We study the large time asymptotic speeds (turbulent flame speeds sT) of the simplified HamiltonJacobi (HJ) models arising in turbulent combustion. One HJ model is G-equation describing the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are HJ equations with convex (L1 type) but non-coercive Hamiltonians. The other is the quadratically nonlinear (L2 type) inviscid HJ model of MajdaSouganidis derived from the KolmogorovPetrovskyPiskunov reactive fronts. Motivated by a question posed by Embid, Majda and Souganidis (1995)[10], we compare the turbulent flame speeds sTs from these inviscid HJ models in two-dimensional cellular flows or a periodic array of steady vortices via sharp asymptotic estimates in the regime of large amplitude. The estimates are obtained by analyzing the action minimizing trajectories in the Lagrangian representation of solutions (Lax formula and its extension) in combination with delicate gradient bound of viscosity solutions to the associated cell problem of homogenization. Though the inviscid turbulent flame speeds share the same leading order asymptotics, their difference due to nonlinearities is identified as a subtle double logarithm in the large flow amplitude from the sharp growth laws. The turbulent flame speeds differ much more significantly in the corresponding viscous HJ models. 2013 Elsevier Masson SAS. All rights reserved.

Dispersive instability appears in time-domain solutions of classical cochlear models. In this letter, a derivation of optimal initial data is presented to minimize the effect of instability. A second-order accurate implicit boundary integral method is introduced. Numerical solutions of two-dimensional models show that the optimal initial conditions work successfully in time-domain steady-state computations for both the zero Neumann and zero Dirichlet fluid pressure boundary conditions at the helicotrema.

G-equations are well-known front propagation models in turbulent combustion which describe the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are HamiltonJacobi equations with convex (<i>L</i> ¹ type) but non-coercive Hamiltonians. Viscous G-equations arise from either numerical approximations or regularizations by small diffusion. The nonlinear eigenvalue $${\bar H}$$ from the cell problem of the viscous G-equation can be viewed as an approximation of the inviscid turbulent flame speed <i>s</i> _T. An important problem in turbulent combustion theory is to study properties of <i>s</i> _T, in particular how <i>s</i> _T depends on the flow amplitude <i>A</i>. In this paper, we study the behavior of $${\bar H=\bar H(A,d)}$$ as <i>A</i> + at any fixed diffusion constant <i>d</i> &gt; 0. For cellular flow, we show that$$\bar H(A,d)\leqq

We study the large-time asymptotic shock-front speed in an inviscid Burgers equation with a spatially random flux function. This equation is a prototype for a class of scalar conservation laws with spatial random coefficients such as the well-known BuckleyLeverett equation for two-phase flows, and the contaminant transport equation in groundwater flows. The initial condition is a shock located at the origin (the indicator function of the negative real line). We first regularize the equation by a special random viscous term so that the resulting equation can be solved explicitly by a ColeHopf formula. Using the invariance principle of the underlying random processes and the Laplace method, we prove that for large times the solutions behave like fronts moving at averaged constant speeds in the sense of distribution. However, the front locations are random, and we show explicitly the probability of observing the

The shape stability of the reaction interface for reactive flow in a layered porous medium is studied. This is done using a complete linearized stability analysis in the setting of a free boundary model of this phenomenon. The spectrum of the linearized problem is obtained in the general case, and it is compared with that obtained for homogeneous media in two typical cases.