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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.

The compressible Euler equations, which govern the gas flow surrounding a solid ball with mass M and frictional damping in n dimensions, t+(u)= 0,(u) t+ (u u)+ P ()=-Mx/| x| n-2u, where , u, P and M are the density, velocity, pressure and mass of the gas, respectively, n 3 is the dimension of x, and > 0 is the frictional constant, are examined. The pressure is assumed to satisfy the law and 12 , K is a positive constant. The existence and non-existence of global smooth solutions are studied for the initial boundary problem of the Euler equations.

The hydrodynamic limit for the Boltzmann equation is studied in the case when the limit system, that is, the system of Euler equations contains contact discontinuities. When suitable initial data is chosen to avoid the initial layer, we prove that there exist a family of solutions to the Boltzmann equation globally in time for small Knudsen number. And this family of solutions converge to the local Maxwellian defined by the contact discontinuity of the Euler equations uniformly away from the discontinuity as the Knudsen number <i></i> tends to zero. The proof is obtained by an appropriately chosen scaling and the energy method through the micro-macro decomposition.

As a tool for solving the Neumann problem for divergence-form equations, Kenig and Pipher introduced the space ${\mathcal{X}}$ of functions on the half-space, such that the non-tangential maximal function of their$L$_{2}Whitney averages belongs to$L$_{2}on the boundary. In this paper, answering questions which arose from recent studies of boundary value problems by Auscher and the second author, we find the pre-dual of ${\mathcal{X}}$ , and characterize the pointwise multipliers from ${\mathcal{X}}$ to$L$_{2}on the half-space as the well-known Carleson-type space of functions introduced by Dahlberg. We also extend these results to$L$_{$p$}generalizations of the space ${\mathcal{X}}$ . Our results elaborate on the well-known duality between Carleson measures and non-tangential maximal functions.