(1+2)维斑图方程的动力学性态

研究了在流体的有限层面由浮力或曲面张力梯度诱导的斑图构成方程,界于不良热导体平展边界间的斑图构成方程由Knobloch于1990年导出 ∂u/∂t=αu-μ∇2u-∇4u+K∇·(|∇u|2∇u+β∇2u∇u-γu∇u+δ∇|∇u|2),其中,u是面函数,μ是Rayleigh数,K=1,α表示热传递效益,是有限Biot数,当界面顶部和底部条件不相同时,β≠0,δ≠0,当出现非Boussinesq效应时,γ≠0,考虑α0,μ0,β=δ=0情形,即界面顶部和底部条件相同且出现非Boussinesq效应时(1+2)维Knobloch方程解的动力学性态,获得解的局部存在、整体存在以及吸引子的存在性.  

Dynamical behavior of the (1+2) D pattern formation

The equation of pattern formation induced by buoyancy or by surface tension gradient in thin layer is considered,the equation confined between horizontal poor heat conductors is introduced by Knobloch (1990) ∂_u/∂_t=αu-μ∇²u-∇⁴u+K∇·(|∇u|²∇u+β∇²u∇u-γu∇u+δ∇|∇u|²), where u is the planform function,μ is the scaled Raifleigh number,K =1 and α represents the effects of a heat transfer finite Biot number.The β,δ and γ represent the boundary condition at top and bottom or non Boussinesg effects respectively.The Knobloch equation with α<0,μ>0,β=δ=0(the boundary condition at top and bottom is the same) is considered,the local existence,global existence in the L²(Ω)-space and the global attractor have been obtained respectively.