We study homogenization of G-equation with a
ow straining term (or the strain G-equation) in two dimensional periodic cellular
ow. The strain G-equation is a highly non-coercive and non-convex level set Hamilton-Jacobi equation. The main objective is to investigate how the
ow induced straining (the nonconvex term) in
uences front propagation as the
ow intensity A increases. Three distinct regimes are identified. When A is below the critical level, homogenization holds and the turbulent
ame speed sT (effective Hamiltonian) is well-defined for any periodic
ow with small divergence and is enhanced by the cellular
ow as $s_T \ge O(A/logA)$. In the second regime where A is slightly above the critical value, homogenization breaks down, and $s_T$ is not well defined along any direction. Solutions become a mixture of
fast moving part and a stagnant part. When $A$ is sufficiently large, the whole
ame front ceases to propagate forward due to the
flow induced straining. In particular, along directions $p = (1; 0)$ and $(0;1)$, $s_T$ is well-defined again with a value of zero (trapping). A partial homogenization result is also proved. If we consider a similar but relatively simpler Hamiltonian, the trapping occurs along all directions. The analysis is based on the two-player dierential game representation of solutions, selection of game strategies and trapping regions, and construction of connecting trajectories.
The diffusive transport of passive tracers or particles can be enhanced by incompressible, turbulent flow fields. Analyzing the effective behavior is a challenging problem with theoretical and practical importance in many areas of science and engineering, ranging from the transport of mass, heat, and pollutants in geophysical flows to sea ice dynamics and turbulent combustion. The long time, large scale behavior of such systems is equivalent to an enhanced diffusion process with an effective diffusivity tensor D*. Two different formulations of integral representations for D* were developed for the case of time-independent fluid velocity fields, involving spectral measures of bounded self-adjoint operators acting on vector fields and scalar fields, respectively. Here, we extend both of these approaches to the case of space-time periodic velocity fields, allowing for chaotic dynamics, providing rigorous integral representations for D* involving spectral measures of unbounded self-adjoint operators.We prove the different formulations are equivalent. Their correspondence follows from a one-to-one isometry between the underlying Hilbert spaces. We also develop a Fourier method for computing D*, which captures the phenomenon of residual diffusion related to Lagrangian chaos of a model flow. This is reflected in the spectral measure by a concentration of mass near the spectral origin.