We consider a one-dimensional model biodegradation system consisting of two reactionadvection equations for nutrient and pollutant concentrations and a rate equation for biomass. The hydrodynamic dispersion is ignored. Under an explicit condition on the decay and growth rates of biomass, the system can be approximated by two component models by setting biomass kinetics to equilibrium. We derive closed form solutions for constant speed traveling fronts for the reduced two component models and compare their profiles in homogeneous media. For a spatially random velocity field, we introduce travel time and study statistics of degradation fronts via representations in terms of the travel time probability density function (<i>pdf</i>) and the traveling front profiles. The travel time <i>pdf</i> does not vary with the nutrient and pollutant concentrations and only depends on the random water velocity. The traveling front profiles

The dynamic stability of vortex solutions to the Ginzburg-Landau and nonlinear Schrdinger equations is the basic assumption of the asymptotic particle plus field description of interacting vortices. For the Ginzburg-Landau dynamics we prove that all vortices are asymptotically nonlinearly stable relative to small radial perturbations. Initially finite energy perturbations of vortices decay to zero in<i>L</i> ^<i>p</i>(²) spaces with an algebraic rate as time tends to infinity. We also prove that under general (nonradial) perturbations, the plus and minus one-vortices are linearly dynamically stable in<i>L</i> ²; the linearized operator has spectrum equal to (, 0] and generates a<i>C</i> ₀ semigroup of contractions on<i>L</i> ²(²). The nature of the zero energy point is clarified; it is<i>resonance</i>, a property related to the infinite energy of planar vortices. Our results on the linearized operator are also used to show that the plus and

We prove the existence of multidimensional traveling-wave solutions to the scalar equation for the transport of solutes (contaminants) with nonlinear adsorption and spatially periodic convection-diffusion-adsorption coefficients under the assumption that the nonlinear adsorption function satisfies the Lax and Oleinik entropy conditions. In the nondegenerate case, we also prove the uniqueness of the traveling waves. These traveling waves are analogues of viscous shock profiles. They propagate with effective speeds that depend on the periodic porous media only up to their mean states, and are given by an averaged Rankine-Hugoniot relation. This is a direct consequence of the fact that the transport equation is in conservation form. We use the sliding domain method, the continuation method, spectral theory, maximum principles, and a priori estimates. In the degenerate case, the traveling waves are weak

We establish the variational principle of Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in temporally random shear flows with sufficiently decaying correlations. A key quantity in the variational principle is the almost sure Lyapunov exponent of a heat operator with random potential. 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. The variational principle then allows us to analytically bound the front speeds. The speed bounds imply the linear growth law in the regime of large root mean square shear amplitude at any fixed temporal correlation length, and the sublinear growth law if the temporal decorrelation is also large enough, the so-called bending phenomenon.

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

We study the asymptotic spreading of KolmogorovPetrovskyPiskunov (KPP) fronts in spacetime random incompressible flows in dimension d&gt; 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. The flow field can become unbounded at large times. The front speeds are characterized by the convex rate function governing large deviations of the associated diffusion in the random flow. Our proofs are based on the Harnack inequality, an application of the sub-additive ergodic theorem, and the construction of comparison functions. Using the variational principles for the front speed, we obtain general lower and upper bounds of front speeds in terms of flow statistics. The bounds show that front speed enhancement in incompressible flows can grow at most linearly in the root mean square amplitude of the flows, and may have much slower growth due to rapid temporal decorrelation of the flows.