One-dimensional Compressible Navier-Stokes Equations Research Articles

Some controllability results for linearized compressible Navier-Stokes system with Maxwell's law

In this article, we study the control aspects of the one-dimensional compressible Navier-Stokes equations with Maxwell's law linearized around a constant steady state with zero velocity. We consider the linearized system with Dirichlet boundary conditions and with interior controls. We prove that the system is not null controllable at any time using localized controls in density and stress equations and even everywhere control in the velocity equation. However, we show that the system is null controllable at any time if the control acting in density or stress equation is everywhere in the domain. This is the best possible null controllability result obtained for this system. Next, we show that the system is approximately controllable at large time using localized controls. Thus, our results give a complete understanding of the system in this direction.

Stability and decay rate of viscous contact wave to one-dimensional compressible Navier-Stokes equations

Stability and decay rate of viscous contact wave to one-dimensional compressible Navier-Stokes equations

Decay rate to contact discontinuities for the one-dimensional compressible Navier–Stokes equations with a reacting mixture

In this paper, we investigate the nonlinear stability of contact waves for the Cauchy problem to the compressible Navier–Stokes equations for a reacting mixture in one dimension. If the corresponding Riemann problem for the compressible Euler system admits a contact discontinuity solution, it is shown that the contact wave is nonlinearly stable, while the strength of the contact discontinuity and the initial perturbation are suitably small. Especially, we obtain the convergence rate by using anti-derivative methods and elaborated energy estimates.

Global Stability and Nonvanishing Vacuum States of 3D Compressible Navier–Stokes Equations

We investigate global stability and nonvanishing vacuum states of large solutions to the compressible Navier–Stokes equations on the torus , and the main purpose of this work is threefold. First, under the assumption that the density verifies , it is shown that the solutions converge to an equilibrium state exponentially in the -norm. In contrast to previous related works where the density has uniform positive lower and upper bounds, this gives the first stability result for large strong solutions of the three-dimensional compressible Navier–Stokes equations in the presence of vacuum. Second, by employing some new thoughts, we also show that the density converges to its equilibrium state exponentially in the -norm if additionally the initial density satisfies . Finally, we prove that the vacuum state will persist for any time provided that the initial density contains vacuum, which is different from the previous work of [H. L. Li, J. Li, and Z. P. Xin, Comm. Math. Phys., 281 (2008), pp. 401–444], where the authors showed that any vacuum state must vanish within finite time for the free boundary problem of the one-dimensional compressible Navier–Stokes equations with density-dependent viscosity with . This phenomenon implies the different behaviors for Navier–Stokes equations with different types of viscous effects, namely, degenerate or not.

Time-asymptotic stability of composite waves of viscous shock and rarefaction for barotropic Navier-Stokes equations

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 [28]. The same authors introduced it officially as an open problem in 1992 in [29] and it was again described as very challenging open problem in 2018 in the survey paper [26]. The main difficulty is due to the incompatibility of the standard anti-derivative method, used to study the stability of viscous shocks, and the energy method used for the stability of rarefactions. Instead of the anti-derivative method, our proof uses the a-contraction with shifts theory recently developed by two of the authors. This method is energy based, and can seamlessly handle the superposition of waves of different kinds.

Asymptotic stability of a composite wave to the one-dimensional compressible Navier-Stokes equations for a reacting mixture

This paper is concerned with the large time behavior of solutions for the Cauchy problem to the compressible Navier-Stokes equations for a reacting mixture with zero viscosity in one dimension. If the corresponding Riemann problem for the compressible Euler system admits the solutions consisting of rarefaction waves and contact discontinuity, it is shown that the composition of two rarefaction waves with a viscous contact wave is asymptotically stable, while the strength of the composite wave and the initial perturbation are suitably small.

The limits of coefficients of angular viscosity and microrotation viscosity to one-dimensional compressible Navier-Stokes equations for micropolar fluids model

In this paper, we are concerned with the initial boundary value problem of the one-dimensional micropolar fluids model, which reacts to miniature rotational movements and spin inertia. The size of the microrotation viscosity and angular viscosity coefficients allows us to measure, in a certain sense, the derivation of flows of micropolar fluids from the flows of the Navier-Stokes model. Here we study vanishing coefficients limit of the above two viscosities, and give the convergence rates.

Formation of singularities for one-dimensional relaxed compressible Navier-Stokes equations

We investigate the formation of singularities in one-dimensional hyperbolic compressible Navier-Stokes equations, a model proposing a relaxation leading to a hyperbolization through a nonlinear Cattaneo law for heat conduction as well as through the constitutive Maxwell type relations for the stress tensor. By using the entropy dissipation inequality, which gives the lower energy estimates of the local solutions without any smallness condition on initial data, and by constructing some useful averaged quantities we show that there are in general no global C1 solutions for the studied system with some large initial data.This appears as a remarkable contrast to the situation without relaxation, i.e. for the classical compressible Navier-Stokes equations, where global large solutions exist. It also contrasts the fact that for the linearized system associated to the classical resp. relaxed compressible Navier-Stokes equations, the qualitative behavior is exactly the same: exponential stability in bounded domains and polynomial decay without loss of regularity for the Cauchy problem.

Decay rates to viscous contact wave for the one-dimensional compressible Navier–Stokes equations with radiation

In this paper, we consider the stability and decay rate of viscous contact wave for the Cauchy problem of one-dimensional compressible Navier–Stokes equations with radiation under the zero mass condition. we consider the case that the constitutive relations for the pressure p, the specific internal energy e, the specific volume v and the absolute temperature θ are given by p=Rθ/v+aθ4/3, e=Cvθ+avθ4, with R>0, Cv>0, and a>0 being the perfect gas constant, the specific heat and the radiation constant, respectively.Our main idea is to use the smallness of the strength of the contact wave to control the possible growth of its solutions induced by the nonlinearity of the system and interactions between the solutions and the contact wave. The key point in our analysis is to obtain the uniform convergence rate of a contact wave pattern (1+t)−58ln12(2+t) by using a new inequality (see Lemma 2.3) and a weighted characteristic energy estimate.

Global Convergence to Compressible Full Navier–Stokes Equations by Approximation with Oldroyd-Type Constitutive Laws

We consider smooth solutions to a relaxed Euler system with Oldroyd-type constitutive laws. This system is derived from the one-dimensional compressible full Navier-Stokes equations for a Newtonian fluid by using the Cattaneo–Christov model and the Oldroyd-B model. In a neighborhood of equilibrium states, we construct an explicit symmetrizer and show that the system is symmetrizable hyperbolic with partial dissipation. Moreover, by establishing uniform estimates with respect to the relaxation times, we prove the uniform global existence of smooth solutions and the global-in-time convergence of the system towards the full Navier–Stokes equations.

Linear stability of viscous shock wave for 1-D compressible Navier-Stokes equations with Maxwell’s law

In this paper, we consider the linear stability of traveling wave solutions for one-dimensional compressible isentropic Navier-Stokes equations with Maxwell’s Law. The global stability of traveling wave solution is established with shock-profile initial data for the linearized system. Anti-derivative and some delicate energy methods are explored to get the desired results. Moreover, the relaxation limit of traveling wave solution is also obtained.

Vanishing viscosity limit of the compressible Navier-Stokes equations with finite energy and total mass

Assume the initial data of compressible Euler equations has finite energy and total mass. We can construct a sequence of solutions of one-dimensional compressible Navier-Stokes equations (density-dependent viscosity) with stress-free boundary conditions, so that, up to a subsequence, the sequence of solutions of compressible Navier-Stokes equations converges to a finite-energy weak solution of compressible Euler equations. Hence the inviscid limit of the compressible Navier-Stokes is justified. It is worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., α=1,γ=2).

Asymptotic Stability of a Viscous Contact Wave for the One-Dimensional Compressible Navier-Stokes Equations for a Reacting Mixture

We consider the large time behavior of solutions of the Cauchy problem for the one-dimensional compressible Navier-Stokes equations for a reacting mixture. When the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, it is shown that the viscous contact wave which corresponds to the contact discontinuity is asymptotically stable, provided the strength of contact discontinuity and the initial perturbation are suitably small. We apply the approach introduced in Huang, Li and Matsumura (2010) [1] and the elementary L2-energy methods.

Exponential boundary-layer approximation space for solving the compressible laminar Navier-Stokes equations

In general, it needs to take about nearly 10 grid points inside a wall boundary layer for low accuracy order methods to get satisfactory wall-normal gradient related results, such as friction coefficient and wall heat flux. If there exist extreme points inside the boundary layer, this situation becomes even worse. In this work, with the help of the analytic solution of the one-dimensional steady compressible Navier-Stokes equations under some restrictions, we show that the flow variables actually vary in the form of an exponential function instead of a polynomial one inside the boundary layer. Then, we propose an exponential space for approximating the solutions inside the boundary layer and numerically implementing it in the frame of direct discontinuous Galerkin (DDG) method. We show that the DDG methods based on the exponential boundary-layer space give much better numerical results for both conservative variables and wall-normal gradients than those with the standard polynomial space. Generally only 1–2 grid points inside the boundary layer are demanded to resolve the wall boundary layer to obtain satisfactory wall-normal gradients under the exponential space. Preliminary extension to two-dimensional laminar boundary-layer flow shows a similar performance of the proposed exponential boundary-layer space, exhibiting its potential applications in high dimensions.

Asymptotics of the 1D compressible Navier-Stokes equations with density-dependent viscosity

We are concerned with the time-asymptotic behavior toward rarefaction waves for strong non-vacuum solutions to the Cauchy problem of the one-dimensional compressible Navier-Stokes equations with degenerate density-dependent viscosity. The case when the pressure p(ρ)=ργ and the viscosity coefficient μ(ρ)=ρα for some parameters α,γ∈R is considered. For α≥0, γ≥max⁡{1,α}, if the initial data is assumed to be sufficiently regular, without vacuum and mass concentrations, we show that the Cauchy problem of the one-dimensional compressible Navier-Stokes equations admits a unique global strong non-vacuum solution, which tends to the rarefaction waves as time goes to infinity. Here both the initial perturbation and the strength of the rarefaction waves can be arbitrarily large. The proof is established via a delicate energy method and the key ingredient in our analysis is to derive the uniform-in-time positive lower and upper bounds on the specific volume.

Existence and stability of partially congested propagation fronts in a one-dimensional Navier–Stokes model

In this paper, we analyze the behavior of viscous shock profiles of one-dimensional compressible Navier-Stokes equations with a singular pressure law which encodes the effects of congestion. As the intensity of the singular pressure tends to 0, we show the convergence of these profiles towards free-congested traveling front solutions of a two-phase compressible-incompressible Navier-Stokes system and we provide a refined description of the profiles in the vicinity of the transition between the free domain and the congested domain. In the second part of the paper, we prove that the profiles are asymptotically nonlinearly stable under small perturbations with zero integral, and we quantify the size of the admissible perturbations in terms of the intensity of the singular pressure.

Entropy bounded solutions to the one-dimensional compressible Navier-Stokes equations with zero heat conduction and far field vacuum

The entropy is one of the fundamental states of a fluid and, in the viscous case, the equation that it satisfies is highly singular in the region close to the vacuum. In spite of its importance in the gas dynamics, the mathematical analyses on the behavior of the entropy near the vacuum region, were rarely carried out; in particular, in the presence of vacuum, either at the far field or at some isolated interior points, it was unknown whether the entropy remains its boundedness. The results obtained in this paper indicate that the ideal gases retain their uniform boundedness of the entropy, locally or globally in time, if the vacuum occurs at the far field only and the density decays slowly enough at the far field. Precisely, we consider the Cauchy problem to the one-dimensional full compressible Navier-Stokes equations without heat conduction, and establish the local and global existence and uniqueness of entropy-bounded solutions, in the presence of vacuum at the far field only. It is also shown that, different from the case that with compactly supported initial density, the compressible Navier-Stokes equations, with slowly decaying initial density, can propagate the regularities in inhomogeneous Sobolev spaces.

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.

Global well-posedness and large-time behavior of 1D compressible Navier-Stokes equations with density-depending viscosity and vacuum in unbounded domains

We consider the Cauchy problem for one-dimensional (1D) barotropic compressible Navier-Stokes equations with density-depending viscosity and large external forces. Under a general assumption on the density-depending viscosity, we prove that the Cauchy problem admits a unique global strong (classical) solution for the large initial data with vacuum. Moreover, the density is proved to be bounded from above time-independently. As a consequence, we obtain the large time behavior of the solution without external forces.

Asymptotic stability of the viscous shock wave for the one-dimensional compressible Navier–Stokes equations with density dependent viscous coefficient on the half space

In this paper, taking the boundary effect into consideration, we investigate the asymptotic stability of a viscous shock wave for one-dimensional isentropic compressible Navier–Stokes equations with density dependent viscous coefficient. Under the assumption that the viscous coefficient is given as a power function of the density, we prove that, for any positive power index of the viscous coefficient, if the amplitude of the corresponding outgoing viscous shock wave is suitably small and the initial data are close to the outgoing viscous shock wave which is far from the boundary, then a global solution exists uniquely in time and tends towards the properly shifted viscous shock wave as the time goes to infinity. The proof is given by an elementary energy method.