Further properties of a continuum of model equations with globally defined flux.

Journal Article · Fri May 01 04:00:00 EDT 1998 · J. Math. Anal. Appl.

DOI:https://doi.org/10.1006/jmaa.1997.5801· OSTI ID:938137

Morlet, A C

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{sub t} = {theta}(H(u)u){sub x} + (1 - {theta})(u)u{sub x} + {nu}u{sub xx}, 0 {le} {theta} {le} 1, {nu} {ge} 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} {le} 0 in finite time.

• Research Organization: Argonne National Laboratory (ANL)
• Sponsoring Organization: ER
• DOE Contract Number: AC02-06CH11357
• OSTI ID: 938137
• Report Number(s): ANL/MCS-P678-0897
• Journal Information: J. Math. Anal. Appl., Journal Name: J. Math. Anal. Appl. Journal Issue: 1 ; May 1998 Vol. 221
• Country of Publication: United States
• Language: ENGLISH

Related Subjects

97 MATHEMATICS AND COMPUTING
99 GENERAL AND MISCELLANEOUS
HILBERT TRANSFORMATION
NUMERICAL SOLUTION
PARTIAL DIFFERENTIAL EQUATIONS
SINGULARITY