We study the global existence and uniqueness of a strong solution to the kinetic thermomechanical Cucker-Smale(for short,TCS) model coupled with Stokes equations in the whole space.The coupled system consists of the k...We study the global existence and uniqueness of a strong solution to the kinetic thermomechanical Cucker-Smale(for short,TCS) model coupled with Stokes equations in the whole space.The coupled system consists of the kinetic TCS equation for a particle ensemble and the Stokes equations for a fluid via a drag force.In this paper,we present a complete analysis of the existence of global-in-time strong solutions to the coupled model without any smallness restrictions on the initial data.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
In this article, we consider the free boundary value problem of 3D isentropic compressible Navier-Stokes equations. A blow-up criterion in terms of the maximum norm of gradients of velocity is obtained for the spheric...In this article, we consider the free boundary value problem of 3D isentropic compressible Navier-Stokes equations. A blow-up criterion in terms of the maximum norm of gradients of velocity is obtained for the spherically symmetric strong solution in terms of the regularity estimates near the symmetric center and the free boundary respectively.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
In this paper, the quadratic nonconforming brick element (MSLK element) intro- duced in [10] is used for the 3D Stokes equations. The instability for the mixed element pair MSLK-P1 is analyzed, where the vector-valu...In this paper, the quadratic nonconforming brick element (MSLK element) intro- duced in [10] is used for the 3D Stokes equations. The instability for the mixed element pair MSLK-P1 is analyzed, where the vector-valued MSLK element approximates the velocity and the piecewise P1 element approximates the pressure. As a cure, we adopt the piecewise P1 macroelement to discretize the pressure instead of the standard piecewise P1 element on cuboid meshes. This new pair is stable and the optimal error estimate is achieved. Numerical examples verify our theoretical analysis.展开更多
Based on the full domain partition, a parallel finite element algorithm for the stationary Stokes equations is proposed and analyzed. In this algorithm, each subproblem is defined in the entire domain. Majority of the...Based on the full domain partition, a parallel finite element algorithm for the stationary Stokes equations is proposed and analyzed. In this algorithm, each subproblem is defined in the entire domain. Majority of the degrees of freedom are associated with the relevant subdomain. Therefore, it can be solved in parallel with other subproblems using an existing sequential solver without extensive recoding. This allows the algorithm to be implemented easily with low communication costs. Numerical results are given showing the high efficiency of the parallel algorithm.展开更多
Li et al. (2015) claim that it is sufficient to use two harmonic functions to express the general solution of Stokes equations. In this paper, we demonstrate that this is not true in a general case and that we in fact...Li et al. (2015) claim that it is sufficient to use two harmonic functions to express the general solution of Stokes equations. In this paper, we demonstrate that this is not true in a general case and that we in fact need three scalar harmonic functions to represent the general solution of Stokes equations (Venkatalaxmi et al., 2004).展开更多
We first prove the existence and uniqueness of solution of quasilinear Stokes equations. Then it is shown when the viscosity vanishes, the solution of the quasilinear Stokes equations tends to the solution of the dege...We first prove the existence and uniqueness of solution of quasilinear Stokes equations. Then it is shown when the viscosity vanishes, the solution of the quasilinear Stokes equations tends to the solution of the degenerate equations, in which the viscous term is omitted from the quasilinear Stokes equations and the boundary condition is weakened. In the end, we obtain the boundary layer estimation, Our result shows that the thickness of the boundary layer is proportional to epsilon(1/4).展开更多
We propose a p-multilevel preconditioner for hybrid high-order(HHO)discretizations of the Stokes equation,numerically assess its performance on two variants of the method,and compare with a classical discontinuous Gal...We propose a p-multilevel preconditioner for hybrid high-order(HHO)discretizations of the Stokes equation,numerically assess its performance on two variants of the method,and compare with a classical discontinuous Galerkin scheme.An efficient implementa-tion is proposed where coarse level operators are inherited using L2-orthogonal projec-tions defined over mesh faces and the restriction of the fine grid operators is performed recursively and matrix-free.Both h-and k-dependency are investigated tackling two-and three-dimensional problems on standard meshes and graded meshes.For the two HHO for-mulations,featuring discontinuous or hybrid pressure,we study how the combination of p-coarsening and static condensation influences the V-cycle iteration.In particular,two dif-ferent static condensation procedures are considered for the discontinuous pressure HHO variant,resulting in global linear systems with a different number of unknowns and matrix non-zero entries.Interestingly,we show that the efficiency of the solution strategy might be impacted by static condensation options in the case of graded meshes.展开更多
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
An Uzawa-type algorithm is designed for the coupled Stokes equations discretized by the mixed finite element method.The velocity solved by the presented algorithm is weakly divergence-free,which is different from the ...An Uzawa-type algorithm is designed for the coupled Stokes equations discretized by the mixed finite element method.The velocity solved by the presented algorithm is weakly divergence-free,which is different from the one solved by the common Uzawa method.Besides,an optimal relaxation parameter of the presented algorithm is provided.展开更多
In this paper, we evaluate the integrals that are solutions of the heat and Stokes’ equations obtained by Fokas’ transform method by deriving exact formulas. Our method is more accurate and efficient than the contou...In this paper, we evaluate the integrals that are solutions of the heat and Stokes’ equations obtained by Fokas’ transform method by deriving exact formulas. Our method is more accurate and efficient than the contour deformation and parametrization used by Fokas to compute these integrals. In fact, for the heat equation, our solution is exact up to the imaginary error function and for the Stokes equation, our solution is exact up to the incomplete Airy function. In addition, our solutions extend to the lateral boundary without convergence issues, allow for asymptotic expansions, and are much faster than those obtained by other methods.展开更多
By making full use of the estimates of solutions to nonstationary Stokes equations and the method discussing global stability, we establish the global existence theorem of strong solutions for Navier-Stokes equations ...By making full use of the estimates of solutions to nonstationary Stokes equations and the method discussing global stability, we establish the global existence theorem of strong solutions for Navier-Stokes equations in arbitrary three dimensional domain with uniformly C3 boundary, under the assumption that ‖a‖L^2(Ω)+‖f‖L^1(o,∞;L^2(Ω)) or‖▽a‖L^2(Ω)+‖f‖L^2(o,∞;L^2(Ω)) small or viscosity, large. Here a is a given initial velocity and f is the external force. This improves on the previous results. Moreover, the solvability of the case with nonhomogeneous boundary conditions is also discussed.展开更多
This paper proposes and analyzes a class of robust globally divergence-free weak Galerkin (WG) finite element methods for Stokes equations. The new methods use the Pk/Pk-1 (k ≥ 1) discontinuous finite element com...This paper proposes and analyzes a class of robust globally divergence-free weak Galerkin (WG) finite element methods for Stokes equations. The new methods use the Pk/Pk-1 (k ≥ 1) discontinuous finite element combination for velocity and pressure in the interior of elements, and piecewise P1/Pk (l = k - 1, k) for the trace approximations of the ve- locity and pressure on the inter-element boundaries. Our methods not only yield globally divergence-free velocity solutions, but also have uniform error estimates with respect to the Reynolds number. Numerical experiments are provided to show the robustness of the proposed methods.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
An optimal control problem governed by the Stokes equations with L^2-norra state constraints is studied. Finite element approximation is constructed. The optimality conditions of both the exact and discretized problem...An optimal control problem governed by the Stokes equations with L^2-norra state constraints is studied. Finite element approximation is constructed. The optimality conditions of both the exact and discretized problems are discussed, and the a priori error estimates of the optimal order accuracy in L^2-norm and H^1-norm are given. Some numerical experiments are presented to verify the theoretical results.展开更多
A triangular spectral method for the Stokes equations is developed in this paper.The main contributions are two-fold:First of all,a spectral method using the rational approximation is constructed and analyzed for the ...A triangular spectral method for the Stokes equations is developed in this paper.The main contributions are two-fold:First of all,a spectral method using the rational approximation is constructed and analyzed for the Stokes equations in a triangular domain.The existence and uniqueness of the solution,together with an error estimate for the velocity,are proved.Secondly,a nodal basis is constructed for the efficient implementation of the method.These new basis functions enjoy the fully tensorial product property as in a tensor-produce domain.The new triangular spectral method makes it easy to treat more complex geometries in the classical spectral-element framework,allowing us to use arbitrary triangular and tetrahedral elements.展开更多
We construct a new stabilized finite volume method on rectangular grids for the Stokes equations. The lowest equal-order conforming finite element pair (piecewise bilinear veloc- ities and pressures) and piecewise c...We construct a new stabilized finite volume method on rectangular grids for the Stokes equations. The lowest equal-order conforming finite element pair (piecewise bilinear veloc- ities and pressures) and piecewise constant test spaces for both the velocity and pressure are employed in this method. We show the stability of this method and prove first optimal rate of convergence for the velocity in the H1 norm and the pressure in the L2 norm. In addition, a second order optimal error estimate for the velocity in the L2 norm is derived. Numerical experiments illustrating the theoretical results are included.展开更多
In this paper,we construct an H1-conforming quadratic finite element on convex polygonal meshes using the generalized barycentric coordinates.The element has optimal approximation rates.Using this quadratic element,tw...In this paper,we construct an H1-conforming quadratic finite element on convex polygonal meshes using the generalized barycentric coordinates.The element has optimal approximation rates.Using this quadratic element,two stable discretizations for the Stokes equations are developed,which can be viewed as the extensions of the P2-P0 and the Q2-(discontinuous)P1 elements,respectively,to polygonal meshes.Numerical results are presented,which support our theoretical claims.展开更多
基金supported by the National Natural Science Foundation of China (12001033)。
文摘We study the global existence and uniqueness of a strong solution to the kinetic thermomechanical Cucker-Smale(for short,TCS) model coupled with Stokes equations in the whole space.The coupled system consists of the kinetic TCS equation for a particle ensemble and the Stokes equations for a fluid via a drag force.In this paper,we present a complete analysis of the existence of global-in-time strong solutions to the coupled model without any smallness restrictions on the initial data.
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.
基金supported by the NNSFC(11171228,11231006,and 11225102)NSFC-RGC Grant 11461161007the Key Project of Beijing Municipal Education Commission No.CIT&TCD20140323
文摘In this article, we consider the free boundary value problem of 3D isentropic compressible Navier-Stokes equations. A blow-up criterion in terms of the maximum norm of gradients of velocity is obtained for the spherically symmetric strong solution in terms of the regularity estimates near the symmetric center and the free boundary respectively.
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.
基金Supported by the National Natural Science Foundation of China(11171052,11301053,61328206 and 61272371)the Fundamental Research Funds for the Central Universities
文摘In this paper, the quadratic nonconforming brick element (MSLK element) intro- duced in [10] is used for the 3D Stokes equations. The instability for the mixed element pair MSLK-P1 is analyzed, where the vector-valued MSLK element approximates the velocity and the piecewise P1 element approximates the pressure. As a cure, we adopt the piecewise P1 macroelement to discretize the pressure instead of the standard piecewise P1 element on cuboid meshes. This new pair is stable and the optimal error estimate is achieved. Numerical examples verify our theoretical analysis.
基金Project supported by the National Natural Science Foundation of China (No.10971166)the National Basic Research Program (No.2005CB321703)the Science and Technology Foundation of Guizhou Province of China (No.[2008]2123)
文摘Based on the full domain partition, a parallel finite element algorithm for the stationary Stokes equations is proposed and analyzed. In this algorithm, each subproblem is defined in the entire domain. Majority of the degrees of freedom are associated with the relevant subdomain. Therefore, it can be solved in parallel with other subproblems using an existing sequential solver without extensive recoding. This allows the algorithm to be implemented easily with low communication costs. Numerical results are given showing the high efficiency of the parallel algorithm.
文摘Li et al. (2015) claim that it is sufficient to use two harmonic functions to express the general solution of Stokes equations. In this paper, we demonstrate that this is not true in a general case and that we in fact need three scalar harmonic functions to represent the general solution of Stokes equations (Venkatalaxmi et al., 2004).
文摘We first prove the existence and uniqueness of solution of quasilinear Stokes equations. Then it is shown when the viscosity vanishes, the solution of the quasilinear Stokes equations tends to the solution of the degenerate equations, in which the viscous term is omitted from the quasilinear Stokes equations and the boundary condition is weakened. In the end, we obtain the boundary layer estimation, Our result shows that the thickness of the boundary layer is proportional to epsilon(1/4).
基金Daniele Di Pietro acknowledges the support of Agence Nationale de la Recherche Grant fast4hho(ANR-17-CE23-0019).
文摘We propose a p-multilevel preconditioner for hybrid high-order(HHO)discretizations of the Stokes equation,numerically assess its performance on two variants of the method,and compare with a classical discontinuous Galerkin scheme.An efficient implementa-tion is proposed where coarse level operators are inherited using L2-orthogonal projec-tions defined over mesh faces and the restriction of the fine grid operators is performed recursively and matrix-free.Both h-and k-dependency are investigated tackling two-and three-dimensional problems on standard meshes and graded meshes.For the two HHO for-mulations,featuring discontinuous or hybrid pressure,we study how the combination of p-coarsening and static condensation influences the V-cycle iteration.In particular,two dif-ferent static condensation procedures are considered for the discontinuous pressure HHO variant,resulting in global linear systems with a different number of unknowns and matrix non-zero entries.Interestingly,we show that the efficiency of the solution strategy might be impacted by static condensation options in the case of graded meshes.
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Project supported by the National Natural Science Foundation of China(No.11861067)。
文摘An Uzawa-type algorithm is designed for the coupled Stokes equations discretized by the mixed finite element method.The velocity solved by the presented algorithm is weakly divergence-free,which is different from the one solved by the common Uzawa method.Besides,an optimal relaxation parameter of the presented algorithm is provided.
文摘In this paper, we evaluate the integrals that are solutions of the heat and Stokes’ equations obtained by Fokas’ transform method by deriving exact formulas. Our method is more accurate and efficient than the contour deformation and parametrization used by Fokas to compute these integrals. In fact, for the heat equation, our solution is exact up to the imaginary error function and for the Stokes equation, our solution is exact up to the incomplete Airy function. In addition, our solutions extend to the lateral boundary without convergence issues, allow for asymptotic expansions, and are much faster than those obtained by other methods.
基金This work is supported by foundation of Institute of Mathematics, Academia Sinica
文摘By making full use of the estimates of solutions to nonstationary Stokes equations and the method discussing global stability, we establish the global existence theorem of strong solutions for Navier-Stokes equations in arbitrary three dimensional domain with uniformly C3 boundary, under the assumption that ‖a‖L^2(Ω)+‖f‖L^1(o,∞;L^2(Ω)) or‖▽a‖L^2(Ω)+‖f‖L^2(o,∞;L^2(Ω)) small or viscosity, large. Here a is a given initial velocity and f is the external force. This improves on the previous results. Moreover, the solvability of the case with nonhomogeneous boundary conditions is also discussed.
文摘This paper proposes and analyzes a class of robust globally divergence-free weak Galerkin (WG) finite element methods for Stokes equations. The new methods use the Pk/Pk-1 (k ≥ 1) discontinuous finite element combination for velocity and pressure in the interior of elements, and piecewise P1/Pk (l = k - 1, k) for the trace approximations of the ve- locity and pressure on the inter-element boundaries. Our methods not only yield globally divergence-free velocity solutions, but also have uniform error estimates with respect to the Reynolds number. Numerical experiments are provided to show the robustness of the proposed methods.
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.
基金Acknowledgments. Research supported partially by National Natural Science Foundation of China, Grant 11071080 Program of Shanghai Subject Chief Scientist, No. 09XD1401600 Fundamental Research Funds for the Central Universities of China and Shanghai Leading Academic Discipline Project: B407.
文摘An optimal control problem governed by the Stokes equations with L^2-norra state constraints is studied. Finite element approximation is constructed. The optimality conditions of both the exact and discretized problems are discussed, and the a priori error estimates of the optimal order accuracy in L^2-norm and H^1-norm are given. Some numerical experiments are presented to verify the theoretical results.
基金supported by NFS grant DMS-0915066supported by the National Natural Scheme Foundation of China(Grant number 11071203).
文摘A triangular spectral method for the Stokes equations is developed in this paper.The main contributions are two-fold:First of all,a spectral method using the rational approximation is constructed and analyzed for the Stokes equations in a triangular domain.The existence and uniqueness of the solution,together with an error estimate for the velocity,are proved.Secondly,a nodal basis is constructed for the efficient implementation of the method.These new basis functions enjoy the fully tensorial product property as in a tensor-produce domain.The new triangular spectral method makes it easy to treat more complex geometries in the classical spectral-element framework,allowing us to use arbitrary triangular and tetrahedral elements.
文摘We construct a new stabilized finite volume method on rectangular grids for the Stokes equations. The lowest equal-order conforming finite element pair (piecewise bilinear veloc- ities and pressures) and piecewise constant test spaces for both the velocity and pressure are employed in this method. We show the stability of this method and prove first optimal rate of convergence for the velocity in the H1 norm and the pressure in the L2 norm. In addition, a second order optimal error estimate for the velocity in the L2 norm is derived. Numerical experiments illustrating the theoretical results are included.
基金supported by the NSFC grant 11671210 and 12171244.
文摘In this paper,we construct an H1-conforming quadratic finite element on convex polygonal meshes using the generalized barycentric coordinates.The element has optimal approximation rates.Using this quadratic element,two stable discretizations for the Stokes equations are developed,which can be viewed as the extensions of the P2-P0 and the Q2-(discontinuous)P1 elements,respectively,to polygonal meshes.Numerical results are presented,which support our theoretical claims.