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On singularity formation of a nonlinear nonlocal system. (English) Zbl 1241.35038

The following Cauchy problem is studied: \[ \begin{aligned} \frac{\partial u}{\partial t}=2uv,\quad \frac{\partial v}{\partial t}=H(u^2),\;t>0,\;x\in \mathbb{R},\\ u(x,0)=u_0(x),\;v(x,0)=v_0(x),\quad x\in \mathbb{R}\end{aligned} \tag{1} \] Here \(u(x,t)\) and \(v(x,t)\) are the unknown functions, \(H\) is the operator of the Hilbert transform along the \(x\) axis, \(u_0(x)\) and \(v_0(x)\) are the given functions from \(H^1\).
It is proved that for initial data with finite energy the solution to the problem (1) must develop a finite time singularity in the \(H^1\) norm. The global regularity of the solution to the problem (1) is proved too. Substantial part of the paper is devoted to numerical results of investigation of the nature of the finite time singularities for both systems (1) and \[ \begin{aligned} \frac{\partial u}{\partial t}=uv+\nu\frac{\partial^2 u}{\partial x^2},\quad \frac{\partial v}{\partial t}=H(u^2)+\nu\frac{\partial^2 v}{\partial x^2}, \end{aligned} \] where \(\nu>0\) is the viscosity.

MSC:

35C06 Self-similar solutions to PDEs
35Q35 PDEs in connection with fluid mechanics
35K45 Initial value problems for second-order parabolic systems
35R09 Integro-partial differential equations
35K58 Semilinear parabolic equations
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