Speed ensemble of bistable (combustion) fronts in mean zero stationary Gaussian shear flows inside two and three dimensional channels is studied with a min-max variational principle. In the small root mean square regime of shear flows, a new class of multi-scale test functions are found to yield speed asymptotics. The quadratic speed enhancement law holds with probability arbitrarily close to one under the almost sure continuity (dimension two) and mean square Hlder regularity (dimension three) of the shear flows. Remarks are made on the quadratic and linear laws of front speed expectation in the small and large root mean square regimes.

In this paper, we establish the optimal error estimate of the particle method for a family of nonlinear evolutionary partial differential equations, or the so-called b-equation. The b-equation, including the Camassa-Holm equation and the Degasperis-Procesi equation, has many applications in diverse scientific fields. The particle method is an approximation of the b-equation in Lagrangian representation. We also prove short-time existence, uniqueness and regularity of the Lagrangian representation of the b-equation.

In this paper, we prove that the celebrated Arnold--Beltrami--Childress (ABC) flow with parameters A=B=C=1, given by $ \dot x =\sin z+\cos y,\ \dot y = \sin x+\cos z,\ \dot z =\sin y + \cos x $, has periodic orbits on (2\pi\mathbb T)^3 with rotation vectors parallel to (1,0,0), (0,1,0), and (0,0,1). Despite ABC flows being studied since the 1960s, this seems to be the first time that existence of nonperturbative periodic orbits has been established for them. The main difficulty here is the lack of a variational structure for these flows, and our proof instead relies on their symmetry properties. As an application of our result, we show that the well-known G-equation model of turbulent combustion with this ABC flow on (2\pi\mathbb T)^3 has a linear (i.e., maximal possible) flame speed enhancement rate as the flow amplitude grows to infinity. To the best of our knowledge, this is the first time an asymptotic flame speed growth law has been established for a

In [9] Anderson’s conjecture was proven by comparing values of Bloch functions with the variation of the function. We extend that result on Bloch functions from two to arbitrary dimension and prove that ∫_{[0,x]}|∇b(ζ)|e^{b(ζ)}d|ζ|<∞.. In the second part of the paper, we show that the area or volume integral ∫_{B^d}|∇u(w)|p(w,θ)dA(w) for positive harmonic functions u is bounded by the value cu(0) for at least one θ. The integral is also transferred to simply connected domains and interpreted from the point of view of stochastics. Several emerging open problems are presented.

In this paper we introduce our preliminary research results for segmentation and labeling of 3-dimensional microscopy neuron image. We segment each of stacked 2-dimensional image slices using a partial differential equation (PDE) based algorithm and project previous slice segmentation result to the next slide as an initialization condition. Then we label neurons using an efficient connected component labeling algorithm. We show sample results obtained from real neuron image data