The well-posedness of the three space dimensional Prandtl equations is studied under some constraint on its flow structure. Together with the instability result given in [28], it gives an almost necessary and sufficient structural condition for the stability of the three-dimensional Prandtl equations. It reveals that the classical Burgers equation plays an important role in determining this type of flow with special structure, that avoids the appearance of the secondary flow, an unstabilizing factor in the three-dimensional Prandtl boundary layers. And the sufficiency of the monotonicity condition on the tangential velocity field for the existence of solutions to the Prandtl boundary layer equations is illustrated in the three-dimensional setting. Moreover, it is shown that this structured flow is linearly stable for any smooth three-dimensional perturbation.

In this paper, we study the multiplicity of positive solutions for the nonhomogeneous elliptic problem: u+ u= f (x) u p 1+ h (x) in R N. We will show how the shape of the graph of f (x) affects the number of positive solutions.

The global existence and uniqueness of classical solutions to the Boltzmann equation without angular cut-off is proved under the condition that the initial data is in some Sobolev space. In addition, the solutions thus obtained are shown to be positive and C in all variables. One of the key observations is the introduction of a new important norm related to the singularity in the cross section of the collision operator. This norm captures the essential property of the singularity and yields precisely the dissipation description of the linearized collision operator through the celebrated H-theorem.

In this paper we study the zero-viscosity limit of 2-D Boussinesq equations with vertical viscosity and zero diffusivity, which is a nonlinear system with partial dissipation arising in atmospheric sciences and oceanic circulation. The domain is taken as R2+ with Navier-type boundary. We prove the nonlinear stability of the approximate solution constructed by boundary layer expansion in conormal Sobolev space. The optimal expansion order and convergence rates for the inviscid limit are also identified in this paper. Our paper extends the partial zero-dissipation limit results of Boussinesq system with full dissipation by Chae D. [Adv.Math.203,no.2,2006] in the whole space to the case with partial dissipation and Navier boundary in the half plane.

In this paper, we study the global well-posedness of the 2D compressible NavierStokes equations with large initial data and vacuum. It is proved that if the shear viscosity<i></i> is a positive constant and the bulk viscosity <i></i> is the power function of the density, that is, <i></i>(<i></i>) = <i></i> ^<i></i> with <i></i> &gt; 3, then the 2D compressible NavierStokes equations with the periodic boundary conditions on the torus T 2 admit a unique global classical solution (<i>, u</i>) which may contain vacuums in an open set of T 2 . Note that the initial data can be arbitrarily large to contain vacuum states.