Jack Xin

We establish the variational principle of Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in temporally random shear flows inside an infinite cylinder, under suitable assumptions of the shear field. A key quantity in the variational principle is the almost sure Lyapunov exponent of a heat operator with random potential. The variational principle then allows us to bound and compute the front speeds. We show the linear and quadratic laws of speed enhancement as well as a resonance-like dependence of front speed on the temporal shear correlation length. To prove the variational principle, we use the comparison principle of solutions, the path integral representation of solutions, and large deviation estimates of the associated stochastic flows.

We study a class of nonlinear nonlocal cochlear models of the transmission line type, describing the motion of basilar membrane (BM) in the cochlea. They are damped dispersive partial differential equations (PDEs) driven by time dependent boundary forcing due to the input sounds. The global well-posedness in time follows from energy estimates. Uniform bounds of solutions hold in case of bounded nonlinear damping. When the input sounds are multi-frequency tones, and the nonlinearity in the PDEs is cubic, we construct smooth quasi-periodic solutions (multi-tone solutions) in the weakly nonlinear regime, where new frequencies are generated due to nonlinear interaction. When the input is two tones at frequencies $f_1$, $f_2$ ($f_1 < f_2$), and high enough intensities, numerical results illustrate the formation of combination tones at $2 f_1 -f_2$ and $2f_2 -f_1$, in agreement with hearing experiments. We visualize the frequency content of solutions through the FFT power spectral density of displacement at selected spatial locations on BM.

We study the two-point correlation function of a freely decaying scalar in Kraichnan's model of advection by a Gaussian random velocity field that is stationary and white noise in time, but fractional Brownian in space with roughness exponent 0ζa≤(d/γ)+1 are statistically realizable, where d is space dimension and γ=2−ζ. An infinite sequence of invariants J p, p=0, 1, 2,..., is pointed out, where J 0 is Corrsin's integral invariant but the higher invariants appear to be new. We show that at least one of the invariants J 0 or J 1 must be nonzero (possibly infinite) for realizable initial data. Initial datum with a finite, nonzero invariant—the first being J p—converges at long times to a scaling solution Φ p with a=(d/γ)+p, p=0, 1. The latter belongs to an exceptional series of self-similar solutions with stretched-exponential decay in space. However, the domain of attraction includes many initial data with power-law decay. When the initial datum has all invariants zero or infinite and also it exhibits power-law decay, then the solution converges at long times to a nonexceptional scaling solution with the same power-law decay. These results support a picture of a “two-scale” decay with breakdown of self-similarity for a range of exponents (d+γ)/γad+2)/γ, analogous to what has recently been found in the decay of Burgers turbulence.