This project aims to study analytically some key problems regarding the G-equation (a noncoercive Hamilton-Jacobi equation), the strain G-equation (a noncoercive and nonconvex Hamilton-Jacobi equation), and the curvature G-equation (a mean-curvature- type equation) in the modeling of turbulent flame speed. Roughly speaking, turbulent flame speed is the averaged flame propagation speed under the effect of the flow (turbulence). The central goal of the project is to understand the dependence of the turbulent flame speed on the turbulence intensity. A particularly important problem is to figure out how the flow-stretching effect contributes to the strong bending and quenching of flame speeds. Another goal of the project is to compare turbulent flame speeds predicted by various G-equations and other models in the mathematics literature, such as the Majda-Souganidis model and the well-studied scalar reaction-diffusion-advection equation model. Some proposed problems have significant connections with the Aubry-Mather theory and weak KAM theory, which are important for understanding nonintegrable dynamical systems. So far there are very few rigorous analytical results on turbulent flame speeds in flows of more than two dimensions. A long-term goal of this project is to understand how chaotic structures in three-dimensional flows (e.g., the Arnold-Beltrami-Childress flow) affect turbulent flame speeds.
Turbulent combustion plays the main role in important industrial issues such as energy production and engine design. The so-called G-equation and its variants are popular models in turbulent combustion due to their simplicity, efficiency, and robustness in fitting experiments. All the research components of the project are closely related to one of the most important unsolved problems in turbulent combustion; namely, to predict the turbulent flame speed and, in particular, to understand how it depends on the turbulence intensity (e.g., think of the relation between the spreading velocity of a wild fire and the strength of the wind). A very important part of the project is its educational component. Besides supervision of graduate students, the principal investigator also plans to participate in several well-established educational programs at UC-Irvine, ranging from the K-12 to the undergraduate levels: California MathCounts, a 6-8th grade math competition; the Irvine Area Math Modelers (IAMM), which is a training program that prepares local high school students for the National High School Mathematical Contest in Modeling (HiMCM)); the SURF program, which is an undergraduate summer research program; the Freshman Seminar Program, which provides introductory lectures on research to undergraduate students at UC-Irvine. The project will generate broad impact on combustion science and engineering, with implications for clean energy, as well as further the education of the next generation of STEM scientists.
