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On Maslov’s conjecture on the structure of weak point singularities of the shallow water equations. (English. Russian original) Zbl 1090.76513
Dokl. Math. 64, No. 1, 127-130 (2001); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 379, No. 2, 173-176 (2001).
From the text: The authors study the following system: $$\eta_t + \nabla\cdot(\eta u) = 0, u_t + (u\cdot\nabla) u - 2\omega T\,u + \nabla \eta = 0,$$ describing a two-dimensional velocity field $$u(x,t)$$ in rotating shallow water for which $$\eta (x,t) >0$$ is the free-surface elevation, $$x=(x_1,x_2)$$, $$\omega = \omega_0 + \beta x_2$$ is the Coriolis frequency with constant coefficients $$\omega_0$$, $$\beta$$, and $T = \left[\begin{matrix} 0&1\\-1&0\end{matrix}\right].$ They outline some points for the proof of the following theorem.
Let $$(u,\eta)$$ be a solution of the form $$u = u_0 (x,t) + U(x,t)\,F(S), \eta = \eta_0 (x,t) + R(x,t)\,F(S)$$, where $$F(S)$$ is a function with the following properties: it is continuous for $$S\geq 0$$, it is smooth for $$S>0$$, $$F(0)=0$$, and $$F'(S)\to\infty$$ as $$S\to 0$$; $$S = S(x,t)\geq 0$$ is a smooth function vanishing for $$x=X(t)$$, and such that $$[ S_{x_ix_j} (X(t),t) ]$$ is a positive definite matrix with two simple eigenvalues. If certain derivatives of $$U$$ and $$R$$ with respect to $$x_1$$ and $$x_2$$ do not vanish, then necessarily $$F(S) = S^{1/2}$$.

##### MSC:
 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76U05 General theory of rotating fluids 35Q35 PDEs in connection with fluid mechanics