Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


IMPACT FACTOR 2018: 6.636

CiteScore 2018: 5.03

SCImago Journal Rank (SJR) 2018: 3.215
Source Normalized Impact per Paper (SNIP) 2018: 3.225

Mathematical Citation Quotient (MCQ) 2018: 3.18

Open Access
Online
ISSN
2191-950X
See all formats and pricing
More options …

Theoretical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach

Olivier Goubet
  • Corresponding author
  • LAMFA CNRS UMR 7352, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens Cedex, France
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Imen Manoubi
  • Unité de recherche: Multifractales et Ondelettes, Faculté des Sciences de Monastir, Av. de l’environnement, 5000 Monastir, Tunisia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-02-21 | DOI: https://doi.org/10.1515/anona-2016-0274

Abstract

In this paper, we study the following water wave model with a nonlocal viscous term:

ut+ux+βuxxx+νπt0tu(s)t-s𝑑s+uux=νuxx,

where 1πt0tu(s)t-s𝑑s is the Riemann–Liouville half-order derivative. We prove the well-posedness of this model using diffusive realization of the half-order derivative, and we discuss the asymptotic convergence of the solution. Also, we compare our mathematical results with those given in [5] and [14] for similar equations.

Keywords: Waterwaves; nonlocal viscous model; decay rate; fractional derivatives; diffusive realization

MSC 2010: 35Q35; 35Q53; 35L05

1 Introduction

1.1 State of the art

The modeling and the mathematical analysis of water wave propagation are challenging issues. When the viscosity is neglected, it is classical to derive the so-called Boussinesq system or the Korteweg–de Vries equation from the Euler equations [3]. Taking into account the effects of viscosity on the propagation of long waves is a very important challenge and many studies have been carried out in the last decade. The pioneering work for this issue is due to Kakutani and Matsuuchi [12]. Recently, Liu and Orfila [13], and Dutykh and Dias [7] have independently derived viscous asymptotic models for transient long-wave propagation on viscous shallow water. These effects appear as nonlocal terms in the form of convolution integrals. A one-dimensional nonlinear system is presented in [6].

In their recent work [5], Chen et al. investigated theoretically and numerically the decay rate of solutions to the following water wave model with a nonlocal viscous dispersive term:

ut+ux+βuxxx+νπ0tut(s)t-s𝑑s+uux=αuxx,(1.1)

where 1π0tut(s)t-s𝑑s represents the Caputo half-derivative in time. Here u is the horizontal velocity of the fluid, -αuxx is the usual diffusion, βuxxx is the geometric dispersion and 1π0tut(s)t-s𝑑s stands for the nonlocal diffusive-dispersive term. We denote by β, ν and α the parameters dedicated to balance or unbalance the effects of viscosity and dispersion against nonlinear effects. Particularly, Chen et al. [5] considered (1.1) with β=0 supplemented with the initial condition u0L1()L2(). They proved that if u0L1() is small enough, then there exists a unique global solution uC(+;Lx2())C1(+;Hx-2()). In addition, u satisfies

t1/4u(t,)Lx2()+t1/2u(t,)Lx()<C(u0).

It is worth pointing out that this result is valid only for small initial data. Moreover, in a recent work [14], the second author investigated a derived model from (1.1), where the fractional term is described by the Riemann–Liouville half derivative instead of that of Caputo, namely,

ut+ux+βuxxx+νπt0tu(s)t-s𝑑s+uux=αuxx.(1.2)

She proved the local and global existence of solutions to problem (1.2) when β=0 using a fixed point theorem. Then she studied theoretically the decay rate of the solutions in this case. She established that if u0L1() is small enough, then there exists a unique global solution uC(+;Lx2())C1/2(+;Hx-2()). Besides, she proved that the solution u satisfies

max(t1/4,t3/4)u(t,)Lx2()+max(t1/2,t)u(t,)Lx()C(u0).

In addition, she performed numerical simulations on the decay rate of the solutions for different values of the parameters α, ν and β.

However, all these results are performed assuming a smallness condition on the initial data. In order to remove this condition and to investigate model (1.2) for a large class of initial data, we introduce here the concept of diffusive realizations for the half-order derivative. This approach was initially developed during the last decade by numerous authors in the automatic community, see [21]. Diffusive realization make possible to represent nonlocal in time operators, and more generally causal pseudo-differential operators, in a state space model formulation where the state belongs to an appropriate Hilbert space. Moreover, this formulation is local in time. Hence, the new system is easier to solve analytically and numerically, see [4]. For more details, we refer the reader to [19, 18, 22, 23]. Different applications of this concept in many scientific domains may be find in [1, 2, 9, 8, 11, 15, 16, 20]. However, this list is by no means exhaustive.

In this paper, we assume that the effects of the geometric dispersion in (1.2) is less important than the viscosity effects (i.e., we take β=0 in (1.2)), and we assume that the other constants are normalized. Thus, our model is reduced as follows:

ut+ux+1πt0tu(s)t-s𝑑s+uux=uxx.(1.3)

We aim to discuss the convergence of the solution u to zero, removing the assumption of smallness of the initial data. To this end, we complete the introduction as follows. We first introduce the diffusive formulation of the half-order Riemann–Liouville derivative. Then we deduce the mathematical model that derives from (1.3) using the diffusive approach. Finally, we state the main results of this article to conclude the introduction.

1.2 Diffusive formulation of the model

We now describe the mathematical framework. We denote by

I1/2(t)=1π0tu(s)t-s𝑑s

the Riemann–Liouville half-order integral and by

D1/2u(t)=ddtI1/2u(t)=1πddt0tu(s)t-s𝑑s

the Riemann–Liouville half-order derivative.

The diffusive realization requires to introduce a new variable y that is not physically relevant. To begin with, we fix x and recall that a realization of the half-order integral I1/2u(t) is given, for all t>0, by

{tΦ(t,y)=Φyy(t,y)+u(t)δy=0,Φ(0,y)=0,y,I1/2u(t)=2δy=0,Φ(t,y)𝒟,𝒟=2Φ(t,0),(1.4)

where δy=0 is the Dirac delta function at y=0 and u(t)δy=0 is the tensorial product in the distributions sense of the applications tu(t) and yδy=0.

In order to get the diffusive realization of the Riemann–Liouville half-order derivative, we derive the half-order derivative in (1.4) with respect to time. Thus, we get

{tΦ(t,y)=Φyy(t,y)+u(t)δy=0,Φ(0,y)=0,y,D1/2u(t)=2δy=0,tΦ(t,y)𝒟,𝒟=2ddtΦ(t,0).(1.5)

Now, we extend the diffusive realization (1.5) for u(t)Lx2(). The previous definition is valid for almost every x. To define the mathematical framework, we define the vector space

X={vL2(2)|(x,y)vyL2(2)}.

We note that X is a Hilbert space for the scalar product

(v,w)X=(v,w)L2(2)+(vy,wy)L2(2)for all v,wX.

The corresponding norm is

vX=vL2(2)2+vyL2(2)2for all vX.

Hence, the application

Γ:XL2(),ϕ(t)Γ(ϕ)(t,x)=ϕ(t,x,y=0),(1.6)

is a linear and continuous operator. Its dual map Γ* is defined as

Γ*:Lx2()X,u(t)Γ*(u)(t,x)=u(t,x)δy=0,(1.7)

where X is the dual space of X.

Taking into account the previous notations and results, we define the diffusive realization of the Riemann–Liouville half-order derivative D1/2u(t,x), for all t>0 and x, as follows:

{tΦ(t,x,y)=Φyy(t,x,y)+Γ*(u)(t,x),Φ(0,x,y)=0,x,y,D1/2u(t,x)=2ddtΓ(Φ)(t,x),(1.8)

where Γ and Γ* are defined in (1.6) and (1.7), respectively. Thanks to the diffusive realization (1.8), problem (1.3) reads as follows:

{ut(t,x)+2t(Γ(Φ))(t,x)+ux(t,x)-uxx(t,x)+u(t,x)ux(t,x)=0,t>0,x,Φt(t,x,y)-Φyy(t,x,y)-Γ*(u)(t,x)=0,t>0,x,y,u(0,x)=u0(x),x,Φ(0,x,y)=0,x,y.(1.9)

1.3 Main results

First, we introduce the following notations:

  • Lx2=L2(x), with (,) being the scalar product and ||Lx2 being the norm.

  • 𝕃x,y2=𝕃2(2)=𝕃2(x×y),with [,] being the scalar product and 𝕃x,y2 being the norm.

  • Hx1=H1() and x,y1=1(2)=1(x×y), with , being the duality product between Hx1 and Hx1 and , being the duality product between x,y1 and x,y-1.

  • Cw([0,T],L2()) is the space of weakly continuous functions on [0,T] that take values in L2().

The diffusive equation (1.9) is only partially diffusive since the Laplace operator acts with respect to the y variable and not to the x variable. For the sake of convenience, we introduce a (completely) parabolic regularization of this system that reads as follows:

{ut(t,x)+2t(Γ(Φ))(t,x)+ux(t,x)-uxx(t,x)+u(t,x)ux(t,x)=0,t>0,x,Φt(t,x,y)-Φyy(t,x,y)-εΦxx(t,x,y)-Γ*(u)(t,x)=0,t>0,x,y,u(0,x)=u0(x),x,Φ(0,x,y)=0,x,y.(1.10)

A first auxiliary result is the following.

Theorem 1.1.

For all initial data u0L2(R), the regularized problem (1.10) has a unique global weak solution (uε,Φε) such that for all T>0,

  • (i)

    uεL(0,T,Lx2())Cw([0,T],Lx2())L2((0,T),Hx1()),

  • (ii)

    ΦεL(0,T,x,y1(2))C([0,T],𝕃x,y2(2)).

The second main result states as follows.

Theorem 1.2.

For all initial data u0L2(R), the problem (1.9) has a unique global weak solution (u,Φ). Moreover, for all T>0,

  • (i)

    uL(+,Lx2())Cw(+,Lx2())L2((0,T),Hx1()),

  • (ii)

    ΦC(+,𝕃x,y2(2))L((0,T),X).

Concerning the issue of convergence of solutions towards 0 when t tends to +, we have the following result.

Proposition 1.3.

The solution u of equation (1.3) satisfies

limt+|ux(t)|L2=0(1.11)

and

u(t)0weakly in L2() as t+.(1.12)

The remaining of this article is organized as follows. In Section 2.1, we prove Theorem 1.1 and Theorem 1.2 using an approximation method based on the Galerkin approximation. In Section 3, we establish Proposition 1.3 which provides the weak convergence of the solution; we complete this last section discussing why one cannot have a stronger convergence to zero.

2 Proofs of the main theorems

2.1 Proof of Theorem 1.1

To begin with, we define a weak solution of (1.10).

Definition 2.1.

By a weak solution of problem (1.10) we mean a pair (u,Φ) such that

uL(+,Lx2())Cw(+,Lx2())L2((0,T),Hx1()),ΦL(+,x,y1(2))C(+,𝕃x,y2(2)),

and which verifies

{ddt[(u(t),u~)L2()+2(ΓΦ(t),u~)L2()]+(ux(t),u~)L2()+(ux(t),u~x)L2()=-(u(t)ux(t),u~)L2()for all u~H1(),ddt[Φ(t),Φ~]𝕃2(2)+[Φy(t),Φ~y]𝕃2(2)+ε[Φx(t),Φ~x]𝕃2(2)=(u(t),Φ~(x,0))L2()for all Φ~1(2),u(0)=u0L2(),Φ(0)=0.

Before going to the proof of Theorem 1.1, we state some preliminary inequalities.

Lemma 2.2.

For all ΦX, we have

|Γ(Φ)|Lx2()Φ𝕃2(2)1/2Φy𝕃2(2)1/2.(2.1)

Moreover, for all ΦH1(R2), we have the straightforward consequence

|Γ(Φ)|Lx2()Φ𝕃2(2)1/2Φ1(2)1/2.

Proof.

We prove (2.1) for Φ𝒟(2) smooth, and then proceed to a limit argument. For such a Φ, we have

0+Φ(x,y)yΦ(x,y)𝑑y=120+y|Φ(x,y)|2dy=-12|Φ(x,0)|2,-0Φ(x,y)yΦ(x,y)𝑑y=12-0y|Φ(x,y)|2dy=12|Φ(x,0)|2.

It follows that

|Φ(x,0)|2=-0+Φ(x,y)yΦ(x,y)dy+-0Φ(x,y)yΦ(x,y)dy|Φ(x,y)||yΦ(x,y)|dy.

Integrating this equation with respect to x and using the Cauchy–Schwarz inequality, we get

|Γ(Φ)(x)|Lx2=|Φ(x,0)|2dx2|Φ(x,y)||yΦ(x,y)|dxdyΦ𝕃x,y2yΦ𝕃x,y2.

Using the density of 𝒟(2) into X completes the proof of the lemma. ∎

We now proceed to the core of the proof of Theorem 1.1. For convenience, we omit the subscript ε throughout this proof. The main idea here is to construct an approximate solution of the perturbed system (1.10) by a suitable Galerkin approximation, and then pass to the limit. Hence, let ((ei)i1) be an orthogonal basis of Lx2(). We suppose that every element ei is compactly supported. We may take, for example, a fractal wavelet basis of Daubechies (see [17]). Let Vn=Span(e1,,en) be the subspace of Lx2() spanned by e1,,en and let Wn=VnVn be the corresponding subspace of 𝕃2(2). We then seek an approximate solution (un,Φn)nVn×Wn that verifies the following approximate problem:

{ddt[(un(t),u~)L2()+2(ΓΦn(t),u~)L2()]+(unx(t),u~)L2()+(unx(t),u~x)L2()=-(un(t)unx(t),u~)L2()for all u~Vn,ddt[Φn(t),Φ~]𝕃2(2)+[Φny(t),Φ~y]𝕃2(2)+ε[Φnx(t),Φ~x]𝕃2(2)=(un(t),Φ~(,0))L2()for all Φ~Wn,un(0)=Pn(u0)L2(),Φn(0)=0,(2.2)

where Pn is the orthogonal projector onto Vn.

Proposition 2.3.

The approximate problem (2.2) has a unique maximal solution defined on the interval [0,Tmax[, where 0<Tmax+.

Proof.

First, we denote by pn(t)=(pkn(t))1kn and qn(t)=(qijn(t))1i,jn, defined, respectively, by un=pknek and Φn=qijneiej. Then the problem (2.2) can be written in the matrix form

(p˙nq˙n)+𝒩(pnqn)=(F1(pn(t))0n2).(2.3)

where

=(In10In2),

is a square matrix of order (n+n2), In is the n×n identity matrix and 1 is a square matrix of order n2. Moreover, 𝒩 is a square matrix of order (n+n2). Finally, F1 is a polynomial vectorial function with respect to pn(t). Since is an invertible matrix, problem (2.3) can be written as

ddt(pnqn)=F(pnqn),

where F:n×n2n+n2 is a locally lipschitzian map. The Cauchy–Lipschitz theorem applies and we deduce that there exists a unique maximal solution (pn,qn) of class C1 from [0,Tmax[ to n+n2, where 0<Tmax+. ∎

We now proceed to the limit as n tends to infinity. For this purpose, we need following a priori estimates.

Lemma 2.4.

Let T>0. Then we have the following a priori estimates, uniformly with respect to n:

(un)n is bounded in L(0,T,Lx2())L2(0,T,Hx1()),(2.4)(Φn)n is bounded in L(0,T,x,y1(2)),(2.5)(tΦn)n is bounded in L2(0,T,𝕃x,y2(2)),(2.6)(Γ(Φn))n is bounded in L(0,T,Lx2()).(2.7)

Proof.

Set (u~,Φ~)=(un,Φnt) in (2.2). We then have

ddt(12|un(t)|Lx22+Φny(t)Lx,y22+εΦnx(t)Lx,y22)+(|unx(t)|Lx22+2Φnt(t)Lx,y22)=0.

Integrating in time, this leads to (2.4), (2.6) and to an upper bound for the gradient of Φn. To complete the proof of (2.5), a bound on the L2 norm of Φn is required. We set Φ~=Φn in the second equation in (2.2) and, thanks to the previous estimates and Lemma 2.2, we obtain

ddtΦn(t)Lx,y22+Φny(t)Lx,y22(un(t),Φ~(,0))L2()CΦn(t)Lx,y2.

Then (2.5) follows promptly. Eventually, (2.7) is a consequence of the previous estimates and Lemma 2.2. ∎

Thanks to Lemma 2.4, we deduce that Tmax=+.

The limit process requires some compactness argument. For this purpose, we state the following lemma.

Lemma 2.5.

Let T>0 and δ(0,T). There exists a constant C>0 that may depends on T but that is independent of δ and n such that

δT-δΦn(t+δ)-Φn(t)𝕃2(2)2Cδ2for all n0,(2.8)δT-δ|un(t+δ)-un(t)|L2()2Cδfor all n0.(2.9)

Proof of Lemma 2.5.

Let T>0 and δ(0,T). Using the mean value theorem, for all n, we get

Φn(t+δ)-Φn(t)=tt+δtΦn(τ)𝑑τ.

Using the Cauchy–Schwarz inequality yields

Φn(t+δ)-Φn(t)𝕃2(2)2δtt+δtΦn(τ)𝕃2(2)2dτ.(2.10)

Now, we integrate (2.10) between 0 and T-δ. Then, using the Fubini–Tonelli theorem, we get

δT-δΦn(t+δ)-Φn(t)𝕃2(2)2dtδδT-δtt+δtΦn(τ)𝕃2(2)2dτdt.

Using the change of variables τ=t+δ, we deduce

δT-δΦn(t+δ)-Φn(t)𝕃2(2)2dtδ2δTtΦn(τ)𝕃2(2)2dτ.

Hence, using (2.6), we deduce (2.8).

In order to establish (2.9), we recall that un satisfies, for any vVn, the following equation:

ddτ(un(τ)v𝑑x+2Γ(Φn)(τ)v𝑑x)+unx(τ)v𝑑x+unx(τ)vx𝑑x=-un(τ)unx(τ)v𝑑x.(2.11)

We now integrate (2.11) with respect to τ from t to t+δ. Then, setting v=un(t+δ)-un(t), we get

|un(t+δ)-un(t)|L2()2+2(Γ(Φn)(t+δ)-Γ(Φn)(t),un(t+δ)-un(t))L2()+tt+δ(unx(τ),un(t+δ)-un(t))L2()𝑑τ+tt+δ(unx(τ),unx(t+δ)-unx(t))L2()𝑑τ=-tt+δ(un(τ)unx(τ),un(t+δ)-un(t))L2()𝑑τ.(2.12)

Using first the Cauchy–Schwarz inequality and then the Young inequality, we get

2δT-δ(Γ(Φn)(t+δ)-Γ(Φn)(t),un(t+δ)-un(t))L2()𝑑tCδT-δ|Γ(Φn(t+δ)-Φn(t+δ))|L2()2dt+δT-δ|un(t+δ)-un(t)|L2()2dt.(2.13)

From Lemma 2.2, we conclude that

δT-δ|Γ(Φn(t+δ)-Φn(t))|L2()2dtCδT-δΦn(t+δ)-Φn(t)𝕃2(2)Φn(t+δ)-Φn(t)1(2)dt.

Moreover, using the Cauchy–Schwarz inequality, and estimations (2.5) and (2.8), we get

δT-δ|Γ(Φn(t+δ)-Φn(t))|L2()2dtCδ.(2.14)

Then, gathering (2.14) and (2.13), we obtain

2δT-δ(Γ(Φn)(t+δ)-Γ(Φn)(t),un(t+δ)-un(t))L2()dtCδ+δT-δ|un(t+δ)-un(t)|L2()2dt.(2.15)

In addition, using the Cauchy–Schwarz inequality and estimation (2.4), we get

tt+δ(unx(τ),un(t+δ)-un(t))L2()𝑑τCδ.(2.16)

Finally, we integrate (2.16) with respect to t from δ to T-δ and get

δT-δtt+δ(unx(τ),un(t+δ)-un(t))L2()𝑑τ𝑑tCδ.(2.17)

In the same way, we establish that

tt+δ(unx(τ),unx(t+δ)-unx(t))L2()𝑑τCδ|unx(t+δ)-unx(t)|L2().

From (2.5), we deduce

δT-δtt+δ(unx(τ),unx(t+δ)-unx(t))L2()𝑑τ𝑑tCδ.(2.18)

Then, using the Hölder and Agmon inequalities, we have

tt+δ(un(τ)unx(τ),un(t+δ)-un(t))L2()dτ(tt+δ|un(τ)|L()|unx(τ)|L2()dt)|un(t+δ)-un(t)|L2()(tt+δ|un(τ)|L2()1/2|unx(τ)|L2()3/2dt)|un(t+δ)-un(t)|L2().

Moreover, using Lemma 2.4, we deduce that there exists a constant C>0 such that

tt+δ(un(τ)unx(τ),un(t+δ)-un(t))L2()𝑑τCδ1/4|un(t+δ)-un(t)|L2().

Finally, using the Young inequality, we get

δT-δtt+δ(un(τ)unx(τ),un(t+δ)-un(t))L2()dτdtCδ+14δT-δ|un(t+δ)-un(t)|L2()2dt.(2.19)

In conclusion, gathering (2.15), (2.17), (2.18), (2.19) and (2.12) yields (2.9). ∎

We rephrase Lemma 2.5 as follows: the sequences un and Φn are, respectively, bounded in H1/4(0,T;L2()) and H1(0,T;L2(2)). Combining this with the a priori bounds in Lemma 2.4 allow us to deduce that un and Φn converge strongly in Lloc2() and Lloc2(2), respectively. Gathering these and the weak convergence that comes from the a priori estimates, it is standard to pass to the limit when n tends towards +. We omit the details.

To complete the proof of the theorem, it remains to establish the continuity of the solution (u,Φ) of (1.10). First, we know that ΦL(0,T,1(2)) and ΦtL2(0,T,𝕃2(2)). Then, due to [24, Lemma 1.1, p. 250], we deduce that

ΦC([0,T],𝕃x,y2(2)).

In the following, we establish new estimations on the approximate solution un in order to establish the weak continuity of the solution over [0,T] with values in Lx2().

Lemma 2.6.

Let δ(0,1). There exists a constant C>0 that may depend on T but that is independent of δ and n such that

|un(t+δ)-un(t)|Hx-1Cδ1/4.

Due to Lemma 2.6, we have that

|u(t+δ)-u(t)|Hx-1lim infn+|un(t+δ)-un(t)|Hx-1Cδ1/4,

It follows that

limδ0|u(t+δ)-u(t)|Hx-1=0.

This implies that u is continuous [0,T] with values in Hx-1(). Also, we know that uL(0,T,Lx2()). Hence, thanks to [24, Lemma 1.4, p. 263] and using the continuous injection L2()H-1(), we deduce that u is weak continuous with values in Lx2().

The proof of the uniqueness of a solution is left to the reader.

2.2 Proof of Theorem 1.2

We now establish Theorem 1.2. To this end, we prove that the initial value problem (1.9) is well posed when passing to the limit as ε0+ in the perturbed problem (1.10). We state the following results, whose proofs are mere consequence of the previous subsection.

Lemma 2.7.

For all T>0, we have the following a priori estimates that are uniform with respect to ε:

  • (i)

    (uε) is bounded in L(0,T,Lx2()) and in L2(0,T,Hx1()),

  • (ii)

    (Φε) is bounded in L(0,T,𝕃x,y2(2)),

  • (iii)

    (yΦε) is bounded in L(0,T,𝕃x,y2(2)),

  • (iv)

    (ε1/2xΦε) is bounded in L(0,T,𝕃x,y2(2)),

  • (v)

    (tΦε) is bounded in L2(0,T,𝕃x,y2(2)).

In addition,

  • (vi)

    (Γ(Φε)) is bounded in L(0,T,Lx2()).

Finally, in order to pass to the limit in the nonlinear term, we state the following result.

Lemma 2.8.

Let T>0 and δ(0,T). There exists a constant C that may depends on T but that is independent of δ, ε such that

δT-δΦε(t+δ)-Φε(t)𝕃2(2)2Cδ2,δT-δ|uε(t+δ)-uε(t)|L2()2Cδ

To establish Theorem 1.2, we pass to the limit when ε0+ in the perturbed problem (1.10). This is standard and left as an exercise to the reader. It remains to check the continuity property.

Let (u,Φ) be a solution to the initial-value problem (1.9). First, applying [24, Lemma 1.1, p. 250] and using the continuous injection X𝕃2(2), we get

ΦC([0,T],𝕃2(2)).

To justify the weak continuity of u, we have the following result.

Lemma 2.9.

Let δ(0,1). There exists a constant C>0 such that

|u(t+δ)-u(t)|Hx-1Cδ1/4.

Due to Lemma 2.9, we conclude that

limδ0|u(t+δ)-u(t)|Hx-1=0.

This implies that u is continuous on [0,T] when it takes values in Hx-1. Moreover, uL(0,T,Lx2()). Thanks to [24, Lemma 1.4, p 263], we conclude that u is weakly continuous on [0,T] when it takes values in Lx2().

3 Proof of Proposition 1.3

3.1 Proof of the proposition

Let t0. In system (1.9), taking the scalar product in L2() of the first equation with u, and the scalar product in 𝕃2(2) of the second equation with Φt, we get

{12ddt|u(t)|Lx22+2(Γ(Φt)(t),u(t))+|ux|Lx22=0,12ddtΦy(t)𝕃x,y22+Φt(t)𝕃x,y22=(u(t),Γ(Φt)(t)).(3.1)

Hence, system (3.1) implies that

ddt(12|u(t)|2+Φy(t)2)+|ux(t)|2+2Φt(t)2=0.(3.2)

We conclude that t12|u(t)|2+Φy(t)2 is decreasing and nonnegative, so there exists l00 such that

limt+12|u(t)|2+Φy(t)2=l00.(3.3)

We now handle estimates for xu. In the equations of system (1.9), we take the derivative with respect to x. Then, letting v=ux and ψ=Φx, it follows that (v,ψ) satisfies

{vt+2Γ(ψt)+vx-vxx=-(uv)x,ψt-ψyy-Γ*(v)=0.(3.4)

In system (3.4), taking the scalar product of the two equations in L2()×𝕃2(2) with v and ψt, respectively, we get

12ddt|v(t)|2+2(Γ(ψt)(t),v(t))+|vx(t)|2=-(u(t)v(t))xv(t)𝑑x,(3.5)ψt(t)2-(ψyy(t),ψt(t))=(Γ*(v)(t),ψt(t)).(3.6)

We observe that

-(ψyy(t),ψt(t))=12ddtψy(t)2.(3.7)

In addition,

(u(t)v(t))xv(t)𝑑x=12v3(t)𝑑x.(3.8)

Gathering (3.5), (3.7) and (3.8) yields

ddt(12|v(t)|2+ψy(t)2)+|vx(t)|2+2ψt(t)2=-12v3(t)𝑑x.

Using the Young and Agmon inequalities, we get

ddt(12|v(t)|2+ψy(t)2)+|vx(t)|2+2ψt(t)2c|v(t)|5/2|vx(t)|1/2c(|v(t)|10/3+|vx(t)|2).

Using the fact that |v(t)|2|v(t)|2+ψy2, we obtain

ddt(|v(t)|2+2ψy(t)2)c|v(t)|2(|v(t)|2+ψy(t)2)2/3.

Hence, we deduce that

ddt((|v(t)|2+2ψy(t)2)1/3)c|v(t)|2.(3.9)

Using (3.2) and (3.9), we get

ddt((|v(t)|2+2ψy(t)2)1/3+c2|u(t)|2+cΦy(t)2)0.

We conclude that the function

t(|v(t)|2+2ψy(t)2)1/3+c2|u(t)|2+cΦy(t)2

is decreasing. Moreover, it is nonnegative, so there exists l10 such that

limt+(|v(t)|2+2ψy(t)2)1/3+c2|u(t)|2+cΦy(t)2=l10.(3.10)

Combining (3.3) and (3.10), we deduce that there exists a nonnegative real number l2 such that

limt+12|v(t)|2+ψy(t)2=l2<+.(3.11)

Moreover, using the Cesàro lemma, we deduce that

limt+1t0t(12|v(s)|2+ψy(s)2ds)=l2.(3.12)

In the following, we establish new estimates for (v,ψ). Hence, taking the scalar product in 𝕃2(2) of the second equation of system (3.4) with ψt, we get

12ddtψ(t)2-(ψyy(t),ψ(t))=(Γ*(v)(t),ψ(t))=(v(t),Γ(ψ)(t)).

Using Lemma 2.2 and the Young inequality, we get

12ddtψ(t)2+34ψy(t)234|v(t)|4/3ψ(t)2/3.(3.13)

We integrate (3.13) from 0 to t, and then use the Hölder inequality to the resulting inequality. Since v=uxL2((0,+),Lx2()), we get

ψ(t)2+340tψy(s)2dsc(0tψ(s)2ds)1/3.

Since ψ(0)=0, there exists a constant c1>0 such that

0tψ(s)2dsc1t3/2,

which implies that

0tψy(s)2dsψ(t)2+340tψy(s)2dsc1t1/2.

We deduce that

1t0t(12|v(s)|2+ψy(s)2)𝑑sct+c1t1/2.(3.14)

Passing to the limit as t+ in (3.14) and using (3.12), we deduce that l2=0. Hence, we conclude, from (3.11), that limt+|v(t)|2=0.

In order to establish (1.12), we equivalently establish that for all sequences (tn) such that tn+, we have

u(tn)0weakly in L2().

First, from Theorem 1.2, we have

uL((0,+),Lx2()).(3.15)

Then, using the Agmon inequality, we deduce that for all t>0, there exists a constant c>0 such that

|u(t)|L()c|ux(t)|L2()1/2.

Hence, using (1.11), we conclude that

|u(t)|L()0as t+.(3.16)

Let (tn) be a sequence which tends to +. Then, from (3.16), we deduce that

u(tn)0in 𝒟().

Moreover, from (3.15) we have that (u(tn)) is bounded in L2(). Then there exists a subsequence (tn) of (tn) such that

u(tn)u*weakly in L2().

This implies that u*=0.

3.2 Miscelleanous remarks

One may wonder why we do not have, by a more direct approach, better results for the convergence towards 0 when t tends towards +. On the one hand, due to the presence of the half-derivative in time, we cannot apply the so-called Schonbek splitting lemma (see [10] and the references therein). We do not have, for instance, an upper bound in L1() (nevertheless, see Appendix A). On the other hand, the nonlinear term 12(u2)x does not amplify the decay rate towards 0. For larger pure power polynomial terms, we expect that the nonlinear equation has the same decay rate as the linear one (the nonlinearity is asymptotically weak, following the classification in [10]). The results for the generalized equation (p>1)

ut+ux+νπt0tu(s)t-s𝑑s+upux=νuxx,

will appear in a forthcoming work.

A Appendix

We state and prove a result for smooth nonnegative solutions of equation (1.3).

Proposition A.1.

Consider a smooth and nonnegative initial data u0. Then u(t,x) remains nonnegative for all times.

Proof.

We first assume that u0>0, the general case follows by a limiting argument. Let us argue by contradiction. We introduce

T=inft>0{x:u(t,x)<0}.

There exists x0 where u achieves its minimum and such that u(T,x0)=ux(T,x0)=0uxx(T,x0).

We introduce

v(t,x)=u(t,x)+1π0tu(s)t-s𝑑s.

Since

vt=uxx-uux-ux,

we have that vt(T,x0)0.

On the other hand, we compute (omitting x0 and writing u(T,x0)=u(T) for simplicity)

Qε=u(T)-u(T-ε)ε+1επ0T-εu(s)(1T-s-1T-ε-s)𝑑s+1επT-εTu(s)T-s𝑑s.

We know that Qε converges towards vt(T)0 when ε goes to zero. We also know that u(T)-u(T-ε)ε0, since u(T)=0. We also have

|1επT-εTu(s)T-s𝑑s|=|1επT-εTu(s)-u(T)T-s𝑑s|Cε1/2.

This implies that

1επ0T-εu(s)(1T-s-1T-ε-s)𝑑s-Qε+O(ε1/2).

Gathering these information and invoking Fatou’s lemma, we have

1π0Tu(s)2(T-s)3/2dslim infε0(1επ0T-εu(s)(1T-ε-s-1T-s)ds)0.

This is a contradiction and the proof is complete. ∎

Corollary A.2.

Assume u0 is nonnegative and in L1(R)L2(R). Then

limt+u(t)L2()=0.

Proof.

Integrating equation (1.3) with respect to x, we have that if

m(t)=u(t,x)𝑑x=uL1(),

then

mt+D1/2m=0.

This leads to u(t)L1()u0L1(). On the other hand, we know that ux converges strongly in L2() to zero. Using an interpolation result completes the proof. ∎

Acknowledgements

The authors would like to thank S. Dumont and E. Zahrouni for challenging discussions about this work.

References

  • [1]

    J. Audounet, V. Giovangigli and J. Roquejoffre, A threshold phenomenon in the propagation of a point-source initiated flame, Phys. D 121 (1998), 295–316.  CrossrefGoogle Scholar

  • [2]

    J. Audounet, D. Matignon and G. Montseny, Opérateurs différentiels fractionnaires, pseudo-différentiels et représentations diffusives, Report Number N-99501, Laboratoire d’analyse et d’architecture des systèmes, Toulouse, 1999.  Google Scholar

  • [3]

    J. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media I: Derivation and the linear theory, J. Nonlinear Sci. 12 (2002), 283–318.  CrossrefGoogle Scholar

  • [4]

    C. Casenave and G. Montseny, Introduction to diffusive représentation, IFAC Proc. Vol. 43 (2010), no. 21, 370–377.  CrossrefGoogle Scholar

  • [5]

    M. Chen, S. Dumont, L. Dupaigne and O. Goubet, Decay of solutions to a water wave model with a nonlocal viscous dispersive term, Discrete Contin. Dyn. Syst. 27 (2010), no. 4, 1473–1492.  Web of ScienceCrossrefGoogle Scholar

  • [6]

    D. Dutykh, Viscous-potential free-surface flows and long wave modelling, Eur. J. Mech. B Fluids 28 (2009), 430–443.  CrossrefGoogle Scholar

  • [7]

    D. Dutykh and F. Dias, Viscous potential free surface flows in a fluid layer of finite depth, C. R. Math. Acad. Sci. Paris 347 (2007), 113–118.  Google Scholar

  • [8]

    H. Haddar and D. Matignon, Well-posedness of non-linear conservative systems when coupled with diffusive systems. Vol. 1, Nonlinear Control Systems (Stuttgart 2004), Elsevier, Amsterdam (2005), 237–242.  Google Scholar

  • [9]

    H. Haddar and D. Matignon, Theoretical and numerical analysis of the Webster Lokshin model, Report Number RR-6558, Institut National de la Recherche en Informatique et Automatique, 2008.  Google Scholar

  • [10]

    N. Hayashi, E. Kaikina, P. Naumkin and I. Shishmarev, Asymptotics For Dissipative Nonlinear Equations, Lecture Notes in Math. 1884, Springer, Berlin, 2006.  Google Scholar

  • [11]

    T. Helie and D. Matignon, Diffusive reprentations for the analysis and simulation of flared acoustic pipes with visco-thermal losse, Math. Models Methods Appl. Sci. 16 (2006), 503–536.  CrossrefGoogle Scholar

  • [12]

    T. Kakutani and M. Matsuuchi, Effect of viscosity of long gravity waves, J. Phys. Soc. Japan 39 (1975), 237–246.  CrossrefGoogle Scholar

  • [13]

    P. Liu and A. Orfila, Viscous effects on transient long wave propagation, J. Fluid Mech. 520 (2004), 83–92.  CrossrefGoogle Scholar

  • [14]

    I. Manoubi, Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann–Liouville half derivative, Discrete Contin. Dyn. Syst. 19 (2014), 2837–2863.  Web of ScienceCrossrefGoogle Scholar

  • [15]

    D. Matignon and B. d’Andréa-Novel, Spectral and time-domain consequences of an integro-differential perturbation of the wave PDE, Mathematical and Numerical Aspects of Wave Propagation (Mandelieul-la-Napoule 1995), SIAM, Philadelphia (1995), 769–771.  Google Scholar

  • [16]

    D. Matignon and C. Prieur, Asymptotic stability of linear conservative systems when coupled with diffusive systems, ESAIM Control Optim. Calc. Var. 11 (2005), 487–507.  CrossrefGoogle Scholar

  • [17]

    Y. Meyer and R. Coifman, Wavelets, Calderón–Zygmund and Multilinear Operators, Cambridge Stud. Adv. Math. 48, Cambridge University, Cambridge, 1997.  Google Scholar

  • [18]

    G. Montseny, Diffusion monodimensionnelle et intégration d’ordre 1/2, Report Number 91232, Laboratoire d’analyse et d’architecture des systèmes, Toulouse, 1991.  Google Scholar

  • [19]

    G. Montseny, Représentation Diffusive, Hermes Science, Paris, 2005.  Google Scholar

  • [20]

    G. Montseny, J. Audounet and D. Matignon, Diffusive representation for pseudo-differentially damped nonlinear systems, Nonlinear Control in the Year 2000. Vol. 2, Lecture Notes in Control and Inform. Sci. 259, Springer, London, (2000), 163–182.  Google Scholar

  • [21]

    G. Montseny, J. Audounet and B. Mbodge, A simple viscoelastic damper model – application to a vibrating string, Analysis and Optimization of Systems: State and Frequency Domain Approaches for Infinite-Dimensional Systems, Lecture Notes in Control and Inform. Sci. 185, Springer, Berlin (1993), 436–446.  Google Scholar

  • [22]

    G. Montseny, J. Audounet and B. Mbodge, Optimal models of fractional integrators and application to systems with fading memory, IEEE International Conference on Systems, Man and Cybernetics (Le Touquet 1993), IEEE Press, Piscataway (1993), 65–70.  Google Scholar

  • [23]

    O. Staffans, Well-posedness and stabilizability of a viscoelastic equation in energy space, Trans. Amer. Math. Soc. 345 (1994), 527–575.  CrossrefGoogle Scholar

  • [24]

    R. Temam, Navier–Stokes Equations Theory and Numerical Analysis, Revised edition, Stud. Math. Appl. 2, North Holland, Amsterdam, 1979.  Google Scholar

About the article

Received: 2016-12-23

Accepted: 2017-01-12

Published Online: 2017-02-21


This work was supported by “PHC Utique ASEO”, KIG1503, and exchange program CNRS/DGRS “modèles non locaux en dynamique des fluides”. A part of this article was performed during the stay of the second author in the LAMFA with the support of the exchange program SSHN2015.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 253–266, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0274.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Serge Dumont and Imen Manoubi
Numerical Methods for Partial Differential Equations, 2018

Comments (0)

Please log in or register to comment.
Log in