Theoretical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach
-
and
Abstract
In this paper, we study the following water wave model with a nonlocal viscous term:
where
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:
where
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,
She proved the local and global existence of solutions to problem (1.2) when
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
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
where
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
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
where
Taking into account the previous notations and results, we define the diffusive realization of the Riemann–Liouville half-order derivative
where Γ and
1.3 Main results
First, we introduce the following notations:
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.
Theorem 1.1.
For all initial data
The second main result states as follows.
Theorem 1.2.
For all initial data
Concerning the issue of convergence of solutions towards 0 when t tends to
Proposition 1.3.
The solution u of equation (1.3) satisfies
and
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
and which verifies
Before going to the proof of Theorem 1.1, we state some preliminary inequalities.
Lemma 2.2.
For all
Moreover, for all
Proof.
We prove (2.1) for
It follows that
Integrating this equation with respect to x and using the Cauchy–Schwarz inequality, we get
Using the density of
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
where
Proposition 2.3.
The approximate problem (2.2) has a unique maximal solution defined on the interval
Proof.
First, we denote by
where
is a square matrix of order
where
We now proceed to the limit as n tends to infinity. For this purpose, we need following a priori estimates.
Lemma 2.4.
Let
Proof.
Set
Integrating in time, this leads to (2.4), (2.6) and to an upper bound
for the gradient of
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
The limit process requires some compactness argument. For this purpose, we state the following lemma.
Lemma 2.5.
Let
Proof of Lemma 2.5.
Let
Using the Cauchy–Schwarz inequality yields
Now, we integrate (2.10) between 0 and
Using the change of variables
Hence, using (2.6), we deduce (2.8).
In order to establish (2.9), we recall that
We now integrate (2.11) with respect to τ from t to
Using first the Cauchy–Schwarz inequality and then the Young inequality, we get
From Lemma 2.2, we conclude that
Moreover, using the Cauchy–Schwarz inequality, and estimations (2.5) and (2.8), we get
Then, gathering (2.14) and (2.13), we obtain
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
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
Finally, using the Young inequality, we get
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
To complete the proof of the theorem, it remains to establish the continuity of the solution
In the following, we establish new estimations on the approximate solution
Lemma 2.6.
Let
Due to Lemma 2.6, we have that
It follows that
This implies that u is continuous
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
Lemma 2.7.
For all
In addition,
Finally, in order to pass to the limit in the nonlinear term, we state the following result.
Lemma 2.8.
Let
To establish Theorem 1.2, we pass to the limit when
Let
To justify the weak continuity of u, we have the following result.
Lemma 2.9.
Let
Due to Lemma 2.9, we conclude that
This implies that u is continuous on
3 Proof of Proposition 1.3
3.1 Proof of the proposition
Let
Hence, system (3.1) implies that
We conclude that
We now handle estimates for
In system (3.4), taking the scalar product of the two equations in
We observe that
In addition,
Gathering (3.5), (3.7) and (3.8) yields
Using the Young and Agmon inequalities, we get
Using the fact that
Hence, we deduce that
We conclude that the function
is decreasing.
Moreover, it is nonnegative, so there exists
Combining (3.3) and (3.10), we deduce that there exists a nonnegative real number
Moreover, using the Cesàro lemma, we deduce that
In the following, we establish new estimates for
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
Since
which implies that
We deduce that
Passing to the limit as
In order to establish (1.12), we equivalently establish that for all sequences
First, from Theorem 1.2, we have
Then, using the Agmon inequality, we deduce that for all
Hence, using (1.11), we conclude that
Let
Moreover, from (3.15) we have 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
will appear in a forthcoming work.
Funding statement: 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.
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
Proof.
We first assume that
There exists
We introduce
Since
we have that
On the other hand, we compute (omitting
We know that
This implies that
Gathering these information and invoking Fatou’s lemma, we have
This is a contradiction and the proof is complete. ∎
Corollary A.2.
Assume
Proof.
Integrating equation (1.3) with respect to x, we have that if
then
This leads to
Acknowledgements
The authors would like to thank S. Dumont and E. Zahrouni for challenging discussions about this work.
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