Fluid dynamic limit to compressible Euler equations from compressible Navier-Stokes equations and Boltzmann equation has been an active topic with limited success so far. In this paper, we consider the case when the solution of the Euler equations is a Riemann solution consisting two rarefaction waves and a contact discontinuity and prove this limit for both Navier-Stokes equations and the Boltzmann equation when the viscosity, heat conductivity coefficients and the Knudsen number tend to zero respectively. In addition, the uniform convergence rates in terms of the above physical parameters are also obtained. It is noted that this is the first rigorous proof of this limit for a Riemann solution with superposition of three waves even though the fluid dynamic limit for a single wave has been proved.
This paper is concerned with the large-time behavior of solutions for the one-dimensional compressible Navier-Stokes system. We show that the combination of viscous contact wave with rarefaction waves for the non-isentropic polytropic gas is stable under \emph{large} initial perturbation without the condition that the adiabatic exponent $γ$ is close to 1, provided the strength of the combination waves is suitably small.
This paper is concerned with the study of nonlinear stability of superposition of boundary layer and rarefaction wave on the two-fluid Navier-Stokes-Poisson system in the half line $\mathbb{R}_{+}=:(0,+\infty)$. On account of the quasineutral assumption and the absence of the electric field for the large time behavior, we successfully construct the boundary layer and rarefaction wave, and then we give the rigorous proofs of the stability theorems on the superposition of boundary layer and rarefaction wave under small perturbations for the corresponding initial boundary value problem of the Nav…
Stability of Superposition of Viscous Contact Wave and Rarefaction Waves for Compressible Navier-Stokes System with Degenerate Heat-Conductivity and Large-Data
We study the large-time asymptotic behavior of solutions toward the combination of a viscous contact wave with two rarefaction waves for the compressible non-isentropic Navier-Stokes equations coupling with the Maxwell equations through the Lorentz force (called the Navier-Stokes-Maxwell equations). It includes the electrodynamic effects into the dissipative structure of the hyperbolic-parabolic system and turns out to be more complicated than that in the simpler compressible Navier-Stokes equations. Based on a new observation of the specific structure of the Maxwell equations in the Lagrangia…
Considering the space-periodic perturbations, we prove the time-asymptotic stability of the composite wave of a viscous contact wave and two rarefaction waves for the Cauchy problem of 1-D compressible Navier-Stokes equations in this paper. This kind of perturbations keep oscillating at the far field and are not integrable. The key is to construct a suitable ansatz carrying the same oscillation %eliminating the oscillation of the solution as in \cite{HuangXuYuan2020,HuangYuan2021}, but due to the degeneration of contact discontinuity, the construction is more subtle. We find a way to use the s…
Nonlinear Stability for the Superposition of Viscous Contact Wave and Rarefaction Waves to Non-isentropic Compressible Navier-Stokes System with General Initial Perturbations
Multi-Dimensional Structural Stability of Mixed Riemann Configurations Containing Centered Rarefaction Waves and Surfaces of Discontinuities of Gas Dynamics
School of Mathematics, Hunan University（湖南大学数学学院）；Department of Mathematics, City University of Hong Kong（香港城市大学数学系）
Global stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system with temperature-dependent transport coefficients and large data
We study supersonic flow past a convex corner which is surrounded by quiescent gas. When the pressure of the upstream supersonic flow is larger than that of the quiescent gas, there appears a strong rarefaction wave to rarefy the supersonic gas. Meanwhile, a transonic characteristic discontinuity appears to separate the supersonic flow behind the rarefaction wave from the static gas. In this paper, we employ a wave front tracking method to establish structural stability of such a flow pattern under non-smooth perturbations of the upcoming supersonic flow. It is an initial-value/free-boundary p…
We investigate the nonlinear stability of the superposition of a viscous contact wave and two rarefaction waves for one-dimensional bipolar Vlasov-Poisson-Boltzmann (VPB) system, which can be used to describe the transportation of charged particles under the additional electrostatic potential force. Based on a new micro-macro type decomposition around the local Maxwellian related to the bipolar VPB system in our previous work [26], we prove that the superposition of a viscous contact wave and two rarefaction waves is time-asymptotically stable to 1D bipolar VPB system under some smallness cond…
The Riemann solutions to Chaplygin Euler equations with a scaled pressure are considered. When the pressure vanishes, there are three cases. The Riemann solution containing two shock waves converges to the delta shock wave solution of the transport equations. During this process, both the strength and propagation speed of the delta shock are investigated in detail. We find that there is something different from that for polytropic or isothermal gas. The Riemann solution containing two rarefaction waves tends to the two contact discontinuity solution to the transport equations as the pressure g…
We prove the time-asymptotic stability of composite waves consisting of the superposition of a viscous shock and a rarefaction for the one-dimensional compressible barotropic Navier-Stokes equations. Our result solves a long-standing problem first mentioned in 1986 by Matsumura and Nishihara in [25]. The same authors introduced it officially as an open problem in 1992 in [26] and it was again described as very challenging open problem in 2018 in the survey paper [23]. The main difficulty is due to the incompatibility of the standard anti-derivative method, used to study the stability of viscou…
We are concerned with the large time behavior of solutions to the Cauchy problem of the one-dimensional compressible micropolar fluid model without viscosity, where the far-field states of the initial data are prescribed to be different. If the corresponding Riemann problem of the compressible Euler system admits a contact discontinuity and two rarefaction waves solutions, we show that for such a non-viscous model, the combination of the viscous contact wave with two rarefaction waves is time-asymptotically stable provided that the strength of the composite wave and the initial perturbation ar…
The interaction of elementary waves for isentropic flow in a variable cross-section duct is obtained (\cite{ShengZhang}). The authors have discussed rarefaction wave or shock wave interacts with stationary wave. In this paper, we extend their results to the nonisentropic flow. It can be proved that if one changes $(u,ρ)$ plane in isentropic flow with $(u,p)$ plane in nonisentropic case, the interaction results of rarefaction wave or shock wave with stationary wave can be moved parallel from the isentropic case when contact discontinuity is involved. Thus we mainly focus on the interactions bet…
In this paper, two kinds of occurrence mechanism on the phenomenon of concentration and the formation of delta shock wave in the flux approximation limit of Riemann solutions to the extended Chaplygin gas equations are analyzed and identified. Firstly, the Riemann problem of the extended Chaplygin gas equations is solved completely. Secondly, we rigorously show that, as the pressure vanishes, any two-shock Riemann solution to the extended Chaplygin gas equations tends to a $δ$-shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted $δ$…
By the flux-approximation method, we study limits of Riemann solutions to the Brio system with two independent parameters. The Riemann problem of the perturbed system is solved analytically, and four kinds of solutions are obtained constructively. It is shown that, as the two-parameter flux perturbation vanishes, any two-shock-wave and two-rarefaction-wave solutions of the perturbed Brio system converge to the delta-shock and vacuum solutions of the transport equations, respectively. In addition, we specially pay attention to the Riemann problem of a simplified system of conservation laws deri…
We study the large-time behavior of solutions to the compressible Navier-Stokes equations for a viscous and heat-conducting ideal polytropic gas in the one-dimensional half-space. A rarefaction wave and its superposition with a non-degenerate stationary solution are shown to be asymptotically stable for the outflow problem with large initial perturbation and general adiabatic exponent.
This paper is concerned with the Riemann problem for the isentropic Chaplygin gas magnetogasdynamics equations and the formation of delta shocks and vacuum states as pressure and magnetic field vanish. Firstly, the Riemann problem of the isentropic magnetogasdynamics equations for Chaplygin gas is solved analytically. Secondly, it is rigorously proved that, as both the pressure and the magnetic field vanish, the Riemann solution containing two shock waves tends to a delta shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted $δ$-meas…
In this paper, two kinds of occurrence mechanism on the phenomenon of concentration and the formation of delta shock waves are analyzed and identified in the flux approximation limit of Riemann solutions to the extended Chaplygin gas equations with Coulomb-like friction, whose special case can also be seen as the model of the magnetogasdynamics with Coulomb-like friction. Firstly, by introducing a transformation, the Riemann problem for the extended Chaplygin gas equations with Coulomb-like friction is solved completely. Secondly, we rigorously show that, as the pressure vanishes, any two-shoc…
In 1977 Korchinski presented a new type of shock discontinuity in conservation laws. These singular solutions were coined $δ$-shocks since there is a time dependent Dirac delta involved. A naive description is that such $δ$-shock is of the overcompressive type: a two-family shock wave the four characteristic lines of which impinge into the shock itself. In this work, we open the fan of solutions by studying two-family waves without intermediate constant states but, possessing central rarefactions and also comprising $δ$-shocks.
In this paper, we study the limits of Riemann solutions to the inhomogeneous Euler equations of one-dimensional compressible fluid flow as the adiabatic exponent $γ$ tends to one. Different from the homogeneous equations, the Riemann solutions of the inhomogeneous system are non self-similar. It is rigorously shown that, as $γ$ tends to one, any two-shock Riemann solution tends to a delta shock solution of the pressureless Euler system with a source term, and the intermediate density between the two shocks tends to a weighted $δ$-mesaure which forms the delta shock; while any two-rarefaction-w…
Recently there has been interest in studying a new class of elastic materials, which is described by implicit constitutive relations. Under some basic assumption for elasticity constants, the system of governing equations of motion for this elastic material is strictly hyperbolic but without the convexity property. In this paper, all wave patterns for the nonclassic nonlinearly elastic materials under Riemann data are established completely by separating the phase plane into twelve disjoint regions and by using a nonnegative dissipation rate assumption and the maximally dissipative kinetics at…
We are concerned with the formation of singularities and the existence of global continuous solutions of the Cauchy problem for the one-dimensional non-isentropic Euler equations for compressible fluids. For the isentropic Euler equations, we pinpoint a necessary and sufficient condition for the formation of singularities of solutions with large initial data that allow a far-field vacuum -- there exists a compression in the initial data. For the non-isentropic Euler equations, we identify a sufficient condition for the formation of singularities of solutions with large initial data that allow …
This paper analyzes the vanishing pressure limit of solutions to the Aw-Rascle model and the perturbed Aw-Rascle model for modified Chaplygin gas. Firstly, the Riemann problem of the Aw-Rascle model is solved constructively. A special delta shock wave in the limit of Riemann solutions is obtained. Secondly, the Riemann problem of the perturbed Aw-Rascle model is solved analytically. It is proved that, as the pressure tends to zero, any Riemann solution containing two shock wave tends to a delta shock solution to the transport equations; any Riemann solution containing two rarefaction wave tend…
A fundamental question in fluid dynamics concerns the formation of discontinuous shock waves from smooth initial data. We prove that from smooth initial data, smooth solutions to the 2d Euler equations in azimuthal symmetry form a first singularity, the so-called $C^{\frac{1}{3}} $ pre-shock. The solution in the vicinity of this pre-shock is shown to have a fractional series expansion with coefficients computed from the data. Using this precise description of the pre-shock, we prove that a discontinuous shock instantaneously develops after the pre-shock. This regular shock solution is shown to…
