A global weak solution to the isentropic Navier-Stokes equation with initial data around a constant state in the L^(1)∩BV class was constructed in[1].In the current paper,we will continue to study the uniqueness and ...A global weak solution to the isentropic Navier-Stokes equation with initial data around a constant state in the L^(1)∩BV class was constructed in[1].In the current paper,we will continue to study the uniqueness and regularity of the constructed solution.The key ingredients are the Holder continuity estimates of the heat kernel in both spatial and time variables.With these finer estimates,we obtain higher order regularity of the constructed solution to Navier-Stokes equation,so that all of the derivatives in the equation of conservative form are in the strong sense.Moreover,this regularity also allows us to identify a function space such that the stability of the solutions can be established there,which eventually implies the uniqueness.展开更多
In this paper,we study the controllability of compressible Navier-Stokes equations with density dependent viscosities.For when the shear viscosityμis a positive constant and the bulk viscosityλis a function of the d...In this paper,we study the controllability of compressible Navier-Stokes equations with density dependent viscosities.For when the shear viscosityμis a positive constant and the bulk viscosityλis a function of the density,it is proven that the system is exactly locally controllable to a constant target trajectory by using boundary control functions.展开更多
The analytical solution of the multi-dimensional,time-fractional model of Navier-Stokes equation using the triple and quadruple Elzaki transformdecompositionmethod is presented in this article.The aforesaidmodel is an...The analytical solution of the multi-dimensional,time-fractional model of Navier-Stokes equation using the triple and quadruple Elzaki transformdecompositionmethod is presented in this article.The aforesaidmodel is analyzed by employing Caputo fractional derivative.We deliberated three stimulating examples that correspond to the triple and quadruple Elzaki transform decomposition methods,respectively.The findings illustrate that the established approaches are extremely helpful in obtaining exact and approximate solutions to the problems.The exact and estimated solutions are delineated via numerical simulation.The proposed analysis indicates that the projected configuration is extremely meticulous,highly efficient,and precise in understanding the behavior of complex evolutionary problems of both fractional and integer order that classify affiliated scientific fields and technology.展开更多
We consider the global well-posedness of strong solutions to the Cauchy problem of compressible barotropic Navier-Stokes equations in R^(2). By exploiting the global-in-time estimate to the two-dimensional(2D for shor...We consider the global well-posedness of strong solutions to the Cauchy problem of compressible barotropic Navier-Stokes equations in R^(2). By exploiting the global-in-time estimate to the two-dimensional(2D for short) classical incompressible Navier-Stokes equations and using techniques developed in(SIAM J Math Anal, 2020, 52(2): 1806–1843), we derive the global existence of solutions provided that the initial data satisfies some smallness condition. In particular, the initial velocity with some arbitrary Besov norm of potential part Pu_0 and large high oscillation are allowed in our results. Moreover, we also construct an example with the initial data involving such a smallness condition, but with a norm that is arbitrarily large.展开更多
We investigate the low Mach number limit for the isentropic compressible NavierStokes equations with a revised Maxwell's law(with Galilean invariance) in R^(3). By applying the uniform estimates of the error syste...We investigate the low Mach number limit for the isentropic compressible NavierStokes equations with a revised Maxwell's law(with Galilean invariance) in R^(3). By applying the uniform estimates of the error system, it is proven that the solutions of the isentropic Navier-Stokes equations with a revised Maxwell's law converge to that of the incompressible Navier-Stokes equations as the Mach number tends to zero. Moreover, the convergence rates are also obtained.展开更多
Many applications in fluid mechanics require the numerical solution of sequences of linear systems typically issued from finite element discretization of the Navier-Stokes equations. The resulting matrices then exhibi...Many applications in fluid mechanics require the numerical solution of sequences of linear systems typically issued from finite element discretization of the Navier-Stokes equations. The resulting matrices then exhibit a saddle point structure. To achieve this task, a Newton-based root-finding algorithm is usually employed which in turn necessitates to solve a saddle point system at every Newton iteration. The involved linear systems being large scale and ill-conditioned, effective linear solvers must be implemented. Here, we develop and test several methods for solving the saddle point systems, considering in particular the LU factorization, as direct approach, and the preconditioned generalized minimal residual (ΡGMRES) solver, an iterative approach. We apply the various solvers within the root-finding algorithm for Flow over backward facing step systems. The particularity of Flow over backward facing step system is an interesting case for studying the performance and solution strategy of a turbulence model. In this case, the flow is subjected to a sudden increase of cross-sectional area, resulting in a separation of flow starting at the point of expansion, making the system of differential equations particularly stiff. We assess the performance of the direct and iterative solvers in terms of computational time, numbers of Newton iterations and time steps.展开更多
This is a continuation of the article(Comm.Partial Differential Equations 26(2001)965).In this article,the authors consider the one-dimensional compressible isentropic Navier-Stokes equations with gravitational fo...This is a continuation of the article(Comm.Partial Differential Equations 26(2001)965).In this article,the authors consider the one-dimensional compressible isentropic Navier-Stokes equations with gravitational force,fixed boundary condition,a general pressure and the density-dependent viscosity coefficient when the viscous gas connects to vacuum state with a jump in density.Precisely,the viscosity coefficient μ is proportional to ρ^θ and 0〈θ〈1/2,where ρ is the density,and the pressure P=P(ρ)is a general pressure.The global existence and the uniqueness of weak solution are proved.展开更多
We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the 3-D compressible Navier-Stokes equations under the assumption that the initial density ||po||L∞ is appropriate...We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the 3-D compressible Navier-Stokes equations under the assumption that the initial density ||po||L∞ is appropriate small and 1 〈 γ 〈 6/5. Here the initial density could have vacuum and we do not require that the initial energy is small.展开更多
For the viscous and heat-conductive fluids governed by the compressible Navier- Stokes equations with external force of general form in R^3, there exist nontrivial stationary solutions provided the external forces are...For the viscous and heat-conductive fluids governed by the compressible Navier- Stokes equations with external force of general form in R^3, there exist nontrivial stationary solutions provided the external forces are small in suitable norms, which was studied in article [15], and there we also proved the global in time stability of the stationary solutions with respect to initial data in H^3-framework. In this article, the authors investigate the rates of convergence of nonstationary solutions to the corresponding stationary solutions when the initial data are small in H^3 and bounded in L6/5.展开更多
In this paper, we investigate the mixed spectral method using generalized Laguerre functions for exterior problems of fourth order partial differential equations. A mixed spectral scheme is provided for the stream fun...In this paper, we investigate the mixed spectral method using generalized Laguerre functions for exterior problems of fourth order partial differential equations. A mixed spectral scheme is provided for the stream function form of the Navier-Stokes equations outside a disc. Numerical results demonstrate the spectral accuracy in space.展开更多
The generalized Navier-Stokes equations with damping are considered.We will show that the generalized Navier-Stokes equations with damping |u|^β-1u have weak solutions for anyβ>1 and 0<α<5/4,and we will us...The generalized Navier-Stokes equations with damping are considered.We will show that the generalized Navier-Stokes equations with damping |u|^β-1u have weak solutions for anyβ>1 and 0<α<5/4,and we will use the Fourier splitting method to prove the L2 decay of weak solutions forβ>2 and 0<α<3/4.展开更多
In this paper,we study the one-dimensional motion of viscous gas near a vacuum,with the gas connecting to a vacuum state with a jump in density.The interface behavior,the pointwise decay rates of the density function ...In this paper,we study the one-dimensional motion of viscous gas near a vacuum,with the gas connecting to a vacuum state with a jump in density.The interface behavior,the pointwise decay rates of the density function and the expanding rates of the interface are obtained with the viscosity coefficientμ(ρ)=ρ^(α)for any 0<α<1;this includes the timeweighted boundedness from below and above.The smoothness of the solution is discussed.Moreover,we construct a class of self-similar classical solutions which exhibit some interesting properties,such as optimal estimates.The present paper extends the results in[Luo T,Xin Z P,Yang T.SIAM J Math Anal,2000,31(6):1175-1191]to the jump boundary conditions case with density-dependent viscosity.展开更多
In this paper,we study a one-dimensional motion of viscous gas near vacuum. We are interested in the case that the gas is in contact with the vacuum at a finite interval. This is a free boundary problem for the one-di...In this paper,we study a one-dimensional motion of viscous gas near vacuum. We are interested in the case that the gas is in contact with the vacuum at a finite interval. This is a free boundary problem for the one-dimensional isentropic Navier-Stokes equations, and the free boundaries are the interfaces separating the gas from vacuum,across which the density changes discontinuosly.Smoothness of the solutions and the uniqueness of the weak solutions are also discussed.The present paper extends results in Luo-Xin-Yang[12] to the jump boundary conditions case.展开更多
The zero dissipation limit of the compressible heat-conducting Navier–Stokes equations in the presence of the shock is investigated. It is shown that when the heat conduction coefficient κ and the viscosity coeffici...The zero dissipation limit of the compressible heat-conducting Navier–Stokes equations in the presence of the shock is investigated. It is shown that when the heat conduction coefficient κ and the viscosity coefficient ε satisfy κ = O(ε), κ/ε≥ c 〉 0, as ε→ 0 (see (1.3)), if the solution of the corresponding Euler equations is piecewise smooth with shock wave satisfying the Lax entropy condition, then there exists a smooth solution to the Navier–Stokes equations, which converges to the piecewise smooth shock solution of the Euler equations away from the shock discontinuity at a rate of ε. The proof is given by a combination of the energy estimates and the matched asymptotic analysis introduced in [3].展开更多
The forward flight of a model butterfly was stud- ied by simulation using the equations of motion coupled with the Navier-Stokes equations. The model butterfly moved under the action of aerodynamic and gravitational f...The forward flight of a model butterfly was stud- ied by simulation using the equations of motion coupled with the Navier-Stokes equations. The model butterfly moved under the action of aerodynamic and gravitational forces, where the aerodynamic forces were generated by flapping wings which moved with the body, allowing the body os- cillations of the model butterfly to be simulated. The main results are as follows: (1) The aerodynamic force produced by the wings is approximately perpendicular to the long-axis of body and is much larger in the downstroke than in the up- stroke. In the downstroke the body pitch angle is small and the large aerodynamic force points up and slightly backward, giving the weight-supporting vertical force and a small neg- ative horizontal force, whilst in the upstroke, the body an- gle is large and the relatively small aerodynamic force points forward and slightly downward, giving a positive horizon- tal force which overcomes the body drag and the negative horizontal force generated in the downstroke. (2) Pitching oscillation of the butterfly body plays an equivalent role of the wing-rotation of many other insects. (3) The body-mass- specific power of the model butterfly is 33.3 W/kg, not very different from that of many other insects, e.g., fruitflies and dragonflies.展开更多
In this article, a new stable nonconforming mixed finite element scheme is proposed for the stationary Navier-Stokes equations, in which a new low order Crouzeix- Raviart type nonconforming rectangular element is take...In this article, a new stable nonconforming mixed finite element scheme is proposed for the stationary Navier-Stokes equations, in which a new low order Crouzeix- Raviart type nonconforming rectangular element is taken for approximating space for the velocity and the piecewise constant element for the pressure. The optimal order error estimates for the approximation of both the velocity and the pressure in L2-norm are established, as well as one in broken H1-norm for the velocity. Numerical experiments are given which are consistent with our theoretical analysis.展开更多
This article is concerned with the time periodic solution to the isentropic compressible Navier-Stokes equations in a periodic domain. Using an approach of parabolic regularization, we first obtain the existence of th...This article is concerned with the time periodic solution to the isentropic compressible Navier-Stokes equations in a periodic domain. Using an approach of parabolic regularization, we first obtain the existence of the time periodic solution to a regularized problem under some smallness and symmetry assumptions on the external force. The result for the original compressible Navier-Stokes equations is then obtained by a limiting process. The uniqueness of the periodic solution is also given.展开更多
In this article a new principle of geometric design for blade's surface of an impeller is provided.This is an optimal control problem for the boundary geometric shape of flow and the control variable is the surfac...In this article a new principle of geometric design for blade's surface of an impeller is provided.This is an optimal control problem for the boundary geometric shape of flow and the control variable is the surface of the blade.We give a minimal functional depending on the geometry of the blade's surface and such that the flow's loss achieves minimum.The existence of the solution of the optimal control problem is proved and the Euler-Lagrange equations for the surface of the blade are derived.In addition,under a new curvilinear coordinate system,the flow domain between the two blades becomes a fixed hexahedron,and the surface as a mapping from a bounded domain in R2 into R3,is explicitly appearing in the objective functional.The Navier-Stokes equations,which include the mapping in their coefficients,can be computed by using operator splitting algorithm.Furthermore,derivatives of the solution of Navier-Stokes equations with respect to the mapping satisfy linearized Navier-Stokes equations which can be solved by using operator splitting algorithms too.Hence,a conjugate gradient method can be used to solve the optimal control problem.展开更多
In this paper, we consider the global existence of classical solution to the 3-D compressible Navier-Stokes equations with a density-dependent viscosity coefficient λ(ρ)provided that the initial energy is small in s...In this paper, we consider the global existence of classical solution to the 3-D compressible Navier-Stokes equations with a density-dependent viscosity coefficient λ(ρ)provided that the initial energy is small in some sense. In our result, we give a relation between the initial energy and the viscosity coefficient μ, and it shows that the initial energy can be large if the coefficient of the viscosity μ is taken to be large, which implies that large viscosity μ means large solution.展开更多
基金partially the National Key R&D Program of China(2022YFA1007300)the NSFC(11901386,12031013)+2 种基金the Strategic Priority Research Program of the Chinese Academy of Sciences(XDA25010403)the NSFC(11801194,11971188)the Hubei Key Laboratory of Engineering Modeling and Scientific Computing。
文摘A global weak solution to the isentropic Navier-Stokes equation with initial data around a constant state in the L^(1)∩BV class was constructed in[1].In the current paper,we will continue to study the uniqueness and regularity of the constructed solution.The key ingredients are the Holder continuity estimates of the heat kernel in both spatial and time variables.With these finer estimates,we obtain higher order regularity of the constructed solution to Navier-Stokes equation,so that all of the derivatives in the equation of conservative form are in the strong sense.Moreover,this regularity also allows us to identify a function space such that the stability of the solutions can be established there,which eventually implies the uniqueness.
基金partially supported by the National Science Foundation of China(11971320,11971496)the National Key R&D Program of China(2020YFA0712500)the Guangdong Basic and Applied Basic Research Foundation(2020A1515010530)。
文摘In this paper,we study the controllability of compressible Navier-Stokes equations with density dependent viscosities.For when the shear viscosityμis a positive constant and the bulk viscosityλis a function of the density,it is proven that the system is exactly locally controllable to a constant target trajectory by using boundary control functions.
基金supported by the Natural Science Foundation of China(GrantNos.61673169,11301127,11701176,11626101,11601485).
文摘The analytical solution of the multi-dimensional,time-fractional model of Navier-Stokes equation using the triple and quadruple Elzaki transformdecompositionmethod is presented in this article.The aforesaidmodel is analyzed by employing Caputo fractional derivative.We deliberated three stimulating examples that correspond to the triple and quadruple Elzaki transform decomposition methods,respectively.The findings illustrate that the established approaches are extremely helpful in obtaining exact and approximate solutions to the problems.The exact and estimated solutions are delineated via numerical simulation.The proposed analysis indicates that the projected configuration is extremely meticulous,highly efficient,and precise in understanding the behavior of complex evolutionary problems of both fractional and integer order that classify affiliated scientific fields and technology.
基金Zhai was partially supported by the Guangdong Provincial Natural Science Foundation (2022A1515011977)the Science and Technology Program of Shenzhen(20200806104726001)+1 种基金Zhong was partially supported by the NNSF of China (11901474, 12071359)the Exceptional Young Talents Project of Chongqing Talent (cstc2021ycjh-bgzxm0153)。
文摘We consider the global well-posedness of strong solutions to the Cauchy problem of compressible barotropic Navier-Stokes equations in R^(2). By exploiting the global-in-time estimate to the two-dimensional(2D for short) classical incompressible Navier-Stokes equations and using techniques developed in(SIAM J Math Anal, 2020, 52(2): 1806–1843), we derive the global existence of solutions provided that the initial data satisfies some smallness condition. In particular, the initial velocity with some arbitrary Besov norm of potential part Pu_0 and large high oscillation are allowed in our results. Moreover, we also construct an example with the initial data involving such a smallness condition, but with a norm that is arbitrarily large.
基金Yuxi HU was supported by the NNSFC (11701556)the Yue Qi Young Scholar ProjectChina University of Mining and Technology (Beijing)。
文摘We investigate the low Mach number limit for the isentropic compressible NavierStokes equations with a revised Maxwell's law(with Galilean invariance) in R^(3). By applying the uniform estimates of the error system, it is proven that the solutions of the isentropic Navier-Stokes equations with a revised Maxwell's law converge to that of the incompressible Navier-Stokes equations as the Mach number tends to zero. Moreover, the convergence rates are also obtained.
文摘Many applications in fluid mechanics require the numerical solution of sequences of linear systems typically issued from finite element discretization of the Navier-Stokes equations. The resulting matrices then exhibit a saddle point structure. To achieve this task, a Newton-based root-finding algorithm is usually employed which in turn necessitates to solve a saddle point system at every Newton iteration. The involved linear systems being large scale and ill-conditioned, effective linear solvers must be implemented. Here, we develop and test several methods for solving the saddle point systems, considering in particular the LU factorization, as direct approach, and the preconditioned generalized minimal residual (ΡGMRES) solver, an iterative approach. We apply the various solvers within the root-finding algorithm for Flow over backward facing step systems. The particularity of Flow over backward facing step system is an interesting case for studying the performance and solution strategy of a turbulence model. In this case, the flow is subjected to a sudden increase of cross-sectional area, resulting in a separation of flow starting at the point of expansion, making the system of differential equations particularly stiff. We assess the performance of the direct and iterative solvers in terms of computational time, numbers of Newton iterations and time steps.
基金Program for New Century ExcellentTalents in University(NCET-04-0745)the Key Project of the National Natural Science Foundation of China(10431060)
文摘This is a continuation of the article(Comm.Partial Differential Equations 26(2001)965).In this article,the authors consider the one-dimensional compressible isentropic Navier-Stokes equations with gravitational force,fixed boundary condition,a general pressure and the density-dependent viscosity coefficient when the viscous gas connects to vacuum state with a jump in density.Precisely,the viscosity coefficient μ is proportional to ρ^θ and 0〈θ〈1/2,where ρ is the density,and the pressure P=P(ρ)is a general pressure.The global existence and the uniqueness of weak solution are proved.
基金supported by National Natural Science Foundation of China (11001090)the Fundamental Research Funds for the Central Universities(11QZR16)
文摘We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the 3-D compressible Navier-Stokes equations under the assumption that the initial density ||po||L∞ is appropriate small and 1 〈 γ 〈 6/5. Here the initial density could have vacuum and we do not require that the initial energy is small.
基金Sponsored by National Natural Science Foundation of China (10431060, 10329101)
文摘For the viscous and heat-conductive fluids governed by the compressible Navier- Stokes equations with external force of general form in R^3, there exist nontrivial stationary solutions provided the external forces are small in suitable norms, which was studied in article [15], and there we also proved the global in time stability of the stationary solutions with respect to initial data in H^3-framework. In this article, the authors investigate the rates of convergence of nonstationary solutions to the corresponding stationary solutions when the initial data are small in H^3 and bounded in L6/5.
基金supported by the National Natural Science Foundation of China (No.10871131)the Science and Technology Commission of Shanghai Municipality (No.075105118)+1 种基金the Shanghai Leading Academic Discipline Project (No.S30405)the Fund for E-institutes of Shanghai Universities(No.E03004)
文摘In this paper, we investigate the mixed spectral method using generalized Laguerre functions for exterior problems of fourth order partial differential equations. A mixed spectral scheme is provided for the stream function form of the Navier-Stokes equations outside a disc. Numerical results demonstrate the spectral accuracy in space.
文摘The generalized Navier-Stokes equations with damping are considered.We will show that the generalized Navier-Stokes equations with damping |u|^β-1u have weak solutions for anyβ>1 and 0<α<5/4,and we will use the Fourier splitting method to prove the L2 decay of weak solutions forβ>2 and 0<α<3/4.
基金supported by the NSFC(11931013)the GXNSF(2022GXNSFDA035078)。
文摘In this paper,we study the one-dimensional motion of viscous gas near a vacuum,with the gas connecting to a vacuum state with a jump in density.The interface behavior,the pointwise decay rates of the density function and the expanding rates of the interface are obtained with the viscosity coefficientμ(ρ)=ρ^(α)for any 0<α<1;this includes the timeweighted boundedness from below and above.The smoothness of the solution is discussed.Moreover,we construct a class of self-similar classical solutions which exhibit some interesting properties,such as optimal estimates.The present paper extends the results in[Luo T,Xin Z P,Yang T.SIAM J Math Anal,2000,31(6):1175-1191]to the jump boundary conditions case with density-dependent viscosity.
文摘In this paper,we study a one-dimensional motion of viscous gas near vacuum. We are interested in the case that the gas is in contact with the vacuum at a finite interval. This is a free boundary problem for the one-dimensional isentropic Navier-Stokes equations, and the free boundaries are the interfaces separating the gas from vacuum,across which the density changes discontinuosly.Smoothness of the solutions and the uniqueness of the weak solutions are also discussed.The present paper extends results in Luo-Xin-Yang[12] to the jump boundary conditions case.
基金the Knowledge Innovation Program of the Chinese Academy of Sciences
文摘The zero dissipation limit of the compressible heat-conducting Navier–Stokes equations in the presence of the shock is investigated. It is shown that when the heat conduction coefficient κ and the viscosity coefficient ε satisfy κ = O(ε), κ/ε≥ c 〉 0, as ε→ 0 (see (1.3)), if the solution of the corresponding Euler equations is piecewise smooth with shock wave satisfying the Lax entropy condition, then there exists a smooth solution to the Navier–Stokes equations, which converges to the piecewise smooth shock solution of the Euler equations away from the shock discontinuity at a rate of ε. The proof is given by a combination of the energy estimates and the matched asymptotic analysis introduced in [3].
基金supported by the National Natural Science Foundation of China(11232002)the Ph.D.Student Foundation of Chinese Ministry of Education(30400002011105001)
文摘The forward flight of a model butterfly was stud- ied by simulation using the equations of motion coupled with the Navier-Stokes equations. The model butterfly moved under the action of aerodynamic and gravitational forces, where the aerodynamic forces were generated by flapping wings which moved with the body, allowing the body os- cillations of the model butterfly to be simulated. The main results are as follows: (1) The aerodynamic force produced by the wings is approximately perpendicular to the long-axis of body and is much larger in the downstroke than in the up- stroke. In the downstroke the body pitch angle is small and the large aerodynamic force points up and slightly backward, giving the weight-supporting vertical force and a small neg- ative horizontal force, whilst in the upstroke, the body an- gle is large and the relatively small aerodynamic force points forward and slightly downward, giving a positive horizon- tal force which overcomes the body drag and the negative horizontal force generated in the downstroke. (2) Pitching oscillation of the butterfly body plays an equivalent role of the wing-rotation of many other insects. (3) The body-mass- specific power of the model butterfly is 33.3 W/kg, not very different from that of many other insects, e.g., fruitflies and dragonflies.
文摘In this article, a new stable nonconforming mixed finite element scheme is proposed for the stationary Navier-Stokes equations, in which a new low order Crouzeix- Raviart type nonconforming rectangular element is taken for approximating space for the velocity and the piecewise constant element for the pressure. The optimal order error estimates for the approximation of both the velocity and the pressure in L2-norm are established, as well as one in broken H1-norm for the velocity. Numerical experiments are given which are consistent with our theoretical analysis.
基金supported by the Program for New Century Excellent Talents in University of the Ministry of Education(NCET-13-0804)NSFC(11471127)+3 种基金Guangdong Natural Science Funds for Distinguished Young Scholar(2015A030306029)The Excellent Young Teachers Program of Guangdong Province(HS2015007)Pearl River S&T Nova Program of Guangzhou(2013J2200064)supported by the General Research Fund of Hong Kong,City U 104511
文摘This article is concerned with the time periodic solution to the isentropic compressible Navier-Stokes equations in a periodic domain. Using an approach of parabolic regularization, we first obtain the existence of the time periodic solution to a regularized problem under some smallness and symmetry assumptions on the external force. The result for the original compressible Navier-Stokes equations is then obtained by a limiting process. The uniqueness of the periodic solution is also given.
基金This work was supported bythe National Natural Science Foundation of China(No.50306019,40375010,10471110,10471109).
文摘In this article a new principle of geometric design for blade's surface of an impeller is provided.This is an optimal control problem for the boundary geometric shape of flow and the control variable is the surface of the blade.We give a minimal functional depending on the geometry of the blade's surface and such that the flow's loss achieves minimum.The existence of the solution of the optimal control problem is proved and the Euler-Lagrange equations for the surface of the blade are derived.In addition,under a new curvilinear coordinate system,the flow domain between the two blades becomes a fixed hexahedron,and the surface as a mapping from a bounded domain in R2 into R3,is explicitly appearing in the objective functional.The Navier-Stokes equations,which include the mapping in their coefficients,can be computed by using operator splitting algorithm.Furthermore,derivatives of the solution of Navier-Stokes equations with respect to the mapping satisfy linearized Navier-Stokes equations which can be solved by using operator splitting algorithms too.Hence,a conjugate gradient method can be used to solve the optimal control problem.
文摘In this paper, we consider the global existence of classical solution to the 3-D compressible Navier-Stokes equations with a density-dependent viscosity coefficient λ(ρ)provided that the initial energy is small in some sense. In our result, we give a relation between the initial energy and the viscosity coefficient μ, and it shows that the initial energy can be large if the coefficient of the viscosity μ is taken to be large, which implies that large viscosity μ means large solution.