The aim of this paper is to show how a weakly dispersive perturbation of the inviscid Burgers equation improve (enlarge) the space of resolution of the local Cauchy problem. More generally we will review several problems arising from weak dispersive perturbations of nonlinear hyperbolic equations or systems.1 for any β ∈ R. It is established in [11] (see also [29] for the case β = 1 2 ) that for 0 ≤ β < 1, there exist initial data u 0 ∈ L 2 (R) ∩ C 1+δ (R), 0 < δ < 1, and T (u 0 ) such that the corresponding solution u of (1.5) satisfiesThis rules out the case −1 < α < 0 in our notation. As observed in [49], the proof in [11] extends easily to non pure power dispersions, such as (1.3) and thus to the Whitham equation (1.1). Note however that it is not clear whether or not the blow-up displayed in the aforementioned papers is shocklike. The solution is proven (by contradiction) to blow-up in a C 1+δ norm and the sup norm of the solution and of its derivative might blow-up at the same time.The case 0 < α < 1 is much more delicate (see the discussion in the final Section).Remark 1.1. Similar issues have been addressed in [43] for the Burgers equation with fractionary dissipation.

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Abstract. The aim of this paper is to study, via theoretical analysis and numerical simulations, the dynamics of Whitham and related equations. In particular we establish rigorous bounds between solutions of the Whitham and KdV equations and provide some insights into the dynamics of the Whitham equation in different regimes, some of them being outside the range of validity of the Whitham equation as a water waves model.

This article is concerned with the Zakharov-Kuznetsov equation (0.1) ∂tu + ∂x∆u + u∂xu = 0.We prove that the associated initial value problem is locally well-posed in H s (R 2 ) for s > 1 2 and globally well-posed in H 1 (R × T) and in H s (R 3 ) for s > 1. Our main new ingredient is a bilinear Strichartz estimate in the context of Bourgain's spaces which allows to control the high-low frequency interactions appearing in the nonlinearity of (0.1). In the R 2 case, we also need to use a recent result by Carbery, Kenig and Ziesler on sharp Strichartz estimates for homogeneous dispersive operators. Finally, to prove the global well-posedness result in R 3 , we need to use the atomic spaces introduced by Koch and Tataru. Theorem 1.3. The initial value problem associated to the Zakharov-Kuznetsov equation is globally well-posed in H 1 (R × T).Remark 1.2. Theorem 1.3 provides a good setting to apply the techniques of Rousset and Tzvetkov [16], [17] and prove the transverse instability of the KdV soliton for the ZK equation. Finally, we combine the conserved quantities M and H with a well-posedness result in the Besov space B 1,1 2 and interpolation arguments to prove : Theorem 1.4. The initial value problem associated to the Zakharov-Kuznetsov equation is globally well-posed in H s (R 3 ) for any s > 1. 2 +δ .Proof. By duality, it suffices to prove thatu, v and w are nonnegative functions, and we used the following notations(5.3) By using dyadic decompositions on the spatial frequencies of u, v and w, we rewrite J as (5.4) J = N1,N2,N J N,N1,N2 , where J N,N1,N2 = q,q1∈Z 2 R 4 Γ ξ1,q1,τ1 ξ,q,τ P N w(ξ, q, τ ) P N1 u(ξ 1 , q 1 , τ 1 ) P N2 v(ξ 2 , q 2 , τ 2 )dν. Now, we use the decomposition (5.5) J = J LL→L + J LH→H + J HL→H + J HH→L + J HH→H , where J LL→L , J LH→H , J HL→H , J HH→L , respectively J HH→H , denote the Low × Low → Low, Low × High → High, High × Low → High, High × High → Low, respectively High × High → High contributions for J as defined in the proof of Proposition 4.1.1. Estimate for J LH→H + J HL→H + J HH→L . Since Proposition 3.6 also holds in the R × T case, we deduce arguing as in (4.17), (4.18) and (4.22) that (5.6)2. Estimate for J HH→H . We recall that N ∼ N 1 ∼ N 2 in this case. We divide the integration domain in several regions.2.1 Estimate for J HH→H in the region |ξ| ≤ 100. We denote by J 1 HH→H the restriction of J HH→H to the region |ξ| ≤ 100 and use dyadic decompositions on the variables σ, σ 1 , σ 2 and ξ, so that (5.7) J N,N1,N2 = k≥0 L,L1,L2 J L,L1,L2 N,N1,N2,k ,