We consider the Cauchy problem for the three-dimensional pressureless Navier-Stokes/Navier-Stokes system,which consists of the pressureless Navier-Stokes equations for(n,w)coupled with the isentropic compressible Navi...We consider the Cauchy problem for the three-dimensional pressureless Navier-Stokes/Navier-Stokes system,which consists of the pressureless Navier-Stokes equations for(n,w)coupled with the isentropic compressible Navier-Stokes equations for(ρ,u)through a drag force term n(w−u).We prove the global existence of strong solutions to the coupled system when the initial data are small perturbations of the constant equilibrium state.However,due to the pressureless structure,one can only deal with the density n of the pressureless flow through the transport equation and it is crucial to obtain the exact time-decay rates for the corresponding velocity w of the pressureless flow.To this end,we make use of the spectral analysis,low-high frequency decomposition and time-weighted energy method to deduce the large time behavior of(w,ρ,u)and consequently establish the Lyapunov stability of the density n in Sobolev space.展开更多
In this paper,we consider the Cauchy problem of the isentropic compressible Navier-Stokes equations with degenerate viscosity and vacuum inℝ,where the viscosity depends on the density in a super-linear power law(i.e.,...In this paper,we consider the Cauchy problem of the isentropic compressible Navier-Stokes equations with degenerate viscosity and vacuum inℝ,where the viscosity depends on the density in a super-linear power law(i.e.,μ(ρ)=ρ^(δ),δ>1).We first obtain the local existence of the regular solution,then show that the regular solution will blow up in finite time if initial data have an isolated mass group,no matter how small and smooth the initial data are.It is worth mentioning that based on the transport structure of some intrinsic variables,we obtain the L^(∞)bound of the density,which helps to remove the restrictionδ≤γin Li-Pan-Zhu[21]and Huang-Wang-Zhu[13].展开更多
本文证明带有临界型阻尼项的Navier-Stokes方程在Lei-Lin-Gevrey空间Xa,σ0(ℝ3)中存在唯一的局部解。文章利用不动点定理和热方程解的有关性质来证明这一主要结论。In this paper, it is proved that the Navier-Stokes equation with c...本文证明带有临界型阻尼项的Navier-Stokes方程在Lei-Lin-Gevrey空间Xa,σ0(ℝ3)中存在唯一的局部解。文章利用不动点定理和热方程解的有关性质来证明这一主要结论。In this paper, it is proved that the Navier-Stokes equation with critical damping terms has a unique local solution in the Lei-Lin-Gevrey space Xa,σ0(ℝ3). In this paper, the main conclusion is proved by using the fixed point theorem and the related properties of the solution of the heat equation.展开更多
本文主要考虑T×R上的二维修正的超粘性Navier-Stokes方程,通过对方程进行线性化处理,揭示了其无粘阻尼特性以及增强耗散现象。进一步地,借助构造合适的权重函数,并运用Bootstrap论证方法,研究发现,当Couette流受到足够小的扰动时,...本文主要考虑T×R上的二维修正的超粘性Navier-Stokes方程,通过对方程进行线性化处理,揭示了其无粘阻尼特性以及增强耗散现象。进一步地,借助构造合适的权重函数,并运用Bootstrap论证方法,研究发现,当Couette流受到足够小的扰动时,混合增强耗散效应将显著发挥作用,解在时间t≫ν15时收敛(其中ν表示运动粘度系数)。因此,可以得出结论:具有初值的二维修正的超粘性Navier-Stokes方程的稳定性阈值不比ν12差。This paper primarily investigates the two-dimensional modified hyperviscous Navier-Stokes equations on T×R. By linearizing the equations, we reveal their inviscid damping properties and enhanced dissipation phenomena. Furthermore, through the construction of appropriate weight functions and the application of the Bootstrap argument, we find that when the Couette flow is subjected to sufficiently small perturbations, the enhanced dissipation effect due to mixing becomes significant, and the solution converges in time at a rate of t≫ν15(where νdenotes the kinematic viscosity coefficient). Therefore, we can conclude that the stability threshold for the two-dimensional modified hyperviscous Navier-Stokes equations with initial values is no worse than that of ν12.展开更多
本文研究了三维粘性系数依赖于密度的非齐次不可压缩热传导Navier-Stokes方程。首先,当粘性系数的梯度的范数满足‖ ∇μ(ρ) ‖L∞(0,T;Lp)∞时,存在一个整体强解,此外,如果初始能量适当小,证明了三维粘性非齐次热传导变粘性Navier-Sto...本文研究了三维粘性系数依赖于密度的非齐次不可压缩热传导Navier-Stokes方程。首先,当粘性系数的梯度的范数满足‖ ∇μ(ρ) ‖L∞(0,T;Lp)∞时,存在一个整体强解,此外,如果初始能量适当小,证明了三维粘性非齐次热传导变粘性Navier-Stokes方程整体强解的唯一性。In this paper, we investigate an 3D viscosity incompressible heat conducting Navier-Stokes equations with density-dependent viscosity. First, we obtain that there exists a global strong solution provided the norm of the gradient of viscosity satisfies ‖ ∇μ(ρ) ‖L∞(0,T;Lp)∞. Moreover, if energy is suitably small, we show the uniqueness of the global strong solution to the three-dimensional viscous non-homogeneous heat conducting Navier-Stokes equations with variable viscosity.展开更多
本文采用物理信息神经网络(PINN)来求解不可压缩湍流Navier-Stokes方程。本研究引入了动态权重调整策略,使得各项误差在训练过程中得到适当的平衡,从而避免了某些误差项主导整个训练过程的问题。此外,为了加速训练收敛并提高精度,本研...本文采用物理信息神经网络(PINN)来求解不可压缩湍流Navier-Stokes方程。本研究引入了动态权重调整策略,使得各项误差在训练过程中得到适当的平衡,从而避免了某些误差项主导整个训练过程的问题。此外,为了加速训练收敛并提高精度,本研究还对网络结构进行了优化,结合物理约束优化过程,改变了优化方法,提高了模型的训练效率。In this paper, physical information neural networks (PINN) are used to solve the Navier-Stokes equations of incompressible turbulence. In this study, the dynamic weighting adjustment strategy is presented to make the errors properly balanced in the training process, so as to avoid the problem that some error terms dominate the whole training process. In addition, in order to accelerate the training convergence and improve the accuracy, this study also optimized the network structure, combining with the physical constraint optimization process and changing the optimization method to improve the training efficiency of the model.展开更多
This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depen...This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form ofη(ρ)=ρ^(α).The existence of unique global H^(2m)-solutions(m∈N)to the free boundary problem is proven for when 0<α<1/4.Furthermore,we obtain the global C^(∞)-solutions if the initial data is smooth.展开更多
In this paper,we study the one-dimensional motion of viscous gas near a vacuum,with the gas connecting to a vacuum state with a jump in density.The interface behavior,the pointwise decay rates of the density function ...In this paper,we study the one-dimensional motion of viscous gas near a vacuum,with the gas connecting to a vacuum state with a jump in density.The interface behavior,the pointwise decay rates of the density function and the expanding rates of the interface are obtained with the viscosity coefficientμ(ρ)=ρ^(α)for any 0<α<1;this includes the timeweighted boundedness from below and above.The smoothness of the solution is discussed.Moreover,we construct a class of self-similar classical solutions which exhibit some interesting properties,such as optimal estimates.The present paper extends the results in[Luo T,Xin Z P,Yang T.SIAM J Math Anal,2000,31(6):1175-1191]to the jump boundary conditions case with density-dependent viscosity.展开更多
This paper is concerned with the global well-posedness of the solution to the compressible Navier-Stokes/Allen-Cahn system and its sharp interface limit in one-dimensional space.For the perturbations with small energy...This paper is concerned with the global well-posedness of the solution to the compressible Navier-Stokes/Allen-Cahn system and its sharp interface limit in one-dimensional space.For the perturbations with small energy but possibly large oscillations of rarefaction wave solutions near phase separation,and where the strength of the initial phase field could be arbitrarily large,we prove that the solution of the Cauchy problem exists for all time,and converges to the centered rarefaction wave solution of the corresponding standard two-phase Euler equation as the viscosity and the thickness of the interface tend to zero.The proof is mainly based on a scaling argument and a basic energy method.展开更多
基金supported by the National Natural Science Foundation of China(11931010,12226326,12226327)the Key Research Project of Academy for Multidisciplinary Studies,Capital Normal Universitysupported by the Anhui Provincial Natural Science Foundation(2408085QA031).
文摘We consider the Cauchy problem for the three-dimensional pressureless Navier-Stokes/Navier-Stokes system,which consists of the pressureless Navier-Stokes equations for(n,w)coupled with the isentropic compressible Navier-Stokes equations for(ρ,u)through a drag force term n(w−u).We prove the global existence of strong solutions to the coupled system when the initial data are small perturbations of the constant equilibrium state.However,due to the pressureless structure,one can only deal with the density n of the pressureless flow through the transport equation and it is crucial to obtain the exact time-decay rates for the corresponding velocity w of the pressureless flow.To this end,we make use of the spectral analysis,low-high frequency decomposition and time-weighted energy method to deduce the large time behavior of(w,ρ,u)and consequently establish the Lyapunov stability of the density n in Sobolev space.
基金supported by the National Natural Science Foundation of China(12371221,12161141004,11831011)the Fundamental Research Funds for the Central Universities and Shanghai Frontiers Science Center of Modern Analysis.
文摘In this paper,we consider the Cauchy problem of the isentropic compressible Navier-Stokes equations with degenerate viscosity and vacuum inℝ,where the viscosity depends on the density in a super-linear power law(i.e.,μ(ρ)=ρ^(δ),δ>1).We first obtain the local existence of the regular solution,then show that the regular solution will blow up in finite time if initial data have an isolated mass group,no matter how small and smooth the initial data are.It is worth mentioning that based on the transport structure of some intrinsic variables,we obtain the L^(∞)bound of the density,which helps to remove the restrictionδ≤γin Li-Pan-Zhu[21]and Huang-Wang-Zhu[13].
文摘本文证明带有临界型阻尼项的Navier-Stokes方程在Lei-Lin-Gevrey空间Xa,σ0(ℝ3)中存在唯一的局部解。文章利用不动点定理和热方程解的有关性质来证明这一主要结论。In this paper, it is proved that the Navier-Stokes equation with critical damping terms has a unique local solution in the Lei-Lin-Gevrey space Xa,σ0(ℝ3). In this paper, the main conclusion is proved by using the fixed point theorem and the related properties of the solution of the heat equation.
文摘本文主要考虑T×R上的二维修正的超粘性Navier-Stokes方程,通过对方程进行线性化处理,揭示了其无粘阻尼特性以及增强耗散现象。进一步地,借助构造合适的权重函数,并运用Bootstrap论证方法,研究发现,当Couette流受到足够小的扰动时,混合增强耗散效应将显著发挥作用,解在时间t≫ν15时收敛(其中ν表示运动粘度系数)。因此,可以得出结论:具有初值的二维修正的超粘性Navier-Stokes方程的稳定性阈值不比ν12差。This paper primarily investigates the two-dimensional modified hyperviscous Navier-Stokes equations on T×R. By linearizing the equations, we reveal their inviscid damping properties and enhanced dissipation phenomena. Furthermore, through the construction of appropriate weight functions and the application of the Bootstrap argument, we find that when the Couette flow is subjected to sufficiently small perturbations, the enhanced dissipation effect due to mixing becomes significant, and the solution converges in time at a rate of t≫ν15(where νdenotes the kinematic viscosity coefficient). Therefore, we can conclude that the stability threshold for the two-dimensional modified hyperviscous Navier-Stokes equations with initial values is no worse than that of ν12.
文摘本文研究了三维粘性系数依赖于密度的非齐次不可压缩热传导Navier-Stokes方程。首先,当粘性系数的梯度的范数满足‖ ∇μ(ρ) ‖L∞(0,T;Lp)∞时,存在一个整体强解,此外,如果初始能量适当小,证明了三维粘性非齐次热传导变粘性Navier-Stokes方程整体强解的唯一性。In this paper, we investigate an 3D viscosity incompressible heat conducting Navier-Stokes equations with density-dependent viscosity. First, we obtain that there exists a global strong solution provided the norm of the gradient of viscosity satisfies ‖ ∇μ(ρ) ‖L∞(0,T;Lp)∞. Moreover, if energy is suitably small, we show the uniqueness of the global strong solution to the three-dimensional viscous non-homogeneous heat conducting Navier-Stokes equations with variable viscosity.
文摘本文采用物理信息神经网络(PINN)来求解不可压缩湍流Navier-Stokes方程。本研究引入了动态权重调整策略,使得各项误差在训练过程中得到适当的平衡,从而避免了某些误差项主导整个训练过程的问题。此外,为了加速训练收敛并提高精度,本研究还对网络结构进行了优化,结合物理约束优化过程,改变了优化方法,提高了模型的训练效率。In this paper, physical information neural networks (PINN) are used to solve the Navier-Stokes equations of incompressible turbulence. In this study, the dynamic weighting adjustment strategy is presented to make the errors properly balanced in the training process, so as to avoid the problem that some error terms dominate the whole training process. In addition, in order to accelerate the training convergence and improve the accuracy, this study also optimized the network structure, combining with the physical constraint optimization process and changing the optimization method to improve the training efficiency of the model.
基金supported by the Key Project of the NSFC(12131010)the NSFC(11771155,12271032)+1 种基金the NSF of Guangdong Province(2021A1515010249,2021A1515010303)supported by the NSFC(11971179,12371205)。
文摘This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form ofη(ρ)=ρ^(α).The existence of unique global H^(2m)-solutions(m∈N)to the free boundary problem is proven for when 0<α<1/4.Furthermore,we obtain the global C^(∞)-solutions if the initial data is smooth.
基金supported by the NSFC(11931013)the GXNSF(2022GXNSFDA035078)。
文摘In this paper,we study the one-dimensional motion of viscous gas near a vacuum,with the gas connecting to a vacuum state with a jump in density.The interface behavior,the pointwise decay rates of the density function and the expanding rates of the interface are obtained with the viscosity coefficientμ(ρ)=ρ^(α)for any 0<α<1;this includes the timeweighted boundedness from below and above.The smoothness of the solution is discussed.Moreover,we construct a class of self-similar classical solutions which exhibit some interesting properties,such as optimal estimates.The present paper extends the results in[Luo T,Xin Z P,Yang T.SIAM J Math Anal,2000,31(6):1175-1191]to the jump boundary conditions case with density-dependent viscosity.
基金supported by the National Natural Science Foundation of China(12361044)supported by the National Natural Science Foundation of China(12171024,11971217,11971020)supported by the Academic and Technical Leaders Training Plan of Jiangxi Province(20212BCJ23027)。
文摘This paper is concerned with the global well-posedness of the solution to the compressible Navier-Stokes/Allen-Cahn system and its sharp interface limit in one-dimensional space.For the perturbations with small energy but possibly large oscillations of rarefaction wave solutions near phase separation,and where the strength of the initial phase field could be arbitrarily large,we prove that the solution of the Cauchy problem exists for all time,and converges to the centered rarefaction wave solution of the corresponding standard two-phase Euler equation as the viscosity and the thickness of the interface tend to zero.The proof is mainly based on a scaling argument and a basic energy method.
