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 . 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 . Recently, Liu and Orfila , and Dutykh and Dias  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 .
In their recent work , Chen et al. investigated theoretically and numerically the decay rate of solutions to the following water wave model with a nonlocal viscous dispersive term:
where represents the Caputo half-derivative in time. Here u is the horizontal velocity of the fluid, is the usual diffusion, is the geometric dispersion and 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.  considered (1.1) with supplemented with the initial condition . They proved that if is small enough, then there exists a unique global solution . In addition, u satisfies
It is worth pointing out that this result is valid only for small initial data. Moreover, in a recent work , 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,
She proved the local and global existence of solutions to problem (1.2) when using a fixed point theorem. Then she studied theoretically the decay rate of the solutions in this case. She established that if is small enough, then there exists a unique global solution . Besides, she proved that the solution u satisfies
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 . 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 . 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 in (1.2)), and we assume that the other constants are normalized. Thus, our model is reduced as follows:
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
the Riemann–Liouville half-order integral and by
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 and recall that a realization of the half-order integral is given, for all , by
where is the Dirac delta function at and is the tensorial product in the distributions sense of the applications and .
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
Now, we extend the diffusive realization (1.5) for . The previous definition is valid for almost every x. To define the mathematical framework, we define the vector space
We note that X is a Hilbert space for the scalar product
The corresponding norm is
Hence, the application
is a linear and continuous operator. Its dual map is defined as
where 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 , for all and , as follows:
1.3 Main results
First, we introduce the following notations:
, with being the scalar product and being the norm.
,with being the scalar product and being the norm.
and , with being the duality product between and and being the duality product between and .
is the space of weakly continuous functions on that take values in .
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:
A first auxiliary result is the following.
For all initial data , the regularized problem (1.10) has a unique global weak solution such that for all ,
The second main result states as follows.
For all initial data , the problem (1.9) has a unique global weak solution . Moreover, for all ,
Concerning the issue of convergence of solutions towards 0 when t tends to , we have the following result.
The solution u of equation (1.3) satisfies
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).
By a weak solution of problem (1.10) we mean a pair such that
and which verifies
Before going to the proof of Theorem 1.1, we state some preliminary inequalities.
For all , we have
Moreover, for all , we have the straightforward consequence
We prove (2.1) for smooth, and then proceed to a limit argument. For such a Φ, we have
It follows that
Integrating this equation with respect to x and using the Cauchy–Schwarz inequality, we get
Using the density of 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 be an orthogonal basis of . We suppose that every element is compactly supported. We may take, for example, a fractal wavelet basis of Daubechies (see ). Let be the subspace of spanned by and let be the corresponding subspace of . We then seek an approximate solution that verifies the following approximate problem:
where is the orthogonal projector onto .
The approximate problem (2.2) has a unique maximal solution defined on the interval , where .
First, we denote by and , defined, respectively, by and . Then the problem (2.2) can be written in the matrix form
is a square matrix of order , is the identity matrix and is a square matrix of order . Moreover, is a square matrix of order . Finally, is a polynomial vectorial function with respect to . Since is an invertible matrix, problem (2.3) can be written as
where is a locally lipschitzian map. The Cauchy–Lipschitz theorem applies and we deduce that there exists a unique maximal solution of class from to , where . ∎
We now proceed to the limit as n tends to infinity. For this purpose, we need following a priori estimates.
Let . Then we have the following a priori estimates, uniformly with respect to n:
Set in (2.2). We then have
Integrating in time, this leads to (2.4), (2.6) and to an upper bound for the gradient of . To complete the proof of (2.5), a bound on the norm of is required. We set in the second equation in (2.2) and, thanks to the previous estimates and Lemma 2.2, we obtain
Thanks to Lemma 2.4, we deduce that .
The limit process requires some compactness argument. For this purpose, we state the following lemma.
Let and . There exists a constant that may depends on T but that is independent of δ and n such that
Proof of Lemma 2.5.
Let and . Using the mean value theorem, for all , we get
Using the Cauchy–Schwarz inequality yields
Now, we integrate (2.10) between 0 and . Then, using the Fubini–Tonelli theorem, we get
Using the change of variables , we deduce
In order to establish (2.9), we recall that satisfies, for any , the following equation:
We now integrate (2.11) with respect to τ from t to . Then, setting , we get
Using first the Cauchy–Schwarz inequality and then the Young inequality, we get
From Lemma 2.2, we conclude that
In addition, using the Cauchy–Schwarz inequality and estimation (2.4), we get
Finally, we integrate (2.16) with respect to t from δ to and get
In the same way, we establish that
From (2.5), we deduce
Then, using the Hölder and Agmon inequalities, we have
Moreover, using Lemma 2.4, we deduce that there exists a constant such that
Finally, using the Young inequality, we get
We rephrase Lemma 2.5 as follows: the sequences and are, respectively, bounded in and . Combining this with the a priori bounds in Lemma 2.4 allow us to deduce that and converge strongly in and , 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.
In the following, we establish new estimations on the approximate solution in order to establish the weak continuity of the solution over with values in .
Let . There exists a constant that may depend on T but that is independent of δ and n such that
Due to Lemma 2.6, we have that
It follows that
This implies that u is continuous with values in . Also, we know that . Hence, thanks to [24, Lemma 1.4, p. 263] and using the continuous injection , we deduce that u is weak continuous with values in .
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 in the perturbed problem (1.10). We state the following results, whose proofs are mere consequence of the previous subsection.
For all , we have the following a priori estimates that are uniform with respect to ε:
is bounded in and in ,
is bounded in ,
is bounded in ,
is bounded in ,
is bounded in .
is bounded in .
Finally, in order to pass to the limit in the nonlinear term, we state the following result.
Let and . There exists a constant C that may depends on T but that is independent of δ, ε such that
To justify the weak continuity of u, we have the following result.
Let . There exists a constant such that
Due to Lemma 2.9, we conclude that
This implies that u is continuous on when it takes values in . Moreover, . Thanks to [24, Lemma 1.4, p 263], we conclude that u is weakly continuous on when it takes values in .
3 Proof of Proposition 1.3
3.1 Proof of the proposition
Let . In system (1.9), taking the scalar product in of the first equation with u, and the scalar product in of the second equation with , we get
Hence, system (3.1) implies that
We conclude that is decreasing and nonnegative, so there exists such that
We now handle estimates for . In the equations of system (1.9), we take the derivative with respect to x. Then, letting and , it follows that satisfies
In system (3.4), taking the scalar product of the two equations in with v and , respectively, we get
We observe that
Using the Young and Agmon inequalities, we get
Using the fact that , we obtain
Hence, we deduce that
We conclude that the function
is decreasing. Moreover, it is nonnegative, so there exists such that
Moreover, using the Cesàro lemma, we deduce that
In the following, we establish new estimates for . Hence, taking the scalar product in of the second equation of system (3.4) with , we get
Using Lemma 2.2 and the Young inequality, we get
We integrate (3.13) from 0 to t, and then use the Hölder inequality to the resulting inequality. Since , we get
Since , there exists a constant such that
which implies that
We deduce that
In order to establish (1.12), we equivalently establish that for all sequences such that , we have
First, from Theorem 1.2, we have
Then, using the Agmon inequality, we deduce that for all , there exists a constant such that
Hence, using (1.11), we conclude that
Let be a sequence which tends to . Then, from (3.16), we deduce that
Moreover, from (3.15) we have that is bounded in . Then there exists a subsequence of such that
This implies that .
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  and the references therein). We do not have, for instance, an upper bound in (nevertheless, see Appendix A). On the other hand, the nonlinear term 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 ). The results for the generalized equation ()
will appear in a forthcoming work.
We state and prove a result for smooth nonnegative solutions of equation (1.3).
Consider a smooth and nonnegative initial data . Then remains nonnegative for all times.
We first assume that , the general case follows by a limiting argument. Let us argue by contradiction. We introduce
There exists where u achieves its minimum and such that .
we have that .
On the other hand, we compute (omitting and writing for simplicity)
We know that converges towards when ε goes to zero. We also know that , since . We also have
This implies that
Gathering these information and invoking Fatou’s lemma, we have
This is a contradiction and the proof is complete. ∎
Assume is nonnegative and in . Then
Integrating equation (1.3) with respect to x, we have that if
This leads to . On the other hand, we know that converges strongly in to zero. Using an interpolation result completes the proof. ∎
The authors would like to thank S. Dumont and E. Zahrouni for challenging discussions about this work.
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
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
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
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
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
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
N. Hayashi, E. Kaikina, P. Naumkin and I. Shishmarev, Asymptotics For Dissipative Nonlinear Equations, Lecture Notes in Math. 1884, Springer, Berlin, 2006. Google Scholar
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
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
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
Y. Meyer and R. Coifman, Wavelets, Calderón–Zygmund and Multilinear Operators, Cambridge Stud. Adv. Math. 48, Cambridge University, Cambridge, 1997. Google Scholar
G. Montseny, Diffusion monodimensionnelle et intégration d’ordre , Report Number 91232, Laboratoire d’analyse et d’architecture des systèmes, Toulouse, 1991. Google Scholar
G. Montseny, Représentation Diffusive, Hermes Science, Paris, 2005. Google Scholar
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
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
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
R. Temam, Navier–Stokes Equations Theory and Numerical Analysis, Revised edition, Stud. Math. Appl. 2, North Holland, Amsterdam, 1979. Google Scholar
About the article
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.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0