Well-posedness and computation of solutions of a regularized Benjamin-Ono system

El buen planteamiento y el cálculo de soluciones de un sistema regularizado de Benjamin-Ono

In this paper we will consider the nonlinear integro-differential system written in dimensionless variables

under the initial conditions

and periodic boundary conditions

where the expression p.v.ʃ stands for the integration in the principal value sense. We refer the reader to the works by Duoandikoetxea [8] and Butzer and Nesser [3] for more information about the Hilbert Transform. In particular, the following property of this linear operator is important in the present paper:

Where

The system above was deduced by Muñoz [12] and it describes the propagation of a weakly nonlinear (α << 1) internal wave propagating at the interface of two immiscible fluids with constant densities, which are contained at rest in a long channel (ǫ << 1) with a horizontal rigid top and bottom, and the thickness of the lower layer is assumed to be effectively infinite, i.e., h₂ >> h₁ (deep water limit). In Figure 1, we sketch the physical setting of the problem. In system (1), the constant ρ is given by ρ2/ρ1 −1, where ρ1 and ρ2 represent the densities of the fluids and ρ2/ρ1 > 1 (for stable stratification). The constants α and ǫ are small positive real numbers that measure the intensity of nonlinear and dispersive effects, respectively, α = a/h1 and ǫ = h1/L, where h1 denotes the thickness of the upper fluid layer and the parameters L and a correspond to the characteristic wavelength and characteristic wave amplitude, respectively. The dimensionless variable x represents the spatial position and t the propagation time. The function u = u(x, t) is the velocity monitored at the normalized depth z = 1−√2/3, and ζ = ζ(x, t) is the wave amplitude at the point x and time t, measured with respect to the rest level of the two- fluid interface. The phenomenon of propagation of waves on the interface between two layers of immiscible fluids of different densities is attracting the interest of many physicians and mathematicians, for both well-posedness theory and asymptotic theory, due to the challenging modelling and mathematical and numerical difficulties involved in the analysis of this physical system. Previous mathematical models for describing physically different scaling regimes of this phenomenon are those by Choi and Camassa [5], [6], [7]. On the other hand, Bona et al. [2] proposed a general method to derive systematically different weakly-dispersive, weakly nonlinear asymptotic models for the propagation of internal waves at a two-fluid interface. The system of variables used in [2] further enable the establishment of analytical results on consistency and convergence of the approximate models to the full Euler equations for an ideal, incompressible, irrotational fluid.

Figure 1
A typical periodic internal wave propagating at the interface of the two-fluid system which are contained at rest in a long channel with a horizontal rigid top and bottom.

In [12], Muñoz also established the existence and uniqueness of solutions in the non-periodic case, and formulated a spectral scheme for approximating solutions of the corresponding Cauchy problem. Pipicano and Muñoz in [14] studied the existence of periodic travelling wave solutions of system (1) by using the topological-degree theory of positive operators on Banach spaces earlier developed in the works by Krasnosel'skii and Granas wave solutions of the Korteweg-de Vries (KdV) equation and Chen [4] in the framework of periodic travelling wave solutions. Quintero and Muñoz [15] also studied existence of non periodic travelling waves (solitary waves) of system (1) by adapting the theory on positive operators on cones applied in [1] to the KdV equation.

In order to continue the study on the properties of system (1), the first objective of the present paper is to show that system (1) is locally well posed in the product periodic Sobolev space H^S_per X H^S_per, provided that s ≥ O, by using semigroup theory and the Banach fixed point principle. This theory can be used to solve a wide class of problems commonly known as evolution equations that arise in many areas of application such as physics, chemistry, biology, engineering and economics, among others. Existence and uniqueness of system (1) is necessary in order to study, in a future research, the stability of travelling wave solutions of system (1).

The second purpose of this paper is to conduct some numerical experiments to analyze the accuracy of the fully discrete solver, introduced by Muñoz in [12], for approximating solutions of system (1) in both the linear and nonlinear cases on a spatial periodic domain. In [12] was only considered the error of the corresponding semi-discrete formulation. It is important to notice that the presence of the nonlocal dispersive operator H in system (1) makes the numerical investigation harder than the study of equations with only local terrns. In this numerical method, the system is discretized in space by the Fourier spectral method and in time by a second-order accurate scherne. A potential application of this numerical tool is to explore the admissible range of velocity to guarantee the existence of solitary waves of the system and to establish whether they are orbitally stable/unstable under small disturbances.

The paper is organized as follows. In Section 2, we introduce notation, definitions and results necessary for the theory developed in the paper. In Section 3, we establish the local well-posedness of system (1) in the periodic case by using semigroup theory and Banach's fixed point theorem. Finally in Section 4, we present the numerical results obtained with a scheme which uses spectral discretization in space and a second-order finite difference approximation for time stepping of the initial value problem associated to system (1).

Throughout this paper we will work with L-periodic functions, where L is a positive real number. The symbol K will denote a positive constant that is updated according to the context.

Let N, Z, R and C be the sets of naturals, integers, reals and complex numbers, respectively. We will denote by L²(0, L) = L² the Banach space of the all Lebesgue-measurable functions on C which are 2-integrable on [0, L]. The usual norm defined on L² is

Additionally,

is an inner product in L² and ||f||_L2 =(f,f)^1/2.

Let C^kper(0, L) = C^k_per, k = 0, 1, 2, ... be the space of all k times continuously-differentiable functions (or class C^k) of period L. Further, C_per = C_per(0, L) = C⁰_per (0, L) is the space of all continuous functions that are L-periodic and C^∞_per = C^∞_per(0, L) = ­ ⋂_k C^k_per. We will denote by P the space of all functions φ : R → C of class C∞, that are L-periodic. We say that T : P → C defines a periodic distribution, i.e., T ∈ P′ , if T is linear and there exist a sequence (Ψ_n)_n∈N ⊂ P such that

Let s ∈ R. The Sobolev space, denoted by H^s_per = H^s_per(0, L), is defined as

where f : Z → C represents the coefficient of the Fourier Transform of f define by

If f ∈ Cper, f(n) can be written as

We recall that for s > 1/2, the space H^s_per is an algebra, i.e.,||fg||_s ≤K|| f||_s ||g||_s.

We will denote by X_s the product space H^s_per × H^s_per. It is easy to show that the expression

defines a norm in X_s with Y = (ζ,u)T . Sometimes we will also use the equivalent norm

Finally, we recall some definitions about semigroup theory. Let X be a Banach space. A one parameter family (T (t))_t≥0 of bounded linear operators from X into X is a semigroup of bounded linear operators on X if

• i. T (0) = I, (I is the identity operator on X);

  ii. T (t + s) = T (t)T (s), ∀t, s ≥ 0 (the semigroup property).

The linear operator A defined by

with domain

is called the infinitesimal generator of the semigroup T (t). A semigroup (T (t))_t≥0 of bounded linear operators on X is called a C0 semigroup if

If C₀ is a semigroup, D(A) is dense on X. We refer the reader to [13] for a detailed information about semigroups of bounded linear operators.

In this section we will investigate the local well-posedness for the regularized BenjaminOno system

where ρ = ρ2/ρ1 − 1, under the initial conditions

and periodic boundary conditions

The system in (3) can be written as

Where

If we denote

we conclude that Y formally must satisfy the following initial value problem:

Remark 3.1. The operators A, B : H^s_per → H^s−2_per are linear and bounded. In fact, let φ ∈ H^s_per and let us denote w_n = 2πn/L . Observe that

And

Let us define the operators A⁻¹, B⁻¹ : H^s−2_per → H^s_per as

Note that the operators A⁻¹ and B⁻¹ are also linear and bounded. It follows that the operator A : X_s−1 → X_s is linear and bounded. In fact, the explicit form of A is

Where

For our purposes, in this article, we consider the operator A from X_s into X_s, s ≥ 0.

3.1. Linear problema

In this part, we consider the linear case of system (4):

To construct the linear semigroup of operators, for s ≥ 0, we consider the expansions

By replacing the expressions of (6) into (5) we obtain

Therefore, for each n ∈ Z we have the system

under the initial conditions

A direct calculation shows that the explicit solutions of this system are given by

for n ≠ 0, and

Here

T0 = Θ₀ = 0, where sign(x) denotes the sign function of x. Here we used the property of the Hilbert Transform given in (2).

Let us define for t ≥ 0 and

the family of linear operators

Where

Note that if Y = ( ζ u) is an element of Xs, s ≥ 0, then

Since

and due to the terms Tn and Θn are controlled by a constant. Thus,

Therefore (T (t))_t≥0 is well defined from X_s into X_s, s ≥ 0.

Theorem 3.2. The family (T (t))_t≥0 is a C₀-semigroup of linear and bounded operators in X_s.

Proof. A direct calculation shows that (T (t))_t≥0 is a family of linear operators and each T (t) is bounded due to (9).

Note that

Thus, T (0) = I. The semigroup property T (t + s) = T (t)T (s), is a consequence of the equality:

Hence, (T (t))_t≥0 is a semigroup of linear and bounded operators on Xs.

On the other hand, let Y = (ζ u) be an element in X_s. Then

Note that the series at the right side above converge uniformly in t ≥ 0 with upper bounds similar to (8). Thus, as a consequence of the dominated convergence theorem for sums, we have

Thus lim_t→0+ T (t)Y = Y . This shows that (T (t))_t≥0 is a C0-semigroup.

Theorem 3.3. The linear operator A : X_s → X_s, with s ≥ 0, is the infinitesimal generator of the semigroup (T (t))_t≥0.

Proof. Note that

Note that the terms

are controlled by a constant. This implies that

And

Therefore, the dominated convergence theorem for sums implies that

Therefore A(Y ) = limt→0+ (T (t)−I/t) (Y ) for any Y ∈ X_s, which proves that the operator A is the infinitesimal generator of the semigroup (T (t))_t≥0.

3.2. Nonlinear problema

In this part, we study the existence, uniqueness and behavior of the solutions for the problem (4), under changes of the initial data.

Lemma 3.4. Let L1 = A−1D and L2 = B−1D, u ∈ Hrper and v ∈ Hr′per with 0 ≤ r ≤ s, 0 ≤ r′ ≤ s, 0 ≤ 2s − r − r′ < 1/4. Then,

And

where C_r,r′,s is a constant depending on r, r′ , s.

Proof. Let us observe that

And

Therefore, the proof is analogous to Lemma 3.1 in [16].

Lemma 3.5. If Y, Y ∈ C([0, T ], X_s) with s ≥ 0, then, for Y = (ζ,u)^T and Y = (ζ, u)^T we have

where L is a linear polynomial depending on ||u||_s, ||ū||_s and ­||ζ||_s.

Proof. First of all, suppose that s > 1/2 . Since H^s_per is an algebra for s > 1/2, we have

where L(||u||_s ,||ū||_s , ­|| ­ζ||_s) = K (||u||_s +||ū||_s + ||ζ||_s).

On the other hand, we recall that

Thus, for 0 ≤ s < 1, we can apply Lemma 3.4 with r = 3s/4 < s, r′ = s, to get again the inequality (10).

Theorem 3.6. If Y ∈ C([0, T ]; X_s) with s ≥ 0 is a solution of (4), then Y satisfies the integral equation

Analogously, if Y ∈ C([0, T ]; X_s), s ≥ 0, is a solution of (11), then Y ∈ C¹([0, T ]; X_s) and satisfies (4) in the following sense:

Proof. Let Y (t) ∈ C([0, T ]; Xs) be a solution of the IVP (4). Then, for 0 ≤ ξ ≤ t we have

Note that

Thus,

Integrating on both sides of the equation (12), we obtain that Y satisfies the integral equation

On the other hand, suppose that Y (t) ∈ C([0, T ]; X_s) is a solution of the integral equation (11). Consider the expression

Replacing the expression (11) into (13) and applying the triangle inequality, we obtain

Note that

The last term can be controlled by using the mean value theorem in the following way:

for some t ≤ ξ ≤ t + h. Observe that the last term in (15) satisfies

since h → 0+ implies ξ → 0⁺. Furthermore, by using Lemma 3.5, we have that the first term in (15) can be bounded as

as h → 0⁺. Hence,

Theorem 3.7. Let Y0 ∈ X_s, s ≥ 0. Then there exist T^∗ > 0 and Y ∈ C([0, T ]; X_s) satisfying the integral equation (11).

Proof. Let us define the set

with the norm ||Y||_C([0,T],Xs) = sup_t∈[0,T] (||ζ(t)||_s + ||u(t)||_s), where Y = (ζ,u)^T . Observe that endowed with this norm, S_M is a complete set in C([0, T ]; X_s) and F is continuous in S_M as a consequence of Lemma 3.5.

For Y ∈ S_M, we define the operator

Observe that

Note that the first term at the right side above satisfies

since (T (t))t≥0 is a C0-semigroup in X_s. The second and third terms at the right side are controlled using the Lebesgue’s dominated convergence theorem. Note that

And

Thus,

Finally,

which is a consequence of the estimate

Since

it follows that

Hence, if Y ∈ SM , then Ψ(Y ) ∈ C([0, T ]; Xs). On the other hand,

Thus, choosing

we obtain that Ψ(Y (t)) ∈ SM provided Y (t) ∈ S_M.

To see that there exists T₂ such that Ψ is a contraction in SM for T < T₂, let us observe that

Thus, choosing

we obtain that Ψ is a contraction on S_M . Thus, with T^∗ < min{T₁, T₂} and by applying the Banach Fixed-Point Theorem on SM , we get the desired result.

Theorem 3.8. The solution obtained in Theorem 3.7 is unique and depends continuously on the initial condition Y0.

Proof. Let Y = (ζ/u) and Y = (ζ/ū) be elements in C([0, T ]; X_s) solutions of the integral equation (11) with initial data Y₀ = (ζ0 u0)and Y₀ = (ζ0/u0), respectively. Then,

Let L := sup_t∈[0,T] L(||u(t)||_s ,||ū(t)||_s ,||ζ(t)||_s). The Gronwall inequality implies that

for all t ∈ [0, T ]. Hence, we conclude the uniqueness and continuous dependence of the solutions on initial data.

In this section we will analyze the error of fully discrete numerical scheme for approximating the initial value problem for system (1) introduced by Muñoz in [12]. In this numerical scheme, the spatial computational domain [0, L] is discretized by N ∈ 2Z equidistant points, with spacing ∆x = L/N. Then, we approximate the unknowns u and ζ with spatial period L as truncated Fourier series in space with time-dependent coefficients:

With

The time-dependent coefficients u_j(t) for j = −N/2+1, ...0, ..., N/2 are calculated by means of the equation

and similarly for ζj (t).

Projecting equations (1) with respect to the orthonormal basis φ_j = L^−1/2e^iwjx and the inner product

we derive that

Now substituting the Fourier expansions (16) into equations (17), using the orthogonal property of the basis φ_j and, integration by parts, we obtain

where P_j[.] is the operator defined by

Then, using the properties of the Hilbert Transform, system (18) reduces to

Finally, we reach expressions for the Fourier coefficients of the unknowns u and ζ:

subject to ζ_j (0) = ζ_0j , u_j (0) = u_0j . Equations (20) can be considered as a system of ordinary differential equations for each frequency w_j, which we discretize numerically by the following second-order scheme:

Here ∆t denotes the time step and û_­j⁽ⁿ⁾, ζj⁽ⁿ⁾ denote the numerical approximations of the Fourier coefficients û_­j (t), ζ_j (t), respectively, at time t = n∆t. Also the notation P_j[g]⁽ⁿ⁾ means the value of Pj [g] when g is evaluated at time t = n∆t.

4.1. Description of the numerical experiments

In this section, we wish to analyze the accuracy of the numerical scheme (21) described above in some numerical experiments.

Linear regime: First we solve system (1) with initial conditions

and subject to spatial periodic conditions on an interval [0, L]. The parameters of the model are α = 0 (linear regime), ǫ = 1, ρr = ρ2/ρ1 = 1.1, L = 100, and the numerical parameters are N = 2⁷, ∆t = 0.01. In Figure 2, we superimpose the exact solution computed in (7) in the linear case together with the output of the numerical scheme (21) evaluated at t = 100. The difference between the profiles is about 1e − 8 showing that the scheme reproduces very well the dispersive characteristics of model (1).

In Figure 3, we repeat the previous numerical experiment but using instead a weak level of dispersion ǫ = 0.01 and ρr = 3. Other parameters and initial conditions are left unchanged. The result is presented in Figure 3. We see again a good agreement (of order 1e − 8) between the prediction of the numerical scheme and the exact solution given in (7). Neither numerical dispersion nor attenuation were observed in the simulations here presented and in many others performed for other values of the modelling parameters.

Figure 2
Solid line: Numerical solution of system (1) in the linear case (a =0), Є = 1 and P_r = 1.1 Dotted line: exact solution (7).

Figure 3
Solid line: Numerical solution of system (1) in the linear case (a =0), Є = 0.01 and Pr = 3. Dotted line: exact solution (7).

Nonlinear regime: In the next numerical simulation, we conduct a computer experiment using an approximate periodic travelling wave with speed c of system (1) in the form

with ζ₀, u₀ being L-periodic functions showed in Figure 4, computed through the numerical scheme introduced in [14] with initial conditions

and L = 4, c = 1.5, a = ε = 0.1 pr = 2. We run the numerical solver (21) using the profiles in Figure 4 as initial conditions and ∆t = 5e — 4, N 2⁸ L 4. In Figure 5, the resulting profiles are displayed at t = 20. We observe that the two profiles coincide with accuracy of roughly 3e — 4, and the travelling wave propagates with the expected speed without changing its shape.

Figure 4
Approximate travelling wave of system (1) computed using the numerical method in [14] after 10 Newton's iterations. Period L = 4, wave speed c = 1.5, pr = 2, a= ε = 0.1, N = 2⁸ FFT points in space.

4.2. Order in space of the Fourier method

We wish to analyze the order of convergence in space of the fully discrete numerical method (21). In the experiment displayed in Figure 6, we fix a small time step ∆t = 1e — 4 and increase gradually the number of FF T points in space. We use the approximate periodic travelling wave in Figure 4 with the same parameters. We start with N = 2⁵ and increase by 2 until we get N = 2⁸. In each simulation, we run the numerical solver (21) within the time interval [O, 1], for every value of N. From Figure 6, we can see that the error in space of scheme (21) decreases very rapidly approximately as N^-7.8, in accordance with the spectral accuracy of the semidiscrete approximation established by Muñoz in [12].

Figure 5
Solid line: Numerical solution with method (21) of system (1) for a =0.1 and Є = 0.1, Pr = 2. Dotted line: Approximate periodic trevelling wave computed in [14] at t = 20 (5 times its period).

Figure 6
Plot of the decimal logarithm of the maximum error against log₁₀ N. The time step is fixed at ∆t = 1e — 4. We see that the plot is approximately a line (see dashed line) with slope —7.8.

4.3. Order in time of the Fourier method

Now, we want to check numerically the order in time for the numerical scheme (21). We use the same periodic travelling wave of system (1) as in the previous experiment. We choose N 2¹⁸ (∆x L/ N = 1.5e — 4), the error in space does not dominate the total error. We start with ∆t 1/2 and decreasing the time step by 1/2 until ∆t 1/2⁶. In each computer simulation, we run the numerical solver (21) within the time interval [0, 1], for every value of ∆t. In Figure 7, we can see that the error of scheme (21) is approximately of order 2 in time.

Figure 7
Plot of the decimal logarithm of the maximum error against log₁₀ N. The number of points in space is fixed at N 2¹⁸. We see that the plot is approximately a line with slope 2.

This work was supported by Universidad del Valle, Calle 13 No. 100-00, Cali, Colombia, under the internal research project C.I. 71020, and Colciencias. The authors would like to thank the referees for their valuable comments which helped to improve the manuscript.