The linear stability problem for the Hocking-Stewartson pulse, obtained by linearizing the complex Ginzburg-Landau (cGL) equation, is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to stability exponents. A numerical algorithm based on the compound matrix method is developed for computing the Evans function. Using values in the cGL equation associated with spanwise modulation of plane Poiseuille flow, we show that the Hocking-Stewartson pulse associated with points along the neutral curve is always linearly unstable due to a real positive eigenvalue. Implications for the spanwise structure of nonlinear Poiseuille problem between parallel plates are also discussed.

The main result of this paper is the global well-posedness of the Cauchy problem to the 2D Navier-Stokes system with the initial data u 0 ∈ BUC(O) and the external force, the fluid flow is supposed to be periodic in one of the spatial directions whereas in the unbounded direction only uniform boundedness is assumed. However, to obtain uniqueness we need to make assumptions which suppresses additional pressure gradients. For this aim Riesz operators on L ∞ (O) are used to define p(t) ∈ BMO(O). For time-independent forces the solutions are shown to grow at most cubically in the time t.Mathematics Subject Classification (2000). 35Q30, 76D03, 76D05.

We study reversible, SO(2)-invariant vector fields in R 4 depending on a real parameter = which possess for ==0 a primary family of homoclinic orbits T : H 0 , : # S 1 . Under a transversality condition with respect to = the existence of homoclinic n-pulse solutions is demonstrated for a sequence of parameter values = (n)The existence of cascades of 2 l 3 m -pulse solutions follows by showing their transversality and then using induction. The method relies on the construction of an SO(2)-equivariant Poincare map which, after factorization, is a composition of two involutions: A logarithmic twist map and a smooth global map. Reversible periodic orbits of this map corresponds to reversible periodic or homoclinic solutions of the original problem. As an application we treat the steady complex Ginzburg Landau equation for which a primary homoclinic solution is known explicitly. Academic Press

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