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.

Photoacoustic imaging is a promising method for visualizing biological tissues with optical absorbers. This article provides an overview of the rapidly developing field of photoacoustic imaging. Photoacoustics, the physical basis of photoacoustic imaging, is analyzed briefly. The merits of photoacoustic technology, compared with optical imaging and ultrasonic imaging, are described. Various imaging techniques are also discussed, including scanning tomography, computed tomography and original detection of photoacoustic imaging. Finally, some biomedical applications of photoacoustic imaging are summarized.

Photoacoustic imaging in biomedicine

This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.

Variational Methods for Nonlocal Fractional Problems

Multiple integrals in the calculus of variations and nonlinear elliptic systems

Second‐Order Parabolic Differential Equations

The solvability in Sobolev spaces is proved for divergence form second order elliptic equations in the whole space, a half space, and a bounded Lipschitz domain. For equations in the whole space or a half space, the leading coefficients aij are assumed to be only measurable in one direction and have locally small BMO semi-norms in the other directions. For equations in a bounded domain, additionally we assume that aij have small BMO semi-norms in a neighborhood of the boundary of the domain. We give a unified approach of both the Dirichlet boundary problem and the conormal derivative problem. We also investigate elliptic equations in Sobolev spaces with mixed norms under the same assumptions on the coefficients.

Elliptic Equations in Divergence Form with Partially BMO Coefficients

We prove the solvability in Sobolev spaces for both divergence and non-divergence form higher order parabolic and elliptic systems in the whole space, on a half space, and on a bounded domain. The leading coefficients are assumed to be merely measurable in the time variable and have small mean oscillations with respect to the spatial variables in small balls or cylinders. For the proof, we develop a set of new techniques to produce mean oscillation estimates for systems on a half space.

On the L_p-solvability of higher order parabolic and elliptic systems with BMO coefficients

We prove generalized Fefferman-Stein type theorems on sharp functions with $A_p$ weights in spaces of homogeneous type with either finite or infinite underlying measure. We then apply these results to establish mixed-norm weighted $L_p$-estimates for elliptic and parabolic equations/systems with (partially) BMO coefficients in regular or irregular domains.

On _{}-estimates for elliptic and parabolic equations with _{} weights

On Lp-estimates for a class of non-local elliptic equations

By adapting a method in [11] with a suitable modification, we show
that the critical dissipative quasi-geostrophic equations in
$R^2$ has global well-posedness with arbitrary $H^1$
initial data. A decay in time estimate for homogeneous Sobolev norms
of solutions is also discussed.

Hongjie Dong

Papers

Elliptic Equations in Divergence Form with Partially BMO Coefficients

On the L_p-solvability of higher order parabolic and elliptic systems with BMO coefficients

On _{}-estimates for elliptic and parabolic equations with _{} weights

On Lp-estimates for a class of non-local elliptic equations

Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space