This paper is devoted to various considerations on a family of sharp interpo- lation inequalities on the sphere, which in dimension greater than 1 interpolate between Poincare, logarithmic Sobolev and critical Sobolev...This paper is devoted to various considerations on a family of sharp interpo- lation inequalities on the sphere, which in dimension greater than 1 interpolate between Poincare, logarithmic Sobolev and critical Sobolev （Onofri in dimension two） inequalities. The connection between optimal constants and spectral properties of the Laplace-Beltrami operator on the sphere is emphasized. The authors address a series of related observations and give proofs based on symmetrization and the ultraspherical setting.展开更多
The well-posedness of smooth solution to a 3-D simplified Energy-Transport model is discussed in this paper. We prove the local existence, uniqueness, and asymp- totic behavior of solution to the equations with hybrid...The well-posedness of smooth solution to a 3-D simplified Energy-Transport model is discussed in this paper. We prove the local existence, uniqueness, and asymp- totic behavior of solution to the equations with hybrid cross-diffusion. The smooth solution convergences to a stationary solution with an exponential rate as time tends to infinity when the initial date is a small perturbation of the stationary solution.展开更多
In this paper, we study a class of coupled fractional nonlinear Schr^dinger system with periodic boundary condition. Using Galerkin method, the existence of global smooth solution is obtained. We also prove the unique...In this paper, we study a class of coupled fractional nonlinear Schr^dinger system with periodic boundary condition. Using Galerkin method, the existence of global smooth solution is obtained. We also prove the uniqueness of the solution.展开更多
Consider the n-dimensional incompressible Navier-Stokes equations δ/(δt)u-α△u +(u ·△↓)u + △↓p = f(x, t), △↓· u = 0,△↓· f = 0,u(x, 0) = u0(x), △↓·u0=0.There exists a global weak soluti...Consider the n-dimensional incompressible Navier-Stokes equations δ/(δt)u-α△u +(u ·△↓)u + △↓p = f(x, t), △↓· u = 0,△↓· f = 0,u(x, 0) = u0(x), △↓·u0=0.There exists a global weak solution under some assumptions on the initial function and the external force. It is well known that the global weak solutions become sufficiently small and smooth after a long time. Here are several very interesting questions about the global weak solutions of the Cauchy problems for the n-dimensional incompressible Navier-Stokes equations.· Can we establish better decay estimates with sharp rates not only for the global weak solutions but also for all order derivatives of the global weak solutions?· Can we accomplish the exact limits of all order derivatives of the global weak solutions in terms of the given information?· Can we use the global smooth solution of the linear heat equation, with the same initial function and the external force, to approximate the global weak solutions of the Navier-Stokes equations?· If we drop the nonlinear terms in the Navier-Stokes equations, will the exact limits reduce to the exact limits of the solutions of the linear heat equation?· Will the exact limits of the derivatives of the global weak solutions of the Navier-Stokes equations and the exact limits of the derivatives of the global smooth solution of the heat equation increase at the same rate as the order m of the derivative increases? In another word, will the ratio of the exact limits for the derivatives of the global weak solutions of the Navier-Stokes equations be the same as the ratio of the exact limits for the derivatives of the global smooth solutions for the linear heat equation?The positive solutions to these questions obtained in this paper will definitely help us to better understand the properties of the global weak solutions of the incompressible Navier-Stokes equations and hopefully to discover new special structures of the Navier-Stokes equations.展开更多
基金supported by the NSFC(No.10926034,No.10771151)the Scientific Research Fund of Sichuan Provincial Education Department(No.08ZA041)the Scientific Research Found of Science and Technology Bureau of Sichuan Province
基金Project supported by ANR grants CBDif and NoNAP,the ECOS project (No. C11E07)the Chileanresearch grants Fondecyt (No. 1090103)Fondo Basal CMM Chile,Project Anillo ACT 125 CAPDEand the National Science Foundation (No.DMS 0901304)
文摘This paper is devoted to various considerations on a family of sharp interpo- lation inequalities on the sphere, which in dimension greater than 1 interpolate between Poincare, logarithmic Sobolev and critical Sobolev （Onofri in dimension two） inequalities. The connection between optimal constants and spectral properties of the Laplace-Beltrami operator on the sphere is emphasized. The authors address a series of related observations and give proofs based on symmetrization and the ultraspherical setting.
基金Acknowledgement The authors were partially supported by the National Natural Science Foundation of China （No.11071009, No.11371042, No.11471027）, and the Beijing Natural Science Foundation （No.1142001）.
文摘The well-posedness of smooth solution to a 3-D simplified Energy-Transport model is discussed in this paper. We prove the local existence, uniqueness, and asymp- totic behavior of solution to the equations with hybrid cross-diffusion. The smooth solution convergences to a stationary solution with an exponential rate as time tends to infinity when the initial date is a small perturbation of the stationary solution.
文摘In this paper, we study a class of coupled fractional nonlinear Schr^dinger system with periodic boundary condition. Using Galerkin method, the existence of global smooth solution is obtained. We also prove the uniqueness of the solution.
文摘Consider the n-dimensional incompressible Navier-Stokes equations δ/(δt)u-α△u +(u ·△↓)u + △↓p = f(x, t), △↓· u = 0,△↓· f = 0,u(x, 0) = u0(x), △↓·u0=0.There exists a global weak solution under some assumptions on the initial function and the external force. It is well known that the global weak solutions become sufficiently small and smooth after a long time. Here are several very interesting questions about the global weak solutions of the Cauchy problems for the n-dimensional incompressible Navier-Stokes equations.· Can we establish better decay estimates with sharp rates not only for the global weak solutions but also for all order derivatives of the global weak solutions?· Can we accomplish the exact limits of all order derivatives of the global weak solutions in terms of the given information?· Can we use the global smooth solution of the linear heat equation, with the same initial function and the external force, to approximate the global weak solutions of the Navier-Stokes equations?· If we drop the nonlinear terms in the Navier-Stokes equations, will the exact limits reduce to the exact limits of the solutions of the linear heat equation?· Will the exact limits of the derivatives of the global weak solutions of the Navier-Stokes equations and the exact limits of the derivatives of the global smooth solution of the heat equation increase at the same rate as the order m of the derivative increases? In another word, will the ratio of the exact limits for the derivatives of the global weak solutions of the Navier-Stokes equations be the same as the ratio of the exact limits for the derivatives of the global smooth solutions for the linear heat equation?The positive solutions to these questions obtained in this paper will definitely help us to better understand the properties of the global weak solutions of the incompressible Navier-Stokes equations and hopefully to discover new special structures of the Navier-Stokes equations.
