We study the incompressible limit of classical solutions to compressible ideal magneto-hydrodynamics in a domain with a flat boundary.The boundary condition is characteristic and the initial data is general.We first e...We study the incompressible limit of classical solutions to compressible ideal magneto-hydrodynamics in a domain with a flat boundary.The boundary condition is characteristic and the initial data is general.We first establish the uniform existence of classical solutions with respect to the Mach number.Then,we prove that the solutions converge to the solution of the incompressible MHD system.In particular,we obtain a stronger convergence result by using the dispersion of acoustic waves in the half space.展开更多
This paper studies the existence and uniqueness of local strong solutions to an Oldroyd-B model with density-dependent viscosity in a bounded domain Ω ⊂ R<sup>d</sup>, d = 2 or 3 via incompressible limit,...This paper studies the existence and uniqueness of local strong solutions to an Oldroyd-B model with density-dependent viscosity in a bounded domain Ω ⊂ R<sup>d</sup>, d = 2 or 3 via incompressible limit, in which the initial data is “well-prepared” and the velocity field enjoys the slip boundary conditions. The main idea is to derive the uniform energy estimates for nonlinear systems and corresponding incompressible limit.展开更多
We study the connection between the compressible Navier-Stokes equations coupled by the Qtensor equation for liquid crystals with the incompressible system in the periodic case, when the Mach number is low. To be more...We study the connection between the compressible Navier-Stokes equations coupled by the Qtensor equation for liquid crystals with the incompressible system in the periodic case, when the Mach number is low. To be more specific, the convergence of the weak solutions of the compressible nematic liquid crystal model to the incompressible one is proved as the Mach number approaches zero, and we also obtain the similar results in the stochastic setting when the equations are driven by a stochastic force. Our approach is based on the uniform estimates of the weak solutions and the martingale solutions, then we justify the limits using various compactness criteria.展开更多
In this paper, we prove local and global existence of classical solutions for a system of equations concerning an incompressible viscoelastic fluid of Oldroyd-B type via the incompressible limit when the initial data ...In this paper, we prove local and global existence of classical solutions for a system of equations concerning an incompressible viscoelastic fluid of Oldroyd-B type via the incompressible limit when the initial data are sufficiently small.展开更多
We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditi...We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition （without the incompressibility condition）, which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in R^n（n ≥ 3） with nonhomogeneous vorticity boundary condition converge in L^2 to the corresponding Euler equations satisfying the kinematic condition.展开更多
The computation of compressible flows becomesmore challengingwhen the Mach number has different orders of magnitude.When the Mach number is of order one,modern shock capturing methods are able to capture shocks and ot...The computation of compressible flows becomesmore challengingwhen the Mach number has different orders of magnitude.When the Mach number is of order one,modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions.However,if theMach number is small,the acoustic waves lead to stiffness in time and excessively large numerical viscosity,thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation.In this paper,we develop an all-speed asymptotic preserving(AP)numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers.Our idea is to split the system into two parts:one involves a slow,nonlinear and conservative hyperbolic system adequate for the use of modern shock capturing methods and the other a linear hyperbolic system which contains the stiff acoustic dynamics,to be solved implicitly.This implicit part is reformulated into a standard pressure Poisson projection system and thus possesses sufficient structure for efficient fast Fourier transform solution techniques.In the zero Mach number limit,the scheme automatically becomes a projection method-like incompressible solver.We present numerical results in one and two dimensions in both compressible and incompressible regimes.展开更多
The combined quasi-neutral and non-relativistic limit of compressible Navier-Stokes-Maxwell equations for plasmas is studied.For well-prepared initial data,it is shown that the smooth solution of compressible Navier-S...The combined quasi-neutral and non-relativistic limit of compressible Navier-Stokes-Maxwell equations for plasmas is studied.For well-prepared initial data,it is shown that the smooth solution of compressible Navier-Stokes-Maxwell equations converges to the smooth solution of incompressible Navier-Stokes equations by introducing new modulated energy functional.展开更多
We investigate the uniform regularity and zero kinematic viscosity-magnetic diffusion limit for the incompressible viscous magnetohydrodynamic equations with the Navier boundary conditions on the velocity and perfectl...We investigate the uniform regularity and zero kinematic viscosity-magnetic diffusion limit for the incompressible viscous magnetohydrodynamic equations with the Navier boundary conditions on the velocity and perfectly conducting conditions on the magnetic field in a smooth bounded domain Ω⊂R^(3).It is shown that there exists a unique strong solution to the incompressible viscous magnetohydrodynamic equations in a finite time interval which is independent of the viscosity coefficient and the magnetic diffusivity coefficient.The solution is uniformly bounded in a conormal Sobolev space and W^(1,∞)(Ω)which allows us to take the zero kinematic viscosity-magnetic diffusion limit.Moreover,we also get the rates of convergence in L^(∞)(0,T;L^(2)),L^(∞)(0,T;W^(1,p))(2≤p<∞),and L^(∞)((0,T)×Ω)for some T>0.展开更多
In this paper,we introduce a nonlocal model for linear steady Stokes system with physical no-slip boundary condition.We use the idea of volume constraint to enforce the no-slip boundary condition and prove that the no...In this paper,we introduce a nonlocal model for linear steady Stokes system with physical no-slip boundary condition.We use the idea of volume constraint to enforce the no-slip boundary condition and prove that the nonlocal model is wellposed.We also show that and the solution of the nonlocal system converges to the solution of the original Stokes system as the nonlocality vanishes.展开更多
An all speed scheme for the Isentropic Euler equations is presented in thispaper. When the Mach number tends to zero, the compressible Euler equations converge to their incompressible counterpart, in which the density...An all speed scheme for the Isentropic Euler equations is presented in thispaper. When the Mach number tends to zero, the compressible Euler equations converge to their incompressible counterpart, in which the density becomes a constant. Increasing approximation errors and severe stability constraints are the main difficultyin the low Mach regime. The key idea of our all speed scheme is the special semiimplicit time discretization, in which the low Mach number stiff term is divided intotwo parts, one being treated explicitly and the other one implicitly. Moreover, the fluxof the density equation is also treated implicitly and an elliptic type equation is derivedto obtain the density. In this way, the correct limit can be captured without requesting the mesh size and time step to be smaller than the Mach number. Compared withprevious semi-implicit methods [11,13,29], firstly, nonphysical oscillations can be suppressed by choosing proper parameter, besides, only a linear elliptic equation needs tobe solved implicitly which reduces much computational cost. We develop this semiimplicit time discretization in the framework of a first order Local Lax-Friedrichs (orRusanov) scheme and numerical tests are displayed to demonstrate its performances.展开更多
In this paper,we study the zero viscosity-diffusivity limit for the incompressible Boussinesq equations in a periodic domain in the framework of Gevrey class.We first prove that there exists an interval of time,indepe...In this paper,we study the zero viscosity-diffusivity limit for the incompressible Boussinesq equations in a periodic domain in the framework of Gevrey class.We first prove that there exists an interval of time,independent of the viscosity coefficient and the diffusivity coefficient,for the solutions to the viscous incompressible Boussinesq equations.Then,based on these uniform estimates,we show that the solutions of the viscous incompressible Boussinesq equations converge to that of the ideal incompressible Boussinesq equations as the viscosity and diffusivity coefficients go to zero.Moreover,the convergence rate is alsogiven.展开更多
We consider the inhomogeneous incompressible Navier-Stokes equation on thin domains T^(2)×∈T,∈→0.It is shown that the weak solutions on T^(2)×∈T converge to the strong/weak solutions of the 2D inhomogene...We consider the inhomogeneous incompressible Navier-Stokes equation on thin domains T^(2)×∈T,∈→0.It is shown that the weak solutions on T^(2)×∈T converge to the strong/weak solutions of the 2D inhomogeneous incompressible Navier-Stokes equations on T^(2)as∈→0 on arbitrary time interval.展开更多
It is showed that, as the Mach number goes to zero, the weak solution of the compressible Navier-Stokes equations in the whole space with general initial data converges to the strong solution of the incompressible Nav...It is showed that, as the Mach number goes to zero, the weak solution of the compressible Navier-Stokes equations in the whole space with general initial data converges to the strong solution of the incompressible Navier-Stokes equations as long as the later exists. The proof of the result relies on the new modulated energy functional and the Strichartz's estimate of linear wave equation.展开更多
In this paper, we consider the viscous incompressible magnetohydrodynamic （MHD） system with a new boundary condition for a general smooth domain in R^3. We obtain the well-posedness of the system and the vanishing v...In this paper, we consider the viscous incompressible magnetohydrodynamic （MHD） system with a new boundary condition for a general smooth domain in R^3. We obtain the well-posedness of the system and the vanishing viscosity limit result.展开更多
This paper studies the incompressible limit and stability of global strong solutions to the threedimensional full compressible Navier-Stokes equations, where the initial data satisfy the "well-prepared" cond...This paper studies the incompressible limit and stability of global strong solutions to the threedimensional full compressible Navier-Stokes equations, where the initial data satisfy the "well-prepared" conditions and the velocity field and temperature enjoy the slip boundary condition and convective boundary condition, respectively. The uniform estimates with respect to both the Mach number ∈(0, ∈] and time t ∈ [0, ∞) are established by deriving a differential inequality with decay property, where ∈∈(0, 1] is a constant.As the Mach number vanishes, the global solution to full compressible Navier-Stokes equations converges to the one of isentropic incompressible Navier-Stokes equations in t ∈ [0, +∞). Moreover, we prove the exponentially asymptotic stability for the global solutions of both the compressible system and its limiting incompressible system.展开更多
In this paper,the inviscid and non-resistive limit is justified for the local-in-time solutions to the equations of nonhomogeneous incompressible magneto-hydrodynamics （MHD）in R3.We prove that as the viscosity and r...In this paper,the inviscid and non-resistive limit is justified for the local-in-time solutions to the equations of nonhomogeneous incompressible magneto-hydrodynamics （MHD）in R3.We prove that as the viscosity and resistivity go to zero,the solution of the Cauchy problem for the nonhomogeneous incompressible MHD system converges to the solution of the ideal MHD system.The convergence rate is also obtained simultaneously.展开更多
We consider dynamics system with damping, which are obtained by some transformations from the system of incompressible Navier-Stokes equations. These have similar properties to original Navier-Stokes equations the sca...We consider dynamics system with damping, which are obtained by some transformations from the system of incompressible Navier-Stokes equations. These have similar properties to original Navier-Stokes equations the scaling invariance. Due to the presence of the damping term, conclusions are different with proving the origin of the incompressible Navier-Stokes equations and get some new conclusions. For one form of dynamics system with damping we prove the existence of solution, and get the existence of the attractors. Moreover, we discuss with limit-behavior the deformations of the Navier-Stokes equation.展开更多
文摘We study the incompressible limit of classical solutions to compressible ideal magneto-hydrodynamics in a domain with a flat boundary.The boundary condition is characteristic and the initial data is general.We first establish the uniform existence of classical solutions with respect to the Mach number.Then,we prove that the solutions converge to the solution of the incompressible MHD system.In particular,we obtain a stronger convergence result by using the dispersion of acoustic waves in the half space.
文摘This paper studies the existence and uniqueness of local strong solutions to an Oldroyd-B model with density-dependent viscosity in a bounded domain Ω ⊂ R<sup>d</sup>, d = 2 or 3 via incompressible limit, in which the initial data is “well-prepared” and the velocity field enjoys the slip boundary conditions. The main idea is to derive the uniform energy estimates for nonlinear systems and corresponding incompressible limit.
文摘We study the connection between the compressible Navier-Stokes equations coupled by the Qtensor equation for liquid crystals with the incompressible system in the periodic case, when the Mach number is low. To be more specific, the convergence of the weak solutions of the compressible nematic liquid crystal model to the incompressible one is proved as the Mach number approaches zero, and we also obtain the similar results in the stochastic setting when the equations are driven by a stochastic force. Our approach is based on the uniform estimates of the weak solutions and the martingale solutions, then we justify the limits using various compactness criteria.
基金Project supported by the National Natural Science Foundation of China (No. 10225102)the 973 Project of the Ministry of Science and Technology of China and the Doctoral Program Foundation of the Ministry of Education of China.
文摘In this paper, we prove local and global existence of classical solutions for a system of equations concerning an incompressible viscoelastic fluid of Oldroyd-B type via the incompressible limit when the initial data are sufficiently small.
基金supported in part by the National Science Foundation under Grants DMS-0807551, DMS-0720925, and DMS-0505473the Natural Science Foundationof China (10728101)supported in part by EPSRC grant EP/F029578/1
文摘We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition （without the incompressibility condition）, which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in R^n（n ≥ 3） with nonhomogeneous vorticity boundary condition converge in L^2 to the corresponding Euler equations satisfying the kinematic condition.
基金J.-G.Liu was supported by NSF grant DMS 10-11738.J.Haack and S.Jin were supported by NSF grant DMS-0608720the NSF FRG grant”Collaborative research on Kinetic Description of Multiscale Phenomena:Modeling,Theory and Computation”(NSF DMS-0757285).
文摘The computation of compressible flows becomesmore challengingwhen the Mach number has different orders of magnitude.When the Mach number is of order one,modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions.However,if theMach number is small,the acoustic waves lead to stiffness in time and excessively large numerical viscosity,thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation.In this paper,we develop an all-speed asymptotic preserving(AP)numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers.Our idea is to split the system into two parts:one involves a slow,nonlinear and conservative hyperbolic system adequate for the use of modern shock capturing methods and the other a linear hyperbolic system which contains the stiff acoustic dynamics,to be solved implicitly.This implicit part is reformulated into a standard pressure Poisson projection system and thus possesses sufficient structure for efficient fast Fourier transform solution techniques.In the zero Mach number limit,the scheme automatically becomes a projection method-like incompressible solver.We present numerical results in one and two dimensions in both compressible and incompressible regimes.
基金supported by the Joint Funds of National Natural Science Foundation of China(Grant No.U1204103)China Postdoctoral Science Foundation Funded Project(Grant No.2013M530032)the Science and Technology Research Projects of Education Department of Henan Province(Grant No.13A110731)
文摘The combined quasi-neutral and non-relativistic limit of compressible Navier-Stokes-Maxwell equations for plasmas is studied.For well-prepared initial data,it is shown that the smooth solution of compressible Navier-Stokes-Maxwell equations converges to the smooth solution of incompressible Navier-Stokes equations by introducing new modulated energy functional.
基金supported partially by NSFC(11671193,11971234)supported partially by the China Postdoctoral Science Foundation(2019M650581).
文摘We investigate the uniform regularity and zero kinematic viscosity-magnetic diffusion limit for the incompressible viscous magnetohydrodynamic equations with the Navier boundary conditions on the velocity and perfectly conducting conditions on the magnetic field in a smooth bounded domain Ω⊂R^(3).It is shown that there exists a unique strong solution to the incompressible viscous magnetohydrodynamic equations in a finite time interval which is independent of the viscosity coefficient and the magnetic diffusivity coefficient.The solution is uniformly bounded in a conormal Sobolev space and W^(1,∞)(Ω)which allows us to take the zero kinematic viscosity-magnetic diffusion limit.Moreover,we also get the rates of convergence in L^(∞)(0,T;L^(2)),L^(∞)(0,T;W^(1,p))(2≤p<∞),and L^(∞)((0,T)×Ω)for some T>0.
基金The research of Z.S.was supported in part by grant NSFC 12071244The research of Q.D.was supported in part by NSF DMS-2012562 and DMS-1937254.
文摘In this paper,we introduce a nonlocal model for linear steady Stokes system with physical no-slip boundary condition.We use the idea of volume constraint to enforce the no-slip boundary condition and prove that the nonlocal model is wellposed.We also show that and the solution of the nonlocal system converges to the solution of the original Stokes system as the nonlocality vanishes.
基金supported by the French"Commissariat´a l’Energie Atomique(CEA)"(Centre de Saclay)in the frame of the contract"ASTRE",#SAV 34160the Marie Curie Actions of the European Commission in the frame of the DEASE project(MESTCT-2005-021122)。
文摘An all speed scheme for the Isentropic Euler equations is presented in thispaper. When the Mach number tends to zero, the compressible Euler equations converge to their incompressible counterpart, in which the density becomes a constant. Increasing approximation errors and severe stability constraints are the main difficultyin the low Mach regime. The key idea of our all speed scheme is the special semiimplicit time discretization, in which the low Mach number stiff term is divided intotwo parts, one being treated explicitly and the other one implicitly. Moreover, the fluxof the density equation is also treated implicitly and an elliptic type equation is derivedto obtain the density. In this way, the correct limit can be captured without requesting the mesh size and time step to be smaller than the Mach number. Compared withprevious semi-implicit methods [11,13,29], firstly, nonphysical oscillations can be suppressed by choosing proper parameter, besides, only a linear elliptic equation needs tobe solved implicitly which reduces much computational cost. We develop this semiimplicit time discretization in the framework of a first order Local Lax-Friedrichs (orRusanov) scheme and numerical tests are displayed to demonstrate its performances.
文摘In this paper,we study the zero viscosity-diffusivity limit for the incompressible Boussinesq equations in a periodic domain in the framework of Gevrey class.We first prove that there exists an interval of time,independent of the viscosity coefficient and the diffusivity coefficient,for the solutions to the viscous incompressible Boussinesq equations.Then,based on these uniform estimates,we show that the solutions of the viscous incompressible Boussinesq equations converge to that of the ideal incompressible Boussinesq equations as the viscosity and diffusivity coefficients go to zero.Moreover,the convergence rate is alsogiven.
基金The research is supported by NSFC underGrant Nos.11571167,11771395,11771206 and PAPD of Jiangsu Higher Education Institutions.
文摘We consider the inhomogeneous incompressible Navier-Stokes equation on thin domains T^(2)×∈T,∈→0.It is shown that the weak solutions on T^(2)×∈T converge to the strong/weak solutions of the 2D inhomogeneous incompressible Navier-Stokes equations on T^(2)as∈→0 on arbitrary time interval.
基金supported by the National Natural Science Foundation of China (Nos. 10431060, 10701011,10501047) the Nanjing University Talent Development Foundation
文摘It is showed that, as the Mach number goes to zero, the weak solution of the compressible Navier-Stokes equations in the whole space with general initial data converges to the strong solution of the incompressible Navier-Stokes equations as long as the later exists. The proof of the result relies on the new modulated energy functional and the Strichartz's estimate of linear wave equation.
基金The authors were partially supported by the National Natural Science Foundation of China （No.11371042）.
文摘In this paper, we consider the viscous incompressible magnetohydrodynamic （MHD） system with a new boundary condition for a general smooth domain in R^3. We obtain the well-posedness of the system and the vanishing viscosity limit result.
基金supported by National Natural Science Foundation of China (Grant No. 11471334)Program for New Century Excellent Talents in University (Grant No. NCET-12-0085)
文摘This paper studies the incompressible limit and stability of global strong solutions to the threedimensional full compressible Navier-Stokes equations, where the initial data satisfy the "well-prepared" conditions and the velocity field and temperature enjoy the slip boundary condition and convective boundary condition, respectively. The uniform estimates with respect to both the Mach number ∈(0, ∈] and time t ∈ [0, ∞) are established by deriving a differential inequality with decay property, where ∈∈(0, 1] is a constant.As the Mach number vanishes, the global solution to full compressible Navier-Stokes equations converges to the one of isentropic incompressible Navier-Stokes equations in t ∈ [0, +∞). Moreover, we prove the exponentially asymptotic stability for the global solutions of both the compressible system and its limiting incompressible system.
基金partly supported by NSFC(1080111110971171)+1 种基金the Natural Science Foundation of Fujian Province of China(2010J05011)the Fundamental Research Funds for the Central Universities(2010121006)
文摘In this paper,the inviscid and non-resistive limit is justified for the local-in-time solutions to the equations of nonhomogeneous incompressible magneto-hydrodynamics （MHD）in R3.We prove that as the viscosity and resistivity go to zero,the solution of the Cauchy problem for the nonhomogeneous incompressible MHD system converges to the solution of the ideal MHD system.The convergence rate is also obtained simultaneously.
文摘We consider dynamics system with damping, which are obtained by some transformations from the system of incompressible Navier-Stokes equations. These have similar properties to original Navier-Stokes equations the scaling invariance. Due to the presence of the damping term, conclusions are different with proving the origin of the incompressible Navier-Stokes equations and get some new conclusions. For one form of dynamics system with damping we prove the existence of solution, and get the existence of the attractors. Moreover, we discuss with limit-behavior the deformations of the Navier-Stokes equation.
