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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 56 (5 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 29 (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
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 29 (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:
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 23 (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.
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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2004: Large Scale Dynamics of Precipitation Fronts in the Tropical Atmosphere: A Novel Relaxation Limit
 Commun. Math. Sci
"... Abstract. 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 s ..."
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Abstract. 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.
Onedimensional stability of viscous strong detonation waves
"... Abstract. Building on Evans function techniques developed to study the stability of viscous shocks, we examine the stability of viscous strong detonation wave solutions of the reacting NavierStokes equations. The primary result, following [1, 17], is the calculation of a stability index whose sign ..."
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Abstract. Building on Evans function techniques developed to study the stability of viscous shocks, we examine the stability of viscous strong detonation wave solutions of the reacting NavierStokes equations. The primary result, following [1, 17], is the calculation of a stability index whose sign determines a necessary condition for spectral stability. We show that for an ideal gas this index can be evaluated in the ZND limit of vanishing dissipative effects. Moreover, when the heat of reaction is sufficiently small, we prove that strong detonations are spectrally stable provided the underlying shock is stable. Finally, for completeness, the stability index calculations for the nonreacting NavierStokes equations are included.
Guidelines for Authoring Comprehensible Web Pages and Evaluating Their Success
 Technical Communication
, 2000
"... r Presents guidelines that should enable authors to create Web pages that their readers can understand r Enables evaluators to judge the comprehensibility of Web pages ..."
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Cited by 15 (2 self)
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r Presents guidelines that should enable authors to create Web pages that their readers can understand r Enables evaluators to judge the comprehensibility of Web pages
Solutions of Semilinear Elliptic Equations in R^N with ConicalShaped Level Sets
"... This article deals with the questions of the existence, of the uniqueness and of the qualitative properties of solutions of semilinear elliptic equations in R N . Three types of conical conditions at infinity are successively considered. This defines three frameworks: the weak framework, the strong ..."
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Cited by 14 (1 self)
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This article deals with the questions of the existence, of the uniqueness and of the qualitative properties of solutions of semilinear elliptic equations in R N . Three types of conical conditions at infinity are successively considered. This defines three frameworks: the weak framework, the strong framework and the framework of solutions with asymptots. The results are based on different kinds of sliding methods and, following the ideas of Berestycki, Nirenberg and Vega, on comparison principles in cones or in R N .