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
This paper develops a generalized scalar auxiliary variable(SAV)method for the time-dependent Ginzburg-Landau equations.The backward Euler method is used for discretizing the temporal derivative of the time-dependent ...This paper develops a generalized scalar auxiliary variable(SAV)method for the time-dependent Ginzburg-Landau equations.The backward Euler method is used for discretizing the temporal derivative of the time-dependent Ginzburg-Landau equations.In this method,the system is decoupled and linearized to avoid solving the non-linear equation at each step.The theoretical analysis proves that the generalized SAV method can preserve the maximum bound principle and energy stability,and this is confirmed by the numerical result,and also shows that the numerical algorithm is stable.展开更多
This article gives a general model using specific periodic special functions, that is, degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through...This article gives a general model using specific periodic special functions, that is, degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through the forcing terms simplify the periodic Navier Stokes equations (PNS) at the centers of arbitrary r balls of the 3-Torus. The continuity equation is satisfied together with spatially periodic boundary conditions. The yicomponent forcing terms consist of a function F as part of its expression that is arbitrarily small in an r ball where it is associated with a singular forcing expression both for inviscid and viscous cases. As a result, a significant simplification occurs with a v3(vifor all velocity components) only governing PDE resulting. The extension of three restricted subspaces in each of the principal directions in the Cartesian plane is shown as the Cartesian product ℋ=Jx,t×Jy,t×Jz,t. On each of these subspaces vi,i=1,2,3is continuous and there exists a linear independent subspace associated with the argument of the W function. Here the 3-Torus is built up from each compact segment of length 2R on each of the axes on the 3 principal directions x, y, and z. The form of the scaled velocities for non zero scaled δis related to the definition of the W function such that e−W(ξ)=W(ξ)ξwhere ξdepends on t and proportional to δ→0for infinite time t. The ratio Wξis equal to 1, making the limit δ→0finite and well defined. Considering r balls where the function F=(x−ai)2+(y−bi)2+(z−ci)2−ηset equal to −1e+rwhere r>0. is such that the forcing is singular at every distance r of centres of cubes each containing an r-ball. At the centre of the balls, the forcing is infinite. The main idea is that a system of singular initial value problems with infinite forcing is to be solved for where the velocities are shown to be locally Hölder continuous. It is proven that the limit of these singular problems shifts the finite time blowup time ti∗for first and higher derivatives to t=∞thereby indicating that there is no finite time blowup. Results in the literature can provide a systematic approach to study both large space and time behaviour for singular solutions to the Navier Stokes equations. Among the references, it has been shown that mathematical tools can be applied to study the asymptotic properties of solutions.展开更多
This paper mainly focus on the global existence of the strong solutions for the generalized NavierStokes equations with damping. We obtain the global existence and uniqueness when α ≥5/4 for β ≥ 1 and when 1/2+β/...This paper mainly focus on the global existence of the strong solutions for the generalized NavierStokes equations with damping. We obtain the global existence and uniqueness when α ≥5/4 for β ≥ 1 and when 1/2+β/2≤ α ≤5/4 for 8/3≤ β<+∞.展开更多
This paper is concerned with the existence of pullback attractors for three dimensional generalized Navier-Stokes equations with delay.According to compact argument,the existence and uniqueness of weak solutions are p...This paper is concerned with the existence of pullback attractors for three dimensional generalized Navier-Stokes equations with delay.According to compact argument,the existence and uniqueness of weak solutions are proved by using Galerkin method,and the continuous dependence of solutions on initial values is also shown.Based on the asymptotic compactness via weak convergence method and pullback absorbing set on appropriate functional phase spaces,we get the existence of pullback attractors.展开更多
In this study,we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov(EFK)problem.The temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolso...In this study,we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov(EFK)problem.The temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolson scheme.Following temporal discretization,the generalized finite difference method(GFDM)with supplementary nodes is utilized to address the nonlinear boundary value problems at each time node.These supplementary nodes are distributed along the boundary to match the number of boundary nodes.By incorporating supplementary nodes,the resulting nonlinear algebraic equations can effectively satisfy the governing equation and boundary conditions of the EFK equation.To demonstrate the efficacy of our approach,we present three numerical examples showcasing its performance in solving this nonlinear problem.展开更多
We study exact solutions to (1 + 1)-dimensional generalized Boussinesq equation with time-space dispersion term by making use of improved sub-equation method, and analyse the dynamical behavior and exact solutions of ...We study exact solutions to (1 + 1)-dimensional generalized Boussinesq equation with time-space dispersion term by making use of improved sub-equation method, and analyse the dynamical behavior and exact solutions of the sub-equation after constructing the nonlinear transformation and constraint conditions. Accordingly, we obtain twenty families of exact solutions such as analytical and singular solitons and singular periodic waves. In addition, we discuss the impact of system parameters on wave propagation.展开更多
In this paper, the initial boundary value problem of a class of nonlinear generalized Kolmogorov-Petrovlkii-Piskunov equations is studied. The existence and uniqueness of the solution and the bounded absorption set ar...In this paper, the initial boundary value problem of a class of nonlinear generalized Kolmogorov-Petrovlkii-Piskunov equations is studied. The existence and uniqueness of the solution and the bounded absorption set are proved by the prior estimation and the Galerkin finite element method, thus the existence of the global attractor is proved and the upper bound estimate of the global attractor is obtained.展开更多
In this paper, we consider the generalized Korteweg-de-Vries (KdV) equations which are remarkable models of the water waves mechanics, the shallow water waves, the quantum mechanics, the ion acoustic waves in plasma, ...In this paper, we consider the generalized Korteweg-de-Vries (KdV) equations which are remarkable models of the water waves mechanics, the shallow water waves, the quantum mechanics, the ion acoustic waves in plasma, the electro-hydro-dynamical model for local electric field, signal processing waves through optical fibers, etc. We determine the useful and further general exact traveling wave solutions of the above mentioned NLDEs by applying the exp(−τ(ξ))-expansion method by aid of traveling wave transformations. Furthermore, we explain the physical significance of the obtained solutions of its definite values of the involved parameters with graphic representations in order to know the physical phenomena. Finally, we show that the exp(−τ(ξ))-expansion method is convenient, powerful, straightforward and provide more general solutions and can be helping to examine vast amount of travelling wave solutions to the other different kinds of NLDEs.展开更多
We establish a resonance type existence theorem for generalized evolutional Navier-Stokes equations with periodic boundary conditions and external force depending on velocity, using the Faedo-Galerkin method.
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
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
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
This paper is concerned with the global well-posedness of the solution to the compressible Navier-Stokes/Allen-Cahn system and its sharp interface limit in one-dimensional space.For the perturbations with small energy...This paper is concerned with the global well-posedness of the solution to the compressible Navier-Stokes/Allen-Cahn system and its sharp interface limit in one-dimensional space.For the perturbations with small energy but possibly large oscillations of rarefaction wave solutions near phase separation,and where the strength of the initial phase field could be arbitrarily large,we prove that the solution of the Cauchy problem exists for all time,and converges to the centered rarefaction wave solution of the corresponding standard two-phase Euler equation as the viscosity and the thickness of the interface tend to zero.The proof is mainly based on a scaling argument and a basic energy method.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
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 National Natural Science Foundation of China(12126318,12126302).
文摘This paper develops a generalized scalar auxiliary variable(SAV)method for the time-dependent Ginzburg-Landau equations.The backward Euler method is used for discretizing the temporal derivative of the time-dependent Ginzburg-Landau equations.In this method,the system is decoupled and linearized to avoid solving the non-linear equation at each step.The theoretical analysis proves that the generalized SAV method can preserve the maximum bound principle and energy stability,and this is confirmed by the numerical result,and also shows that the numerical algorithm is stable.
文摘This article gives a general model using specific periodic special functions, that is, degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through the forcing terms simplify the periodic Navier Stokes equations (PNS) at the centers of arbitrary r balls of the 3-Torus. The continuity equation is satisfied together with spatially periodic boundary conditions. The yicomponent forcing terms consist of a function F as part of its expression that is arbitrarily small in an r ball where it is associated with a singular forcing expression both for inviscid and viscous cases. As a result, a significant simplification occurs with a v3(vifor all velocity components) only governing PDE resulting. The extension of three restricted subspaces in each of the principal directions in the Cartesian plane is shown as the Cartesian product ℋ=Jx,t×Jy,t×Jz,t. On each of these subspaces vi,i=1,2,3is continuous and there exists a linear independent subspace associated with the argument of the W function. Here the 3-Torus is built up from each compact segment of length 2R on each of the axes on the 3 principal directions x, y, and z. The form of the scaled velocities for non zero scaled δis related to the definition of the W function such that e−W(ξ)=W(ξ)ξwhere ξdepends on t and proportional to δ→0for infinite time t. The ratio Wξis equal to 1, making the limit δ→0finite and well defined. Considering r balls where the function F=(x−ai)2+(y−bi)2+(z−ci)2−ηset equal to −1e+rwhere r>0. is such that the forcing is singular at every distance r of centres of cubes each containing an r-ball. At the centre of the balls, the forcing is infinite. The main idea is that a system of singular initial value problems with infinite forcing is to be solved for where the velocities are shown to be locally Hölder continuous. It is proven that the limit of these singular problems shifts the finite time blowup time ti∗for first and higher derivatives to t=∞thereby indicating that there is no finite time blowup. Results in the literature can provide a systematic approach to study both large space and time behaviour for singular solutions to the Navier Stokes equations. Among the references, it has been shown that mathematical tools can be applied to study the asymptotic properties of solutions.
基金supported by NSFC(No.12171014)by the National Key Research and Development Project of China Grant(No.2017YFC1500301)+1 种基金Joint key project of the National Science Foundation of China and the China earthquake administration(No.U1839206)by Research Foundation for Advanced Talents of Beijing Technology and Business University(No.19008020161)。
文摘This paper mainly focus on the global existence of the strong solutions for the generalized NavierStokes equations with damping. We obtain the global existence and uniqueness when α ≥5/4 for β ≥ 1 and when 1/2+β/2≤ α ≤5/4 for 8/3≤ β<+∞.
基金supported by NSFC of China(Grant 11771444)the Yue Qi Young Scholar Project,China University of Mining and Technology(Beijing)+1 种基金the Fund of Young Backbone Teachers in Henan Province(No.2018GGJS039)Incubation Fund Project of Henan Normal University(No.2020PL17).
文摘This paper is concerned with the existence of pullback attractors for three dimensional generalized Navier-Stokes equations with delay.According to compact argument,the existence and uniqueness of weak solutions are proved by using Galerkin method,and the continuous dependence of solutions on initial values is also shown.Based on the asymptotic compactness via weak convergence method and pullback absorbing set on appropriate functional phase spaces,we get the existence of pullback attractors.
基金supported by the Key Laboratory of Road Construction Technology and Equipment(Chang’an University,No.300102253502)the Natural Science Foundation of Shandong Province of China(GrantNo.ZR2022YQ06)the Development Plan of Youth Innovation Team in Colleges and Universities of Shandong Province(Grant No.2022KJ140).
文摘In this study,we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov(EFK)problem.The temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolson scheme.Following temporal discretization,the generalized finite difference method(GFDM)with supplementary nodes is utilized to address the nonlinear boundary value problems at each time node.These supplementary nodes are distributed along the boundary to match the number of boundary nodes.By incorporating supplementary nodes,the resulting nonlinear algebraic equations can effectively satisfy the governing equation and boundary conditions of the EFK equation.To demonstrate the efficacy of our approach,we present three numerical examples showcasing its performance in solving this nonlinear problem.
文摘We study exact solutions to (1 + 1)-dimensional generalized Boussinesq equation with time-space dispersion term by making use of improved sub-equation method, and analyse the dynamical behavior and exact solutions of the sub-equation after constructing the nonlinear transformation and constraint conditions. Accordingly, we obtain twenty families of exact solutions such as analytical and singular solitons and singular periodic waves. In addition, we discuss the impact of system parameters on wave propagation.
文摘In this paper, the initial boundary value problem of a class of nonlinear generalized Kolmogorov-Petrovlkii-Piskunov equations is studied. The existence and uniqueness of the solution and the bounded absorption set are proved by the prior estimation and the Galerkin finite element method, thus the existence of the global attractor is proved and the upper bound estimate of the global attractor is obtained.
文摘In this paper, we consider the generalized Korteweg-de-Vries (KdV) equations which are remarkable models of the water waves mechanics, the shallow water waves, the quantum mechanics, the ion acoustic waves in plasma, the electro-hydro-dynamical model for local electric field, signal processing waves through optical fibers, etc. We determine the useful and further general exact traveling wave solutions of the above mentioned NLDEs by applying the exp(−τ(ξ))-expansion method by aid of traveling wave transformations. Furthermore, we explain the physical significance of the obtained solutions of its definite values of the involved parameters with graphic representations in order to know the physical phenomena. Finally, we show that the exp(−τ(ξ))-expansion method is convenient, powerful, straightforward and provide more general solutions and can be helping to examine vast amount of travelling wave solutions to the other different kinds of NLDEs.
文摘We establish a resonance type existence theorem for generalized evolutional Navier-Stokes equations with periodic boundary conditions and external force depending on velocity, using the Faedo-Galerkin method.
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.
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(12361044)supported by the National Natural Science Foundation of China(12171024,11971217,11971020)supported by the Academic and Technical Leaders Training Plan of Jiangxi Province(20212BCJ23027)。
文摘This paper is concerned with the global well-posedness of the solution to the compressible Navier-Stokes/Allen-Cahn system and its sharp interface limit in one-dimensional space.For the perturbations with small energy but possibly large oscillations of rarefaction wave solutions near phase separation,and where the strength of the initial phase field could be arbitrarily large,we prove that the solution of the Cauchy problem exists for all time,and converges to the centered rarefaction wave solution of the corresponding standard two-phase Euler equation as the viscosity and the thickness of the interface tend to zero.The proof is mainly based on a scaling argument and a basic energy method.
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.