This paper studies stability of the Ekman boundary layer. We utilize a new approach, developed by the authors in a precedent paper, based on Fourier transformed finite vector Radon measures, which yields exponential stability of the Ekman spiral. By this method we can also derive very explicit bounds for solutions of the linearized and the nonlinear Ekman systems. For example, we can prove these bounds to be uniform with respect to the angular velocity of rotation, which has proved to be relevant for several aspects. Another advantage of this approach is that we obtain well-posedness in classes containing nondecaying vector fields such as almost periodic functions. These outcomes give respect to the nature of boundary layer problems and cannot be obtained by approaches in standard function spaces such as Lebesgue, Bessel-potential, Hölder or Besov spaces.

Two nonlinear diffusion equations for thin film epitaxy, with or without slope selection, are studied in this work. The nonlinearity models the Ehrlich–Schwoebel effect – the kinetic asymmetry in attachment and detachment of adatoms to and from terrace boundaries. Both perturbation analysis and numerical simulation are presented to show that such an atomistic effect is the origin of a nonlinear morphological instability, in a rough-smooth-rough pattern, that has been experimentally observed as transient in an early stage of epitaxial growth on rough surfaces. Initial-boundary-value problems for both equations are proven to be well-posed, and the solution regularity is also obtained. Galerkin spectral approximations are studied to provide both a priori bounds for proving the well-posedness and numerical schemes for simulation. Numerical results are presented to confirm part of the analysis and to explore the difference between the two models on coarsening dynamics.

In this paper, we give an instability criterion for the Prandtl equations in three-dimensional space, which shows that the monotonicity condition on tangential velocity fields is not sufficient for the well-posedness of the three-dimensional Prandtl equations, in contrast to the classical well-posedness theory of the two-dimensional Prandtl equations under the Oleinik monotonicity assumption. Both linear stability and nonlinear stability are considered. This criterion shows that the monotonic shear flow is linearly stable for the three-dimensional Prandtl equations if and only if the tangential velocity field direction is invariant with respect to the normal variable, and this result is an exact complement to our recent work (A well-posedness theory for the Prandtl equations in three space variables. arXiv:1405.5308 , 2014) on the well-posedness theory for the three-dimensional Prandtl

In this paper, we study a class of analytical solutions to the 3-D compressible Navier-Stokes equations with density-dependent viscosity coefficients, where the shear viscosity h ( ) = 0 , and the bulk viscosity g ( ) = ( > 0 ). By constructing a class of radial symmetric and self-similar analytical solutions in R N ( N 2 ) with both the continuous density condition and the stress free condition across the free boundaries separating the fluid from vacuum, we derive that the free boundary expands outward in the radial direction at an algebraic rate in time and we also have shown that such solutions exhibit interesting new information such as the formation of a vacuum at the center of the symmetry as time tends to infinity and explicit regularities, and we have large time decay estimates of the velocity

Motivated by the paper Grard-Varet and Dormy (2010) [6] [JAMS, 2010] about the linear ill-posedness for the Prandtl equations around a shear flow with exponential decay in normal variable, and the recent study of well-posedness on the Prandtl equations in Sobolev spaces, this paper aims to extend the result in [6] to the case when the shear flow has general decay. The key observation is to construct an approximate solution that captures the initial layer to the linearized problem motivated by the precise formulation of solutions to the inviscid Prandtl equations.