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].

Abstract. We consider the generalization of the Navier-Stokes equations from R n to the Riemannian manifolds. There are inequivalent formulations of the Navier-Stokes equations on manifolds due to the different possibilities for the Laplacian operator acting on vector fields on a Riemannian manifold. We present several distinct arguments that indicate that the form of the equations proposed by Ebin and Marsden in 1970 should be adopted as the correct generalization of the Navier-Stokes to the Riemannian manifolds.

We prove that a weak solution of a slightly supercritical fractional Burgers equation becomes Hölder continuous for large time.

We consider the Navier-Stokes equation on H 2 (−a 2 ), the two dimensional hyperbolic space with constant sectional curvature −a 2 . We prove an ill-posedness result in the sense that the uniqueness of the Leray-Hopf weak solutions to the Navier-Stokes equation breaks down on H 2 (−a 2 ). We also obtain a corresponding result on a more general negatively curved manifold for a modified geometric version of the Navier-Stokes equation. Finally, as a corollary we also show a lack of the Liouville theorem in the hyperbolic setting both in two and three dimensions.

We consider the fractional Burgers' equation on R N with the critical dissipation term. We follow the parabolic De-Giorgi's method of Caffarelli and Vasseur and show existence of smooth solutions given any initial datum in L 2 (R N ).