A central problem in the mathematical analysis of fluid dynamics is the asymptotic limit of the fluid flow as viscosity goes to zero.This is particularly important when boundaries are present since vorticitv is typica...A central problem in the mathematical analysis of fluid dynamics is the asymptotic limit of the fluid flow as viscosity goes to zero.This is particularly important when boundaries are present since vorticitv is typically generated at the boundary as a result of boundary layer separation.The boundary laver theory,developed by Prandtl about a hundred years ago,has become a standard tool in addressing these questions.Yet at the mathematical level,there is still a lack of fundamental understanding of these questions and the validity of the boundary layer theory.In this article,we review recent progresses on the analysis of Prandtl’s equation and the related issue of the zero-viscosity limit for the solutions of the Navier-Stokes equation.We also discuss some directions where progress is expected in the near future.展开更多
In this paper,we study the dynamical stability of a family of explicit blowup solutions of the threedimensional(3D)incompressible Navier-Stokes(NS)equations with smooth initial values,which is constructed in Guo et al...In this paper,we study the dynamical stability of a family of explicit blowup solutions of the threedimensional(3D)incompressible Navier-Stokes(NS)equations with smooth initial values,which is constructed in Guo et al.(2008).This family of solutions has finite energy in any bounded domain of R3,but unbounded energy in R3.Based on similarity coordinates,energy estimates and the Nash-Moser-H?rmander iteration scheme,we show that these solutions are asymptotically stable in the backward light-cone of the singularity.Furthermore,the result shows the existence of local energy blowup solutions to the 3D incompressible NS equations with growing data.Finally,the result also shows that in the absence of physical boundaries,the viscous vanishing limit of the solutions does not satisfy the 3D incompressible Euler equations.展开更多
In this paper,we consider the zero-viscosity limit of the 2D steady Navier-Stokes equations in(0,L)×R+with no-slip boundary conditions.By estimating the stream-function of the remainder,we justify the validity of...In this paper,we consider the zero-viscosity limit of the 2D steady Navier-Stokes equations in(0,L)×R+with no-slip boundary conditions.By estimating the stream-function of the remainder,we justify the validity of the Prandtl boundary layer expansions.Specially,we show the global stability under the concavity condition of the Prandtl profile for an arbitrarily large constant L when the Euler flow is shear.展开更多
文摘A central problem in the mathematical analysis of fluid dynamics is the asymptotic limit of the fluid flow as viscosity goes to zero.This is particularly important when boundaries are present since vorticitv is typically generated at the boundary as a result of boundary layer separation.The boundary laver theory,developed by Prandtl about a hundred years ago,has become a standard tool in addressing these questions.Yet at the mathematical level,there is still a lack of fundamental understanding of these questions and the validity of the boundary layer theory.In this article,we review recent progresses on the analysis of Prandtl’s equation and the related issue of the zero-viscosity limit for the solutions of the Navier-Stokes equation.We also discuss some directions where progress is expected in the near future.
基金supported by National Natural Science Foundation of China(Grant Nos.12231016 and 12071391)Guangdong Basic and Applied Basic Research Foundation(Grant No.2022A1515010860)。
文摘In this paper,we study the dynamical stability of a family of explicit blowup solutions of the threedimensional(3D)incompressible Navier-Stokes(NS)equations with smooth initial values,which is constructed in Guo et al.(2008).This family of solutions has finite energy in any bounded domain of R3,but unbounded energy in R3.Based on similarity coordinates,energy estimates and the Nash-Moser-H?rmander iteration scheme,we show that these solutions are asymptotically stable in the backward light-cone of the singularity.Furthermore,the result shows the existence of local energy blowup solutions to the 3D incompressible NS equations with growing data.Finally,the result also shows that in the absence of physical boundaries,the viscous vanishing limit of the solutions does not satisfy the 3D incompressible Euler equations.
基金supported by National Natural Science Foundation of China (Grant Nos. 11471320 and 11631008)。
文摘In this paper,we consider the zero-viscosity limit of the 2D steady Navier-Stokes equations in(0,L)×R+with no-slip boundary conditions.By estimating the stream-function of the remainder,we justify the validity of the Prandtl boundary layer expansions.Specially,we show the global stability under the concavity condition of the Prandtl profile for an arbitrarily large constant L when the Euler flow is shear.
