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

Abstract

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: $$u_{yyt}-\gamma u_{xxx}-\alpha u_{yyyy}-\beta u_{yy}+{\left(u^2\right)}_{xyy}=0$$ for $(x,y)\in [0,2\pi ]\times [0,\pi ]$, $t>0$, 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 $T_{s,b}$. We also show that the energy $\Vert u{\Vert }_{L^2}^2$ and cumulative dissipation ${\int }_0^t\Vert u_y{\Vert }_{L^2}^{2}dt$ are globally controlled in time.

21 pages

References in corpus (2)

• On the low regularity of the fifth order Kadomtsev-Petviashvili I equation
• On the well-posedness for Kadomtsev-Petviashvili-Burgers I equation