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Given any suitably small, localized, and smooth initial data, in this paper, we prove global regularity for the 3D finite depth gravity water wave system. As a byproduct, we rule out the small, localized traveling waves in 3D, which do exist for the same system in 2D.

We prove global existence and a modified scattering property for the solutions of the 2D gravity water waves system in the infinite depth setting for a class of initial data, which is only required to be small above the level \dot{H}^{1/5}\times \dot{H}^{1/2+1/5}. No assumption is made below this level. Therefore, the nonlinear solution can have infinite energy. As a direct consequence, the momentum condition assumed on the physical velocity in all previous small energy results by Ionescu‐Pusateri, Alazard‐Delort, and Ifrim‐Tataru is removed.

In this paper, we consider the Cauchy problem to the heat conductive compressible Navier-Stokes equations in the presence of vacuum and with vacuum far field. Global well-posedness of strong solutions is established under the assumption, among some other regularity and compatibility conditions, that the scaling invariant quantity $\|\rho_0\|_\infty(\|\rho_0\|_3+\|\rho_0\|_\infty^2\|\sqrt{\rho_0}u_0\|_2^2)(\|\nabla u_0\|_2^2+ \|\rho_0\|_\infty\|\sqrt{\rho_0}E_0\|_2^2)$ is sufficiently small, with the smallness depending only on the parameters $R, \gamma, \mu, \lambda,$ and $\kappa$ in the system. Notably, the smallness assumption is imposed on the above scaling invariant quantity exclusively, and it is independent of any norms of the initial data, which is different from the existing papers, see, e.g., \cite{HLX12,HUANGLI11,WENZHU17}. The total mass can be either finite or infinite. A new equation for the density, more precisely for the cubic of the density, derived from combing the continuity and momentum equations, is employed to get the $L^\infty_t(L^3)$ type estimate of the density.