Global Existence Of Entropy Solutions Research Articles

Global entropy solutions for systems modelling polymer flooding in enhanced oil recovery

In this paper, we obtain the existence of global entropy solutions for the Cauchy problem of the nonstrictly hyperbolic systems modelling polymer flooding in enhanced oil recovery, under a more flexible condition on the function β(T), which models the adsorption of the polymer on rock. This work improves the previous results in the paper Lu (2013), where β(T) is limited to meas{T:β′′(T)=0}=0 or β(T)=bT for a nonnegative constant b.

Stability of Conical Shocks in the Three-Dimensional Steady Supersonic Isothermal Flows Past Lipschitz Perturbed Cones

We are concerned with the structural stability of conical shocks in the three-dimensional steady supersonic flows past Lipschitz perturbed cones whose vertex angles are less than the critical angle. The flows under consideration are governed by the steady isothermal Euler equations for potential flow with axisymmetry so that the equations contain a singular geometric source term. We first formulate the shock stability problem as an initial-boundary value problem with the leading conical shock-front as a free boundary, and then establish the existence and asymptotic behavior of global entropy solutions of bounded variation (BV) of the problem. To achieve this, we first develop a modified Glimm scheme to construct approximate solutions via self-similar solutions as building blocks in order to incorporate with the geometric source term. Then we introduce the Glimm-type functional, based on the local interaction estimates between weak waves, the strong leading conical shock, and self-similar solutions, as well as the estimates of the center changes of the self-similar solutions. To make sure the decreasing of the Glimm-type functional, we choose appropriate weights by careful asymptotic analysis of the reflection coefficients in the interaction estimates, when the Mach number of the incoming flow is sufficiently large. Finally, we establish the existence of global entropy solutions involving a strong leading conical shock-front, besides weak waves, under the conditions that the Mach number of the incoming flow is sufficiently large and the weighted total variation of the slopes of the generating curve of the Lipschitz perturbed cone is sufficiently small. Furthermore, the entropy solution is shown to approach asymptotically the self-similar solution that is determined by the incoming flow and the asymptotic tangent of the cone boundary at infinity.

Existence of global solutions for isentropic gas flow with friction

In this paper, we study the existence of global entropy solutions for the Cauchy problem of the extended river flow system with a friction α(x) depending on the space variable x. First, under the boundedness of , we apply the viscosity-flux approximation method coupled with the maximum principle to obtain the a priori L∞ estimates for the approximation solutions. Second, we prove the pointwise convergence of the approximation solutions by using the compactness framework from the compensated compactness theory. Finally, as a by-product, we obtain the existence of global solutions for isentropic gas flow in the Laval nozzle with friction.

Hypersonic similarity for the two dimensional steady potential flow with large data

In this paper, we establish the first rigorous mathematical result on the validation of the hypersonic similarity globally, which is also called the Mach-number independence principle, for the two dimensional steady potential flow. The hypersonic similarity is equivalent to the Van Dyke's similarity theory, that is, if the hypersonic similarity parameter K is fixed, the shock solution structures (after scaling) are consistent, when the Mach number of the flow is sufficiently large. One of the difficulty is that after scaling, the solutions are usually of large data since the perturbation of the hypersonic flow is usually not small related to the sonic speed. In order to make it, we first develop a modified Glimm scheme to construct the approximate solutions with large data and find fine structure of the elementary wave curves to obtain the global existence of entropy solutions with large data, for fixed K and sufficiently large Mach number of the incoming flow M∞. Finally, we further show that for a fixed hypersonic similarity parameter K, if the Mach number M∞→∞, the solutions obtained above approach to the solution of the corresponding initial-boundary value problem of the hypersonic small-disturbance equations. Therefore, the Van Dyke's similarity theory is verified rigorously for the first time.

Global existence and stability to the polytropic gas dynamics with an outer force

In this paper, we apply the viscosity-flux approximation method coupled with the maximum principle to obtain the a-priori L∞ estimates for the approximation solutions of the compressible polytropic gas dynamics system with an outer force, and prove the global existence of entropy solutions for any adiabatic exponent γ>1.The present problem was classical and the global existence was known long ago. However, the L∞ estimates grow larger with respect to the time variable. As a result, it does not guarantee the stability of solutions. We thus prove the uniformly L∞ estimate with respect to the time variable. This means that our solutions are stable. The key idea is to employ an invariant region depending on the space and time variables. This method enables us to investigate the behavior of solutions in detail and deduce the desired estimates.

Stability of Rarefaction Wave to the 1-D Piston Problem for the Pressure-Gradient Equations

The 1-D piston problem for the pressure gradient equations arising from the flux-splitting of the compressible Euler equations is considered. When the total variations of the initial data and the velocity of the piston are both sufficiently small, the author establishes the global existence of entropy solutions including a strong rarefaction wave without restriction on the strength by employing a modified wave front tracking method.

Global Entropy Solutions to the Gas Flow in General Nozzle

We are concerned with the global existence of entropy solutions for the compressible Euler equations describing the gas flow in a nozzle with general cross-sectional area, for both isentropic and isothermal fluids. New viscosities are delicately designed to obtain the uniform bound of approximate solutions. The vanishing viscosity method and compensated compactness framework are used to prove the convergence of approximate solutions. Moreover, the entropy solutions for both cases are uniformly bounded independent of time. No smallness condition is assumed on initial data. The techniques developed here can be applied to compressible Euler equations with general source terms.

Global Existence of Shock Front Solution to 1-D Piston Problem for Compressible Euler Equations

We study the 1-D piston problem for compressible Euler equations under the assumptions that the total variations of the initial data and the piston velocity are both sufficiently small. By a modified Glimm scheme, we establish the global existence of entropy solutions including a strong shock wave without restriction on the strength.

Global solutions to isothermal system in a divergent nozzle with friction

In this paper, we remove the restriction A′(x)≥0 in the paper ‘Lu (2011)’, the restriction z0(x)≤0 or w0(x)≤0 in the paper ‘Klingenberg and Lu (1997)’, and obtain the global existence of entropy solutions to the isothermal gas dynamics system in a divergent nozzle with friction.

Large time behavior of entropy solutions to one-dimensional unipolar hydrodynamic model for semiconductor devices

We are concerned with the global existence and large time behavior of entropy solutions to the one dimensional unipolar hydrodynamic model for semiconductors in the form of Euler-Poisson equations in a bounded interval. In this paper, we first prove the global existence of entropy solution by vanishing viscosity and compensated compactness framework. In particular, the solutions are uniformly bounded with respect to space and time variables by introducing modified Riemann invariants and the theory of invariant region. Based on the uniform estimates of density, we further show that the entropy solution converges to the corresponding unique stationary solution exponentially in time. No any smallness condition is assumed on the initial data and doping profile. Moreover, the novelty in this paper is about the unform bound with respect to time for the weak solutions of the isentropic Euler-Possion system.

Global bounded variation solutions describing Fanno–Rayleigh fluid flows in nozzles

In this paper, we investigate the initial-boundary value problem of compressible Euler equations including friction and heating that model the transonic Fanno–Rayleigh flows through symmetric variable area nozzles. In particular, the case of contracting nozzles is considered. A new version of a generalized Glimm scheme (GGS) is presented for establishing the global existence of entropy solutions with bounded variation. Modified Riemann and boundary Riemann solutions are applied to design this GGS, which is constructed using the contraction matrices acting on the homogeneous Riemann (or boundary-Riemann) solutions. The extended Glimm–Goodman’s type of wave interaction estimates are investigated to determine the stability of the scheme and the positivity of gas velocity that results in the existence of the weak solution. The limit of approximation solutions serves as an entropy solution. Moreover, a quantitative relation between the shape of the nozzle, friction, and heat is proposed for the global existence result in the contracting nozzle. Numerical simulations of the contraction-expansion and expansion-contraction nozzles are presented to validate the scheme.

Global existence of solutions to system of polytropic gas dynamics with friction

In this paper, we study the existence of global entropy solutions for the Cauchy problem of system of polytropic gas dynamics in a divergent nozzle with a friction (1.1) with bounded L∞ initial data (1.2).

Non-relativistic limits of rarefaction wave to the 1-D piston problem for the isentropic relativistic Euler equations

We consider the 1-D piston problem for the isentropic relativistic Euler equations when the total variations of the initial data and the speed of the piston are both sufficiently small. By a modified wave front tracking method, we establish the global existence of entropy solutions including a strong rarefaction wave without restriction on the strength. Meanwhile, we study the convergence of the entropy solutions to the corresponding entropy solutions of the classical non-relativistic isentropic Euler equations as the light speed c→+∞.

Concentration of source terms in generalized Glimm scheme for global entropy solutions of nonlinear hyperbolic balance laws

In this paper, we investigate the Cauchy problem for a nonlinear hyperbolic system of balance laws with sources axg and ath. To get the approximate solutions of our problem, we consider a version of generalized Riemann problem that concentrates the variation of a on a thin T-shaped region of each grid. A new version of Glimm scheme is introduced to construct the approximate solutions and its stability is proved by considering two types of conditions on a. Finally, we verify the consistency of the scheme and the entropy inequality to establish the global existence of entropy solutions.

Existence and stability of rarefaction wave to 1-D piston problem for the relativistic full Euler equations

We consider the 1-D piston problem for the compressible relativistic full Euler equations when the total variations of the initial data and the piston velocity are both sufficiently small. By using a modified wave front tracking method, we establish the global existence of entropy solutions including a strong rarefaction wave without restriction on the strength. Moreover, we study the asymptotic behavior of the solutions as t→+∞.

Stability and non-relativistic limits of rarefaction wave to the 1-D piston problem for the relativistic Euler equations

We study the 1-D piston problem for the relativistic Euler equations under the assumption that the total variations of both the initial data and the velocity of the piston are sufficiently small. By a modified wave front tracking method, we establish the global existence of entropy solutions including a strong rarefaction wave without restriction on the strength. Meanwhile, we consider the convergence of the entropy solutions to the corresponding entropy solutions of the classical non-relativistic Euler equations as the light speed \(c\rightarrow +\infty \).

Two-Dimensional Steady Supersonic Exothermically Reacting Euler Flow past Lipschitz Bending Walls

We are concerned with the two-dimensional steady supersonic reacting Euler flow past Lipschitz bending walls that are small perturbations of a convex one, and establish the existence of global entropy solutions when the total variation of both the initial data and the slope of the boundary is sufficiently small. The flow is governed by an ideal polytropic gas and undergoes a one-step exothermic chemical reaction under the reaction rate function that is Lipschtiz and has a positive lower bound. The heat released by the reaction may cause the total variation of the solution to increase along the flow direction. We employ the modified wavefront tracking scheme to construct approximate solutions and develop a Glimm-type functional by incorporating the approximate strong rarefaction waves and Lipschitz bending walls to obtain the uniform bound on the total variation of the approximate solutions. Then we employ this bound to prove the convergence of the approximate solutions to a global entropy solution that contains a strong rarefaction wave generated by the Lipschitz bending wall. In addition, the asymptotic behavior of the entropy solution in the flow direction is also analyzed.

Global stability of rarefaction wave to the 1-D piston problem for the Compressible Full Euler Equations

In this paper, we study the 1-D piston problem for compressible full Euler equations under the assumptions that the total variation of the initial data and the perturbation of the piston velocity are sufficiently small. By a modified wave front tracking method, we establish the global existence of entropy solutions including a strong rarefaction wave without restriction on the strength. Moreover, we establish the asymptotic behavior of the solutions as t→∞.

Existence of Entropy Solutions to Two-Dimensional Steady Exothermically Reacting Euler Equations

We are concerned with the global existence of entropy solutions of the two-dimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics. The reaction rate function ϕ(T) is assumed to have a positive lower bound. We first consider the Cauchy problem (the initial value problem), that is, seek a supersonic downstream reacting flow when the incoming flow is supersonic, and establish the global existence of entropy solutions when the total variation of the initial data is sufficiently small. Then we analyze the problem of steady supersonic, exothermically reacting Euler flow past a Lipschitz wedge, generating an additional detonation wave attached to the wedge vertex, which can be then formulated as an initial-boundary value problem. We establish the global existence of entropy solutions containing the additional detonation wave (weak or strong, determined by the wedge angle at the wedge vertex) when the total variation of both the slope of the wedge boundary and the incoming flow is suitably small. The downstream asymptotic behavior of the global solutions is also obtained.

Existence of global entropy solutions to general system of Keyfitz–Kranzer type

In this paper, we apply the div–curl lemma in the compensated compactness theory (Tartar, 1979 [38], Murat, 1978 [34]) to the special pair of functions (c,wiε(x,t)) to obtain a very short proof of the existence of global entropy solutions for quite general system (1.1) of Keyfitz–Kranzer type (Keyfitz and Kranzer, 1980 [24]), where c is a constant and wiε(x,t) are variants in (1.1). This work extends in some sense the previous work by the author (Lu, 2011 [32]) for a special system of Keyfitz–Kranzer or Aw–Rascle type. As a by-product, a simple proof of the existence and stability of entropy solutions is also obtained for the hyperbolic system of isentropic gas dynamics in Eulerian coordinates under the compactness assumption of uxε(x,t) or ρxε(x,t) in Wloc−1,α(R×R+), α∈(1,2), where uε(x,t), ρε(x,t) are viscosity velocity and density.