation. In Section 4, we extend the estimations developed in Section 3 of [3] for arbitrary order, which reduce some computation in the proof of the Carleman estimate presented in Section 5.

Some calculations are postponed and computed in Section 6.

2. Calculus formulas for uniform meshes

In this section, we state the elementary notions concerning discrete calculus formulas. First, we set some useful identities that shall be used in what follows. Then, we present the integration by parts for the difference and average operator. For the sake of presentation, we set the results in the discrete space variable framework instead of the semi-discrete setting.

The following Lemma gives us a calculus rule for finite difference operators. Its proof can be found in [3].

Lemma 2.1. LetWh⊂ Mh be a regular mesh. Foruh, vh∈C(Mh), we have the following identities onW_h^∗ Dh(uhvh) =Dh(uh)Ah(vh) +Ah(uh)Dh(vh),

Ah(uhvh) =Ah(uh)Ah(vh) +h²

4 Dh(uh)Dh(vh).

(2.1)

As a direct consequence of Lemma2.1, we point out some identities to develop the rest of this paper.

Corollary 2.2. Let Wh⊆ Mh be a regular mesh.

– Foru∈C(Wh),

Ah(u²_h) = (Ahuh)²+h²

4 (Dhuh)²,on W_h^∗. (2.2)

In particular, for allu∈C(Wh),

Ah(u²_h)≥(Ahuh)², onW_h^∗. (2.3) – Foru∈(Wh)

Dh(u²_h) = 2DhuhAhuh. (2.4)

Note that as a consequence of the first identity in (2.1) of Lemma 2.1, it can be proved by induction the following identity.

Corollary 2.3. Suppose that for n∈ N and h >0, the set (nh/2,1−nh/2) is not empty. Then, for each u, v ∈C(Wh)we have

D_hⁿ(u v) =

n

X

k=0

n k

D^n−k_h A^k_hu A^n−k_h D^k_hv onX_h, (2.5)

where

X_h=

(M^∗_h∩[nh/2,1−nh/2], if nis odd, Mh∩[nh/2,1−nh/2], if nis even.

Now, we state the following integration by parts presented for discrete space variable. We note that the same identities could be considered in the semi-discrete setting since the temporal variable does not play any significant role.

Proposition 2.4. LetWh⊆ Mh be a regular mesh. Foruh∈C(Wh)andvh∈C(W_h^∗)we have Z

W_h

u_hD_h(v_h) =− Z

W_h^∗

D_h(u_h)v_h+ Z

∂W_h

u_ht_r(v_h)n_h

and

Z

Wh

uhAh(vh) = Z

W_h^∗

Ah(uh)vh−h 2 Z

∂Wh

uhtr(vh).

Proof. We have

Z

Wh

u_hτ₊(v_h) = Z

τ₊(Wh)

τ₋(u_h)v_h

= Z

W^∗_h

τ₋(u_h)v_h− Z

W_h^∗\τ+(W_h)

τ₋(u_h)v_h

= Z

W^∗_h

τ−(uh)vh− Z

τ₋²(Wh)\Wh

uhτ+(vh).

(2.6)

and

Z

Wh

uhτ−(vh) = Z

τ−(Wh)

τ+(uh)vh

= Z

W_h^∗

τ₊(u_h)v_h− Z

W_h^∗\τ−(W_h)

τ₊(u_h)v_h

= Z

W_h^∗

τ+(uh)vh− Z

τ₊²Wh\Wh

uhτ₋(vh).

(2.7)

Combining (2.6) and (2.7) it follows that Z

Wh

uhDh(vh) =− Z

W_h^∗

Dh(uh)vh−1 h

Z

τ₋²Wh\Wh

uhτ+(vh) +1

h Z

τ₊²Wh\Wh

uhτ−(vh)

=− Z

W_h^∗

D_h(u_h)v_h+ Z

∂W

u_ht_r(v_h)n_h. Similarly, averaging the equations (2.6) and (2.7) we obtain

Z

Wh

uhAh(vh) = Z

W_h^∗

Ah(uh)vh−1 2

Z

τ₋²(Wh)\Wh

uhτ+(vh)

−1 2

Z

τ₊²W_h\W_h

u_hτ₋(v_h)

= Z

W_h^∗

Ah(uh)vh−h 2 Z

∂Wh

uhtr(vh), which completes the proof.

Remark 2.5. One can consider the previous result for the meshesW_h⁰, see for instance a similar result by S.

Ervedoza and F. de Gournay in [7]. In that case we should change the definition of the boundary nodes of our mesh. As we are interested in the boundary that kind of integral-by-parts formulas does not fulfill our goal. In contrast with the integration-by-part formulas from F. Boyeret al. [3], we do not make any distinction on the difference and average operators, then we need to specify that difference in the respective meshes.

3. Stability estimate for space semi-discrete BMM equation

This section is devoted to proof Theorem 1.2. We follow as close as possible the ideas from its continuous formulation. For this reason we state a stability estimate for (1.3). The main tool for the proof is a Carleman estimate for Laplacian operator. For sake of exposition we postpone that proof, see Section 5. It is worth to mention that we refined the result presented in [15] (see Sect.5.1).

3.1. The continuous case

We considerQ:= (0,1)×(0, T), forT >0, and we define the classical inner product (u, v)_L2(Q):=

Z

Q

u vdxdt

and its respective L²-normkuk²_L2(Q)= (u, v)_L2(Q).

Following the methodology from [15] we can obtain a stability estimate for (1.3). The proof is based on the Carleman estimate (5.2) and Lemma 6.4.2 from V. Isakov [11].

Theorem 3.1. Let ∂_x^j∂_t^ku∈C([0,1]×[0, T]) with j= 0,1,2 andk= 0,1. For λ₀>0 sufficiently large, there exist constants s₀≥0andC(s₀, λ₀, ψ)>0, such that

s³ke^sϕuk²_L2(Q)+s³ke^sϕ∂tuk²_L2(Q)+ske^sϕ∂x∂tuk²_L2(Q)≤Cs³ Z T

0

(∂xψ)³ϕ³e^2sϕ|∂tu|² (1, t) dt +Cs

Z T 0

∂_xψϕe^2sϕ|∂x∂_tu|²

(1, t) dt,

(3.1)

for all s≥s0, anduverify (1.3)with u(x,0) = 0for all x∈(0,1).

As a Corollary of Theorem3.1, it follows the main result presented in [15].

Corollary 3.2. Let ∂_x^j∂_t^ku∈C([0,1]×[0, T]) with 0,1,2 and k = 0,1. If u is solution of (1.3) such that u(1, t) =∂xu(1, t) = 0 for allt∈(0, T)andu(x,0) = 0in (0,1), thenu(x, t) = 0 in(0,1)×(0, T).

3.2. Proof of Theorem 1.2

As we mentioned above the proof of Theorem1.2is based on the continuous setting strategy. Then, we write down a space semi-discrete version of Lemma 6.4.2 from [11], which is a Poincar´e weighted inequality.

Lemma 3.3. Letϕ∈C¹(Qh)be such that ∂ϕ

∂t ≤0, then Z

Q_h

Z t 0

u_h(x_h, σ)dσ

2

e^2sϕ(xh,t)≤T² Z

Q_h

(u_h(x_h, t))²e^2sϕ(xh,t),

for all uh∈L²_h(Qh).

Proof. We have

Z T 0

Z t 0

u(xh, σ)dσ

2

e^2sϕ(xh,t)dt≤ Z T

0

t Z t

0

(u(xh, σ))²e^2sϕ(xh,t)dσdt

= Z T

0

Z T σ

te^2sϕ(xh,t)(u(xh, σ))²dtdσ

≤T² Z T

0

(u(xh, σ))²e^2sϕ(xh,σ)dσ,

and integrating over ˚Mh we complete the proof.

3.2.1. Proof of Theorem 1.2

In this section, we shall give the proof of the Theorem 1.2.

Proof. Note that from (1.10) we have

D_h²∂tuh=∂tuh−phDhAhuh−qhuh inQh.

Then, applying the Carleman estimate (1.9) tovh=∂tuh we obtain s³ke^sϕ∂_tu_hk²_L2

h(Q_h)+ske^sϕD_h∂_tu_hk²_L2

h(Q^∗_h)≤s Z

∂M˚⁺_h

ϕ∂_xψt_r(e^2sϕ)t_r(|D_h∂_tu_h|²) +s³

Z

∂M˚⁺_h

(ϕ∂xψ)³tr(e^2sϕ)tr(Ah(|∂tuh|²)) +ke^sϕ(∂_tu_h−p_hD_hA_hu_h−q_hu_h)k²_L2

h(Q_h),

(3.2)

for 0< h≤h₀,s≥s₀andsh < ε₀. On the other hand, we note that

D_hA_hu_h(x_h, t) = Z t

0

D_hA_h∂_tu_h(x_h, σ)dσ

and

u_h(x_h, t) = Z t

0

∂_tu_h(x_h, σ)dσ,

sinceu_h(x_h,0) = 0 in ˚Mh. Thus, by Lemma3.3it follows that ke^sϕ(∂tuh−phDhAhuh−qhuh)k²_L2

h(Q_h)≤CT²kphk²_L∞

h(Q_h)ke^sϕDhAh∂tuhk²_L2 h(Q_h)

+C(1 +T²kq_hk²_L∞

h(Qh))ke^sϕ∂_tu_hk²_L2 h(Qh)

≤CT²M²ke^sϕD_hA_h∂_tu_hk²_L2 h(Q_h)

+C(1 +T²M²)ke^sϕ∂tuhk²_L2 h(Q_h).

(3.3)

Now, we focus on the first term of the right-hand side above. Using (2.3) and a discrete integration by parts for the discrete average operator we get

ke^sϕDhAh∂tuhk²_L2 h(Qh)=

Z

Qh

e^2sϕ(DhAh∂tuh)²

≤ Z

Qh

e^2sϕAh

(Dh∂tuh)²

= Z

Q^∗_h

A_h(e^2sϕ) (D_h∂_tu_h)²−h 2 Z

∂Q_h

e^2sϕt_r

(D_h∂_tu_h)²

≤ Z

Q^∗_h

Ah(e^2sϕ) (Dh∂tuh)².

From Proposition4.5we have Ah(e^2sϕ)≤Cλe^2sϕ, we thus obtain ke^sϕDhAh∂tuhk²_L2

h(Q_h)≤Cλke^sϕDh∂tuhk_L2

h(Q^∗). (3.4)

Combining (3.2), (3.3) and (3.4) we get s³ke^sϕ∂tuhk²_L2

h(Q_h)+ske^sϕDh∂tuhk²_L2

h(Q^∗_h)≤s Z

∂Q⁺_h

ϕ∂xψtr(e^2sϕ)tr(|Dh∂tuh|²) +s³

Z

∂Q⁺_h

(ϕ∂_xψ)³t_r(e^2sϕ)t_r(A_h(|∂tu_h|²)) + (1 +T²M²)ke^sϕ∂tuhk²_L2

h(Q_h)

+C_λT²M²ke^sϕD_h∂_tu_hk²_L2 h(Q^∗_h).

(3.5)

We note that by choosing somes≥T^2/3(1 +M²)^1/3+T²M, the last term on the right-hand side from (3.5) can be absorbs by its left-hand side. Thus, recalling the hypothesis on sfrom the Carleman estimates, by choosing s₁:= max{s0, k(M, T)} ≥s₀ large enough we obtain

s³ke^sϕ∂_tu_hk²_L2

h(Q_h)+ske^sϕD_h∂_tu_hk²_L2

h(Q^∗_h)≤s Z

∂Q⁺_h

ϕ∂_xψt_r(e^2sϕ)t_r(|Dh∂_tu_h|²) +s³

Z

∂Q⁺_h

(ϕ∂xψ)³tr(e^2sϕ)tr(Ah(|∂tuh|²)),

provided s≥s1, where k(M, T) :=T^2/3 1 +M^21/3

+T²M². Now, we need to connect the condition over the Carleman parameterswith the mesh sizeh. Defining

h₁:= ε₀ s1

,

it follows that for 0< h≤min{h0, h1} we havesh≤ε0provided s∈(s1, ε0/h), which conclude the proof.

As a consequence, we obtain the UCP presented in Corollary1.3for semi-discrete BBM equation (1.10).

4. Some preliminary discrete calculus results

In this section, we establish some previous estimates for the Carleman weight function that shall be used in the next section to obtain the discrete Carleman estimate, see Theorem 1.1. Recall that our weight function is defined ase^sϕfors≥1, withϕ=e^λψ, whereψ∈C^k forksufficiently large andλ≥1. Our goal is to generalize the results presented previously in Section 3, obtained by F. Boyeret al.in [3], related to discrete operations performed on the Carleman weight functions, considering estimates and expansions for higher order discrete operators.

For easier comparison, we use the same notation by settingr=e^sϕandρ=r⁻¹, these positive parameterss andhshall be large and small respectively and limited by the conditionsh≤1. The proofs are similar in spirit to those given in [3].

We denote byOλ(sh) the functions that verifykOλ(sh)k_L∞(Q_h)≤Cλshfor some constantCλ depending on λ. ByO(1) we denote bounded functions and byOλ(1) a bounded function onceλis fixed.

We say thatαis a multi-index ifα= (α1, α2, . . . , αn)∈Nⁿ and fory∈Rⁿ we write:

|α|=α1+α2+· · ·+αn, ∂^α=∂_x^α₁¹. . . ∂^α_xnⁿ, y^α=y₁^α1. . . y_n^αn.

Proposition 4.1. Let us consider n∈N. Letf be a (n+ 2)−times differentiable andg a twice differentiable functions on R. Then

Aⁿ_hg=g+RAⁿ_h(g), Dⁿ_hf =f⁽ⁿ⁾+R_Dⁿ

h(f).

where R_Dⁿ

h andR_Aⁿ

h are given by

R_Dⁿ

h(f) :=h²

n

X

k=0

n k

(−1)^k

(n−2k) 2

ⁿ⁺²Z 1 0

(1−σ)ⁿ⁺¹

(n+ 1)! f⁽ⁿ⁺²⁾(·+(n−2k)h

2 σ)dσ

and

RAⁿ_h(g) := h² 2ⁿ⁺²

n

X

k=0

n k

(n−2k)² Z 1

0

(1−σ)g⁽²⁾(·+(n−2k)h 2 σ)dσ.

Proof. The proof of this proposition follows from Taylor expansion

g(x+y) =

i−1

X

j=0

y^j

j!h^(j)(x) +yⁱ Z 1

0

(1−σ)ⁱ⁻¹

(i−1)! g⁽ⁱ⁾(x+σy)dσ. (4.1)

First, we use (4.1) withi= 2 andy= (n−2k)h

2 to obtain

τ₊^n−2kg=g+(n−2k)h 2 g⁰+

(n−2k)h 2

2Z 1 0

(1−σ)g⁽²⁾(·+(n−2k)h 2 σ)dσ.

Then, it follows that

Aⁿ_hg=1

2ⁿ(τ++τ−)ⁿg

=1 2ⁿ

n

X

k=0

n k

τ₊^n−kτ₋^kg

=1 2ⁿ

n

X

k=0

n k

τ₊^n−2kg

=1 2ⁿ

n

X

k=0

n

k g+(n−2k)h 2 g⁰

+RAⁿ(g)

Now, using

n

X

k=0

n k

= 2ⁿ and

n

X

k=0

n k

k=n2ⁿ⁻¹ we write

Aⁿ_hg=g+RAⁿ_h(g).

On the other hand, applying (4.1) forf withi=n+ 2 andy= (n−2k)h

2 , we have τ₊^n−2kf=

n+1

X

j=0

1 j!

(n−2k)h 2

j

f^(j)+

(n−2k)h 2

n+2Z 1

0

(1−σ)ⁿ⁺¹

(n+ 1)! f⁽ⁿ⁺²⁾(·+(n−2k)h 2 σ)dσ.

Thus, for the difference operator we get D_hⁿf = 1

hⁿ(τ₊−τ₋)ⁿf

= 1 hⁿ

n

X

k=0

n k

(−1)^kτ₊^n−kτ₋^kf

= 1 hⁿ

n

X

k=0

n k

(−1)^kτ₊^n−2kf

= 1 hⁿ

n

X

k=0

n k

(−1)^k

n+1

X

j=0

1 j!

(n−2k)h 2

j

f^(j) +R_Dⁿ

h(f)

= 1 hⁿ

n+1

X

j=0

1 j!h^j

n

X

k=0

n k

(−1)^kn 2 −kj

f^(j)+RDⁿ_h(f)

Now, using

n

X

k=0

(−1)^k n

k

(x−k)ⁿ=n! and

n

X

k=0

(−1)^k n

k

(n

2 −k)ⁿ⁺¹= 0, we obtain

Dⁿ_hf = 1 hⁿ

1 (n)!hⁿ

n

X

k=0

n k

(−1)^kn 2 −kn

f⁽ⁿ⁾

+ 1 hⁿ

1

(n+ 1)!hⁿ⁺¹

n

X

k=0

n k

(−1)^kn

2 −kn+1

f⁽ⁿ⁺¹⁾+R_Dn(f)

=f⁽ⁿ⁾+RDⁿ_h(f), and the proof is complete.

Corollary 4.2. Let f be a (n+ 4)−times differentiable function defined onRandm, n∈N. Then A^m_hDⁿ_hf =f⁽ⁿ⁾+R_A^m

h(f⁽ⁿ⁾) +R_Dⁿ

h(f) +R_A^m

hDⁿ_h(f), where

R_A^m

hD_hⁿ(f) :=

n,m

X

k,k⁰=0

a_k,k⁰ Z 1

0

Z 1 0

(1−σ)ⁿ⁺¹

(n+ 1)! (1−σ⁰)f⁽ⁿ⁺⁴⁾(x+ (n−2k)h

2σ+(m−2k⁰)h

2 σ⁰)dσ⁰dσ with

ak,k⁰:= h⁴ 2^m

m k⁰

n k

(−1)^k

(n−2k) 2

n+2

(m−2k⁰)²

4 .

Proof. It is enough to see that

A^m_h(Dⁿ_hf) =A^m_hf⁽ⁿ⁾+A^m_h(R_Dⁿ

h(f))

=f⁽ⁿ⁾+R_A^m

h(f⁽ⁿ⁾) +R_Dⁿ

h(f) +R_A^m

hD_hⁿ(f).

Note that RA^m_hDⁿ_h =RA^m_h ◦RD_hⁿ=RDⁿ_h◦RA^m_h. Now, we consider two fundamental estimates for our weight function. The proofs of these results can be found in [3]. We consider α= (α_t, α_x)∈N² multi-indices.

Lemma 4.3. Letαandβ be multi-indices. We have

∂^β(r∂^αρ) =|α|^|β|(−sϕ)^|α|λ^|α+β|(∂xψ)^α+β

+|α||β|(sϕ)^|α|λ^|α+β|−1O(1) +s^|α|−1|α|(|α| −1)Oλ(1)

=Oλ(s^|α|).

(4.2)

Moreover, let σ∈[0,1]andsh≤1, then∂^β(r(x)(∂^αρ)(x+σh)) =s^|α|O_λ(1).

Corollary 4.4. Let α,β andδ be multi-indices. We have

∂^δ r²(∂^αρ)∂^βρ

=|α+β|^|δ|(−sϕ)^|α+β|λ^|α+β+δ|(∂xψ)^α+β+δ +|δ||α+β|(sϕ)^|α+β|λ^{|α+β+δ|−1}O(1) +s^|α+β|−1(|α|(|α| −1) +|β|(|β| −1))Oλ(1)

=O_λ(s^|α+β|).

Corollary4.2 and Lemma4.3yield.

Proposition 4.5. Letαbe a multi-index andn, m∈N. Provided sh≤1, we have rA^m_hD_hⁿ∂^αρ=r∂_xⁿ∂^αρ+s^|α|+nOλ((sh)²) =s^|α|+nOλ(1).

Proof. From Corollary 4.2we write

rA^m_hDⁿ_h∂^αρ=r∂_xⁿ∂^αρ+rR_A^m

h(∂_xⁿ∂^αρ) +rR_Dⁿ

h(∂^αρ) +rR_A^m

hDⁿ_h(∂^αρ) By Lemma4.3we have

r(x)(∂_xⁿ⁺²∂^αρ)(x+ (n−2k)hσ/2) =O_λ(s^|α|+n+2) and

r(x)∂_xⁿ⁺⁴∂^αρ(x+ (n−2k)hσ/2) =Oλ(s^|α|+n+4).

Then

rR_A^m

h(∂_xⁿ∂^αρ) =s^|α|+nOλ((sh)²), rR_Dⁿ

h(∂^αρ) =s^|α|+nO_λ((sh)²), rRA^m_hDⁿ_h(∂^αρ) =s^|α|+nOλ((sh)⁴),

which yields the result.

Lemma 4.6. Letαandβ multi-index and n∈N. Providedsh≤1, we have A^m_hDⁿ_h ∂^β(r∂^αρ)

=∂_xⁿ∂^β(r∂^αρ) +h²Oλ(s^|α|) Let σ∈[0,1], we have A^m_hDⁿ_h∂^β(r(x)∂^αρ(x+σh)) =Oλ(s^|α|).

Proof. By Corollary4.2we write

A^m_hD_hⁿ(∂^β(r∂^αρ)) =∂_xⁿ∂^β(r∂^αρ) +RA^m_h(∂ⁿ_x∂^β(r∂^αρ)) +RD_hⁿ(∂^β(r∂^αρ)) +RA^m_hD_hⁿ(∂^β(r∂^αρ)).

By Lemma4.3, it follows that

(∂_xⁿ∂^β(r∂^αρ))(x+ (n−2k)hσ/2) =Oλ(s^|α|), (∂ⁿ⁺²_x ∂^β(r∂^αρ))(x+ (n−2k)hσ/2) =Oλ(s^|α|), (∂ⁿ⁺⁴_x ∂^β(r∂^αρ))(x+ (n−2k)hσ/2) =Oλ(s^|α|), which concludes the proof of the first result.

On the other hand, we setν(x, σh) :=r(x)ρ(x+σh) andµ_α:=r∂^αρ. Sincerρ= 1 it follows thatr(x)∂^αρ(x+ σh) =ν(x, σh)µ_α(x+σh). Note that, by continuous Leibniz rule,∂_xⁿ∂^β(νµ_α) is a linear combination of terms of the form∂^β0ν∂^β00µ_α, withβ⁰+β⁰⁰=n+β and by Lemma4.3we write∂^β0ν =O_λ(1) and∂^β00µ_α=O_λ(s^|α|).

Besides, it holds for the terms∂_xⁿ⁺²∂^β(νµ_α) and∂_xⁿ⁺⁴∂^β(νµ_α) as well. Therefore, applying Corollary4.2toνµ_α we obtain

A^m_hD_hⁿ(∂^β(r(x)∂^αρ(x+σh))) =Oλ(s^|α|) +h²Oλ(s^|α|) +h⁴Oλ(s^|α|).

Lemma 4.7. Letα,β,δ be multi-indices andn, m∈N. Provided sh≤1, we have:

A^m_hDⁿ_h∂^δ r²(∂^αρ)∂^βρ

=∂_xⁿ∂^δ r²(∂^αρ)∂^βρ

+h²Oλ(s^|α|+|β|)

=O_λ(s^|α|+|β|).

Let σ, σ⁰ ∈[0,1]. We have

A^m_hD_hⁿ∂^δ r(x)²(∂^αρ(x+σh))∂^βρ(x+σ⁰h)

=Oλ(s^|α|+|β|).

Proof. Applying Corollary4.2to∂^δ r²(∂^αρ)∂^βρ

we obtain A^m_hD_hⁿ∂^δ r²(∂^αρ)∂^βρ

=∂_xⁿ∂^δ r²(∂^αρ)∂^βρ

+RA^m_h ∂_xⁿ∂^δ r²(∂^αρ)∂^βρ +R_Dⁿ

h ∂^δ r²(∂^αρ)∂^βρ +R_A^m

hDⁿ_h ∂^δ r²(∂^αρ)∂^βρ .

Then, the first result follows from Corollary 4.4. For the second one, we proceed similarly as the proof of the second result of Lemma4.6, that is, we apply Corollary4.2toν²µαµβ, then we use continuous Leibniz rule and Lemma 4.6to conclude.