Well-posedness for the initial value problem associated to the Zakharov–Kuznetsov (ZK) equation in asymmetric spaces

We study well-posedness for Zakharov–Kuznetsov and modified Zakharov–Kuznetsov equations in asymmetric spaces. In order to do so, we extend a theory initiated by Kato for the Korteweg-de Vries equation to higher dimensions n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document}. As an application, we prove a result concerning dispersive blow-up for the modified Zakharov–Kuznetsov in dimension 2.


Introduction and main results
The Zakharov-Kuznetsov (ZK) equation was first formally derived by Zakharov and Kuznetsov in [23], as an asymptotic limit of the Euler-Poisson system, in the setting of the "cold plasma" approximation. This equation describes motion of plasma in a uniform magnetic field, in a long wave small-amplitude limit, and can be stated as In [15], this asymptotic limit was rigorously justified. In [7], this equation was shown to be an asymptotic limit for the Vlasov-Poisson system. In the case n = 1, this equation becomes the well-known Korteweg-de Vries (KdV) equation, which describes waves on shallow water surfaces. Thus equation (1) can be seen as a generalization of the KdV equation in higher dimensions. Note that (1) is not integrable (see [5]). However, it possesses conserved quanti-This article is part of the section "Theory of PDEs" edited by Eduardo Teixeira.
The second author was partially supported by CNPq and FAPERJ/Brazil. ties (cf. [5] for instance). These equations belongs to the larger class of nonlinear dispersive equations (see [18] for an introduction to the subject). We will focus on the properties of the initial value problem (IVP) associated to (1), that is ∂ t u + ∂ 1 u + u∂ 1 u = 0 (t, x) ∈ R * + × R n , u| t=0 = u 0 , (2) and to the IVP associated to the generalized Zakharov-Kuznetsov equation which can be written as where k ≥ 2. These IVPs were studied by many authors for an initial data u 0 ∈ H s (R n ). In [5], Faminskii showed local well posedness for (2) in dimension 2, in the setting H s , s ∈ Z + . Ever since, a lot of advancements have been made. Still in the two dimensional case, Linares and Pastor proved local well-posedness of (3) with k = 2 for s > 3/4 by using smoothing effects in [16]. The Fourier restriction method was also used by Molinet and Pilod in [20] and by Grürock and Herr in [6] to extend local well-posedness of (2) to s > 1/2. In dimension 3, Molinet and Pilod [20] and Ribaud and Vento [21] proved local and global well-posedness for (2) when s > 1. We also mention the recent works of Kinoshita [13] and Herr and Kinoshita [8] in which well-posedness for (2) was obtained with the Picard iteration method in the best possible setting: s > −1/4 in dimension 2 and s > (n − 4)/2 when n ≥ 3.
To describe our results, we define the solution of the linear problem associated to the IVPs (2) and (3) by using a group of unitary operators {V (t)} t∈R . This group is given explicitely by the formula V (t)u 0 = exp(−t∂ 1 )u 0 , or with the Fourier transform by V (t)u 0 (t, ξ) = exp(itξ 1 |ξ | 2 ) u 0 (ξ ).
In [11], Kato studied well-posedness of the IVP associated to the KdV equation (dimension n = 1) for an initial datum in The key property that Kato used for this particular space is that, after pointwise multiplication by e bx where b > 0, the unitary group of evolution V (t) becomes parabolic. More precisely, there exists a parabolic semigroup {U b (t)} such that e bx V (t) = U b (t)e bx (cf Sect. 9 in [11]). Among other results, he proved that for initial data φ ∈ H s ∩ L 2 b , s ≥ 2, there exists a unique solution u ∈ C([0, +∞), H s ∩ L 2 b ) of the IVP associated to the KdV equation, with the map φ → u being continuous in the associated topologies. Furthermore, he proved that the KdV equation possesses a smoothing property for solution with initial data in this space: for any u 0 ∈ H 2 ∩ L 2 b ), there exists a unique corresponding solution u of KdV such that (cf. Theorem 11.1 in [11]). In particular, the solution u belongs to C ∞ (R * + × R). For further properties of the solutions of the KdV equation, see [10] and [14].
Here, we generalize these results for the Zakharov-Kuznetsov and modified Zakharov-Kuznetsov equations in dimension n ≥ 2, by using a similar method. Our first result covers well-posedness in H s ∩ L 2 b for (2) and (3): for some s 0 > n/2 and b 1 > ( n k=2 b 2 k /3) 1/2 . Then there is a unique solution to (2) with the map u 0 → u continuous in the associated topologies. Moreover, e b·x u ∈ C((0, ∞), H s ) for any s < s 0 + 2.
We also extend the smoothing property discovered by Kato in this particular setting. For the Zakharov-Kuznetsov and the generalized Zakharov-Kuznetsov equation, we obtain the following result: Theorem 1, then e b·x u ∈ C((0, ∞), H ∞ ) with the following estimates: for any T > 0, s ≥ 0 and β > nk/2, and for every α ≥ 3β, α > β(1 + kn/2), Nonlinear dispersive equations are also known to exhibit what is called a dispersive blowup : a smooth and bounded initial datum with finite energy can result in a solution which develops pointwise singularities in finite time. This focusing phenomenon is caused by the linear operator which possesses an unbounded dispersion speed. In an unbounded domain, it is then possible that infinitely many waves, initially spatially dispersed, come all together at the same point after a finite time, resulting in a blow-up. Bona and Saut initiated the mathematical study of dispersive blow-up for generalized KdV in [3]. We mention [19] for an improvement of their result, and [17] for a more recent study.
Dispersive blow-up was also studied for other nonlinear dispersive equations. In [2], Bona, Ponce, Saut, and Sparber studied dispersive blow-up for the nonlinear Schrödinger equation. In [9], the pointwise notion of dispersive blow-up is extended in higher dimension n ≥ 2 to larger sets such as lines or spheres for the nonlinear Schrödinger equation.
As an application of our previous results, we exhibit an example of dispersive blow-up in the setting n = 2, k = 2. This example extends Theorem 1.3 from [17]. Theorem 3 Let p > 2 and s ∈ N, s ≥ 2. For any t * ∈ R * , there exists u 0 ∈ H s (R 2 ) ∩ W s, p (R 2 ) ∩ C ∞ (R 2 ) such that the corresponding solution of (3) with k = 2 is global in time and satisfies Note that dispersive blow-up was initially defined for the W s,∞ (R 2 ) norm caused by the highest derivative, cf for instance [3]. However, our proof here only works in the setting W s, p (R 2 ) with p < ∞. In fact, we can almost prove the same theorem in the W s,∞ (R 2 ) setting. The missing part is that in the latter case, the smoothing effect on the nonlinear term defined hereafter in the proof is not sufficient to prove that the blow-up is caused by the linear part of the solution (third property of the solution in Theorem 3).
The paper is organised as follows: in Sect. 2, we give the notations and state a set of useful results that we will need. In Sect. 3, we prove some preliminary results concerning the space L 2 b . In particular, we show that the linear group of evolution operators {V (t)} becomes parabolic after multiplication by an exponential function. Section 4 is devoted to the proof of Theorem 1. In Sect. 5, we prove Theorem 2. Sections 3, 4 and 5 are greatly inspired of Sects. 9, 10 and 11 in [11]. In Sect. 6, we give two examples of linear dispersive blow-up for the group {V (t)}, and then we prove Theorem 3.

Notations and some helpful results
We denote the Laplacian operator by = ∂ 2 1 + · · · + ∂ 2 n and the gradient operator Preliminary results: we use the following propositions to estimate products in Sobolev spaces (see [1] and [22], [12] respectively): The following lemma is an oscillatory integral estimate (see Lemma 2.3 in [16]).
Lemma 1 Let n = 2. For any t = 0 and u 0 ∈ L 1 (R 2 ), the following estimate holds: Finally, this nonlinear smoothing effect comes from Proposition 1.4 of [17]: Here we follow the proof of Kato in [11] for the dimension 1 and try to adapt it for higher dimensions. Kato's proof has three main steps. The first step uses a commutation property (see below) combined with Duhamel's principle to derive smoothing effects for the solutions of the KdV equation. The second step combines these smoothing properties with an energy estimate to obtain the well-posedness for the KdV equation in H s ∩ L 2 b (see Sect. 4). Finally, the third step improves the well-posedness result, by using again the smoothing effects, to show that the solution has an higher regularity (see Sect. 5).
Then one has e b·x u ∈ C([0, T ], L 2 ) ∩ C((0, T ], H s ) for every s < 1, Proof By multiplying the equation (11) by e b·x and using the commutation property, we obtain This gives the integral form of the equation. We can then use Lemma 2 to obtain that We can then bound the nonlinear integral term by

e b·x a(t )∂ 1 u(t )dt
and we obtain the estimate for every s < 1. Now for fix s ∈ [1/2, s 0 + 3/2), we will show that if the result is true for s − 1/2, then it is also true for s. We first note that t s/2 u solves We want to apply again Lemma 3 with The term e b·x f is not bounded in H −1 because of the factor t s/2−1 in front of u (recall that s could be less than 2). However, for every > 0, we see that e b·x f ∈ L ∞ ([ , T ], H −1 ). We apply Lemma 3 on the interval [ , T ] to get that We then let go to zero to obtain the following integral equation, for 0 < t ≤ T , valid a priori in H −1 (note that s ≥ 1/2, hence the unbounded term (t ) s/2−1 is integrable): Hence by using the properties of the semigroup U b we obtain

/4 (t ) s/2−1 e b·x u H s−1/2 +(t − t ) −3/4 (t ) s 0 /2 a(t )e b·x ∂ 1 u(t ) H s−3/2 dt .
Now by hypothesis the first term can be estimated by which is integrable, with an integral bounded for t ∈ [0, T ], and the second one by which is also integrable, with a bounded integral. Here we used again Proposition 1 and the hypothesis on e b·x u H s−1/2 . This concludes the proof.

Proof of Theorem 1
Let s 0 > n/2 and b ∈ R n such that b 1 > 0, and b 1 > ( n k=2 b 2 k /3) 1/2 . In this section, we show well-posedness of (2) and (3) in H s 0 ∩ L 2 b . As a consequence of the well posedness theory in H s 0 , we already know that there exists a unique solution in C([0, T ], H s 0 ), which is global if the norm of the initial data is sufficiently small. To simplify computations, we will restrict ourselves to the setting of global solutions. However, one could adapt the following proof to show the local well-posedness in H s 0 ∩ L 2 b when the solution of (2) or (3) in H s 0 is only local in time. According to Lemma 4, it is enough to show that e b·x u(t) ∈ L 2 . In the following, we fix k ≥ 1.
Again, we follow Kato [11] and introduce the bounded weight functions depending on a parameter > 0. Both q and r are L ∞ functions with the L ∞ norm proportional to −1/2 , and both tend monotonically to e b·x as ↓ 0. We note several properties of these functions required in the sequel: We now take u the solution of the problem in H s , multiply equation (1) (or (3)) by 2 pu and integrate over R n to obtain Integrations by parts give that and Now, using (13) leads to Note that b 1 |∇u| 2 + 2(∂ 1 u)b · ∇u = θ(∇u) ≥ C|∇u| 2 , where C > 0, in virtue of the condition on b (see the proof of Lemma 2 for the definition and properties of θ ). Using again (13) and putting everything together yields Since u ∈ H s 0 with s 0 > n/2, u ∈ L ∞ . Finally, we get Since K is independent of , going to the limit ↓ 0 gives where is an operator valued kernel such that W (t, t ) B(L 2 ) ≤ C(t − t ) −1/2 (because u ∈ H s 0 and s 0 > n/2). This equation is obtained by Lemma 1 with f = −u k ∂ 1 u. It should be noted that W (t, t ) depends on u and hence on u 0 , but the dependence is known to be continuous in the H s 0 norm which is weaker than the H s 0 ∩ L 2 b norm.

Proof of Theorem 2
We present here the proof of Theorem 2.
Proof We start by proving the first inequality. By Theorem 1 and the estimate of Lemma 4, we already know that it is true for any s < s 0 + 2 (note that β > nk/2 ≥ 1, hence the estimate of Lemma 4 for s < s 0 + 2 is stronger than the one that we need to prove). Now we fix δ > 0 and show that if the estimate holds for some s − δ with s ≥ 1/2, then it also holds for s. We write again the integral equation satisfied by t βs/2 e b·x u(t): Hence by using the properties of the semigroup U b we obtain Now by hypothesis the first term can be estimated by and the integral of this term is finite and bounded in t ≤ T whenever β ≥ 1 and 0 < δ < 2 (to prove this, one can again make the change of variables r = t /t).
For the second term, we write that To estimate the norm of e b·x u k+1 = (e b·x/(k+1) u) k+1 we use Proposition 2. By induction on k, we obtain the following generalized version: for We use this last inequality with v = e b·x/(k+1) u and s = s − δ, combined with the Sobolev embedding theorem, to obtain with s 1 > n/2. Now we use the hypothesis for s − δ, and the estimate of Lemma 4 for s 1 , to obtain that The integral of this expression is finite and bounded in t ≤ T if we take 1 > δ > 0 and s 1 > n/2 such that β ≥ 1 + (ks 1 − 1)/δ (once again, this bound comes from the change of variables r = t /t). The hypothesis β > nk/2 ensures that we can find such s 1 and δ. Note that we are allowed to use the property for e b·x/(k+1) u instead of e b·x u because e b·x/(k+1) u 0 ∈ L 2 , with e b·x/(k+1) u 0 (Hölder). The homogeneous condition verified by b is also true for b/(k+1), hence we can use the hypothesis for e b·x/(k+1) u instead of e b·x u. Hence the decay (5) is valid for every s ≥ 0. Now we prove the second inequality by induction on l. For l = 0, it is known by (5). Assuming that it has been proved for all s ≥ 0 up to a given l, we prove it for l + 1. Again using (12) with f = −u k ∂ 1 u, we obtain on differentiation The H s norm of the first term on the right is dominated by (d/dt) k e b·x u H s+3 ≤ Ct −(β(s+3)+αl)/2 ≤ Ct −(βs+α(l+1))/2 by induction hypothesis. This gives the required estimate.
For the second term in (15), we have as above where we have written v = e b·x/(k+1) u and d = d/dt for simplicity. Using Proposition 2 multiple times again, we obtain By induction hypothesis and H s 1 → L ∞ , where s 1 = (α/β − 1)/k (since we know that α > β(kn/2 + 1)), this is dominated by Ct −m/2 , where This is the required estimate.
6 Application: dispersive blow-up in dimension n ≥ 2

Linear dispersive blow-up
In this section, we construct an initial datum for the linear problem associated to (1) such that the linear evolution exhibits a singularity at a given time, on a linear subspace of dimension d < n. More precisely, we state the following Proposition 4 Let n ≥ 2 and 1 ≤ d ≤ n. For t ∈ R, let V (t) = e −t∂ 1 . Recall that (V (t)) t∈R is a group of evolution operators that preserves H s norms. For any t * ∈ R * , there exists u 0 ∈ H 1 (R n ) ∩ C ∞ (R n ) such that: Proof For x ∈ R n , let us write x = (y, z) where y = (x 1 , . . . , x d ) and z = (x d+1 , . . . , x n ). Let φ(x) = e −2π |y| e −π |z| 2 . Note that φ has an exponential decay, which will enable to use the smoothing properties of Lemma 2. Take any b ∈ R n such that 1 ≥ b 1 > 0 and n k=2 b 2 k < 3b 2 1 . Then e b·x φ belongs to L 2 (R n ). Note that e b·x V (t)φ = U b (t)e b·x φ, where U b (t) is defined as in Lemma 2. By the smoothing properties of U b (t) stated in Lemma 2, for any t > 0, the function U b (t)e b·x φ belong to H ∞ (R n ), hence is smooth. Thus V (t)φ is also a smooth function.
For negative times, use the fact that e −b·x φ also belongs to L 2 (R n ), and Reversing the proof of Lemma 2 shows that U −b is parabolic backwards in time. Hence here again e −b·x V (−t)φ and then V (−t)φ are smooth functions, for any t > 0.
The candidate for Proposition 4 is thus u 0 = V (−t * )φ. By the previous arguments, V (t)u 0 is smooth for any t = t * . We then show that u 0 ∈ H 1 (R n ). By the properties of V (t), it is enough to check that φ ∈ H 1 (R n ). The Fourier transform of φ is given bŷ where C > 0 is a constant. Since g ∈ S(R n−d ) and f ∈ H 1+d/2− (R d ), φ ∈ H 1+d/2− (R n ). Note that, for any s ≥ 1, We also state the following example in the case n = 2: Proposition 5 Let s ∈ N * . For any t * ∈ R * , there exists u 0 ∈ H s (R 2 ) ∩ C ∞ (R 2 ) such that: Proof Let p > 2 and φ p (x, y) = |x| s−1/ p e −x 2 −y 2 . Then φ p ∈ H s ∩ L 2 b for any b ∈ R 2 . The proof of the previous proposition shows that V (t)φ p is smooth for any t = 0. Note that ∂ s x φ p (x, y) = Csgn(x) s |x| −1/ p e −x 2 −y 2 + g(x, y), where g is a continuous function with exponential decay. In particular, ∂ s x φ p ∈ L 2 and |∂ s x φ p (x, y)| → ∞ for any y when x goes to zero. Taking u 0 = V (−t * )φ p again enables to end the proof.

Non linear dispersive blow-up on a line
We give here the proof of Theorem 4.
Proof We use here a proof very similar to the one of Theorem 1.3 in [17]. Consider φ p as in the proof of Proposition 5 and define u 0 = V (−t * )φ p . We write the solution of (3) with n = k = 2. Since {V (t)} t∈R is an unitary group in H s and φ p ∈ H s (R 2 ), u 0 ∈ H s (R 2 ). Up to multiplying φ p by a small constant, we can suppose that u(t) is globally defined and u ∈ C(R, H s ). By the Proposition 1.4 of [17] (cf. Proposition 3), z(t) ∈ H s+1 (R 2 ) ⊂ W s, p (R 2 ) for all times. By the proof of Proposition 5, V (t * )u 0 = φ p ∈ W s,1 (R 2 ). By Lemma 1, for any t = t * , Hence for any t = t * , V (t)u 0 ∈ W s,∞ (R 2 )∩ H s (R 2 ) ⊂ W s, p (R 2 ). Thus the solution u(t) = V (t)u 0 + z(t) belongs to W s, p whenever t = t * . Finally, by Proposition 5, u 0 ∈ C ∞ (R 2 ).
But for any y ∈ R, |∂ s x φ p (x, y)| ∼ C|x| −1/ p e −x 2 −y 2 as x goes to zero. Hence the L p norm of ∂ s x u(t * ) = ∂ s x φ p + ∂ s x z(t * ) blows up on any open subset U ⊂ R 2 such that U ∩ ({0} × R) = ∅.