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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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