In this paper,the authors will prove the global existence of solutions to the three dimensional axially symmetric Prandtl boundary layer equations with small initial data,which lies in H1Sobolev space with respect to ...In this paper,the authors will prove the global existence of solutions to the three dimensional axially symmetric Prandtl boundary layer equations with small initial data,which lies in H1Sobolev space with respect to the normal variable and is analytical with respect to the tangential variables.The main novelty of this paper relies on careful constructions of a tangentially weighted analytic energy functional and a specially designed good unknown for the reformulated system.The result extends that of Paicu-Zhang in[Paicu,M.and Zhang,P.,Global existence and the decay of solutions to the Prandtl system with small analytic data,Arch.Ration.Mech.Anal.,241(1),2021,403–446].from the two dimensional case to the three dimensional axially symmetric case,but the method used here is a direct energy estimates rather than Fourier analysis techniques applied there.展开更多
In this note,we study the Cauchy problem of the linear spatially homogeneous Landau equation with soft potentials.We prove that the solution to the Cauchy problem enjoys the analytic regularizing effect of the time va...In this note,we study the Cauchy problem of the linear spatially homogeneous Landau equation with soft potentials.We prove that the solution to the Cauchy problem enjoys the analytic regularizing effect of the time variable with an L2 initial datum for positive time.So that the smoothing effect of Cauchy problem for the linear spatially homogeneous Landau equation with soft potentials is similar to the heat equation.展开更多
基金supported by the National Natural Science Foundation of China(Nos.12031006,11801268)the Fundamental Research Funds for the Central Universities of China(No.NS2023039)。
文摘In this paper,the authors will prove the global existence of solutions to the three dimensional axially symmetric Prandtl boundary layer equations with small initial data,which lies in H1Sobolev space with respect to the normal variable and is analytical with respect to the tangential variables.The main novelty of this paper relies on careful constructions of a tangentially weighted analytic energy functional and a specially designed good unknown for the reformulated system.The result extends that of Paicu-Zhang in[Paicu,M.and Zhang,P.,Global existence and the decay of solutions to the Prandtl system with small analytic data,Arch.Ration.Mech.Anal.,241(1),2021,403–446].from the two dimensional case to the three dimensional axially symmetric case,but the method used here is a direct energy estimates rather than Fourier analysis techniques applied there.
基金supported by the NSFC(No.12031006)the Fundamental Research Funds for the Central Universities of China.
文摘In this note,we study the Cauchy problem of the linear spatially homogeneous Landau equation with soft potentials.We prove that the solution to the Cauchy problem enjoys the analytic regularizing effect of the time variable with an L2 initial datum for positive time.So that the smoothing effect of Cauchy problem for the linear spatially homogeneous Landau equation with soft potentials is similar to the heat equation.
