Perturbed Riemann Problem for a Scalar Chapman-Jouguet Combustion Model

The author considers the perturbed Riemann problem for a scalar ChapmanJouguet combustion model which comes from Majda’s model with a modified, bump-type ignition function proposed in the results of Lyng and Zumbrun in 2004. Under the entropy conditions, the unique solution in a neighborhood of the origin on the(x, t) plane(t > 0) is obtained. It is found that, for some cases, the perturbed Riemann solutions are essentially different from the corresponding Riemann solutions. The perturbation may transform a strong detonation into a weak deflagration in the neighborhood of the origin. Especially, it can be observed that burning happens although the corresponding Riemann solution does not contain combustion wave, which exhibits the instability for the unburnt state.     

Construction of Solutions and L^1-error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.     

A NOTE ON THE EXISTENCE OF STATIONARY SOLUTIONS OF THE COMPRESSIBLE 6 EULER-POISSON EQUATIONS WITH 5/6〈γ 〈 2

This paper is concerned with the system of Euler-Poisson equations as a model to describe the motion of the self-induced gravitational gaseous stars. When ~ 〈 7 〈 2, under the weak smoothness of entropy function, we find a sufficient condition to guarantee the existence of stationary solutions for some velocity fields and entropy function that solve the conservation of mass and energy.     

ON THE REGULARITY AND EXISTENCE OF SOLUTIONS FOR SEMILINEAR ELLIPTIC EQUATIONS OF SECOND ORDER

Let Ω(?)R~n be a bounded domain with a smooth boundary (?)Ω L a strictly elliptic operator and c(x)≥0 in Ω. In this paper we are concerned with the following Dirichlet problem with the growth condition (P_1): a<2, for n=2. It is proved that if p(x, t) has all derivatives up to order l which are locally Hlder continuous in (?)×R. and if a_(ij)(x) ∈C_(l 1,α)(Ω) and c(x)∈C_(l,α)(Ω), then any weak solution in W_0~(1,2) of (1) lies in C_(l 2,α)(Ω). Moreover, under the growth condition (P_1) and some additional assumptions, the existence of nontrivial solution of (1) is proved. The main difficulity here is that the simple bootstrapping procedure fails to apply directly to the case of the growth condition (P_1).     

OPTIMAL INTERIOR PARTIAL REGULARITY FOR NONLINEAR ELLIPTIC SYSTEMS UNDER THE NATURAL GROWTH CONDITION： THE METHOD OF A-HARMONIC APPROXIMATION

In this article,the authors consider the nonlinear elliptic systems under the natural growth condition.They use a new method introduced by Duzaar and Grotowski, for proving partial regularity for weak solutions,based on a generalization of the technique of harmonic approximation.And directly establish the optimal Holder exponent for the derivative of a weak solution.     

THE FIRST BOUNDARY VALUE PROBLEM FOR A CLASS OF QUASILINEAR DEGENERATE ELLIPTIC EQUATIONS

In this paper, the first boundary value problem for quasilinear equation of the form △A（u,x）＋∑i=1^m δb^i（u,x）/δxi＋c（u,x）=0,Au（u,x） ≥0is studied. By using the compensated compactness theory, some results on the existence of weak solution are established. In addition, under certain condition the uniqueness of solution is proved.     

THE GLOBAL EXISTENCE OF STRONG SOLUTIONS TO THE 3D COMPRESSIBLE ISOTHERMAL NAVIER-STOKES EQUATIONS

This paper concerns the global existence of strong solutions to the 3 D compressible isothermal Navier-Stokes equations with a vacuum at infinity.Based on the special structure of the Zlotnik inequality,the time uniform upper bounds for density are established through some time-dependant a priori estimates under the assumption that the total mass is suitably small.     

THE GLOBAL EXISTENCE OF STRONG SOLUTIONS TO THE 3D COMPRESSIBLE ISOTHERMAL NAVIER-STOKES EQUATIONS

This paper concerns the global existence of strong solutions to the 3 D compressible isothermal Navier-Stokes equations with a vacuum at infinity.Based on the special structure of the Zlotnik inequality,the time uniform upper bounds for density are established through some time-dependant a priori estimates under the assumption that the total mass is suitably small.     

SOME PROPERTIES OF THE SOLUTION OF A FILTRATION PROBLEM IN PARTIALLY SATURATED POROUS MEDIA

A filtration problem with second initial-boundary value in partially saturated porous media isconsidered,In addition to discussion of the existence and uniqueness of weak solution of the problem,it is demonstrated that the interface between the saturated and unsaturated regions exists andcontinues under certain conditions and the solution possesses some properties,e.g.,the balance ofwater content,the time-limit existence of weak solution ect.,which differ from those of the solutionsof the first initial-boundary value problem.     

THE ENERGY CONSERVATION OF THE LANDAU-LIFSHITZ-BLOCH EQUATION

In this paper,we consider the three-dimensional Landau-Lifshitz-Bloch equation in the whole space,which can describe the micromagnetic dynamic behavior of material at all temperatures,especially near the Curie temperature.We establish a sufficient condition of energy conservation for when weak solutions of the Landau-Lifshitz-Bloch equation with the temperature higher than the Curie temperature and its gradient belong to the Besov space■ for some α∈(1/2,1) and p=9/3α+1.Moreover,we also use the d...     

Convergence of a finite difference method for combustion model problems

We study a finite difference scheme for a combustion model problem. A projection scheme near the combustion wave, and the standard upwind finite difference scheme away from the combustion wave are applied. Convergence to weak solutions with a combustion wave is proved under the normal Courant-Friedrichs-Lewy condition. Some con-     

Convergence of a finite difference method for combustion model problems

We study a finite difference scheme for a combustion model problem. A projection scheme near the combustion wave, and the standard upwind finite difference scheme away from the combustion wave are applied. Convergence to weak solutions with a combustion wave is proved under the normal Courant-Friedrichs-Lewy condition. Some conditions on the ignition temperature are given to guarantee the solution containing a strong detonation wave or a weak detonation wave. Convergence to strong detonation wave solutions for the random projection method is also proved.     

Finite difference method for a combustion model

We study a projection and upwind finite difference scheme for a combustion model problem. Convergence to weak solutions is proved under the Courant-Friedrichs-Lewy condition. More assumptions are given on the ignition temperature; then convergence to strong detonation wave solutions or to weak detonation wave solutions is proved.

     

GUARANTEED BOUNDS FOR THE SOLUTION OF THE WAVE EQUATION

Abstract

Numerical solution of the wave equation in the form of close lower and upper bounds provides a secure a posteriori error estimate that can be used for efficient accuracy control. The method considered in this paper uses some monotone properties of the differential operator in the wave equation to construct bounds for the solution in the form of trigonometric polynomials of x. Aspects of the numerical implementation, the accuracy of the computed bounds and some numerical examples are discussed.     

On supersonic combustion

Some basic concepts and features of supersonic combustion are explained from the view point of macroscopic aerodynamics. Two kinds of interpretations of supersonic combustion are proposed. The difference between supersonic combustion and subsonic combustion is discussed, and the mechanism of supersonic combustion propagation and the limitation of heat addition in supersonic flow are pointed out. The results of the calculation of deflagration in supersonic flow show that the entropy increment and the total pressure loss of the combustion products may decrease with the increase of combustion velocity. It is also demonstrated that the oblique detonation wave angle may not be controlled by the wedge angle under weak underdriven solution conditions and be determined only by combustion velocity. Therefore, the weak underdriven solution may become self-sustaining oblique detonation waves with a constant wave angle. Project supported by the National Natural Science Foundation of China (Grant Nos. 9482010, 59686005).     

Detailed simulation of the pulsating detonation wave in the shock-attached frame

The paper is devoted to the numerical investigation of the stability of propagation of pulsating gas detonation waves. For various values of the mixture activation energy, detailed propagation patterns of the stable, weakly unstable, irregular, and strongly unstable detonation are obtained. The mathematical model is based on the Euler system of equations and the one-stage model of chemical reaction kinetics. The distinctive feature of the paper is the use of a specially developed computational algorithm of the second approximation order for simulating detonation wave in the shock-attached frame. In distinction from shock capturing schemes, the statement used in the paper is free of computational artifacts caused by the numerical smearing of the leading wave front. The key point of the computational algorithm is the solution of the equation for the evolution of the leading wave velocity using the second-order grid-characteristic method. The regimes of the pulsating detonation wave propagation thus obtained qualitatively match the computational data obtained in other studies and their numerical quality is superior when compared with known analytical solutions due to the use of a highly accurate computational algorithm.     

The perturbation on initial binding energy for a Majda-CJ combustion model

We investigate the perturbed Riemann problem for a scalar Chapman–Jouguet combustion model – the perturbation on initial binding energy. Under the entropy conditions, we obtain the unique solutions in a neighbourhood of the origin (t?>?0) on the (x,?t) plane. It is found that, for some cases, the perturbed Riemann solutions are essentially different from the corresponding Riemann solutions. That is, the perturbation may transform a Chapman–Jouguet detonation into a strong detonation or a weak deflagration following a shock wave; a strong detonation into a weak deflagration following a shock wave; a Chapman–Jouguet deflagration into a weak deflagration.     

Shock-induced ignition of thermally sensitive explosives

The process of planar detonation ignition, induced by a constant-velocitypiston or equivalently by a shock reflected from a stationarywall, is investigated using high-resolution one-dimensionalnumerical simulations. The standard one-step model with Arrheniuskinetics, which models thermally sensitive explosives, is employed.Emphasis is on comparing and contrasting the results of thefinite activation temperature simulations with high activationtemperature asymptotic predictions and previous simulations.During the induction phase, it is shown that the asymptoticresults give qualitatively good predictions. However, for parametersrepresentative of gaseous explosives, subsequent to thermalrunaway at the piston and the formation of a reaction wave,the high activation temperature asymptotic theory is qualitativelyincorrect for moderately high activation temperatures. It isshown that the results are very sensitive to the value of theactivation temperature, especially the distance from the pistonat which a secondary shock forms and the degree of unsteadinessin the reaction wave which moves away from the piston. The dependenceof the ignition evolution on the other parameters (initial shockMach number, heat of reaction and polytropic index) is alsoinvestigated. It is shown that qualitative predictions regardingthe dependence of the ignition evolution on each of the parameterscan be elucidated from finite activation temperature homogeneousexplosion calculations together with the high activation temperatureasymptotic shock ignition results. It is found that for sufficientlystrong initiating shocks the ignition evolution is qualitativelydifferent from cases studied previously in that no secondaryshock forms. For a high polytropic index, corresponding to asimple equation of state model for condensed phase explosives,the results are in much better qualitative agreement with theasymptotic theory.