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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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