We consider the ill-posedness issue for the nonlinear Schrödinger equation with a quadratic nonlinearity. We refine the Bejenaru-Tao result by constructing an example in the following sense. There exist a sequence of time T N → 0 T_N\to 0 and solution u N ( t ) u_N(t) such that u N ( T N ) → ∞ u_N(T_N)\to \infty in the Besov space B 2 , σ − 1 ( R ) B_{2,\sigma }^{-1}(\mathbb {R}) ( σ > 2 \sigma >2 ) for one space dimension. We also construct a similar ill-posed sequence of solutions in two space dimensions in the scaling critical Sobolev space H − 1 ( R 2 ) H^{-1}(\mathbb {R}^2) . We systematically utilize the modulation space M 2 , 1 0 M_{2,1}^0 for one dimension and the scaled modulation space ( M 2 , 1 0 ) N (M_{2,1}^0)_N for two dimensions.

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We consider the initial value problem of the 3D incompressible Boussinesq equations for rotating stratified fluids. We establish the dispersive and the Strichartz estimates for the linear propagator associated with both the rotation and the stable stratification. As an application, we give a simple proof of a unique existence of global in time solutions to our system using just a contraction mapping principle.

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The Cauchy problems for Navier-Stokes equations and nonlinear heat equations are studied in modulation spaces M s q,σ (R n ). Though the case of the derivative index s = 0 has been treated in our previous work, the case s = 0 is also treated in this paper. Our aim is to reveal the conditions of s, q and σ of M s q,σ (R n ) for the existence of local and global solutions for initial data u 0 ∈ M s q,σ (R n ).