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Further properties of a continuum of model equations with globally defined flux. (English) Zbl 0916.35049
Author’s summary: 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 equations \(u_t= \theta(H(u)u)_x+ (1-\theta)H(u) u_x+ \nu u_{xx}\), \(0\leq \theta\leq 1\), \(\nu\geq 0\), where \(H(u)\) is the Hilbert transform of \(u\). We show that when \(\theta= 1/2\), for \(\nu>0\), 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 \(\theta= 1/2\) and \(\nu>0\) 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 \(\theta= 1/2\) for various viscosities. The results confirm that for \(\nu>0\), the solution is well behaved and analytic. The numerical results also confirm that for \(\nu>0\) and \(\theta= 1/2\), 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 \(0<\theta< 1/3\) and \(\theta= 1\), that become infinite for \(\nu\geq 0\) in finite time.

MSC:
35K55 Nonlinear parabolic equations
35Q53 KdV equations (Korteweg-de Vries equations)
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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