On Maslov’s conjecture on the structure of weak point singularities of the shallow water equations. (English. Russian original) Zbl 1090.76513

Dobrokhotov, S. Yu.; Pankrashkin, K. V.; Semenov, E. S.

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