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On Hugoniot-Maslov conditions for vortex singular solutions of a system of shallow water equations. (English. Russian original) Zbl 1029.76007
Math. Notes 71, No. 6, 825-835 (2002); translation from Mat. Zametki 71, No. 6, 902-913 (2002).
This paper examines shallow water equations with variable Coriolis force \({\partial\eta \over\partial t}+ (\nabla,\eta u)=0\), \({\partial u\over\partial t}+(u,\nabla u)- \omega Tu+\nabla \eta=0\), where \(x= (x_1,x_2)^t \in\mathbb{R}^2\), the function \(\eta(x,t)\) is the elevation above the free surface, \(u(x,t)= {^t(u_1(x,t)}, u_2(x,t))\) is the velocity vector, \(T= \left(\begin{smallmatrix} 0 & 1\\ -1 & 0\end{smallmatrix}\right)\), \(\nabla={^t({\partial \over \partial x_1}},{\partial \over\partial x_2})\), \(\omega(x,t)= \widetilde \omega+\beta x_2\) is the doubled Coriolis frequency on \(\beta\)-plane, and \(\widetilde\omega,\beta\) are some physical constants. The goal of the author is to study solutions of the above system with singularities having the form of the square root of a quadratic form (the so-called solutions of vortex structure). Moreover, the author obtains the next correction to Cauchy-Riemann conditions describing how the singular part of the solution affects the smooth background.
MSC:
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76U05 General theory of rotating fluids
76B47 Vortex flows for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
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