In this paper,we study the time-asymptotically nonlinear stability of rarefaction waves for the Cauchy problem of the compressible Navier-Stokes equations for a reacting mixture with zero heat conductivity in one dime...In this paper,we study the time-asymptotically nonlinear stability of rarefaction waves for the Cauchy problem of the compressible Navier-Stokes equations for a reacting mixture with zero heat conductivity in one dimension.If the corresponding Riemann problem for the compressible Euler system admits the solutions consisting of rarefaction waves only,it is shown that its Cauchy problem has a unique global solution which tends time-asymptotically towards the rarefaction waves,while the initial perturbation and the strength of rarefaction waves are suitably small.展开更多
We investigate the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem of a one-dimensional compressible Navier-Stokes type system for a viscous,compressible,radiative and reactive gas,w...We investigate the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem of a one-dimensional compressible Navier-Stokes type system for a viscous,compressible,radiative and reactive gas,where the constitutive relations for the pressure p,the speci c internal energy e,the speci c volume v,the absolute temperature θ,and the specific entropy s are given by p=Rθv+aθ^(4)/3,e=C_(v)θ+avθ^(4),and s=C_(v)lnθ+4avθ^(3)/3+Rln v with R>0,C_(v)>0 and a>0 being the perfect gas constant,the speci c heat and the radiation constant,respectively.For such a specific gas motion,a somewhat surprising fact is that,generally speaking,the pressure 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.展开更多
The authors study a 3×3 rate-type viscoelastic system, which is a relaxation approximationto a 2×2 quasi-linear hyperbolic system, including the well-known p-system. It is shown thatthe rarefaction waves are...The authors study a 3×3 rate-type viscoelastic system, which is a relaxation approximationto a 2×2 quasi-linear hyperbolic system, including the well-known p-system. It is shown thatthe rarefaction waves are nonlinear asymptotically stable in this relaxation approximation.展开更多
In this paper, author considers a 3 x 3 system for a reacting flow model propesed by [9]. Since this model has source term, it can be considered as a relaxation approximation to 2 x 2 systems of conservation laws, whi...In this paper, author considers a 3 x 3 system for a reacting flow model propesed by [9]. Since this model has source term, it can be considered as a relaxation approximation to 2 x 2 systems of conservation laws, which include the well-known p-system. From this viewpoint, the author establishes the global existence and the nonlinear stability of travelling wave solutions by L-2 energy method.展开更多
The authors study a 3 x 3 rate-type viscoelastic system, which is a relaxation approximation to a 2 x 2 quasi-linear hroerbolic system, including the well-known p-system. The nonlinear stability of two-mode shock wave...The authors study a 3 x 3 rate-type viscoelastic system, which is a relaxation approximation to a 2 x 2 quasi-linear hroerbolic system, including the well-known p-system. The nonlinear stability of two-mode shock waves in this relaxation approximation is proved.展开更多
This paper is concerned with the stability of the rarefaction wave for the Burgers equationwhere 0 ≤ a < 1/4p (q is determined by (2.2)). Roughly speaking, under the assumption that u_ < u+, the authors prove t...This paper is concerned with the stability of the rarefaction wave for the Burgers equationwhere 0 ≤ a < 1/4p (q is determined by (2.2)). Roughly speaking, under the assumption that u_ < u+, the authors prove the existence of the global smooth solution to the Cauchy problem (I), also find the solution u(x, t) to the Cauchy problem (I) satisfying sup |u(x, t) -uR(x/t)| → 0 as t → ∞, where uR(x/t) is the rarefaction wave of the non-viscous Burgersequation ut + f(u)x = 0 with Riemann initial data u(x, 0) =展开更多
In this article, we investigate the global stability of the wave patterns with the superposition of viscous contact wave and rarefaction wave for the one-dimensional compressible Navier-Stokes equations with a free bo...In this article, we investigate the global stability of the wave patterns with the superposition of viscous contact wave and rarefaction wave for the one-dimensional compressible Navier-Stokes equations with a free boundary. It is shown that for the ideal polytropic gas, the superposition of the viscous contact wave with rarefaction wave is nonlinearly stable for the free boundary problem under the large initial perturbations for any γ 〉 1 with V being the adiabatic exponent provided that the wave strength is suitably small.展开更多
This paper is a review on the results inspired by the publication “Hyperbolic conservation laws with relaxation” by Tai-Ping Liu [1], with emphasis on the topic of nonlinear waves （specifically, rarefaction and sho...This paper is a review on the results inspired by the publication “Hyperbolic conservation laws with relaxation” by Tai-Ping Liu [1], with emphasis on the topic of nonlinear waves （specifically, rarefaction and shock waves）. The aim is twofold： firstly, to report in details the impact of the article on the subsequent research in the area; secondly, to detect research trends which merit attention in the （near） future.展开更多
基金supported by the Beijing Natural Science Foundation(1182004,Z180007,1192001).
文摘In this paper,we study the time-asymptotically nonlinear stability of rarefaction waves for the Cauchy problem of the compressible Navier-Stokes equations for a reacting mixture with zero heat conductivity in one dimension.If the corresponding Riemann problem for the compressible Euler system admits the solutions consisting of rarefaction waves only,it is shown that its Cauchy problem has a unique global solution which tends time-asymptotically towards the rarefaction waves,while the initial perturbation and the strength of rarefaction waves are suitably small.
基金supported by the Fundamental Research Funds for the Central Universities and National Natural Science Foundation of China(Grant Nos.11731008 and 11671309)supported by the Fundamental Research Funds for the Central Universities(Grant No.YJ201962)supported by National Postdoctoral Program for Innovative Talents of China(Grant No.BX20180054).
文摘We investigate the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem of a one-dimensional compressible Navier-Stokes type system for a viscous,compressible,radiative and reactive gas,where the constitutive relations for the pressure p,the speci c internal energy e,the speci c volume v,the absolute temperature θ,and the specific entropy s are given by p=Rθv+aθ^(4)/3,e=C_(v)θ+avθ^(4),and s=C_(v)lnθ+4avθ^(3)/3+Rln v with R>0,C_(v)>0 and a>0 being the perfect gas constant,the speci c heat and the radiation constant,respectively.For such a specific gas motion,a somewhat surprising fact is that,generally speaking,the pressure 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.
文摘The authors study a 3×3 rate-type viscoelastic system, which is a relaxation approximationto a 2×2 quasi-linear hyperbolic system, including the well-known p-system. It is shown thatthe rarefaction waves are nonlinear asymptotically stable in this relaxation approximation.
文摘In this paper, author considers a 3 x 3 system for a reacting flow model propesed by [9]. Since this model has source term, it can be considered as a relaxation approximation to 2 x 2 systems of conservation laws, which include the well-known p-system. From this viewpoint, the author establishes the global existence and the nonlinear stability of travelling wave solutions by L-2 energy method.
文摘The authors study a 3 x 3 rate-type viscoelastic system, which is a relaxation approximation to a 2 x 2 quasi-linear hroerbolic system, including the well-known p-system. The nonlinear stability of two-mode shock waves in this relaxation approximation is proved.
文摘This paper is concerned with the stability of the rarefaction wave for the Burgers equationwhere 0 ≤ a < 1/4p (q is determined by (2.2)). Roughly speaking, under the assumption that u_ < u+, the authors prove the existence of the global smooth solution to the Cauchy problem (I), also find the solution u(x, t) to the Cauchy problem (I) satisfying sup |u(x, t) -uR(x/t)| → 0 as t → ∞, where uR(x/t) is the rarefaction wave of the non-viscous Burgersequation ut + f(u)x = 0 with Riemann initial data u(x, 0) =
基金supported by NSFC Grant No.11171153supported by NSFC Grant No.11322106supported by the Fundamental Research Funds for the Central Universities No.2015ZCQ-LY-01 and No.BLX2015-27
文摘In this article, we investigate the global stability of the wave patterns with the superposition of viscous contact wave and rarefaction wave for the one-dimensional compressible Navier-Stokes equations with a free boundary. It is shown that for the ideal polytropic gas, the superposition of the viscous contact wave with rarefaction wave is nonlinearly stable for the free boundary problem under the large initial perturbations for any γ 〉 1 with V being the adiabatic exponent provided that the wave strength is suitably small.
基金supported partially supported by the italian Project FIRB 2012 "Dispersive dynamics:Fourier Analysis and Variational Methods"
文摘This paper is a review on the results inspired by the publication “Hyperbolic conservation laws with relaxation” by Tai-Ping Liu [1], with emphasis on the topic of nonlinear waves （specifically, rarefaction and shock waves）. The aim is twofold： firstly, to report in details the impact of the article on the subsequent research in the area; secondly, to detect research trends which merit attention in the （near） future.
