The initial boundary value problem for a viscoelastic equation ｜ u t ｜ ρ u tt △u-△u tt ＋ t 0 g（ts）△u（s）ds ＋ ｜ u t ｜ m u t = ｜ u ｜ p u in a bounded domain is considered, where ρ, m, p 〉 0 and g is a n...The initial boundary value problem for a viscoelastic equation ｜ u t ｜ ρ u tt △u-△u tt ＋ t 0 g（ts）△u（s）ds ＋ ｜ u t ｜ m u t = ｜ u ｜ p u in a bounded domain is considered, where ρ, m, p 〉 0 and g is a nonnegative and decaying function. The general uniform decay of solution energy is discussed under some conditions on the relaxation function g and the initial data by adopting the method of [14, 15, 19]. This work generalizes and improves earlier results in the literature.展开更多
We investigate the global existence and asymptotic behavior of classical solutions for the 3D compressible non-isentropic damped Euler equations on a periodic domain. The global existence and uniqueness of classical s...We investigate the global existence and asymptotic behavior of classical solutions for the 3D compressible non-isentropic damped Euler equations on a periodic domain. The global existence and uniqueness of classical solutions are obtained when the initial data is near an equilibrium. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.展开更多
In this paper, we consider the global existence of solutions for the Cauchy problem of the generalized sixth order bad Boussinesq equation. Moreover, we show that the supremum norm of the solution decays algebraically...In this paper, we consider the global existence of solutions for the Cauchy problem of the generalized sixth order bad Boussinesq equation. Moreover, we show that the supremum norm of the solution decays algebraically to zero as （1 ＋ t）-（1/7） when t approaches to infinity, provided the initial data are sufficiently small and regular.展开更多
This paper is concerned with the global classical solution and the asymptotic behavior to a kind of linearly degenerate quasilinear hyperbolic system in several space variables.When the semilinear terms contain at lea...This paper is concerned with the global classical solution and the asymptotic behavior to a kind of linearly degenerate quasilinear hyperbolic system in several space variables.When the semilinear terms contain at least two waves with different propagation speeds,we can prove that the system considered admits a global classical solution by the weighted energy estimate under the small and suitable decay assumptions on the initial data.Furthermore,we can show that the solution converges to a solution of the linearized system based on the decay property of the nonlinearties.展开更多
In this paper,we consider 3D compressible magneto-micropolar fluids without resistivity and spin viscosity in a strip domain.The prominent character of the governing model is the presence of the microstructure,a linea...In this paper,we consider 3D compressible magneto-micropolar fluids without resistivity and spin viscosity in a strip domain.The prominent character of the governing model is the presence of the microstructure,a linear coupling structure involving derivatives of the velocity fields,which along with the lack of spin viscosity brings several challenges to the analysis.By exploiting the two-tier energy method developed in Guo and Tice(Arch Ration Mech Anal,2013),we prove the global existence of classical solutions to the governing model around a uniform magnetic field that is non-parallel to the horizontal boundary.Moreover,we show that the solution converges to the steady state at an almost exponential rate as time goes to infinity.One of the main ingredients in our analysis,compared with previous works on micropolar fluids,is that we deal with the microstructure by establishing some delicate estimates based on the analysis of the div-curl decomposition,and the coupling between the fluid velocity and the vorticity of angular velocity.展开更多
This paper is concerned with the large time behavior of the Cauchy problem for Navier-Stokes/AllenCahn system describing the interface motion of immiscible two-phase flow in 3-D. The existence and uniqueness of global...This paper is concerned with the large time behavior of the Cauchy problem for Navier-Stokes/AllenCahn system describing the interface motion of immiscible two-phase flow in 3-D. The existence and uniqueness of global solutions and the stability of the phase separation state are proved under the small initial perturbations. Moreover, the optimal time decay rates are obtained for higher-order spatial derivatives of density,velocity and phase. Our results imply that if the immiscible two-phase flow is initially located near the phase separation state, then under small perturbation conditions, the solution exists globally and decays algebraically to the complete separation state of the two-phase flow, that is, there will be no interface fracture, vacuum, shock wave, mass concentration at any time, and the interface thickness tends to zero as the time t → +∞.展开更多
This paper is concerned with the following logarithmic Schrodinger system:{-Δu_(1)+ω_(1)u_(1)=u_(1)u_(1)logu_(1)^(2)+2p/p+q|u_(2)|^(q)|u_(1)|^(p-2)u_(1),-Δu_(2)+ω_(2)u_(2)=u_(2)u_(2)log u_(2)^(2)+2q/p+q|u_(1)|^(p)...This paper is concerned with the following logarithmic Schrodinger system:{-Δu_(1)+ω_(1)u_(1)=u_(1)u_(1)logu_(1)^(2)+2p/p+q|u_(2)|^(q)|u_(1)|^(p-2)u_(1),-Δu_(2)+ω_(2)u_(2)=u_(2)u_(2)log u_(2)^(2)+2q/p+q|u_(1)|^(p)|u_(2)|^(q-2)u_(2),∫_(Ω)|u_(i)|^(2)dx=ρ_(i),i=1,2,(u_(1),u_(2))∈H_(0)^(1)(Ω;R^(2)),where Ω=R^(N)or Ω■R^(N)(N≥3)is a bounded smooth domain,andω_(i)R,μ_(i),ρ_(i)>0 for i=1,2.Moreover,p,q≥1,and 2≤p+q≤2^(*),where 2^(*):=2N/N-2.By using a Gagliardo-Nirenberg inequality and a careful estimation of u log u^(2),firstly,we provide a unified proof of the existence of the normalized ground state solution for all 2≤p+q≤2^(*).Secondly,we consider the stability of normalized ground state solutions.Finally,we analyze the behavior of solutions for the Sobolev-subcritical case and pass to the limit as the exponent p+q approaches 2^(*).Notably,the uncertainty of the sign of u log u^(2)in(0,+∞)is one of the difficulties of this paper,and also one of the motivations we are interested in.In particular,we can establish the existence of positive normalized ground state solutions for the Brézis-Nirenberg type problem with logarithmic perturbations(i.e.,p+q=2^(*)).In addition,our study includes proving the existence of solutions to the logarithmic type Bréis-Nirenberg problem with and without the L^(2)-mass.constraint ∫_(Ω)|u_(i)|^(2)dx=ρ_(i)(i=1,2)by two different methods,respectively.Our results seem to be the first result of the normalized solution of the coupled nonlinear Schrodinger system with logarithmic perturbations.展开更多
文摘The initial boundary value problem for a viscoelastic equation ｜ u t ｜ ρ u tt △u-△u tt ＋ t 0 g（ts）△u（s）ds ＋ ｜ u t ｜ m u t = ｜ u ｜ p u in a bounded domain is considered, where ρ, m, p 〉 0 and g is a nonnegative and decaying function. The general uniform decay of solution energy is discussed under some conditions on the relaxation function g and the initial data by adopting the method of [14, 15, 19]. This work generalizes and improves earlier results in the literature.
基金supported by the National Natural Science Foundation of China(11301172,11226170)China Postdoctoral Science Foundation funded project(2012M511640)Hunan Provincial Natural Science Foundation of China(13JJ4095)
文摘We investigate the global existence and asymptotic behavior of classical solutions for the 3D compressible non-isentropic damped Euler equations on a periodic domain. The global existence and uniqueness of classical solutions are obtained when the initial data is near an equilibrium. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.
文摘In this paper, we consider the global existence of solutions for the Cauchy problem of the generalized sixth order bad Boussinesq equation. Moreover, we show that the supremum norm of the solution decays algebraically to zero as （1 ＋ t）-（1/7） when t approaches to infinity, provided the initial data are sufficiently small and regular.
基金partially supported by the Outstanding Youth Fund of Zhejiang Province (Grant No. LR22A010004)the NSFC (Grant No. 12071435)。
文摘This paper is concerned with the global classical solution and the asymptotic behavior to a kind of linearly degenerate quasilinear hyperbolic system in several space variables.When the semilinear terms contain at least two waves with different propagation speeds,we can prove that the system considered admits a global classical solution by the weighted energy estimate under the small and suitable decay assumptions on the initial data.Furthermore,we can show that the solution converges to a solution of the linearized system based on the decay property of the nonlinearties.
基金supported by National Natural Science Foundation of China(Grant No.12101095)the Science and Technology Research Program of Chongqing Municipal Education Commission(Grant No.KJQN202100517)+6 种基金the Research Project of Chongqing Education Commission(Grant No.CXQT21014)the Natural Science Foundation of Chongqing(Grant No.cstc2021jcyj-msxm X0224)the Grant of Chongqing Young Experts’Workshopsupported by National Natural Science Foundation of China(Grant No.12201221)the Guangdong Basic and Applied Basic Research Foundation(Grant No.2021A1515111038)Changjiang Zhu was supported by National Natural Science Foundation of China(Grant Nos.12171160 and 11831003)the Guangdong Provincial Key Laboratory of Human Digital Twin(Grant No.2022B1212010004)。
文摘In this paper,we consider 3D compressible magneto-micropolar fluids without resistivity and spin viscosity in a strip domain.The prominent character of the governing model is the presence of the microstructure,a linear coupling structure involving derivatives of the velocity fields,which along with the lack of spin viscosity brings several challenges to the analysis.By exploiting the two-tier energy method developed in Guo and Tice(Arch Ration Mech Anal,2013),we prove the global existence of classical solutions to the governing model around a uniform magnetic field that is non-parallel to the horizontal boundary.Moreover,we show that the solution converges to the steady state at an almost exponential rate as time goes to infinity.One of the main ingredients in our analysis,compared with previous works on micropolar fluids,is that we deal with the microstructure by establishing some delicate estimates based on the analysis of the div-curl decomposition,and the coupling between the fluid velocity and the vorticity of angular velocity.
基金supported by the National Natural Science Foundation of China (Nos. 12171024, 11901025,11971217, 11971020)Academic and Technical Leaders Training Plan of Jiangxi Province (No. 20212BCJ23027)。
文摘This paper is concerned with the large time behavior of the Cauchy problem for Navier-Stokes/AllenCahn system describing the interface motion of immiscible two-phase flow in 3-D. The existence and uniqueness of global solutions and the stability of the phase separation state are proved under the small initial perturbations. Moreover, the optimal time decay rates are obtained for higher-order spatial derivatives of density,velocity and phase. Our results imply that if the immiscible two-phase flow is initially located near the phase separation state, then under small perturbation conditions, the solution exists globally and decays algebraically to the complete separation state of the two-phase flow, that is, there will be no interface fracture, vacuum, shock wave, mass concentration at any time, and the interface thickness tends to zero as the time t → +∞.
文摘This paper is concerned with the following logarithmic Schrodinger system:{-Δu_(1)+ω_(1)u_(1)=u_(1)u_(1)logu_(1)^(2)+2p/p+q|u_(2)|^(q)|u_(1)|^(p-2)u_(1),-Δu_(2)+ω_(2)u_(2)=u_(2)u_(2)log u_(2)^(2)+2q/p+q|u_(1)|^(p)|u_(2)|^(q-2)u_(2),∫_(Ω)|u_(i)|^(2)dx=ρ_(i),i=1,2,(u_(1),u_(2))∈H_(0)^(1)(Ω;R^(2)),where Ω=R^(N)or Ω■R^(N)(N≥3)is a bounded smooth domain,andω_(i)R,μ_(i),ρ_(i)>0 for i=1,2.Moreover,p,q≥1,and 2≤p+q≤2^(*),where 2^(*):=2N/N-2.By using a Gagliardo-Nirenberg inequality and a careful estimation of u log u^(2),firstly,we provide a unified proof of the existence of the normalized ground state solution for all 2≤p+q≤2^(*).Secondly,we consider the stability of normalized ground state solutions.Finally,we analyze the behavior of solutions for the Sobolev-subcritical case and pass to the limit as the exponent p+q approaches 2^(*).Notably,the uncertainty of the sign of u log u^(2)in(0,+∞)is one of the difficulties of this paper,and also one of the motivations we are interested in.In particular,we can establish the existence of positive normalized ground state solutions for the Brézis-Nirenberg type problem with logarithmic perturbations(i.e.,p+q=2^(*)).In addition,our study includes proving the existence of solutions to the logarithmic type Bréis-Nirenberg problem with and without the L^(2)-mass.constraint ∫_(Ω)|u_(i)|^(2)dx=ρ_(i)(i=1,2)by two different methods,respectively.Our results seem to be the first result of the normalized solution of the coupled nonlinear Schrodinger system with logarithmic perturbations.
