Analysis of 2+1 diffusive-dispersive PDE arising in river braiding

We present local existence and uniqueness results for the following $2+1$ dispersive diffusive equation due to P. Hall arising in modeling of river braiding: $u_{yyt} - γu_{xxx} -αu_{yyyy} - βu_{yy} + \left (u^2 \right)_{xyy} = 0$ for $(x,y) \in [0, 2π] \times [0, π]$, $t> 0$, 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 $\| u \|^2_{L^2}$ and cumulative dissipation $\int_0^t \| u_y \|_{L^2}^2 dt$ are globally controlled in time.

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