Damping to prevent the blow-up of the Korteweg-de Vries equation

We study the behavior of the solution of a generalized damped KdV equation $u_t + u_x + u_{xxx} + u^p u_x + \mathscr{L}_{\gamma}(u)= 0$. We first state results on the local well-posedness. Then when $p \geq 4$, conditions on $\mathscr{L}_{\gamma}$ are given to prevent the blow-up of the solution. Finally, we numerically build such sequences of damping.


Introduction
The Korteweg-de Vries (KdV) equation is a model of one-way propagation of small amplitude, long wave [KdV95]. It is written as In [BDKM96], Bona et al. consider the initial-and periodic-boundary-value problem for the generalized Korteweg-de-Vries equation u t + u xxx + u p u x = 0 and study the effect of a dissipative term on the global well-posedness of the solution. Actually, they consider two different dissipative terms, a Burgers-type one −δu xx and a zeroth-order term σu. For both these terms, they show that for p ≥ 4, there exist critical values δ c and σ c such that if δ > δ c or σ > σ c the solution is globally well-defined. However, the solution blows-up when the damping is too weak as for the KdV equation [MM02].The literature is full of work concerning the dampen KdV equation with p = 1 [ABS89, CR04, CS13b, CS13a, Ghi88, Ghi94, Gou00, GR02], but few are concerning more general nonlinearities.
In our paper, we consider a more general damping term denoted by L γ (u). Our purpose is to find similar results as above, both theorically and numerically. So the KdV equation becomes a damped KdV (dKdV) equation and is written u t + u x + u xxx + u p u x + L γ (u) = 0.
The KdV equation has an infinite number of invariants such that the L 2 -norm. But, for the dKdV equation, the L 2 -norm decreases. Indeed, for all t ∈ R, where the natural space of study is and the associated norm is An other property of the KdV equation is that the solution can blow-up as soon as p ≥ 4 . The blow-up is caracterized by lim t→T u H 1 = +∞.
In this paper, we first establish the local well-posedness of the dKdV equation. Then we study the global well-posedness. More precisely, we focus on the behavior of the H 1 -norm with respect to p and we obtain conditions on γ so there is no blow-up. Finally, we illustrate the results using some numerical simulations. We first find a constant damping (γ(ξ) =constant) such that there is no blow-up and then the damping is weaken in such a way lim |ξ|→+∞ γ(ξ) = 0.

Preliminary results
Some results of injection concerning the space H γ (R) are given. Then We assumed that γ(ξ) > 0. Hence, the Cauchy-Schwarz inequality involves for all x ∈ R : Proposition 1.2. Let γ and β be such that for all ξ ∈ R, γ(ξ) > β(ξ). We define .
Proof. The condition is necessary. Indeed, if there exists α > 0 such that ρ(N ) > α, ∀N , then the norms |u| β and |u| γ are equivalent, the injection cannot be compact. Let us prove now that the condition is sufficient. First, we have for u ∈ H γ (R) : This shows that the injection is continuous. Now we prove that the injection is compact. We use finite rank operators and we take the limit. Let I N be the orthogonal operator on the polynomials of frequencies ξ such that −N ≤ ξ ≤ N . We have Thus Therefore Id is a compact operator and consequently the injection is compact.

Local well-posedness
We study the following Cauchy problem : ∀x ∈ R, ∀t > 0, The semi-group generated by the linear part is written as In the rest of the section, f (u) denotes the non-linear part of the equation, i.e., f (u) = u p u x . We first state a result of regularization.
Lemma 2.1. Assume that s, r ∈ R + . Then there exists a constant C r > 0, depending only on r, such that ∀u ∈ H γ s (R) and ∀t > 0 we have Proof. Let r ∈ R + , u ∈ H γ s (R) and t > 0. Then we have Theorem 2.2. Assume that there exists r ∈]0, 2[ and for all ξ ∈ R, γ(ξ) ≥ ξ 2 r . We also assume that R 1 γ(ξ) s < +∞ and there exists a constant C > 0 such that ∀ξ, η ∈ R and s ∈ R + we have

Then there exists a unique solution in
Moreover, for all M > 0 with |u 0 | γ s ≤ M and |v 0 | γ s ≤ M , there exists a constant C 1 > 0 such that the solution u and v, associated with the initial data u 0 and v 0 respectively, satisfy for Proof. Thanks to Duhamel's formula, Φ(u) is solution of the Cauchy problem, where Let show that u is the unique fixed-point of Φ. We introduce the closed ballB(T ) defined for We apply the Picard fixed-point theorem. We first show that Φ B (T ) ⊂B(T ). Let us take u ∈B(T ) and show that Φ(u(t)) ∈B(T ). We have On the one hand, we have On the other hand, we apply Lemma 2.1 But u ∈B(T ), then we have That involves Using the equality and the injection results, we obtain The map Φ is strictly contracting if It remains to prove the continuity with respect to the initial data. Duhamel's formula gives Remark 2.3. Actually we can proove the local well-posedness for every γ using a parabolic regularisation Using lemma 2.1 with γ(ξ) = ξ 2 , the same computations as theorem 2.2 and taking the limit → 0 give the result [Iór90,BS75].

Global well-posedness
We work here under the hypothesis of the local theorem and study the global well-posedness of the damped KdV equation. We use here an energy method [BS75,BS74] We first multiply (1) by u and we integrate with respect to x. Then we have Integrating with respect to time, we obtain Which can also be written as We deduce from that expression that N (u) is a decreasing function and t 0 |u| 2 γ dτ is bounded independently of t by N (u 0 ). Now, we multiply (1) by u xx + u p+1 p+1 and we integrate with respect to x. Then we have d dt Integrating with respect to time, we obtain

From this expression, we have
Using the inequality u 2 L ∞ ≤ 2 u L 2 u x L 2 and because R |L γ (u)u| = R L γ (u)u, Then sup If there exists T > 0 such that lim L 2 → +∞ since p < 4 and this is impossible because of (3). Consequently, u x L 2 is bounded for all t and so is the H 1 -norm.

Case p ≥ 4:
We estimate the L 2 -norm of u xx . We multiply (1) with u xxxx and we integrate with respect to x. Then we have Using two integrations by part, we have Let us work on the last term. Using integrations by part, we have It follows that But, from the Cauchy-Schwarz inequality Then we have Using the inequalty u 2 ∞ ≤ 2 u L 2 u x L 2 , we obtain From (4), it leads to the inequality The function Ω is increasing for its two arguments. We previously notice that u(·, t) L 2 is an decreasing function with respect to the time. Then, if γ(ξ) − Ω| t=0 ≥ 0, u xx (·, t) L 2 does not increase for t ≥ 0. Particularly, if γ(ξ) ≥ Ω ( u 0 L 2 , u 0xx L 2 ) =: θ, the semi-norm u xx (·, t) L 2 is bounded by its values at t = 0. Hereû k is the k−th Fourier coefficient of u and (γ k ) k∈Z are positive real numbers chosen such that

Numerical results
In this part, we illustrate the theorem 3.1 numerically. Our purpose is first to find similar results as in [BDKM96] i.e. find a γ k constant such that the solution does not blow up. Then build a sequence of γ k , still preventing the blow-up, such that lim |k|→+∞ γ k = 0. Since dKdV is a low frequencies problem, we do not need to damp all the frequencies.

Computation of the damping
In order to find the suitable damping, one may use the dichotomy. We remind that our goal is to prevent the blow-up, i.e., avoid that lim t→+∞ u H 1 = +∞. Let us begin finding a constant damping L γ (u) = γu as weak as possible. We mean by weak that γ has to be as lower as possible to prevents the blow up. Let γ a respectively γ e be the damping which prevents the explosion and which does not respectively. To initialize the dichotomy, we give a value to γ and we determine the initial values of γ a and γ e . Then from these two initial values, we bring them closer by using dichotomy. The method is detailed in algorithm 1 and illustrated in Figures 1  and 2.  2 -Dichotomy W extend the method to frequencies bands in order to build sequencies γ k decreasing with respect to |k| and tending to 0 when |k| tends to the infinity. So we begin by defining the frequencies bands (N 1 < N 2 < . . .) and we proceed as previously but only on the frequencies |k| ≥ N i . The method is described in algorithm 2 and illustrated in Figure 3

Numerical scheme
Numerous schemes were introduced in [CheSad]. Here we chose a Sanz-Serna scheme for the discretisation in time. In space, we use the FFT. Actually, the scheme is written, for all k, as We find u (n+1) k with a fixed-point method. In order to have a good look of the blow-up, we also use an adaptative time step.

Simulations
We consider the domain [−L, L] where L = 50. We take as initial datum a disturbed soliton, written as u 0 (x) = 1.01 × (p + 1)(p + 2)(c − 1) 2 where p = 5, c = 1.5 and d = 0.2L. We discretise the space in 2 11 points. The Figure 5 shows the solution whithout damping, i.e., γ k = 0, ∀k. We observe that the L 2 -norm of u x increases strongly and the solution tends to a wavefront (as in [BDKM96]). Using the methods introduced previously, we first find two optimal constant dampings γ e = 0.0025 and γ a = 0.0027. As we can see in Figure 6, γ e does not prevent the blow up. In the opposite in Figure 7 γ a does. And we also notice that the two dampings are quite close. Considering more general sequences, particularly such that lim |k|→+∞ γ k = 0. Using algorithm 2, Figure 8 shows that the sequence (γ a ) as a frontier between the dampings which prevent the blow up and the other which do not. To illustrate this, we take two dampings written as gaussians. The first (denoted by γ 1 ) is build to be always above the sequence γ a and the second (denoted by γ 2 ) to be always below. In Figures 9 and 10 we observe the damping γ = γ 1 prevents the blow up. But if we take γ = γ 2 , the solution blows-up.

Conclusion
We studied the behavior of the damped generalized KdV equation. If p < 4, the solution does not blow-up whereas if p ≥ 4, it can. To prevent the blow-up, the term γ defining the damping has to be large enough. In particular, we build a sequence of γ which vanishes for high frequencies.
This frequential approach for the damping seems useful for low frequencies problem.