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Front propagation in periodic excitable media
"... This paper is devoted to the study of pulsating travelling fronts for reactiondiffusionadvection equations in a general class of periodic domains with underlying periodic diffusion and velocity fields. Such fronts move in some arbitrarily given direction with an unknown effective speed. The notio ..."
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Cited by 121 (17 self)
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This paper is devoted to the study of pulsating travelling fronts for reactiondiffusionadvection equations in a general class of periodic domains with underlying periodic diffusion and velocity fields. Such fronts move in some arbitrarily given direction with an unknown effective speed. The notion of pulsating travelling fronts generalizes that of travelling fronts for planar or shear flows. Various existence, uniqueness and monotonicity results are proved for two classes of reaction terms. Firstly, for a combustiontype nonlinearity, it is proved that the pulsating travelling front exists and that its speed is unique. Moreover, the front is increasing with respect to the time variable and unique up to translation in time. We also consider one class of monostable nonlinearity which arises either in combustion or biological models. Then, the set of possible speeds is a semiinfinite interval, closed and bounded from below. For each possible speed, there exists a pulsating travelling front which is increasing in time. This result extends the classical KolmogorovPetrovskyPiskunov case. Our study covers in particular the case of flows in all of space with periodic advections such as periodic shear flows or a periodic array of vortical cells. These results are also obtained for cylinders with oscillating boundaries or domains with a periodic array of holes.
The Influence of Advection on the Propagation of Fronts in ReactionDiffusion Equations
, 2002
"... This paper is intended as a survey of some of these developments. It is the written form of a series of lectures given at the NATO Advanced Scientific Institute in Cargese in the summer of 1999. Some of this material was also presented in a course at the Pacific Institute of Mathematical Sciences (U ..."
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Cited by 51 (6 self)
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This paper is intended as a survey of some of these developments. It is the written form of a series of lectures given at the NATO Advanced Scientific Institute in Cargese in the summer of 1999. Some of this material was also presented in a course at the Pacific Institute of Mathematical Sciences (UBC) in Vancouver in the summer 2001. I have also included some recent results on the subject that have appeared since
Stability and Asymptotic Analysis of a FluidParticle Interaction Model
"... We are interested in coupled microscopic/macroscopic models describing the evolution of particles dispersed in a fluid. The system consists in a VlasovFokkerPlanck equation to describe the microscopic motion of the particles coupled to the Euler equations for a compressible fluid. We investigate d ..."
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Cited by 30 (6 self)
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We are interested in coupled microscopic/macroscopic models describing the evolution of particles dispersed in a fluid. The system consists in a VlasovFokkerPlanck equation to describe the microscopic motion of the particles coupled to the Euler equations for a compressible fluid. We investigate dissipative quantities, equilibria and their stability properties and the role of external forces. We also study some asymptotic problems, their equilibria and stability and the derivation of macroscopic twophase models. Key words. FluidParticles Interaction. VlasovEuler system. Stability. Hydrodynamic Limit. AMS Subject classification. 35Q99 35B25. 1
Global solutions to the compressible NavierStokes equations for a reacting mixture
 SIAM J. Math. Anal
, 1992
"... We prove the global existence of weak solutions to the NavierStokes equations for compressible, heatconducting flow in one space dimension with large, discontinuous initial data, and we obtain apriori estimates for these solutions which are independent of time, sufficient to determine their asymp ..."
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Cited by 26 (7 self)
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We prove the global existence of weak solutions to the NavierStokes equations for compressible, heatconducting flow in one space dimension with large, discontinuous initial data, and we obtain apriori estimates for these solutions which are independent of time, sufficient to determine their asymptotic behavior. In particular, we show that, as time goes to infinity, the solution tends to a constant state determined by the initial mass and the initial energy, and that the magnitudes of singularities in the solution decay to zero. 1991 Mathematics Subject Classification. 35B40, 35D05, 76N10, 35B45. Key words and phrases. NavierStokes equations, compressible flow, global discontinuous solutions, largetime behavior, large discontinuous initial data, uniform bounds.
Existence of nonplanar solutions of a simple model of premixed Bunsen flames
 SIAM J. Math. Anal
, 1999
"... This work deals with the existence of solutions of a reactiondiffusion equation in the plane IR2. The problem, whose unknowns are the real c and the function u, is the following: ..."
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Cited by 24 (7 self)
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This work deals with the existence of solutions of a reactiondiffusion equation in the plane IR2. The problem, whose unknowns are the real c and the function u, is the following:
The speed of propagation for KPP type problems. I  Periodic framework
"... This paper is devoted to some nonlinear propagation phenomena in periodic and more general domains, for reactiondiffusion equations with KolmogorovPetrovskyPiskunov (KPP) type nonlinearities. The case of periodic domains with periodic underlying excitable media is a followup of the article [7]. ..."
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Cited by 18 (5 self)
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This paper is devoted to some nonlinear propagation phenomena in periodic and more general domains, for reactiondiffusion equations with KolmogorovPetrovskyPiskunov (KPP) type nonlinearities. The case of periodic domains with periodic underlying excitable media is a followup of the article [7]. It is proved that the minimal speed of pulsating fronts is given by a variational formula involving linear eigenvalue problems. Some consequences concerning the influence of the geometry of the domain, of the reaction, advection and diffusion coefficients are given. The last section deals with the notion of asymptotic spreading speed. The main properties of the spreading speed are given. Some of them are based on some new Liouville type results for nonlinear elliptic equations in unbounded domains.
A roetype Riemann solver for hyperbolic systems with relaxation based on timedependent wave decomposition
, 1997
"... this paper we shall only consider systems with relaxation. Although very efficient and accurate methods have been developped for both hyperbolic systems and systems of ordinary differential equations, many numerical schemes for hyperbolic systems with relaxation are unsatisfactory and the main diffi ..."
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Cited by 17 (0 self)
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this paper we shall only consider systems with relaxation. Although very efficient and accurate methods have been developped for both hyperbolic systems and systems of ordinary differential equations, many numerical schemes for hyperbolic systems with relaxation are unsatisfactory and the main difficulty arises from the need to handle very different relaxation times with the same scheme. For instance solid particles are usually added in rocket engines in order to damp the combustion instabilities. The particles burn inside the rocket so that the stiffness of the drag terms range from nonstiff to very stiff. On the other hand the computation of an initial value problem for an hyperbolic system with relaxation also involves a wide range of stiffness of the source terms: if the initial data is away from equilibrium, there is a boundary layer in time of order # after which the solution is close to equilibrium. During a time interval of order # the relaxation terms are thus stiff while they become nonstiff after a time of order # . The chalenge is thus to construct a numerical scheme that Numerische Mathematik Electronic Edition page 144 of Numer. Math. 77: 143185 (1997) A Riemann solver for hyperbolic systems with relaxation 145 may handle any stiffness and whose computational cost is of the same order as the cost of usual methods such as the Strang splitting for instance: in order that the computational cost of the method be of the same order as the cost of usual methods for hyperbolic systems of conservation laws, we want to chose the time step only on the CFL condition relative to the convection terms and the source terms should be underresolved in the stiff case. The construction of numerical schemes for hyperbolic systems with relaxation has attracted a lot of atte...
The NavierStokesVlasovFokkerPlanck system near equilibrium. Preprint 2009. and Simulation of FluidParticles Flows 25
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Large Scale Dynamics of Precipitation Fronts in the Tropical Atmosphere: A Novel Relaxation Limit
 COMMUN. MATH. SCI
, 2004
"... A simplified set of equations is derived systematically below for the interaction of large scale flow fields and precipitation in the tropical atmosphere. These equations, the Tropical Climate Model, have the form of a shallow water equation and an equation for moisture coupled through a strongly ..."
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Cited by 16 (6 self)
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A simplified set of equations is derived systematically below for the interaction of large scale flow fields and precipitation in the tropical atmosphere. These equations, the Tropical Climate Model, have the form of a shallow water equation and an equation for moisture coupled through a strongly nonlinear source term. This source term, the precipitation, is of relaxation type in one region of state space for the temperature and moisture, and vanishes identically elsewhere in the state space of these variables. In addition, the equations are coupled nonlinearly to the equations for barotropic incompressible flow. Several mathematical features of this system are developed below including energy principles for solutions and their first derivatives independent of relaxation time. With these estimates, the formal infinitely fast relaxation limit converges to a novel hyperbolic free boundary problem for the motion of precipitation fronts from a large scale dynamical perspective. Elementary exact solutions of the limiting dynamics involving precipitation fronts are developed below and include three families of waves: fast drying fronts as well as slow and fast moistening fronts. The last two families of waves violate Lax’s Shock Inequalities; nevertheless, numerical experiments presented below confirm their robust realizability with realistic finite relaxation times. From the viewpoint of tropical atmospheric dynamics, the theory developed here provides a new perspective on the fashion in which the prominent large scale regions of moisture in the tropics associated with deep convection can move and interact with large scale dynamics in the quasiequilibrium approximation.