Further Properties of a Continuum of Model Equations with Globally Defined Flux

• Published in: Journal of mathematical analysis and applications 1998-05, Vol.221 (1), p.132-160
• Main Author: Morlet, Anne C
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Mathematical analysis; Mathematics; Partial differential equations; Sciences and techniques of general use
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1006/jmaa.1997.5801

To develop an understanding of singularity formation in vortex sheets, we consider model equations that exhibit shared characteristics with the vortex sheet equation but are slightly easier to analyze. A model equation is obtained by replacing the flux term in Burgers' equation by alternatives that contain contributions depending globally on the solution. We consider the continuum of partial differential equationsut=θ(H(u)u)x+(1−θ)(u)ux+νuxx, 0≤θ≤1, ν≥0, whereH(u) is the Hilbert transform ofu. We show that when θ=1/2, for ν>0, the solution of the equation exists for all time and is unique. We also show with a combination of analytical and numerical means that the solution when θ=1/2 and ν>0 is analytic. Using a pseudo-spectral method in space and the Adams–Moulton fourth-order predictor-corrector in time, we compute the numerical solution of the equation with θ=1/2 for various viscosities. The results confirm that for ν>0, the solution is well behaved and analytic. The numerical results also confirm that for ν>0 and θ=1/2, the solution becomes singular in finite time and finite viscosity prevents singularity formation. We also present, for a certain class of initial conditions, solutions of the equation, with 0