We prove the existence of global solutions to a coupled system of NavierStokes, and reaction-diffusion equations (for temperature and mass fraction) with prescribed front data on an infinite vertical strip or tube. This system models a one-step exothermic chemical reaction. The heat release induced volume expansion is accounted for via the Boussinesq approximation. The solutions are time dependent moving fronts in the presence of fluid convection. In the general setting, the fronts are subject to intensive Rayleigh-Taylor and thermal-diffusive instabilities. Various physical quantities, such as fluid velocity, temperature, and front speed, can grow in time. We show that the growth is at most for large time <i>t</i> by constructing a nonlinear functional on the temperature and mass fraction components. These results hold for arbitrary order reactions in two space dimensions and for quadratic and cubic reactions in

Abstract We study the GinzburgLandau equation on the plane with initial data being the product of n wellseparated+ 1 vortices and spatially decaying perturbations. If the separation distances are O ( 1), l, we prove that the n vortices do not move on the time scale O(^2),=o(1\over); instead, they move on the time scale O(^-21\over) according to the law j= xj W, W= l j log| xl xj|, xj=(j, j) 2, the location of the jth vortex. The main ingredients of our proof consist of estimating the large space behavior of solutions, a monotonicity inequality for the energy density of solutions, and energy comparisons. Combining these, we overcome the infinite energy difficulty of the planar vortices to establish the dynamical law. John &amp; Wiley Sons, Inc.

We study the front dynamics of the bistable reaction-diffusion equations with periodic diffusion and/or convection coefficients in several space dimensions. When traveling wave solutions exist, the solutions of the initial value problem behave as wave fronts propagating with the effective speeds of traveling waves under various initial conditions. Yet due to the bistable nature of the nonlinearity, traveling waves may not always exist when the medium variations from the mean states are large enough. Their existence is closely related to the detailed forms of diffusion and convection coefficients, more so in multidimension than in one. We present a simple sufficient condition for the nonexistence of traveling waves (quenching) using perturbation method. Our two dimensional finite difference numerical computations show a variety of front behaviors, such as: the propagation, quenching and retreat of fronts. We found

Diffusion-generated motion by mean curvature is a simple algorithm for producing motion by mean curvature of a surface, in which the motion is generated by alternately diffusing and renormalizing a characteristic function. In this paper, we generalize diffusion-generated motion to a procedure that can be applied to the curvature motion of filaments, i.e., curves in <i>R</i> ^3, that may initially consist of a complex configuration of links. The method consists of applying diffusion to a complex-valued function whose values wind around the filament, followed by normalization. We motivate this approach by considering the essential features of the complex Ginzburg-Landau equation, which is a reaction-diffusion PDE that describes the formation and propagation of filamentary structures. The new algorithm naturally captures topological merging and breaking of filaments without fattening curves. We justify the new

G-equations are popular front propagation models in combustion literature and describe the front motion law of normal velocity equal to a constant plus the normal projection of fluid velocity. G-equations are Hamilton-Jacobi equations with convex but non-coercive Hamiltonians. We prove homogenization of the inviscid G-equation for space periodic incompressible flows. This extends a two space dimensional result in" Periodic homogenization of G-equations and viscosity effects," Nonlinearity, to appear. We construct approximate correctors to bypass the lack of compactness due to the non-coercive Hamiltonian. The existence of approximate correctors rely on a local reachability property of the controlled flow trajectory as well as incompressibility of the flow. Homogenization then follows from the comparison principle and the perturbed test function method. The effective Hamiltonian is convex and homogeneous of