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Formation of singularities for a transport equation with nonlocal velocity. (English) Zbl 1101.35052
The authors study the nature of solutions of the equation \[ \theta_t -(H\theta)\theta_x =-\nu\Lambda^\alpha \theta \tag{1} \] on \(\mathbb{R}\). Here \(H\) is the Hilbert transform of the velocity, \(\Lambda\) is the square root of negative Laplacian and \(0\leq \nu \mathbb{R}\). This equation can be thought of as a generalization of the Burgers equation; it is also related to Birkhoff-Rott equation modelling the evolution of a vortex sheet. The authors show that the solutions to (1) with \(\nu=0\) blow up in finite time for a large class of initial data. On the other hand, adding the viscosity term with \(\nu>0\) and \(1<\alpha\leq 2\) ensures the global existence of solutions.

MSC:
35L60 First-order nonlinear hyperbolic equations
35L67 Shocks and singularities for hyperbolic equations
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