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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.

Abstract. Ill-posedness is established for the initial value problem (IVP) associated to the derivative nonlinear Schrödinger equation for data in H s (R), s < 1/2. This result implies that best result concerning local well-posedness

We consider in this paper the rigorous justification of the Zakharov-Kuznetsov equation from the Euler-Poisson system for uniformly magnetized plasmas. We first provide a proof of the local well-posedness of the Cauchy problem for the aforementioned system in dimensions two and three. Then we prove that the long-wave small-amplitude limit is described by the Zakharov-Kuznetsov equation. This is done first in the case of cold plasma; we then show how to extend this result in presence of the isothermal pressure term with uniform estimates when this latter goes to zero.