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Published online by Cambridge University Press: 16 February 2016
We present local existence and uniqueness results for the following 2 + 1 diffusive–dispersive equation due to P. Hall arising in modelling of river braiding: for (x,y) ∈ [0, 2π] × [0, π], t > 0, with boundary condition u y =0=u yyy at y=0 and y=π and 2π periodicity in x, using a contraction mapping argument in a Bourgain-type space T s,b . We also show that the energy ∥u(·, ·, t)∥2 L2 and cumulative dissipation ∫0 t ∥u y (·, ·, s)∥L2 2 dt are globally controlled in time t.$$\begin{equation*} u_{yyt} - \gamma u_{xxx} -\alpha u_{yyyy} - \beta u_{yy} + \left ( u^2 \right )_{xyy} = 0 \end{equation*}$$