In this paper, the authors first consider the global well-posedness of 3-D Boussinesq system, which has variable kinematic viscosity yet without thermal conductivity and buoyancy force, provided that the viscosity coefficient is sufficiently close to some positive constant in L ^∞ and the initial velocity is small enough in $\dot{B}_{3,1}^0(\mathbb{R}^3)$. With some thermal conductivity in the temperature equation and with linear buoyancy force θe ₃ on the velocity equation in the Boussinesq system, the authors also prove the global well-posedness of such system with initial temperature and initial velocity being sufficiently small in L ¹(ℝ³) and $\dot{B}_{3,1}^0(\mathbb{R}^3)$ respectively.

Models for weather and climate prediction are complex, and each model typically has at least a small number of phenomena that are poorly represented, such as perhaps the Madden-Julian Oscillation (MJO for short) or El Niño-Southern Oscillation (ENSO for short) or sea ice. Furthermore, it is often a very challenging task to modify and improve a complex model without creating new deficiencies. On the other hand, it is sometimes possible to design a low-dimensional model for a particular phenomenon, such as the MJO or ENSO, with significant skill, although the model may not represent the dynamics of the full weather-climate system. Here a strategy is proposed to mitigate these model errors by taking advantage of each model’s strengths. The strategy involves inter-model data assimilation, during a forecast simulation, whereby models can exchange information in order to obtain more faithful representations of the full weather-climate system. As an initial investigation, the method is examined here using a simplified scenario of linear models, involving a system of stochastic partial differential equations (SPDEs for short) as an imperfect tropical climate model and stochastic differential equations (SDEs for short) as a low-dimensional model for the MJO. It is shown that the MJO prediction skill of the imperfect climate model can be enhanced to equal the predictive skill of the low-dimensional model. Such an approach could provide a route to improving global model forecasts in a minimally invasive way, with modifications to the prediction system but without modifying the complex global physical model itself.

An intrinsic property of almost any physical measuring device is that it makes observations which are slightly blurred in time. The authors consider a nudging-based approach for data assimilation that constructs an approximate solution based on a feedback control mechanism that is designed to account for observations that have been blurred by a moving time average. Analysis of this nudging model in the context of the subcritical surface quasi-geostrophic equation shows, provided the time-averaging window is sufficiently small and the resolution of the observations sufficiently fine, that the approximating solution converges exponentially fast to the observed solution over time. In particular, the authors demonstrate that observational data with a small blur in time possess no significant obstructions to data assimilation provided that the nudging properly takes the time averaging into account. Two key ingredients in our analysis are additional bounded-ness properties for the relevant interpolant observation operators and a non-local Gronwall inequality.

The authors study the fluid dynamic behavior of the stochastic Galerkin (SG for short) approximation to the kinetic Fokker-Planck equation with random uncertainty. While the SG system at the kinetic level is hyperbolic, its fluid dynamic limit, as the Knudsen number goes to zero and the underlying kinetic equation approaches to the uncertain isentropic Euler equations, is not necessarily hyperbolic, as will be shown in the case study fashion for various orders of the SG approximations.

This is in the sequel of authors’ paper [Lin, F. H., Pan, X. B. and Wang, C. Y., Phase transition for potentials of high dimensional wells, Comm. Pure Appl. Math., 65(6), 2012, 833-888] in which the authors had set up a program to verify rigorously some formal statements associated with the multiple component phase transitions with higher dimensional wells. The main goal here is to establish a regularity theory for minimizing maps with a rather non-standard boundary condition at the sharp interface of the transition. The authors also present a proof, under simplified geometric assumptions, of existence of local smooth gradient flows under such constraints on interfaces which are in the motion by the mean-curvature. In a forthcoming paper, a general theory for such gradient flows and its relation to Keller-Rubinstein-Sternberg’s work (in 1989) on the fast reaction, slow diffusion and motion by the mean curvature would be addressed.

Data assimilation refers to the methodology of combining dynamical models and observed data with the objective of improving state estimation. Most data assimilation algorithms are viewed as approximations of the Bayesian posterior (filtering distribution) on the signal given the observations. Some of these approximations are controlled, such as particle filters which may be refined to produce the true filtering distribution in the large particle number limit, and some are uncontrolled, such as ensemble Kalman filter methods which do not recover the true filtering distribution in the large ensemble limit. Other data assimilation algorithms, such as cycled 3DVAR methods, may be thought of as controlled estimators of the state, in the small observational noise scenario, but are also uncontrolled in general in relation to the true filtering distribution. For particle filters and ensemble Kalman filters it is of practical importance to understand how and why data assimilation methods can be effective when used with a fixed small number of particles, since for many large-scale applications it is not practical to deploy algorithms close to the large particle limit asymptotic. In this paper, the authors address this question for particle filters and, in particular, study their accuracy (in the small noise limit) and ergodicity (for noisy signal and observation) without appealing to the large particle number limit. The authors first overview the accuracy and minorization properties for the true filtering distribution, working in the setting of conditional Gaussianity for the dynamics-observation model. They then show that these properties are inherited by optimal particle filters for any fixed number of particles, and use the minorization to establish ergodicity of the filters. For completeness we also prove large particle number consistency results for the optimal particle filters, by writing the update equations for the underlying distributions as recursions. In addition to looking at the optimal particle filter with standard resampling, they derive all the above results for (what they term) the Gaussianized optimal particle filter and show that the theoretical properties are favorable for this method, when compared to the standard optimal particle filter.