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

Published online by Cambridge University Press:  16 February 2016

SALEH TANVEER
Affiliation:
Mathematics Department, The Ohio State University, Columbus, OH 43210, USA email: tanveer@math.ohio-state.edu
CHARIS TSIKKOU
Affiliation:
Mathematics Department, West Virginia University, Morgantown, WV 26505, USA email: tsikkou@math.wvu.edu
Corresponding

Abstract

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:

$$\begin{equation*} u_{yyt} - \gamma u_{xxx} -\alpha u_{yyyy} - \beta u_{yy} + \left ( u^2 \right )_{xyy} = 0 \end{equation*}$$
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 L 2 and cumulative dissipation ∫0 t u y (·, ·, s)∥ L 2 2 dt are globally controlled in time t.

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Copyright
Copyright © Cambridge University Press 2016 

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