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- James Nolen
- 2008

We consider solutions of a scalar reaction-diffusion equation of the ignition type with a random, stationary and ergodic reaction rate. We show that solutions of the Cauchy problem spread with a deterministic rate in the long time limit. We also establish existence of generalized random traveling waves and of transition fronts in general heterogeneous media.

We consider Fisher-KPP-type reaction-diffusion equations with spatially inhomogeneous reaction rates. We show that a sufficiently strong localized inhomogeneity may prevent existence of transition-front-type global in time solutions while creating a global in time bump-like solution. This is the first example of a medium in which no reaction-diffusion… (More)

- James Nolen, George Papanicolaou, Olivier Pironneau
- Multiscale Modeling & Simulation
- 2008

We describe a projection framework for developing adaptive multiscale methods for computing approximate solutions to elliptic boundary value problems. The framework is consistent with homogenization when there is scale separation. We introduce an adaptive form of the finite element algorithms for solving problems with no clear scale separation. We present… (More)

We study the qualitative properties of the generalized transition fronts for the reaction-diffusion equations with the spatially inhomogeneous nonlinearity of the ignition type. We show that transition fronts are unique up to translation in time and are globally exponentially stable for the solutions of the Cauchy problem. The results hold for reaction… (More)

- James Nolen, Jack Xin, JAMES NOLEN
- 2005

We prove the existence of reaction-diffusion traveling fronts in mean zero space-time periodic shear flows for nonnegative reactions including the classical KPP (Kolmogorov-Petrovsky-Piskunov) nonlinearity. For the KPP nonlinearity, the minimal front speed is characterized by a variational principle involving the principal eigenvalue of a space-time… (More)

- James Nolen, Jack Xin
- Multiscale Modeling & Simulation
- 2003

We study the asymptotics of two space dimensional reaction-diffusion front speeds through mean zero space-time periodic shears using both analytical and numerical methods. The analysis hinges on traveling fronts and their estimates based on qualitative properties such as mono-tonicity and a priori integral inequalities. The computation uses an explicit… (More)

- James Nolen
- NHM
- 2011

We consider solutions to a nonlinear reaction diffusion equation when the reaction term varies randomly with respect to the spatial coordinate. The nonlinearity is the KPP type nonlinearity. For a stationary and ergodic medium, and for certain initial condition, the solution develops a moving front that has a deterministic asymptotic speed in the large time… (More)

We study the homogenization limit of solutions to the G-equation with random drift. This Hamilton-Jacobi equation is a model for flame propagation in a turbulent fluid in the regime of thin flames. For a fluid velocity field that is statistically stationary and ergodic, we prove sufficient conditions for homogenization to hold with probability one. These… (More)

- James Nolen, Jack Xin
- 2007

We study the asymptotic spreading of Kolmogorov-Petrovsky-Piskunov (KPP) fronts in space-time random incompressible flows in dimension d > 1. We prove that if the flow field is stationary, ergodic, and obeys a suitable moment condition, the large time front speeds (spreading rates) are deterministic in all directions for compactly supported initial data.… (More)

- James Nolen, Matthew Rudd, Jack Xin
- 2005

We prove the existence of Kolmogorov-Petrovsky-Piskunov (KPP) type traveling fronts in space-time periodic and mean zero incompress-ible advection, and establish a variational (minimization) formula for the minimal speeds. We approach the existence by considering limit of a sequence of front solutions to a regularized traveling front equation where the… (More)