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
We investigate a sufficient condition,in terms of the azimuthal componentω^(θ)ofω=curl u in cylindrical coordinates,for the regularity of axisymmetric weak solutions to the 3D incompressible Navier-Stokes equations...We investigate a sufficient condition,in terms of the azimuthal componentω^(θ)ofω=curl u in cylindrical coordinates,for the regularity of axisymmetric weak solutions to the 3D incompressible Navier-Stokes equations.More precisely,we prove that if■,then the weak solution u is actually a regular solution.Similar regularity criterion still holds in the homogeneous Triebel-Lizorkin spaces.展开更多
The existence and uniqueness of stationary solutions to anisotropic Navier-Stokes equations is investigated by a Galerkin technique in this work.Based on this conclusion,we further explore the exponential stability of...The existence and uniqueness of stationary solutions to anisotropic Navier-Stokes equations is investigated by a Galerkin technique in this work.Based on this conclusion,we further explore the exponential stability of weak solutions to stochastic anisotropic NavierStokes equations.We present a relationship among different growth exponents,which is sufficient to guarantee the existence,uniqueness and exponential stability of stationary solutions.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
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.
基金Supported by the National Natural Science Foundation of China(12361034)the Natural Science Foundation of Shaanxi Province(2022JM-034)。
文摘We investigate a sufficient condition,in terms of the azimuthal componentω^(θ)ofω=curl u in cylindrical coordinates,for the regularity of axisymmetric weak solutions to the 3D incompressible Navier-Stokes equations.More precisely,we prove that if■,then the weak solution u is actually a regular solution.Similar regularity criterion still holds in the homogeneous Triebel-Lizorkin spaces.
基金supported by the Natural Science Foundation of Hunan Province of China(2024JJ5123)supported by the Shandong Provincial Natural Science Foundation(ZR2023MA072,ZR2020MA036)。
文摘The existence and uniqueness of stationary solutions to anisotropic Navier-Stokes equations is investigated by a Galerkin technique in this work.Based on this conclusion,we further explore the exponential stability of weak solutions to stochastic anisotropic NavierStokes equations.We present a relationship among different growth exponents,which is sufficient to guarantee the existence,uniqueness and exponential stability of stationary solutions.
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.
