Abstract
We examine periodic solutions to an initial boundary value problem for a Liouville equation with sign-changing weight. A representation formula is derived for both singular and nonsingular boundary data, including data arising from fractional linear maps. In the case of singular boundary data, we study the effects that the induced singularity has on the interior regularity of solutions. Regularity criteria are also found for a generalized form of the equation.
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References
Chuaqui M., Duren P., Osgood B.: The Schwarzian derivative for harmonic mappings. Journal D’Analyse Mathematique 91, 329–351 (2003)
D.G. Crowdy, General solutions to the 2D Liouville equation, Int. J. Engng Sci., 35 2, (1997) 141–149.
Crowdy D.G.: Stuart vortices on a sphere. J. Fluid Mech. 498, 381–402 (2004)
G.P. Dzhordzhadze, Regular solutions of the Liouville equation, Theor. Math. Phys+, 41 1, (1979) 867–871.
G. P. Dzhordzhadze, A. K. Pogrebkov, and M. K. Polivanov, On the solutions with singularities of the Liouville equation, Dokl. Akad. Nauk SSSR, 243, 318 (1978).
Khesin B., Lenells J., Misiolek G.: Generalized Hunter–Saxton equation and the geometry of the group of circle diffeomorphisms. Math. Ann. 342(3), 617–656 (2008)
S. Kichenassamy and W. Littman, Blow-up surfaces for nonlinear wave equations, I, Comm. PDE 18, No. 3&4, 431–452 (1993).
J. Liouville, J. Math. Pures Appl., 18, 71 (1853).
Okamoto H.: Well-posedness of the generalized Proudman–Johnson equation without viscosity. J. Math. Fluid Mech. 11, 46–59 (2009)
A.K. Pogrebkov, Global solutions of Cauchy problems for the Liouville equation \({\phi_{tt}-\phi_{xx}=-\frac{m^2}{2}e^{\phi}}\) in the case of singular initial data, Dokl. Akad. Nauk SSSR, 244, 873 (1979).
A.K. Pogrebkov, Complete integrability of dynamical systems generated by singular solutions of Liouville’s equation, Theor. Math. Phys.+, 45 2, (1980) 951–957.
C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory, Cambridge texts in applied mathematics, (2002).
B. Ruf and P. Ubilla, On a Liouville-type equation with sign-changing weight, Proceedings of the Royal Society of Edinburg, Sect. A, 139 (2009), 183–192.
A. Sarria and R. Saxton, Blow-up of solutions to the generalized inviscid Proudman–Johnson equation, J Math Fluid Mech, 15, 3 (2013), 493–523.
J.T. Stuart, Fluid. Mech., (1967), 29, 417.
Wunsch M.: The generalized Proudman-Johnson equation revisited. J. Math. Fluid Mech. 13((1), 147–154 (2009)
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Sarria, A., Saxton, R. A sign-changing Liouville equation. J. Evol. Equ. 15, 847–867 (2015). https://doi.org/10.1007/s00028-015-0283-5
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DOI: https://doi.org/10.1007/s00028-015-0283-5