Nonlinear instability for nonhomogeneous incompressible viscous fluids  

作　　者：JIANG Fei JIANG Song NI GuoXi 

机构地区：[1]College of Mathematics and Computer Science,Fuzhou University [2]Laboratory of Computational Physics,Institute of Applied Physics and Computational Mathematics

出　　处：《Science China Mathematics》2013年第4期665-686,共22页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant Nos. 11101044,11271051,11229101 and 91130020);National Basic Research Program of China (Grant No.2011CB309705)

摘　　要：We investigate the nonlinear instability of a smooth steady density profile solution to the threedimensional nonhomogeneous incompressible Navier-Stokes equations in the presence of a uniform gravitational field,including a Rayleigh-Taylor steady-state solution with heavier density with increasing height(referred to the Rayleigh-Taylor instability).We first analyze the equations obtained from linearization around the steady density profile solution.Then we construct solutions to the linearized problem that grow in time in the Sobolev space H k,thus leading to a global instability result for the linearized problem.With the help of the constructed unstable solutions and an existence theorem of classical solutions to the original nonlinear equations,we can then demonstrate the instability of the nonlinear problem in some sense.Our analysis shows that the third component of the velocity already induces the instability,which is different from the previous known results.We investigate the nonlinear instability of a smooth steady density profile solution to the three- dimensional nonhomogeneous incompressible Navier-Stokes equations in the presence of a uniform gravitational field, including a Rayleigh-Taylor steady-state solution with heavier density with increasing height （referred to the Rayleigh-Taylor instability）. We first analyze the equations obtained from linearization around the steady density profile solution. Then we construct solutions to the linearized problem that grow in time in the Sobolev space Hk, thus leading to a global instability result for the linearized problem. With the help of the constructed unstable solutions and an existence theorem of classical solutions to the original nonlinear equations, we can then demonstrate the instability of the nonlinear problem in some sense. Our analysis shows that the third component of the velocity already induces the instability, which is different from the previous known results.

关 键 词：nonhomogeneous Navier-Stokes equations steady density profile Rayieigh-Taylor instability incompressible viscous flows 

分 类 号：O357.1[理学—流体力学] P433[理学—力学]