Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

The potential for sea ice-albedo feedback to give rise to nonlinear climate change in the Arctic Ocean – defined as a nonlinear relationship between polar and global temperature change or, equivalently, a time-varying polar amplification – is explored in IPCC AR4 climate models. Five models supplying SRES A1B ensembles for the 21 st century are examined and very linear relationships are found between polar and global temperatures (indicating linear Arctic Ocean climate change), and between polar temperature and albedo (the potential source of nonlinearity). Two of the climate models have Arctic Ocean simulations that become annually sea ice-free under the stronger CO 2 increase to quadrupling forcing. Both of these runs show increases in polar amplification at polar temperatures above-5 o C and one exhibits heat budget changes that are consistent with the small ice cap instability of simple energy balance models. Both models show linear warming up to a polar temperature of-5 o C, well above the disappearance of their September ice covers at about-9 o C. Below-5 o C, surface albedo decreases smoothly as reductions move, progressively, to earlier parts of the sunlit period. Atmospheric heat transport exerts a strong cooling effect during the transition to annually ice-free conditions. Specialized experiments with atmosphere and coupled models show that the main damping mechanism for sea ice region surface temperature is reduced upward heat flux through the adjacent ice-free oceans resulting in reduced atmospheric heat transport into the region.

Geophysical fluid dynamics.

In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.

General Properties of Interpolation Spaces

Hydrodynamic and hydromagnetic stability

Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation

The incompressible Boussinesq equations serve as an important model in geophysics as well as in the study of Rayleigh–Bénard convection. One generalization is to replace the standard Laplacian operator by a fractional Laplacian operator, namely (−Δ)α/2 in the velocity equation and (−Δ)β/2 in the temperature equation. This paper is concerned with the two-dimensional (2D) incompressible Boussinesq equations with critical dissipation (α+β=1) or supercritical dissipation (α+β<1). We prove two main results. This first one establishes the global-in-time existence of classical solutions to the critical Boussinesq equations with α+β=1 and 0.7692≈ 10 13 <α<1. The second one proves the eventual regularity of Leray–Hopf type weak solutions to the Boussinesq equations with supercritical dissipation α+β<1 and 0.7692≈ 10 13 <α<1.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cms/2016/0014/0007/CMS-2016-0014-0007-a009.pdf

Regularity results for the 2D Boussinesq equations with critical or supercritical dissipation

In this paper, we consider the $$n$$
 -dimensional (
 $$n\ge 2$$
 ) damped models of incompressible fluid mechanics in Besov spaces and establish the global (in time) regularity of classical solutions provided that the initial data are suitable small.

/pdf/global-smooth-solutions-to-the-n-dimensional-damped-models-27hfb2l18n.pdf

Global Smooth Solutions to the n-Dimensional Damped Models of Incompressible Fluid Mechanics with Small Initial Datum

Global regularity for a class of 2D tropical climate model

In this paper we establish the regularity criteria of solutions to the three space dimensions generalized viscous Hall-magnetohydrodynamics. Moreover, we also prove that there exist classical solutions globally in time to the generalized viscous Hall-MHD if the initial data is small enough or the fractional Laplacian dissipation powers α and β are suitably large. These results are some improvements of very recent work (Chae and Lee, 2014) established by Chae and Lee.

Zhuan Ye

Papers

Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation

Regularity results for the 2D Boussinesq equations with critical or supercritical dissipation

Global Smooth Solutions to the n-Dimensional Damped Models of Incompressible Fluid Mechanics with Small Initial Datum

Global regularity for a class of 2D tropical climate model

Regularity criteria and small data global existence to the generalized viscous Hall-magnetohydrodynamics