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As a continuation of the previous work [18], we consider the global well-posedness for the derivative nonlinear Schrödinger equation. We prove that it is globally well-posed in energy space, provided that the initial dataLocal well-posedness for the Cauchy problem (1.1) is well-understood. It was proved in the energy space H 1 (R) by Hayashi and Ozawa in [7,8,9], see also Guo and Tan [6] for earlier result in smooth spaces. The problem for rough data below the energy space, see [3,15,16] for local well-posedness and ill-posedness.The global well-posedness for (1.1) has been also widely studied. By using mass and energy conservation laws, and by developing the gauge transformations, Hayashi 2010 Mathematics Subject Classification. Primary 35Q55; Secondary 35A01.

In this paper, we prove that there exists some small ε * > 0, such that the derivative nonlinear Schrödinger equation (DNLS) is global well-posedness in the energy space, provided that the initial data u 0 ∈ H 1 (R) satisfies u 0 L 2 < √ 2π +ε * . This result shows us that there are no blow up solutions whose masses slightly exceed 2π, even if their energies are negative. This phenomenon is much different from the behavior of nonlinear Schrödinger equation with critical nonlinearity. The technique used is a variational argument together with the momentum conservation law. Further, for the DNLS on half-line R + , we show the blow-up for the solution with negative energy. 1 2 u(α 2 t, αx), α > 0. It has the same scaling invariance as the quintic nonlinear Schrödinger equation, i∂ t u + ∂ 2 x u + µ|u| 4 u = 0, t ∈ R, x ∈ R, and the quintic generalized Korteweg-de Vries equation, ∂ t u + ∂ 3 x u + µ∂ x (u 5 ) = 0, t ∈ R, x ∈ R.

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Abstract. This paper studies the global well-posedness of the incompressible magnetohydrodynamic (MHD) system with a velocity damping term. We establish the global existence and uniqueness of smooth solutions when the initial data is close to an equilibrium state. In addition, explicit large-time decay rates for various Sobolev norms of the solutions are also given.

We prove that the derivative nonlinear Schrödinger equation is globally well-posed in H 1 2 (R) when the mass of initial data is strictly less than 4π.2010 Mathematics Subject Classification. Primary 35Q55; Secondary 35A01.

The blowup is studied for the nonlinear Schrödinger equation iu t + ∆u + |u| p−1 u = 0 with p is odd and p ≥ 1 + 4 N −2 (the energy-critical or energy-supercritical case). It is shown that the solution with negative energy E(u 0 ) < 0 blows up in finite or infinite time. A new proof is also presented for the previous result in [9], in which a similar result but more general in a case of energy-subcritical was shown.

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