The zero dissipation limit of the compressible heat-conducting Navier–Stokes equations in the presence of the shock is investigated. It is shown that when the heat conduction coefficient κ and the viscosity coeffici...The zero dissipation limit of the compressible heat-conducting Navier–Stokes equations in the presence of the shock is investigated. It is shown that when the heat conduction coefficient κ and the viscosity coefficient ε satisfy κ = O（ε）, κ/ε≥ c 〉 0, as ε→ 0 （see （1.3））, if the solution of the corresponding Euler equations is piecewise smooth with shock wave satisfying the Lax entropy condition, then there exists a smooth solution to the Navier–Stokes equations, which converges to the piecewise smooth shock solution of the Euler equations away from the shock discontinuity at a rate of ε. The proof is given by a combination of the energy estimates and the matched asymptotic analysis introduced in [3].展开更多
The zero dissipation limit to the contact discontinuities for one-dimensional com- pressible Navier-Stokes equations was recently proved for ideal polytropic gas （see Huang et al. [15, 22] and Ma [31]）, but there is...The zero dissipation limit to the contact discontinuities for one-dimensional com- pressible Navier-Stokes equations was recently proved for ideal polytropic gas （see Huang et al. [15, 22] and Ma [31]）, but there is few result for general gases including ideal polytropic gas. We prove that if the solution to the corresponding Euler system of general gas satisfying （1.4） is piecewise constant with a contact discontinuity, then there exist smooth solutions to Navier-Stokes equations which converge to the inviscid solutions at a rate of k1/4 as the heat-conductivity coefficient k tends to zero. The key is to construct a viscous contact wave of general gas suitable to our proof （see Section 2）. Notice that we have no need to restrict the strength of the contact discontinuity to be small.展开更多
For the general gas including ideal polytropic gas, we study the zero dissipation limit problem of the full 1-D compressible Navier-Stokes equations toward the superposition of contact discontinuity and two rarefactio...For the general gas including ideal polytropic gas, we study the zero dissipation limit problem of the full 1-D compressible Navier-Stokes equations toward the superposition of contact discontinuity and two rarefaction waves. In the case of both smooth and Riemann initial data, we show that if the solutions to the corresponding Euler system consist of the composite wave of two rarefaction wave and contact discontinuity, then there exist solutions to Navier-Stokes equations which converge to the Riemman solutions away from the initial layer with a decay rate in any fixed time interval as the viscosity and the heat-conductivity coefficients tend to zero. The proof is based on scaling arguments, the construction of the approximate profiles and delicate energy estimates. Notice that we have no need to restrict the strengths of the contact discontinuity and rarefaction waves to be small.展开更多
In this paper,we study the zero dissipation limit with a vacuum for the reacting mixture Navier-Stokes equations.For proper smooth initial data that the initial density tends to zero as the relevant physical coefficie...In this paper,we study the zero dissipation limit with a vacuum for the reacting mixture Navier-Stokes equations.For proper smooth initial data that the initial density tends to zero as the relevant physical coefficients tend to zero,we demonstrate that the solution tends to a rarefaction wave connected to a vacuum on the left side coupled with a zero mass fraction of reactant.What is more,the uniform convergence rate is obtained.展开更多
This paper is devoted to studying the zero dissipation limit problem for the one-dimensional compressible Navier-Stokes equations with selected density-dependent viscosity.In particular,we focus our attention on the v...This paper is devoted to studying the zero dissipation limit problem for the one-dimensional compressible Navier-Stokes equations with selected density-dependent viscosity.In particular,we focus our attention on the viscosity taking the formμ(ρ)=ρ^(ϵ)(ϵ>0).For the selected density-dependent viscosity,it is proved that the solutions of the one-dimensional compressible Navier-Stokes equations with centered rarefaction wave initial data exist for all time,and converge to the centered rarefaction waves as the viscosity vanishes,uniformly away from the initial discontinuities.New and subtle analysis is developed to overcome difficulties due to the selected density-dependent viscosity to derive energy estimates,in addition to the scaling argument and elementary energy analysis.Moreover,our results extend the studies in[Xin Z P.Comm Pure Appl Math,1993,46(5):621-665].展开更多
We investigate the zero dissipation limit problem of the one-dimensional compressible isentropic Navier-Stokes equations with Riemann initial data in the case of the composite wave of two shock waves. It is shown that...We investigate the zero dissipation limit problem of the one-dimensional compressible isentropic Navier-Stokes equations with Riemann initial data in the case of the composite wave of two shock waves. It is shown that the unique solution to the Navier-Stokes equations exists for all time, and converges to the Riemann solution to the corresponding Euler equations with the same Riemann initial data uniformly on the set away from the shocks, as the viscosity vanishes. In contrast to previous related works, where either the composite wave is absent or the effects of initial layers are ignored, this gives the first mathematical justification of this limit for the compressible isentropic Navier-Stokes equations in the presence of both composite wave and initial layers. Our method of proof consists of a scaling argument, the construction of the approximate solution and delicate energy estimates.展开更多
基金the Knowledge Innovation Program of the Chinese Academy of Sciences
文摘The zero dissipation limit of the compressible heat-conducting Navier–Stokes equations in the presence of the shock is investigated. It is shown that when the heat conduction coefficient κ and the viscosity coefficient ε satisfy κ = O（ε）, κ/ε≥ c 〉 0, as ε→ 0 （see （1.3））, if the solution of the corresponding Euler equations is piecewise smooth with shock wave satisfying the Lax entropy condition, then there exists a smooth solution to the Navier–Stokes equations, which converges to the piecewise smooth shock solution of the Euler equations away from the shock discontinuity at a rate of ε. The proof is given by a combination of the energy estimates and the matched asymptotic analysis introduced in [3].
文摘The zero dissipation limit to the contact discontinuities for one-dimensional com- pressible Navier-Stokes equations was recently proved for ideal polytropic gas （see Huang et al. [15, 22] and Ma [31]）, but there is few result for general gases including ideal polytropic gas. We prove that if the solution to the corresponding Euler system of general gas satisfying （1.4） is piecewise constant with a contact discontinuity, then there exist smooth solutions to Navier-Stokes equations which converge to the inviscid solutions at a rate of k1/4 as the heat-conductivity coefficient k tends to zero. The key is to construct a viscous contact wave of general gas suitable to our proof （see Section 2）. Notice that we have no need to restrict the strength of the contact discontinuity to be small.
基金Fundamental Research Funds for the Central Universities(2015ZCQ-LY-01 and BLX2015-27)the National Natural Sciences Foundation of China(11601031)
文摘For the general gas including ideal polytropic gas, we study the zero dissipation limit problem of the full 1-D compressible Navier-Stokes equations toward the superposition of contact discontinuity and two rarefaction waves. In the case of both smooth and Riemann initial data, we show that if the solutions to the corresponding Euler system consist of the composite wave of two rarefaction wave and contact discontinuity, then there exist solutions to Navier-Stokes equations which converge to the Riemman solutions away from the initial layer with a decay rate in any fixed time interval as the viscosity and the heat-conductivity coefficients tend to zero. The proof is based on scaling arguments, the construction of the approximate profiles and delicate energy estimates. Notice that we have no need to restrict the strengths of the contact discontinuity and rarefaction waves to be small.
基金supported by the National Natural Science Foundation of China (11971193 and 12171001)。
文摘In this paper,we study the zero dissipation limit with a vacuum for the reacting mixture Navier-Stokes equations.For proper smooth initial data that the initial density tends to zero as the relevant physical coefficients tend to zero,we demonstrate that the solution tends to a rarefaction wave connected to a vacuum on the left side coupled with a zero mass fraction of reactant.What is more,the uniform convergence rate is obtained.
基金supported by the National Natural Science Foundation of China(11671319,11931013).
文摘This paper is devoted to studying the zero dissipation limit problem for the one-dimensional compressible Navier-Stokes equations with selected density-dependent viscosity.In particular,we focus our attention on the viscosity taking the formμ(ρ)=ρ^(ϵ)(ϵ>0).For the selected density-dependent viscosity,it is proved that the solutions of the one-dimensional compressible Navier-Stokes equations with centered rarefaction wave initial data exist for all time,and converge to the centered rarefaction waves as the viscosity vanishes,uniformly away from the initial discontinuities.New and subtle analysis is developed to overcome difficulties due to the selected density-dependent viscosity to derive energy estimates,in addition to the scaling argument and elementary energy analysis.Moreover,our results extend the studies in[Xin Z P.Comm Pure Appl Math,1993,46(5):621-665].
基金supported by National Natural Science Foundation of China(Grant Nos.11226170,10976026 and 11271305)China Postdoctoral Science Foundation Funded Project(Grant No.2012M511640)+1 种基金Hunan Provincial Natural Science Foundation of China(Grant No.13JJ4095)National Science Foundation of USA(Grant Nos.DMS-0807406 and DMS-1108994)
文摘We investigate the zero dissipation limit problem of the one-dimensional compressible isentropic Navier-Stokes equations with Riemann initial data in the case of the composite wave of two shock waves. It is shown that the unique solution to the Navier-Stokes equations exists for all time, and converges to the Riemann solution to the corresponding Euler equations with the same Riemann initial data uniformly on the set away from the shocks, as the viscosity vanishes. In contrast to previous related works, where either the composite wave is absent or the effects of initial layers are ignored, this gives the first mathematical justification of this limit for the compressible isentropic Navier-Stokes equations in the presence of both composite wave and initial layers. Our method of proof consists of a scaling argument, the construction of the approximate solution and delicate energy estimates.
