We consider the three-dimensional compressible Navier-Stokes-Poisson system where the electric field of the internal electrostatic potential force is governed by the self-consistent Poisson equation.If the Fourier mod...We consider the three-dimensional compressible Navier-Stokes-Poisson system where the electric field of the internal electrostatic potential force is governed by the self-consistent Poisson equation.If the Fourier modes of the initial data are degenerate at the low frequency or the initial data decay fast at spatial infinity,we show that the density converges to its equilibrium state at the L 2-rate (1+t)(-7/4) or L ∞-rate (1+t)(-5/2),and the momentum decays at the L 2-rate (1+t)(-5/4) or L ∞-rate (1+t)(-2).These convergence rates are shown to be optimal for the compressible Navier-Stokes-Poisson system.展开更多
The authors study the asymptotic behaviour of solutions of the heat equation and a number of evolution equations using scaling techniques. It is proved that in the framework of bounded data stabilization need not occu...The authors study the asymptotic behaviour of solutions of the heat equation and a number of evolution equations using scaling techniques. It is proved that in the framework of bounded data stabilization need not occur and the general asymptotic behaviour is complex. This behaviour reflects for large times, even on compact sets, the complexity of the initial data at infinity.展开更多
The asymptotic behavior of periodic solutions to fractal nonlinear Burgers equation is considered and the initial data are allowed to be arbitrarily large.The exponential decay estimates of the solutions are obtained ...The asymptotic behavior of periodic solutions to fractal nonlinear Burgers equation is considered and the initial data are allowed to be arbitrarily large.The exponential decay estimates of the solutions are obtained for the power of Laplacian α∈[1/2,1).展开更多
The Cauchy problem of the compressible Euler equations with damping in multi-dimensions is considered when the initial perturbation in H3-norm is small. First, by using two new energy functionals together with the Gre...The Cauchy problem of the compressible Euler equations with damping in multi-dimensions is considered when the initial perturbation in H3-norm is small. First, by using two new energy functionals together with the Green's function and iteration method, we improve the L2-decay rate in Tan and Wang(2013)and Tan and Wu(2012)when(ρ0-ˉρ,m)˙B-s1,∞×˙B-s+11,∞with s∈[0,2]is bounded.In particular,it holds that the density converges to its equilibrium state at the rate(1+t)-34-s2 in L2-norm and the momentum decays at the rate(1+t)-54-s2 in L2-norm.Moreover,under a weaker and more general condition on the initial data,we show that the density and the momentum have different pointwise estimates in dimension d with d 3on both space variable x and time variable t as|Dαx(ρ-ˉρ)|C(1+t)-d2-|α|2(1+|x|21+t)-rwith r>d2and|Dαxm|C(1+t)-d2-|α|+12(1+|x|21+t)-d2 by a more elaborate analysis on the Green’s function.These results improve those in Wang and Yang(2001),where the density and the velocity(the momentum)have the same pointwise estimates.展开更多
We study the long-time behavior of viscosity solutions for time-dependent Hamilton-Jacobi equations by the dynamical approach based on weak KAM(Kolmogorov-Arnold-Moser) theory due to Fathi. We establish a general conv...We study the long-time behavior of viscosity solutions for time-dependent Hamilton-Jacobi equations by the dynamical approach based on weak KAM(Kolmogorov-Arnold-Moser) theory due to Fathi. We establish a general convergence result for viscosity solutions and adherence of the graph as t →∞.展开更多
The initial boundary value problems (IBVP) for the system of compressible adiabatic flow through porous media and the IBVP for its corresponding reduced system through Darcy’ laws on [0, 1] x [0, +] are considered re...The initial boundary value problems (IBVP) for the system of compressible adiabatic flow through porous media and the IBVP for its corresponding reduced system through Darcy’ laws on [0, 1] x [0, +] are considered respectively. The global existence of smooth solutions to the IBVP problems for two systems are proved, and their large-time behavior is analyzed. The time-asymptotic equivalence of these two systems are investigated, the decay rate of the difference of solutions between these two systems are shown to be determined explicitly by the initial perturbations and boundary effects. It is found that the oscillation of the specific volume can be cancelled by that of entropy, i.e., the oscillation of the specific volume and entropy is not required to be small.展开更多
基金partially supported by National Natural Science Foundation of China(Grant Nos.10871134,11011130029)the Huo Ying Dong Foundation (Grant No.111033)+3 种基金the Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (Grant No.PHR201006107)partially supported by National Natural Science Foundation of China (Grant Nos.10871175,10931007,10901137)Zhejiang Provincial Natural Science Foundation of China (Grant No.Z6100217)Specialized Research Fund for the Doctoral Program of Higher Education (Grant No.20090101120005)
文摘We consider the three-dimensional compressible Navier-Stokes-Poisson system where the electric field of the internal electrostatic potential force is governed by the self-consistent Poisson equation.If the Fourier modes of the initial data are degenerate at the low frequency or the initial data decay fast at spatial infinity,we show that the density converges to its equilibrium state at the L 2-rate (1+t)(-7/4) or L ∞-rate (1+t)(-5/2),and the momentum decays at the L 2-rate (1+t)(-5/4) or L ∞-rate (1+t)(-2).These convergence rates are shown to be optimal for the compressible Navier-Stokes-Poisson system.
文摘The authors study the asymptotic behaviour of solutions of the heat equation and a number of evolution equations using scaling techniques. It is proved that in the framework of bounded data stabilization need not occur and the general asymptotic behaviour is complex. This behaviour reflects for large times, even on compact sets, the complexity of the initial data at infinity.
基金Project supported by the National Natural Science Foundation of China (No. 11071162)the Shanghai Jiao Tong University Innovation Fund for Postgraduates (No. WS3220507101)
文摘The asymptotic behavior of periodic solutions to fractal nonlinear Burgers equation is considered and the initial data are allowed to be arbitrarily large.The exponential decay estimates of the solutions are obtained for the power of Laplacian α∈[1/2,1).
基金supported by National Natural Science Foundation of China(Grant Nos.11101112 and 11231006)the Fundamental Research Funds for the Central Universities(Grant No.2232015D3-33)
文摘The Cauchy problem of the compressible Euler equations with damping in multi-dimensions is considered when the initial perturbation in H3-norm is small. First, by using two new energy functionals together with the Green's function and iteration method, we improve the L2-decay rate in Tan and Wang(2013)and Tan and Wu(2012)when(ρ0-ˉρ,m)˙B-s1,∞×˙B-s+11,∞with s∈[0,2]is bounded.In particular,it holds that the density converges to its equilibrium state at the rate(1+t)-34-s2 in L2-norm and the momentum decays at the rate(1+t)-54-s2 in L2-norm.Moreover,under a weaker and more general condition on the initial data,we show that the density and the momentum have different pointwise estimates in dimension d with d 3on both space variable x and time variable t as|Dαx(ρ-ˉρ)|C(1+t)-d2-|α|2(1+|x|21+t)-rwith r>d2and|Dαxm|C(1+t)-d2-|α|+12(1+|x|21+t)-d2 by a more elaborate analysis on the Green’s function.These results improve those in Wang and Yang(2001),where the density and the velocity(the momentum)have the same pointwise estimates.
基金supported by National Natural Science Foundation of China(Grant Nos.1132510311301106 and 11201288)+1 种基金China Postdoctoral Science Foundation(Grant No.2014M550210)Guangxi Experiment Center of Information Science(Grant No.YB1410)
文摘We study the long-time behavior of viscosity solutions for time-dependent Hamilton-Jacobi equations by the dynamical approach based on weak KAM(Kolmogorov-Arnold-Moser) theory due to Fathi. We establish a general convergence result for viscosity solutions and adherence of the graph as t →∞.
基金the MST Grant #1999075107 and the Innovation funds of AMSS, CAS of China.
文摘The initial boundary value problems (IBVP) for the system of compressible adiabatic flow through porous media and the IBVP for its corresponding reduced system through Darcy’ laws on [0, 1] x [0, +] are considered respectively. The global existence of smooth solutions to the IBVP problems for two systems are proved, and their large-time behavior is analyzed. The time-asymptotic equivalence of these two systems are investigated, the decay rate of the difference of solutions between these two systems are shown to be determined explicitly by the initial perturbations and boundary effects. It is found that the oscillation of the specific volume can be cancelled by that of entropy, i.e., the oscillation of the specific volume and entropy is not required to be small.
