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

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: u(yyt) - gamma u(xxx) - alpha u(yyyy) - beta u(yy) + (u(2))(xyy) = 0 for (x, y) is an element of [0, 2 pi] x [0, pi], t > 0, with boundary condition u(y) = 0 = u(yyy) at y = 0 and y = pi and 2 pi periodicity in x, using a contraction mapping argument in a Bourgain-type space T-s,T-b. We also show that the energy parallel to u(center dot, center dot, t)parallel to(2)(L2) and cumulative dissipation integral(t)(0) parallel to u(y)(center dot, center dot, s)parallel to(2)(L2) dt are globally controlled in time t.