关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

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

Stochastic equations in infinite dimensions

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.

1 Barotropic geophysical flows and two-dimensional fluid flows: an elementary introduction 2 The Response to large scale forcing 3 The selective decay principle for basic geophysical flows 4 Nonlinear stability of steady geophysical flows 5 Topographic mean-flow interaction, nonlinear instability, and chaotic dynamics 6 Introduction to empirical statistical theory 7 Equilibrium statistical mechanics for systems of ordinary differential equations 8 Statistical mechanics for the truncated quasi-geostrophic equations 9 Empirical statistical theories for most probable states 10 Assessing the potential applicability of equilibrium statistical theories for geophysical flows: an overview 11 Predictions and comparison of equilibrium statistical theories 12 Equilibrium statistical theories and dynamical modeling of flows with forcing and dissipation 13 Predicting the jets and spots on Jupiter by equilibrium statistical mechanics 14 Statistically relevant and irrelevant conserved quantities for truncated quasi-geostrophic flow and the Burger-Hopf model 15 A mathematical framework for quantifying predictability utilizing relative entropy 16 Barotropic quasi-geostrophic equations on the sphere Bibliography Index

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Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows

We construct unconditionally stable, unconditionally uniquely solvable, and second-order accurate (in time) schemes for gradient flows with energy of the form $\int_\Omega ( F(\nabla\phi({\bf x})) + \frac{\epsilon^2}{2}|\Delta\phi({\bf x})|^2 ) d{\bf x}$. The construction of the schemes involves the appropriate combination and extension of two classical ideas: (i) appropriate convex-concave decomposition of the energy functional and (ii) the secant method. As an application, we derive schemes for epitaxial growth models with slope selection ($F({\bf y})= \frac14(|{\bf y}|^2-1)^2$) or without slope selection ($F({\bf y})=-\frac12\ln(1+|{\bf y}|^2)$). Two types of unconditionally stable uniquely solvable second-order schemes are presented. The first type inherits the variational structure of the original continuous-in-time gradient flow, while the second type does not preserve the variational structure. We present numerical simulations for the case with slope selection which verify well-known physical scaling laws for the long time coarsening process.

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Second-order Convex Splitting Schemes for Gradient Flows with Ehrlich-Schwoebel Type Energy: Application to Thin Film Epitaxy

We investigate the well-posedness of a coupled Stokes-Darcy model with Beavers-Joseph interface boundary conditions. In the steady-state case, the well-posedness is established under the assumption of a small coefficient in the Beavers-Joseph interface boundary condition. In the time-dependent case, the well-posedness is established via an appropriate time discretization of the problem and a novel scaling of the system under an isotropic media assumption. Such coupled systems are crucial to the study of subsurface flow problems, in particular, flows in karst aquifers.

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Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition

We present unconditionally stable and convergent numerical sche- mes for gradient flows with energy of the form $ \int_\Omega( F(\nabla\phi(\x)) + \frac{\epsilon^2}{2}|\Delta\phi(\x)|^2 )$dx. The construction of the schemes involves an appropriate extension of Eyre's idea of convex-concave decomposition of the energy functional. As an application, we derive unconditionally stable and convergent schemes for epitaxial film growth models with slope selection (F(y)= 1/4(|y|2-1)2) and without slope selection (F(y)= -1/2ln(1+|y|2)). We conclude the paper with some preliminary computations that employ the proposed schemes.

Unconditionally stable schemes for equations of thin film epitaxy

Numerical solutions using finite element methods are considered for transient flow in a porous medium coupled to free flow in embedded conduits. Such situations arise, for example, for groundwater flows in karst aquifers. The coupled flow is modeled by the Darcy equation in a porous medium and the Stokes equations in the conduit domain. On the interface between the matrix and conduit, Beavers-Joseph interface conditions, instead of the simplified Beavers-Joseph-Saffman conditions, are imposed. Convergence and error estimates for finite element approximations are obtained. Numerical experiments illustrate the validity of the theoretical results.

/pdf/finite-element-approximations-for-stokes-darcy-flow-with-10cvo30wwb.pdf

Xiaoming Wang

Papers

Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows

Second-order Convex Splitting Schemes for Gradient Flows with Ehrlich-Schwoebel Type Energy: Application to Thin Film Epitaxy

Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition

Unconditionally stable schemes for equations of thin film epitaxy

Finite Element Approximations for Stokes-Darcy Flow with Beavers-Joseph Interface Conditions