We prove that the Cauchy problem for the three-dimensional Navier-Stokes equations is ill-posed iṅ B −1,∞ ∞ in the sense that a "norm inflation" happens in finite time. More precisely, we show that initial data in the Schwartz class S that are arbitrarily small inḂ −1,∞ ∞ can produce solutions arbitrarily large inḂ −1,∞ ∞ after an arbitrarily short time. Such a result implies that the solution map itself is discontinuous inḂ −1,∞ ∞ at the origin.

We investigate the dynamics of a boson gas with three-body interactions in dimensions d = 1, 2. We prove that in the limit of infinite particle number, the BBGKY hierarchy of k-particle marginals converges to a limiting (Gross-Pitaevskii (GP)) hierarchy for which we prove existence and uniqueness of solutions. Factorized solutions of the GP hierarchy are shown to be determined by solutions of a quintic nonlinear Schrödinger equation. Our proof is based on, and extends, methods of Erdös-Schlein-Yau, KlainermanMachedon, and Kirkpatrick-Schlein-Staffilani.

We present a new, simpler proof of the unconditional uniqueness of solutions to the cubic Gross-Pitaevskii hierarchy in R 3 . One of the main tools in our analysis is the quantum de Finetti theorem. Our uniqueness result is equivalent to the one established in the celebrated works of Erdős, Schlein, and Yau.

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We derive the defocusing cubic Gross-Pitaevskii (GP) hierarchy in dimension d = 3, from an N -body Schrödinger equation describing a gas of interacting bosons in the GP scaling, in the limit N → ∞. The main result of this paper is the proof of convergence of the corresponding BBGKY hierarchy to a GP hierarchy in the spaces introduced in our previous work on the well-posedness of the Cauchy problem for GP hierarchies, [7,8,9], which are inspired by the solutions spaces based on space-time norms introduced by Klainerman and Machedon in [25]. We note that in d = 3, this has been a wellknown open problem in the field. While our results do not assume factorization of the solutions, consideration of factorized solutions yields a new derivation of the cubic, defocusing nonlinear Schrödinger equation (NLS) in d = 3.

We prove that for the Navier-Stokes equation with dissipation (−∆) α , where 1 < α < 5/4, and smooth initial data, the Hausdorff dimension of the singular set at time of first blow up is at most 5 − 4α. This unifies two directions from which one might approach the problem of global solvability, though it provides no direct progress on either.

Abstract. We introduce a dyadic model for the Euler equations and the Navier-Stokes equations with hyper-dissipation in three dimensions. For the dyadic Euler equations we prove finite time blow-up. In the context of the dyadic Navier-Stokes equations with hyper-dissipation we prove finite time blow-up in the case when the dissipation degree is sufficiently small.

In this paper we establish a complete local theory for the energycritical nonlinear wave equation (NLW) in high dimensions R × R d with d ≥ 6. We prove the stability of solutions under the weak condition that the perturbation of the linear flow is small in certain space-time norms. As a by-product of our stability analysis, we also prove local well-posedness of solutions for which we only assume the smallness of the linear evolution. These results provide essential technical tools that can be applied towards obtaining the extension to high dimensions of the analysis of Kenig and Merle [17] of the dynamics of the focusing (NLW) below the energy threshold. By employing refined paraproduct estimates we also prove unconditional uniqueness of solutions for d ≥ 5 in the natural energy class. This extends an earlier result by Planchon [26].