Duan, J., Bu, C., Gao, H. and Taboada, M.
On a Coupled Kuramoto-Sivashinsky and Ginzburg-Landau Type Model
for the Marangoni Convection
(198K, PostScript File)
ABSTRACT. The surface tension driven Marangoni convection is an interesting
pattern formation system. The ``primitive" governing equations are too
complicated to be investigated analytically. In this paper, the authors
consider a simplified model for this system. This simplified model is
in the form of coupled Kuramoto-Sivashinsky and Ginzburg-Landau type
partial differential equations. The authors prove the existence and
uniqueness of global solutions of this simplified mathematical model,
under the condition that the Marangoni number $Ma > Ma_c
+\frac{k}{2^5}$, where $Ma_c$ is the critical Marangoni number at which
the trivial stationary state becomes linearly unstable, and $k$ is a
positive constant related to other system parameters. The authors use
the contraction mapping principle in the proof as the usual semigroup
method does not apply directly to this system.
This work sets the foundation for further study of this model.