In this paper,we construct a new class of efficient and high-order schemes for the Cahn-Hilliard-Navier-Stokes equations with periodic boundary conditions.These schemes are based on two types of scalar auxiliary varia...In this paper,we construct a new class of efficient and high-order schemes for the Cahn-Hilliard-Navier-Stokes equations with periodic boundary conditions.These schemes are based on two types of scalar auxiliary variable approaches.By using a new pressure correction method,the accuracy of the pressure has been greatly improved.Furthermore,one only needs to solve a series of fully decoupled linear equations with constant coefficients at each time step.In addition,we prove the unconditional energy stability of the schemes,rigorously.Finally,plenty of numerical simulations are carried out to verify the convergence rates,stability,and effectiveness of the proposed schemes numerically.展开更多
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
Cauchy problem for the linearized bipolar isentropic Navier-Stokes-Poisson system in R^(2) is studied.Through the reformulation of unknown functions,we change the formal system into a linearized Navier-Stokes system a...Cauchy problem for the linearized bipolar isentropic Navier-Stokes-Poisson system in R^(2) is studied.Through the reformulation of unknown functions,we change the formal system into a linearized Navier-Stokes system and a unipolar Navier-Stokes-Poisson system.Based on a delicate analysis of the corresponding Green function,L^(2) decay estimate of the solution is obtained.展开更多
This article gives a general model using specific periodic special functions, that is, degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through...This article gives a general model using specific periodic special functions, that is, degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through the forcing terms simplify the periodic Navier Stokes equations (PNS) at the centers of arbitrary r balls of the 3-Torus. The continuity equation is satisfied together with spatially periodic boundary conditions. The yicomponent forcing terms consist of a function F as part of its expression that is arbitrarily small in an r ball where it is associated with a singular forcing expression both for inviscid and viscous cases. As a result, a significant simplification occurs with a v3(vifor all velocity components) only governing PDE resulting. The extension of three restricted subspaces in each of the principal directions in the Cartesian plane is shown as the Cartesian product ℋ=Jx,t×Jy,t×Jz,t. On each of these subspaces vi,i=1,2,3is continuous and there exists a linear independent subspace associated with the argument of the W function. Here the 3-Torus is built up from each compact segment of length 2R on each of the axes on the 3 principal directions x, y, and z. The form of the scaled velocities for non zero scaled δis related to the definition of the W function such that e−W(ξ)=W(ξ)ξwhere ξdepends on t and proportional to δ→0for infinite time t. The ratio Wξis equal to 1, making the limit δ→0finite and well defined. Considering r balls where the function F=(x−ai)2+(y−bi)2+(z−ci)2−ηset equal to −1e+rwhere r>0. is such that the forcing is singular at every distance r of centres of cubes each containing an r-ball. At the centre of the balls, the forcing is infinite. The main idea is that a system of singular initial value problems with infinite forcing is to be solved for where the velocities are shown to be locally Hölder continuous. It is proven that the limit of these singular problems shifts the finite time blowup time ti∗for first and higher derivatives to t=∞thereby indicating that there is no finite time blowup. Results in the literature can provide a systematic approach to study both large space and time behaviour for singular solutions to the Navier Stokes equations. Among the references, it has been shown that mathematical tools can be applied to study the asymptotic properties of solutions.展开更多
基金Supported by the Research Project Supported of Shanxi Scholarship Council of China(No.2021-029)Shanxi Provincial International Cooperation Base and Platform Project(202104041101019)Shanxi Province Natural Science Research(202203021211129)。
文摘In this paper,we construct a new class of efficient and high-order schemes for the Cahn-Hilliard-Navier-Stokes equations with periodic boundary conditions.These schemes are based on two types of scalar auxiliary variable approaches.By using a new pressure correction method,the accuracy of the pressure has been greatly improved.Furthermore,one only needs to solve a series of fully decoupled linear equations with constant coefficients at each time step.In addition,we prove the unconditional energy stability of the schemes,rigorously.Finally,plenty of numerical simulations are carried out to verify the convergence rates,stability,and effectiveness of the proposed schemes numerically.
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 National Natural Science Foundation of China (12271141)。
文摘Cauchy problem for the linearized bipolar isentropic Navier-Stokes-Poisson system in R^(2) is studied.Through the reformulation of unknown functions,we change the formal system into a linearized Navier-Stokes system and a unipolar Navier-Stokes-Poisson system.Based on a delicate analysis of the corresponding Green function,L^(2) decay estimate of the solution is obtained.
文摘This article gives a general model using specific periodic special functions, that is, degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through the forcing terms simplify the periodic Navier Stokes equations (PNS) at the centers of arbitrary r balls of the 3-Torus. The continuity equation is satisfied together with spatially periodic boundary conditions. The yicomponent forcing terms consist of a function F as part of its expression that is arbitrarily small in an r ball where it is associated with a singular forcing expression both for inviscid and viscous cases. As a result, a significant simplification occurs with a v3(vifor all velocity components) only governing PDE resulting. The extension of three restricted subspaces in each of the principal directions in the Cartesian plane is shown as the Cartesian product ℋ=Jx,t×Jy,t×Jz,t. On each of these subspaces vi,i=1,2,3is continuous and there exists a linear independent subspace associated with the argument of the W function. Here the 3-Torus is built up from each compact segment of length 2R on each of the axes on the 3 principal directions x, y, and z. The form of the scaled velocities for non zero scaled δis related to the definition of the W function such that e−W(ξ)=W(ξ)ξwhere ξdepends on t and proportional to δ→0for infinite time t. The ratio Wξis equal to 1, making the limit δ→0finite and well defined. Considering r balls where the function F=(x−ai)2+(y−bi)2+(z−ci)2−ηset equal to −1e+rwhere r>0. is such that the forcing is singular at every distance r of centres of cubes each containing an r-ball. At the centre of the balls, the forcing is infinite. The main idea is that a system of singular initial value problems with infinite forcing is to be solved for where the velocities are shown to be locally Hölder continuous. It is proven that the limit of these singular problems shifts the finite time blowup time ti∗for first and higher derivatives to t=∞thereby indicating that there is no finite time blowup. Results in the literature can provide a systematic approach to study both large space and time behaviour for singular solutions to the Navier Stokes equations. Among the references, it has been shown that mathematical tools can be applied to study the asymptotic properties of solutions.
