This paper provides a review of the recent results on the stability of vortex sheets in compressible flows.Vortex sheets are contact discontinuities of the underlying flows.The vortex sheet problem is a free boundary ...This paper provides a review of the recent results on the stability of vortex sheets in compressible flows.Vortex sheets are contact discontinuities of the underlying flows.The vortex sheet problem is a free boundary problem with a characteristic boundary and is challenging in analysis.The formulation of the vortex sheet problem will be introduced.The linear stability and nonlinear stability for both the two-dimensional two-phase compressible flows and the two-dimensional elastic flows are summarized.The linear stability of vortex sheets for the three-dimensional elastic flows is also presented.The difficulties of the vortex sheet problems and the ideas of proofs are discussed.展开更多
A free boundary problem for the one-dimensional compressible Navier-Stokes equations is investigated. The asymptotic behavior of solutions toward the superposition of contact discontinuity and shock wave is establishe...A free boundary problem for the one-dimensional compressible Navier-Stokes equations is investigated. The asymptotic behavior of solutions toward the superposition of contact discontinuity and shock wave is established under some smallness conditions. To do this, we first construct a new viscous contact wave such that the momentum equation is satisfied exactly and then determine the shift of the viscous shock wave. By using them together with an inequality concerning the heat kernel in the half space, we obtain the desired a priori estimates. The proof is based on the elementary energy method by the anti-derivative argument.展开更多
In this paper,we are concerned with the asymptotic behavior of L^(∞) weak-entropy solutions to the compressible Euler equations with a vacuum and time-dependent damping-m/(1+t)^(λ).As λ∈(0,l/7],we prove tht the L^...In this paper,we are concerned with the asymptotic behavior of L^(∞) weak-entropy solutions to the compressible Euler equations with a vacuum and time-dependent damping-m/(1+t)^(λ).As λ∈(0,l/7],we prove tht the L^(∞) weak-entropy solution converges to the nonlinear diffusion wave of the generalized porous media equation(GPME)in L^(2)(R).As λ∈(1/7,1),we prove that the L^(∞) weak-entropy solution converges to an expansion around the nonlinear diffusion wave in L^(2)(R),which is the best asymptotic profile.The proof is based on intensive entropy analysis and an energy method.展开更多
The zero dissipation limit for the one-dimensional Navier-Stokes equations of compressible,isentropic gases in the case that the corresponding Euler equations have rarefaction wave solutions is investigated in this pa...The zero dissipation limit for the one-dimensional Navier-Stokes equations of compressible,isentropic gases in the case that the corresponding Euler equations have rarefaction wave solutions is investigated in this paper.In a paper(Comm.Pure Appl.Math.,46,1993,621-665) by Z.P.Xin,the author constructed a sequence of solutions to one-dimensional Navier-Stokes isentropic equations converging to the rarefaction wave as the viscosity tends to zero.Furthermore,he obtained that the convergence rate is ε 1/4 | ln ε|.In this paper,Xin's convergence rate is improved to ε1/3|lnε|2 by different scaling arguments.The new scaling has various applications in related problems.展开更多
It is known that the Boltzmann equation has close relation to the classical systems in fluid dynamics. However, it provides more information on the microscopic level so that some phenomena, like the thermal creep flow...It is known that the Boltzmann equation has close relation to the classical systems in fluid dynamics. However, it provides more information on the microscopic level so that some phenomena, like the thermal creep flow, can not be modeled by the classical systems of fluid dynamics, such as the Euler equations. The author gives an example to show this phenomenon rigorously in a special setting. This paper is completely based on the author's recent work, jointly with Wang and Yang.展开更多
The Silkroad Mathematics Center(SMC)was established in September 2016 by the Chinese Mathematical Society under the support of the China Association for Science and Technology.The main task of the center is to promote...The Silkroad Mathematics Center(SMC)was established in September 2016 by the Chinese Mathematical Society under the support of the China Association for Science and Technology.The main task of the center is to promote mathematics exchanges and cooperation among the countries along the Belt and Road.Professor Ya-xiang Yuan is the current director of SMC.The founding member societies of SMC include Chinese Mathematical Society,Georgian展开更多
The usual heat flow moves along the direction from high temperature place to the low one,as often observed in the daily life.However,when the gas is very rarefied,the gas may move along a different way,that is,the so-...The usual heat flow moves along the direction from high temperature place to the low one,as often observed in the daily life.However,when the gas is very rarefied,the gas may move along a different way,that is,the so-called thermal creep flow moves along the direction from the low temperature place to the high one.In this note,we will survey our recent mathematical works on this topic,mainly based on[27]and[25].展开更多
基金R.M.Chen is supported in part by the NSF grant DMS-1907584F.Huang was supported in part by the National Center for Mathematics and Interdisciplinary Sciences,Academy of Mathematics and Systems Science,Chinese Academy of Sciences and the National Natural Sciences Foundation of China under Grant Nos.11371349 and 11688101+1 种基金D.Wang was supported in part by the NSF under grant DMS-1907519D.Yuan was supported in part by the National Natural Sciences Foundation of China under Grant No.12001045 and the China Postdoctoral Science Foundation under Grant Nos.2020M680428 and 2021T140063.
文摘This paper provides a review of the recent results on the stability of vortex sheets in compressible flows.Vortex sheets are contact discontinuities of the underlying flows.The vortex sheet problem is a free boundary problem with a characteristic boundary and is challenging in analysis.The formulation of the vortex sheet problem will be introduced.The linear stability and nonlinear stability for both the two-dimensional two-phase compressible flows and the two-dimensional elastic flows are summarized.The linear stability of vortex sheets for the three-dimensional elastic flows is also presented.The difficulties of the vortex sheet problems and the ideas of proofs are discussed.
基金partially supported by NSFC (10825102)for distinguished youth scholarsupported by the CAS-TWAS postdoctoral fellowships (FR number:3240223274)AMSS in Chinese Academy of Sciences
文摘A free boundary problem for the one-dimensional compressible Navier-Stokes equations is investigated. The asymptotic behavior of solutions toward the superposition of contact discontinuity and shock wave is established under some smallness conditions. To do this, we first construct a new viscous contact wave such that the momentum equation is satisfied exactly and then determine the shift of the viscous shock wave. By using them together with an inequality concerning the heat kernel in the half space, we obtain the desired a priori estimates. The proof is based on the elementary energy method by the anti-derivative argument.
基金S.Geng's research was supported in part by the National Natural Science Foundation of China(12071397)Excellent Youth Project of Hunan Education Department(21B0165)+1 种基金F.Huang's research was supported in part by the National Key R&D Program of China 2021YFA1000800the National Natural Science Foundation of China(12288201).
文摘In this paper,we are concerned with the asymptotic behavior of L^(∞) weak-entropy solutions to the compressible Euler equations with a vacuum and time-dependent damping-m/(1+t)^(λ).As λ∈(0,l/7],we prove tht the L^(∞) weak-entropy solution converges to the nonlinear diffusion wave of the generalized porous media equation(GPME)in L^(2)(R).As λ∈(1/7,1),we prove that the L^(∞) weak-entropy solution converges to an expansion around the nonlinear diffusion wave in L^(2)(R),which is the best asymptotic profile.The proof is based on intensive entropy analysis and an energy method.
基金supported by the National Natural Science Foundation of China for Outstanding Young Scholars(No. 10825102)the National Basic Research Program of China (973 Program) (No. 2011CB808002)
文摘The zero dissipation limit for the one-dimensional Navier-Stokes equations of compressible,isentropic gases in the case that the corresponding Euler equations have rarefaction wave solutions is investigated in this paper.In a paper(Comm.Pure Appl.Math.,46,1993,621-665) by Z.P.Xin,the author constructed a sequence of solutions to one-dimensional Navier-Stokes isentropic equations converging to the rarefaction wave as the viscosity tends to zero.Furthermore,he obtained that the convergence rate is ε 1/4 | ln ε|.In this paper,Xin's convergence rate is improved to ε1/3|lnε|2 by different scaling arguments.The new scaling has various applications in related problems.
文摘It is known that the Boltzmann equation has close relation to the classical systems in fluid dynamics. However, it provides more information on the microscopic level so that some phenomena, like the thermal creep flow, can not be modeled by the classical systems of fluid dynamics, such as the Euler equations. The author gives an example to show this phenomenon rigorously in a special setting. This paper is completely based on the author's recent work, jointly with Wang and Yang.
文摘The Silkroad Mathematics Center(SMC)was established in September 2016 by the Chinese Mathematical Society under the support of the China Association for Science and Technology.The main task of the center is to promote mathematics exchanges and cooperation among the countries along the Belt and Road.Professor Ya-xiang Yuan is the current director of SMC.The founding member societies of SMC include Chinese Mathematical Society,Georgian
文摘The usual heat flow moves along the direction from high temperature place to the low one,as often observed in the daily life.However,when the gas is very rarefied,the gas may move along a different way,that is,the so-called thermal creep flow moves along the direction from the low temperature place to the high one.In this note,we will survey our recent mathematical works on this topic,mainly based on[27]and[25].
