Nonlinear Stability Of Rarefaction Waves Research Articles

Nonlinear Stability of Rarefaction Waves to the Compressible Navier-Stokes Equations for a Reacting Mixture with Zero Heat Conductivity

Nonlinear Stability of Rarefaction Waves to the Compressible Navier-Stokes Equations for a Reacting Mixture with Zero Heat Conductivity

Nonlinear stability of rarefaction waves for the compressible MHD equations

Nonlinear stability of rarefaction waves for the compressible MHD equations

Nonlinear stability of rarefaction waves for micropolar fluid model with large initial perturbation

In this paper, we consider the nonlinear stability of solutions to one-dimensional compressible micropolar equations. If the Riemann problem of corresponding Euler equations admits a solution which is composed of 1-rarefaction wave and 3-rarefaction wave, we proved that the solution to micropolar equations is nonlinear stable to the composite waves as time goes to infinity. Compared to the previous studies, we established the result under large initial perturbation. The key point in our analysis is how to deduce the positive lower and upper bounds of the specific volume and the absolute temperature.

Stability of rarefaction waves for the bipolar Vlasov-Poisson-Boltzmann system with hard potentials

The purpose of this paper is to study the nonlinear stability of rarefaction waves for the one-dimensional bipolar Vlasov-Poisson-Boltzmann system with hard potentials. To this end, we first construct the global solutions near a local Maxwellian for the bipolar Vlasov-Poisson-Boltzmann system. Then the time asymptotic stability of rarefaction waves to compressible Euler equations is proved for the bipolar Vlasov-Poisson-Boltzmann system. Moreover, the exponential time decay rates of the disparity between two species and the electric field are obtained. Our analysis is based on the two different sets of decompositions of the solutions and a time velocity weight function which is designed to control the large velocity growth in the nonlinear term.

Nonlinear stability of rarefaction waves for a hyperbolic system with Cattaneo's law

<p style="text-indent:20px;">This paper is concerned with the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem and the initial-boundary value problem in the half space with impermeable wall boundary condition for a scalar conservation laws with an artificial heat flux satisfying Cattaneo's law. In our results, although the <inline-formula><tex-math id="M1">\begin{document}$ L^2\cap L^\infty- $\end{document}</tex-math></inline-formula>norm of the initial perturbation is assumed to be small, the <inline-formula><tex-math id="M2">\begin{document}$ H^1- $\end{document}</tex-math></inline-formula>norm of the first order derivative of the initial perturbation with respect to the spatial variable can indeed be large. Moreover the far fields of the artificial heat flux can be different. Our analysis is based on the <inline-formula><tex-math id="M3">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> energy method.</p>

Nonlinear Stability of Rarefaction Waves for a Compressible Micropolar Fluid Model with Zero Heat Conductivity

In 2018, Duan [1] studied the case of zero heat conductivity for a one-dimensional compressible micropolar fluid model. Due to the absence of heat conductivity, it is quite difficult to close the energy estimates. He considered the far-field states of the initial data to be constants; that is, $$\mathop {\lim }\limits_{x \to \pm \infty } ({v_0},{u_0},{\omega _0},{\theta _0})(x) = (1,0,0,1)$$ . He proved that the solution tends asymptotically to those constants. In this article, under the same hypothesis that the heat conductivity is zero, we consider the far-field states of the initial data to be different constants — that is, $$\mathop {\lim }\limits_{x \to \pm \infty } ({v_0},{u_0},{\omega _0},{\theta _0})(x) = ({v_ \pm },{u_ \pm },0,{\theta _ \pm })$$ -and we prove that if both the initial perturbation and the strength of the rarefaction waves are assumed to be suitably small, the Cauchy problem admits a unique global solution that tends time — asymptotically toward the combination of two rarefaction waves from different families.

Nonlinear stability of rarefaction waves for a viscous radiative and reactive gas with large initial perturbation

We investigate the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem of an one-dimensional compressible Navier-Stokes type system for a viscous, compressible, radiative and reactive gas, where the constitutive relations for the pressure $p$, the specific internal energy $e$, the specific volume $v$, the absolute temperature $\theta$, and the specific entropy $s$ are given by $p=R\theta/v +a\theta^4/3$, $e=C_v\theta+av\theta^4$, and $s=C_v\ln \theta+ 4av\theta^3/3+R\ln v$ with $R>0$, $C_{v}>0$, and $a>0$ being the perfect gas constant, the specific heat and the radiation constant, respectively. For such a specific gas motion, a somewhat surprising fact is that, general speaking, the pressure $\widetilde{p}(v,s)$ is not a convex function of the specific volume $v$ and the specific entropy $s$. Even so, we show in this paper that the rarefaction waves are time-asymptotically stable for large initial perturbation provided that the radiation constant $a$ and the strength of the rarefaction waves are sufficiently small. The key point in our analysis is to deduce the positive lower and upper bounds on the specific volume and the absolute temperature, which are uniform with respect to the space and the time variables, but are independent of the radiation constant $a$.

Global stability of rarefaction waves for a viscous radiative and reactive gas with temperature-dependent viscosity

We study the nonlinear stability of rarefaction waves to the Cauchy problem of a one-dimensional viscous radiative and reactive gas when the viscosity and heat conductivity coefficients depend on both density and absolute temperature. Our main idea is to use the smallness of the strength of the rarefaction waves to control the possible growth of its solutions induced by the nonlinearity of the system and the interactions of rarefaction waves from different families. The proof is based on some detailed analysis on uniform positive lower and upper bounds of the specific volume and the absolute temperature.

Nonlinear stability of rarefaction waves for one-dimensional compressible Navier–Stokes equations for a reacting mixture

In this paper, we study the long-time behavior toward rarefaction waves for the Cauchy problem to a one-dimensional Navier–Stokes equations for a reacting mixture. It is shown that under the condition adiabatic exponent $$\gamma $$ is close to 1, the global stability is established. In this paper, the initial perturbation can be large.

Nonlinear stability of rarefaction waves for the compressible Navier–Stokes equations with zero heat conductivity

This paper is concerned with the time-asymptotic nonlinear stability of rarefaction waves to the Cauchy problem of the one-dimensional compressible Navier–Stokes equations with zero heat conductivity. Under the assumption that the unique global entropy solution to the resulting Riemann problem of the corresponding compressible Euler equations consists of rarefaction waves only, then if both the initial perturbation and the strengths of rarefaction waves are assumed to be suitably small, we show that its Cauchy problem admits a unique global solution which tends time-asymptotically toward the rarefaction waves. This result is proved by using the elementary energy method and the argument developed by Kawashima and Okada (1982).

Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients

We study the nonlinear stability of rarefaction waves to the Cauchy problem of one-dimensional compressible Navier-Stokes equations for a viscous and heat conducting ideal polytropic gas when the transport coefficients depend on both temperature and density. When the strength of the rarefaction waves is small or the rarefaction waves of different families are separated far enough initially, we show that rarefaction waves are nonlinear stable provided that $(\gamma- 1)\cdot H^3(\mathbb{R})$-norm of the initial perturbation is suitably small with $\gamma>1$ being the adiabatic gas constant.

Nonlinear stability of rarefaction waves for the compressible Navier-Stokes equations with large initial perturbation

The expansion waves for the compressible Navier-Stokes equations have recently been shown to be nonlinear stable. The nonlinear stability results are called local stability or global stability depending on whether the H1 -norm of the initial perturbation is small or not. Up to now, local stability results have been well established. However, for global stability, only partial results have been obtained. The main purpose of this paper is to study the global stability of rarefaction waves for the compressible Navier-Stokes equations. For this purpose, we introduce a positive parameter to in the construction of smooth approximations of the rarefaction wave solutions for the compressible Euler equations so that the quantity £ = ty (6 denotes the strength of the rarefaction waves) is sufficiently large to control the growth induced by the nonlinearity of the system and the interaction of waves from different families. Then by using the energy method together with the continuation argument, we obtain some nonlinear stability results provided that the initial perturbation satisfies certain growth conditions as £ ► +oo. Notice that the assumption that the quantity £ can be chosen to be sufficiently large implies that either the strength of the rarefaction waves is small or the rarefaction waves of different families are separated far enough initially.

Nonlinear Stability of Rarefaction Waves for the Boltzmann Equation

It is well known that the Boltzmann equation is related to the Euler and Navier-Stokes equations in the field of gas dynamics. The relation is either for small Knudsen number, or, for dissipative waves in the time-asymptotic sense. In this paper, we show that rarefaction waves for the Boltzmann equation are time-asymptotic stable and tend to the rarefaction waves for the Euler and Navier-Stokes equations. Our main tool is the combination of techniques for viscous conservation laws and the energy method based on micro-macro decomposition of the Boltzmann equation. The expansion nature of the rarefaction waves and the suitable microscopic version of the H-theorem are essential elements of our analysis.

Nonlinear Stability of Rarefaction Waves for a Viscoelasticity

Nonlinear Stability of Rarefaction Waves for a Viscoelasticity

Nonlinear Stability of Rarefaction Waves for a Viscoelastic Material with Memory

Nonlinear Stability of Rarefaction Waves for a Viscoelastic Material with Memory

Nonlinear stability of rarefaction waves for a viscoelastic material with memory

In this paper we will discuss the stability of rarefaction waves for a viscoelastic material with memory. The rarefaction waves for which the stability is tested are not themselves solutions to the integrodifferential equations (1.1) governing the viscoelastic material. They are solutions to a related equilibrium system of conservation laws given by (1.11). We shall show that if the forcing term and the past history are small and if the initial data are close to the rarefaction waves, the solutions to (1.1) will approach the rarefaction waves in sup norm as the time goes to infinity.

Nonlinear stability of rarefaction waves for compressible Navier Stokes equations

It is shown that expansion waves for the compressible Navier-Stokes equations are nonlinearly stable. The expansion waves are constructed for the compressible Euler equations based on the inviscid Burgers equation. Our result shows that Navier-Stokes equations and Euler equations are time-asymptotically equivalent on the level of expansion waves. The result is proved using the energy method, making essential use of the expansion of the underlining nonlinear waves and the specific form of the constitutive eqution for a polytropic gas.