A two grid technique for solving the steady incompressible Navier Stokes equations in a penalty method was presented and the convergence of numerical solutions was analyzed. If a coarse size H and a fine size ...A two grid technique for solving the steady incompressible Navier Stokes equations in a penalty method was presented and the convergence of numerical solutions was analyzed. If a coarse size H and a fine size h satisfy H=O(h 13-s )(s=0(n=2);s=12(n=3), where n is a space dimension), this method has the same convergence accuracy as the usual finite element method. But the two grid method can save a lot of computation time for its brief calculation. Moreover, a numerical test was couducted in order to verify the correctness of above theoretical analysis.展开更多
This article is concerned with the existence of maximal attractors in Hi (i = 1, 2, 4) for the compressible Navier-Stokes equations for a polytropic viscous heat conductive ideal gas in bounded annular domains Ωn i...This article is concerned with the existence of maximal attractors in Hi (i = 1, 2, 4) for the compressible Navier-Stokes equations for a polytropic viscous heat conductive ideal gas in bounded annular domains Ωn in Rn(n = 2,3). One of the important features is that the metric spaces H(1), H(2), and H(4) we work with are three incomplete metric spaces, as can be seen from the constraints θ 〉 0 and u 〉 0, with θand u being absolute temperature and specific volume respectively. For any constants δ1, δ2……,δ8 verifying some conditions, a sequence of closed subspaces Hδ(4) H(i) (i = 1, 2, 4) is found, and the existence of maximal (universal) attractors in Hδ(i) (i = 1.2.4) is established.展开更多
We study the nonlinear stability of viscous shock waves for the Cauchy problem of one-dimensional nonisentropic compressible Navier–Stokes equations for a viscous and heat conducting ideal polytropic gas. The viscous...We study the nonlinear stability of viscous shock waves for the Cauchy problem of one-dimensional nonisentropic compressible Navier–Stokes equations for a viscous and heat conducting ideal polytropic gas. The viscous shock waves are shown to be time asymptotically stable under large initial perturbation with no restriction on the range of the adiabatic exponent provided that the strengths of the viscous shock waves are assumed to be sufficiently small.The proofs are based on the nonlinear energy estimates and the crucial step is to obtain the positive lower and upper bounds of the density and the temperature which are uniformly in time and space.展开更多
With the cell vertex finite volume discretization in space and second order backward implicit discretization in time, 2D unsteady Navier Stokes equations are solved by a dual time stepping method to simulate compr...With the cell vertex finite volume discretization in space and second order backward implicit discretization in time, 2D unsteady Navier Stokes equations are solved by a dual time stepping method to simulate compressible viscous flow around rigid airfoils in arbitrary unsteady motion. The selection of physical time step is not restricted by stability condition any more, and most of the successful acceleration techniques used in steady calculations can be implemented to increase the computation efficiency.展开更多
We prove the asymptotic properties of the solutions to the 3D Navier–Stokes system with singular external force, by making use of Fourier localization method, the Littlewood–Paley theory and some subtle estimates in...We prove the asymptotic properties of the solutions to the 3D Navier–Stokes system with singular external force, by making use of Fourier localization method, the Littlewood–Paley theory and some subtle estimates in Fourier–Herz space. The main idea of the proof is motivated by that of Cannone et al. [J. Differential Equations, 314, 316–339(2022)]. We deal either with the nonstationary problem or with the stationary problem where solution may be singular due to singular external force. In this paper, the Fourier–Herz space includes the function space of pseudomeasure type used in Cannone et al. [J. Differential Equations, 314, 316–339(2022)]展开更多
In this paper, we apply Littlewood-Paley theory and Ito integral to get the global existence of stochastic Navier-Stokes equations with Coriolis force in Fourier-Besov spaces. As a comparison, we also give correspondi...In this paper, we apply Littlewood-Paley theory and Ito integral to get the global existence of stochastic Navier-Stokes equations with Coriolis force in Fourier-Besov spaces. As a comparison, we also give corresponding results of the deterministic Navier-Stokes equations with Coriolis force.展开更多
We prove the convergence of the Chorin-Marsden product formula for solving the initial-boundary value problems of the Navier-Stokes equations on convex domains. As a particular case we consider the case of the half pl...We prove the convergence of the Chorin-Marsden product formula for solving the initial-boundary value problems of the Navier-Stokes equations on convex domains. As a particular case we consider the case of the half plane.展开更多
A new type of third\|order upwind finite volume implicit scheme is proposed for solving two/three\|dimensional Euler/Reynolds\|averaged Navier\|Stokes equations for steady flow. The fundamental form of the implicit sc...A new type of third\|order upwind finite volume implicit scheme is proposed for solving two/three\|dimensional Euler/Reynolds\|averaged Navier\|Stokes equations for steady flow. The fundamental form of the implicit scheme is based on the LU\|TVD finite volume scheme with the hybrid flux splitting technique. The third\|order ENN scheme's numerical flux is used to calculate the inviscid terms of Navier\|Stokes equations.A fourth\|order accurate symmetric compact difference is applied to its viscous terms. The Baldwin\|Lomax turbulence model is used to calculate the turbulent viscosity. Numerical experiments suggest that the proposed scheme not only has a fairly rapid convergence rate, but also can generate a highly resolved approximation to the numerical solution.展开更多
We show that the asymptotics of solutions to stationary Navier Stokes equations in 4, 5 or 6 dimensions in the whole space with a smooth compactly supported forcing are given by the linear Stokes equation. We do not n...We show that the asymptotics of solutions to stationary Navier Stokes equations in 4, 5 or 6 dimensions in the whole space with a smooth compactly supported forcing are given by the linear Stokes equation. We do not need to assume any smallness condition. The result is in contrast to three dimensions, where the asymptotics for steady states are different from the linear Stokes equation, even for small data, while the large data case presents an open problem. The case of dimension n = 2 is still harder.展开更多
In the recent work, we have developed a decay framework in general Lp critical spaces and established optimal time-decay estimates for barotropic compressible Navier-Stokes equations. Those decay rates of Lq-Lr type o...In the recent work, we have developed a decay framework in general Lp critical spaces and established optimal time-decay estimates for barotropic compressible Navier-Stokes equations. Those decay rates of Lq-Lr type of the solution and its derivatives are available in the critical regularity framework, which were exactly firstly observed by Matsumura & Nishida, and subsequently generalized by Ponce for solutions with high Sobolev regularity. We would like to mention that our approach is likely to be effective for other hyperbolic/parabolic systems that are encountered in fluid mechanics or mathematical physics. In this paper, a new observation is involved in the high frequency, which enables us to improve decay exponents for the high frequencies of solutions.展开更多
In the present paper, the flow field of dual rectangular jets was numerically simulated by solving the full Reynolds averaged Navier Stokes equations , where the RNG k ε model and the finite volume method were ...In the present paper, the flow field of dual rectangular jets was numerically simulated by solving the full Reynolds averaged Navier Stokes equations , where the RNG k ε model and the finite volume method were used. The flow structure in dual rectangular jets and the effects of the velocity were investigated. The numerical results agree qualitatively with the experimental data.展开更多
The secondary instability theory is used to study the behavior of spatially growingdisturbance in free turbulent shear layer.The numerical results indicate that secondaryinstability of subharmonic mode shows a strong ...The secondary instability theory is used to study the behavior of spatially growingdisturbance in free turbulent shear layer.The numerical results indicate that secondaryinstability of subharmonic mode shows a strong choice of spanwise wavenumber andthe maximum growth two dimensional case.In contrast to thatsecondary instabilities of the fundamental mode occur in a wide scope of spanwisewavenumber.We have found so called translative atβ=0 and bifurcationphenomenon for an amplitude of the KH wave larger than 0.06.Dey words instability,large scale structure,bifurcation展开更多
Flow in pumps is essentially three-dimensional and unsteady, and it has much influence on the pump hydraulic performance and structural vibration. This paper presents a numerical methodology developed for modeling suc...Flow in pumps is essentially three-dimensional and unsteady, and it has much influence on the pump hydraulic performance and structural vibration. This paper presents a numerical methodology developed for modeling such complicated flows. Three-dimensional Reynolds-averaged Navier-Stokes (RANS) equations, together with standard k-Ε equation, describe the unsteady-turbulent flow in the pumps. System characteristics are incorporated into the pump CFD models to allow for fluid acceleration in the piper Arbitrary Sliding Interface (ASI) is used to simulate the relative movement between the impeller and stationary components; a numerical analysis is carried out for the entire circumference to consider the asymmetrical flow physics during the stall condition. Combination of these techniques has captured the realistic unsteady flow physics in the pumps and it permits good prediction for the pump off-design performance.展开更多
To improve the current understanding of the reduction of tsunami-like solitary wave runup by the pile breakwater on a sloping beach, we developed a 3D numerical wave tank based on the CFD tool OpenFOAM in this study. ...To improve the current understanding of the reduction of tsunami-like solitary wave runup by the pile breakwater on a sloping beach, we developed a 3D numerical wave tank based on the CFD tool OpenFOAM in this study. The Navier Stokes equations were applied to solve the two-phase incompressible flow, combined with an LES model to solve the turbulence and a VOF method to capture the free surface. The adopted model was firstly validated with existing empirical formulas for solitary wave runup on the slope without the pile structure. It is then validated using our new laboratory observations of the free surface elevation, the velocity and the pressure around a row of vertical slotted piles subjected to solitary waves, as well as the wave runup on the slope behind the piles. Subsequently, a set of numerical simulations were implemented to analyze the wave reflection, the wave transmission, and the shoreline runup with various offshore wave heights, offshore water depths, adjacent pile spaces and beach slopes. Finally, an improved empirical equation accounting for the maximum wave runup on the slope was proposed by taking the presence of the pile breakwater into consideration.展开更多
Wave forces acting on a vertical cylinder at different locations on a slope beach in the near-shore region are investigated considering solitary waves as incoming waves.Based on the Reynolds-averaged Navier-Stokes equ...Wave forces acting on a vertical cylinder at different locations on a slope beach in the near-shore region are investigated considering solitary waves as incoming waves.Based on the Reynolds-averaged Navier-Stokes equations and the k-ε turbulence model,wave forces due to the interaction between the solitary wave and cylinder are simulated and analyzed with different incident wave heights and cylinder locations.The numerical results are first compared with previous theoretical and experimental results to validate the model accuracy.Then,the wave forces and characteristics around the cylinder are studied,including the velocity field,wave surface elevation and pressure.The effects of relative wave height,Keulegan-Carpenter(KC)number and cylinder locations on the wave forces are also discussed.The results show that the wave forces exerted on a cylinder exponentially increase with the increasing incident wave height and KC number.Before the wave force peaks,the growth rate of the wave force shows an increasing trend as the cylinder moves onshore.The cylinder location has a notable effect on the wave force on the cylinder in the near-shore region.As the cylinder moves onshore,the wave force on the cylinder initially increases and then decreases.For the cases considered here,the maximum wave force appears when the cylinder is located one cylinder diameter below the still-water shoreline.Furthermore,the fluid velocity peaks when the maximum wave force appears at the same location.展开更多
This paper focuses on the rapid time-decay phenomenon of the 3 D incompressible NavierStokes flow in exterior domains.By using the representation of the flow in exterior domains,together with the estimates of the Gaus...This paper focuses on the rapid time-decay phenomenon of the 3 D incompressible NavierStokes flow in exterior domains.By using the representation of the flow in exterior domains,together with the estimates of the Gaussian kernel,the tensor kernel,and the Stokes semigroup,we prove that under the assumption∫0∞∫ΩT[u,p](y,t).νdSydt=0 for the body pressure tensor T[u,p],if u0∈L1(Ω)∩Lσ3(Ω)∩W2/5,5/4(Ω)with‖u0‖3≤ηfor some sufficiently small numberη>0,then rapid time-decay phenomenon of the Navier-Stokes flow appears.If additionally|x|αu0∈Lr0(Ω)for some0<α<1 and 1<r0<(1-α/3)-1 orα=1 and r0=1,then the flow exhibits higher decay rates as t→∞.展开更多
In this paper,we investigate the time-periodic solution to a coupled compressible Navier–Stokes/Allen–Cahn system which describes the motion of a mixture of two viscous compressible fluids with a time periodic exter...In this paper,we investigate the time-periodic solution to a coupled compressible Navier–Stokes/Allen–Cahn system which describes the motion of a mixture of two viscous compressible fluids with a time periodic external force in a periodic domain in R^N.The existence of the time-periodic solution to the system is established by using an approach of parabolic regularization and combining with the topology degree theory,and then the uniqueness of the period solution is obtained under some smallness and symmetry assumptions on the external force.展开更多
The Perturbational Finite Difference (PFD) method is a kind of high-order-accurate compact difference method, but its idea is different from the normal compact method and the multi-nodes method. This method can get a ...The Perturbational Finite Difference (PFD) method is a kind of high-order-accurate compact difference method, but its idea is different from the normal compact method and the multi-nodes method. This method can get a Perturbational Exact Numerical Solution P(PENS) scheme for locally linearized Convection-Diffusion (CD) equation. The PENS scheme is similar to the Finite Analytical (FA) scheme and Exact Difference Solution (EDS) scheme, which are all exponential schemes, but PENS scheme is simpler and uses only 3,5 and 7 nodes for 1-, 2- and 3-dimensional problems, respectively. The various approximate schemes of PENs scheme are also called Perturbational-High-order-accurate Difference (PHD) scheme. The PHD schemes can be got by expanding the exponential terms in the PENS scheme into power series of grid Reynolds number, and they are all upwind schemes and remain the concise structure form of first-order upwind scheme. For 1-dimensional (1-D) CD equation and 2-D incompressible Navier-Stokes equation, their PENS and PHD schemes were constituted in this paper, they all gave highly accurate results for the numerical examples of three 1-D CD equations and an incompressible 2-D flow in a square cavity.展开更多
In this paper,we are concerned with the precise relationship between the Hausdorff dimension of possible singular point set S of suitable weak solutions and the parameterαin the nonlinear term in the following parabo...In this paper,we are concerned with the precise relationship between the Hausdorff dimension of possible singular point set S of suitable weak solutions and the parameterαin the nonlinear term in the following parabolic equationh_(t)+h_(xxxx)+∂_(xx)|h_(x)|^(α)=f.It is shown that when 5/3≤α<7/3,the 3α-5/α−1-dimensional parabolic Hausdorff measure of S is zero,which generalizes the recent corresponding work of Ozánski and Robinson in[SIAM J.Math.Anal.,51,228–255(2019)]forα=2 and f=0.The same result is valid for a 3D modified Navier–Stokes system.展开更多
This paper is concerned with the stochastic incompressible Navier–Stokes equations in a layer of fluid between two flat no-slip boundaries.The fluid is driven by the noisy movement of the bottom boundary,where the no...This paper is concerned with the stochastic incompressible Navier–Stokes equations in a layer of fluid between two flat no-slip boundaries.The fluid is driven by the noisy movement of the bottom boundary,where the noise is given by a Lévy process.After establishing existence of a martingale solution,we use the background flow method to derive an upper bound on the turbulent energy dissipation rate.Our estimate recovers one of the basic scaling ideas of turbulence theory,namely,that the dissipation rate is independent of the viscosity at high Reynolds number.展开更多
文摘A two grid technique for solving the steady incompressible Navier Stokes equations in a penalty method was presented and the convergence of numerical solutions was analyzed. If a coarse size H and a fine size h satisfy H=O(h 13-s )(s=0(n=2);s=12(n=3), where n is a space dimension), this method has the same convergence accuracy as the usual finite element method. But the two grid method can save a lot of computation time for its brief calculation. Moreover, a numerical test was couducted in order to verify the correctness of above theoretical analysis.
基金supported in part by the NSF of China (10571024,10871040)the grant of Prominent Youth of Henan Province of China (0412000100)
文摘This article is concerned with the existence of maximal attractors in Hi (i = 1, 2, 4) for the compressible Navier-Stokes equations for a polytropic viscous heat conductive ideal gas in bounded annular domains Ωn in Rn(n = 2,3). One of the important features is that the metric spaces H(1), H(2), and H(4) we work with are three incomplete metric spaces, as can be seen from the constraints θ 〉 0 and u 〉 0, with θand u being absolute temperature and specific volume respectively. For any constants δ1, δ2……,δ8 verifying some conditions, a sequence of closed subspaces Hδ(4) H(i) (i = 1, 2, 4) is found, and the existence of maximal (universal) attractors in Hδ(i) (i = 1.2.4) is established.
文摘We study the nonlinear stability of viscous shock waves for the Cauchy problem of one-dimensional nonisentropic compressible Navier–Stokes equations for a viscous and heat conducting ideal polytropic gas. The viscous shock waves are shown to be time asymptotically stable under large initial perturbation with no restriction on the range of the adiabatic exponent provided that the strengths of the viscous shock waves are assumed to be sufficiently small.The proofs are based on the nonlinear energy estimates and the crucial step is to obtain the positive lower and upper bounds of the density and the temperature which are uniformly in time and space.
文摘With the cell vertex finite volume discretization in space and second order backward implicit discretization in time, 2D unsteady Navier Stokes equations are solved by a dual time stepping method to simulate compressible viscous flow around rigid airfoils in arbitrary unsteady motion. The selection of physical time step is not restricted by stability condition any more, and most of the successful acceleration techniques used in steady calculations can be implemented to increase the computation efficiency.
基金Supported by the National Natural Science Foundation of China (Grant No. 11771423)。
文摘We prove the asymptotic properties of the solutions to the 3D Navier–Stokes system with singular external force, by making use of Fourier localization method, the Littlewood–Paley theory and some subtle estimates in Fourier–Herz space. The main idea of the proof is motivated by that of Cannone et al. [J. Differential Equations, 314, 316–339(2022)]. We deal either with the nonstationary problem or with the stationary problem where solution may be singular due to singular external force. In this paper, the Fourier–Herz space includes the function space of pseudomeasure type used in Cannone et al. [J. Differential Equations, 314, 316–339(2022)]
基金supported by NSFC(Grant Nos.11471309 and 11771423)NSFC of Fujian(Grant No.2017J01564)+1 种基金Teaching Reform Project in Putian University(Grant No.JG201524)supported partly by NSFC(Grant No.11771423)
文摘In this paper, we apply Littlewood-Paley theory and Ito integral to get the global existence of stochastic Navier-Stokes equations with Coriolis force in Fourier-Besov spaces. As a comparison, we also give corresponding results of the deterministic Navier-Stokes equations with Coriolis force.
文摘We prove the convergence of the Chorin-Marsden product formula for solving the initial-boundary value problems of the Navier-Stokes equations on convex domains. As a particular case we consider the case of the half plane.
文摘A new type of third\|order upwind finite volume implicit scheme is proposed for solving two/three\|dimensional Euler/Reynolds\|averaged Navier\|Stokes equations for steady flow. The fundamental form of the implicit scheme is based on the LU\|TVD finite volume scheme with the hybrid flux splitting technique. The third\|order ENN scheme's numerical flux is used to calculate the inviscid terms of Navier\|Stokes equations.A fourth\|order accurate symmetric compact difference is applied to its viscous terms. The Baldwin\|Lomax turbulence model is used to calculate the turbulent viscosity. Numerical experiments suggest that the proposed scheme not only has a fairly rapid convergence rate, but also can generate a highly resolved approximation to the numerical solution.
基金supported in part by grant DMS(Grant No.1362467)from the National Science Foundationthe first author is supported in part by DMS(Grant No.1600779)
文摘We show that the asymptotics of solutions to stationary Navier Stokes equations in 4, 5 or 6 dimensions in the whole space with a smooth compactly supported forcing are given by the linear Stokes equation. We do not need to assume any smallness condition. The result is in contrast to three dimensions, where the asymptotics for steady states are different from the linear Stokes equation, even for small data, while the large data case presents an open problem. The case of dimension n = 2 is still harder.
基金Supported by the National Natural Science Foundation of China(Grant No.11471158)the Program for New Century Excellent Talents in University(Grant No.NCET-13–0857)the Fundamental Research Funds for the Central Universities(Grant No.NE2015005)
文摘In the recent work, we have developed a decay framework in general Lp critical spaces and established optimal time-decay estimates for barotropic compressible Navier-Stokes equations. Those decay rates of Lq-Lr type of the solution and its derivatives are available in the critical regularity framework, which were exactly firstly observed by Matsumura & Nishida, and subsequently generalized by Ponce for solutions with high Sobolev regularity. We would like to mention that our approach is likely to be effective for other hyperbolic/parabolic systems that are encountered in fluid mechanics or mathematical physics. In this paper, a new observation is involved in the high frequency, which enables us to improve decay exponents for the high frequencies of solutions.
文摘In the present paper, the flow field of dual rectangular jets was numerically simulated by solving the full Reynolds averaged Navier Stokes equations , where the RNG k ε model and the finite volume method were used. The flow structure in dual rectangular jets and the effects of the velocity were investigated. The numerical results agree qualitatively with the experimental data.
文摘The secondary instability theory is used to study the behavior of spatially growingdisturbance in free turbulent shear layer.The numerical results indicate that secondaryinstability of subharmonic mode shows a strong choice of spanwise wavenumber andthe maximum growth two dimensional case.In contrast to thatsecondary instabilities of the fundamental mode occur in a wide scope of spanwisewavenumber.We have found so called translative atβ=0 and bifurcationphenomenon for an amplitude of the KH wave larger than 0.06.Dey words instability,large scale structure,bifurcation
文摘Flow in pumps is essentially three-dimensional and unsteady, and it has much influence on the pump hydraulic performance and structural vibration. This paper presents a numerical methodology developed for modeling such complicated flows. Three-dimensional Reynolds-averaged Navier-Stokes (RANS) equations, together with standard k-Ε equation, describe the unsteady-turbulent flow in the pumps. System characteristics are incorporated into the pump CFD models to allow for fluid acceleration in the piper Arbitrary Sliding Interface (ASI) is used to simulate the relative movement between the impeller and stationary components; a numerical analysis is carried out for the entire circumference to consider the asymmetrical flow physics during the stall condition. Combination of these techniques has captured the realistic unsteady flow physics in the pumps and it permits good prediction for the pump off-design performance.
基金financially supported by the National Natural Science Foundation of China(Grant Nos.51679014 and 51839002)the Hunan Science and Technology Plan Program(Grant No.2017RS3035)the Open Foundation of Key Laboratory of Key Technology on Hydropower Development of Hunan Province(Grant No.PKLHD201706)
文摘To improve the current understanding of the reduction of tsunami-like solitary wave runup by the pile breakwater on a sloping beach, we developed a 3D numerical wave tank based on the CFD tool OpenFOAM in this study. The Navier Stokes equations were applied to solve the two-phase incompressible flow, combined with an LES model to solve the turbulence and a VOF method to capture the free surface. The adopted model was firstly validated with existing empirical formulas for solitary wave runup on the slope without the pile structure. It is then validated using our new laboratory observations of the free surface elevation, the velocity and the pressure around a row of vertical slotted piles subjected to solitary waves, as well as the wave runup on the slope behind the piles. Subsequently, a set of numerical simulations were implemented to analyze the wave reflection, the wave transmission, and the shoreline runup with various offshore wave heights, offshore water depths, adjacent pile spaces and beach slopes. Finally, an improved empirical equation accounting for the maximum wave runup on the slope was proposed by taking the presence of the pile breakwater into consideration.
基金financially supported by the National Key R&D Program of China(Grant No.2017YFC1404202)the National Natural Science Foundation of China(Grant No.11572332)the Strategic Priority Research Program of the Chinese Academy of Sciences(Grant Nos.XDB22040203 and XDA22000000)。
文摘Wave forces acting on a vertical cylinder at different locations on a slope beach in the near-shore region are investigated considering solitary waves as incoming waves.Based on the Reynolds-averaged Navier-Stokes equations and the k-ε turbulence model,wave forces due to the interaction between the solitary wave and cylinder are simulated and analyzed with different incident wave heights and cylinder locations.The numerical results are first compared with previous theoretical and experimental results to validate the model accuracy.Then,the wave forces and characteristics around the cylinder are studied,including the velocity field,wave surface elevation and pressure.The effects of relative wave height,Keulegan-Carpenter(KC)number and cylinder locations on the wave forces are also discussed.The results show that the wave forces exerted on a cylinder exponentially increase with the increasing incident wave height and KC number.Before the wave force peaks,the growth rate of the wave force shows an increasing trend as the cylinder moves onshore.The cylinder location has a notable effect on the wave force on the cylinder in the near-shore region.As the cylinder moves onshore,the wave force on the cylinder initially increases and then decreases.For the cases considered here,the maximum wave force appears when the cylinder is located one cylinder diameter below the still-water shoreline.Furthermore,the fluid velocity peaks when the maximum wave force appears at the same location.
基金National Natural Science Foundation of China(Grant No.11771223)。
文摘This paper focuses on the rapid time-decay phenomenon of the 3 D incompressible NavierStokes flow in exterior domains.By using the representation of the flow in exterior domains,together with the estimates of the Gaussian kernel,the tensor kernel,and the Stokes semigroup,we prove that under the assumption∫0∞∫ΩT[u,p](y,t).νdSydt=0 for the body pressure tensor T[u,p],if u0∈L1(Ω)∩Lσ3(Ω)∩W2/5,5/4(Ω)with‖u0‖3≤ηfor some sufficiently small numberη>0,then rapid time-decay phenomenon of the Navier-Stokes flow appears.If additionally|x|αu0∈Lr0(Ω)for some0<α<1 and 1<r0<(1-α/3)-1 orα=1 and r0=1,then the flow exhibits higher decay rates as t→∞.
基金Supported by the NNSF of China(Grant Nos.11671367 and 11801133)the Natural Science Foundation of Henan Province(Grant No.152300410227)the Key Research Projects of Henan Higher Education Institutions(Grant No.18A110038)。
文摘In this paper,we investigate the time-periodic solution to a coupled compressible Navier–Stokes/Allen–Cahn system which describes the motion of a mixture of two viscous compressible fluids with a time periodic external force in a periodic domain in R^N.The existence of the time-periodic solution to the system is established by using an approach of parabolic regularization and combining with the topology degree theory,and then the uniqueness of the period solution is obtained under some smallness and symmetry assumptions on the external force.
文摘The Perturbational Finite Difference (PFD) method is a kind of high-order-accurate compact difference method, but its idea is different from the normal compact method and the multi-nodes method. This method can get a Perturbational Exact Numerical Solution P(PENS) scheme for locally linearized Convection-Diffusion (CD) equation. The PENS scheme is similar to the Finite Analytical (FA) scheme and Exact Difference Solution (EDS) scheme, which are all exponential schemes, but PENS scheme is simpler and uses only 3,5 and 7 nodes for 1-, 2- and 3-dimensional problems, respectively. The various approximate schemes of PENs scheme are also called Perturbational-High-order-accurate Difference (PHD) scheme. The PHD schemes can be got by expanding the exponential terms in the PENS scheme into power series of grid Reynolds number, and they are all upwind schemes and remain the concise structure form of first-order upwind scheme. For 1-dimensional (1-D) CD equation and 2-D incompressible Navier-Stokes equation, their PENS and PHD schemes were constituted in this paper, they all gave highly accurate results for the numerical examples of three 1-D CD equations and an incompressible 2-D flow in a square cavity.
基金the National Natural Science Foundation of China(Grant Nos.11971446,11601492,11771423 and12071113)Natural Science Foundation of He’nan(Grant No.232300421077)Research Start-up Funding Program of Hangzhou Normal University(Grant No.4235C50223204065)。
文摘In this paper,we are concerned with the precise relationship between the Hausdorff dimension of possible singular point set S of suitable weak solutions and the parameterαin the nonlinear term in the following parabolic equationh_(t)+h_(xxxx)+∂_(xx)|h_(x)|^(α)=f.It is shown that when 5/3≤α<7/3,the 3α-5/α−1-dimensional parabolic Hausdorff measure of S is zero,which generalizes the recent corresponding work of Ozánski and Robinson in[SIAM J.Math.Anal.,51,228–255(2019)]forα=2 and f=0.The same result is valid for a 3D modified Navier–Stokes system.
基金supported by the Research Fund of Indiana University.
文摘This paper is concerned with the stochastic incompressible Navier–Stokes equations in a layer of fluid between two flat no-slip boundaries.The fluid is driven by the noisy movement of the bottom boundary,where the noise is given by a Lévy process.After establishing existence of a martingale solution,we use the background flow method to derive an upper bound on the turbulent energy dissipation rate.Our estimate recovers one of the basic scaling ideas of turbulence theory,namely,that the dissipation rate is independent of the viscosity at high Reynolds number.