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Tracing complex singularities with spectral methods. (English) Zbl 0519.76002
Summary: A numerical method for investigating the possibility of blow-up after a finite time is introduced for a large class of nonlinear evolution problems. With initial data analytic in the space variable(s), the solutions have for any \(t > 0\) complex-space singularities at the edge of an analyticity strip of width \(\delta(t)\). Loss of regularity corresponds to the vanishing of \(\delta(t)\). Numerical integration by high resolution spectral methods reveals the large wavenumber behavior of the Fourier transform of the solutions, from which \(\delta(t)\) is readily obtained. Its time evolution can be traced down to about one mesh length. By extrapolation of \(\delta(t)\), such numerical experiments provide evidence suggesting finite-time blow-up or all-time regularity. The method is tested on the inviscid and viscous Burgers equations and is applied to the one-dimensional nonlinear Schrödinger equation with quartic potential and to the two-dimensional incompressible Euler equation, all with periodic boundary conditions. In the latter case evidence is found suggesting that existing all-time regularity results can be substantially sharpened.

MSC:
76M99 Basic methods in fluid mechanics
35Q99 Partial differential equations of mathematical physics and other areas of application
76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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