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Analytic sharp fronts for the surface quasi-geostrophic equation. (English) Zbl 1228.35010
This work is concerned with the evolution of sharp fronts for the quasi-geostrophic surface waves. The nonlinear integro-differential equation governing the evolution wave fronts in such flows was already obtained. The authors consider a simplified version of this equation and study the existence of analytical solutions through extensive calculations. By carefully investigating the evolution of the second space derivative of the unknown function the authors prove that the new system fits well into the abstract version of the celebrated Cauchy-Kovalevskaya theorem.

##### MSC:
 35A10 Cauchy-Kovalevskaya theorems 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 35R11 Fractional partial differential equations
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##### References:
 [1] Constantin P., Majda A., Tabak E.: Singular front formation in a model for quasigesotrophic flow. Phys. Fluids 6(1), 9–11 (1994) · Zbl 0826.76014 [2] Constantin P., Majda A., Tabak E.: Formation of strong fronts in the 2 quasigeostrophic thermal active scalar. Nonlinearity 7(6), 1495–1533 (1994) · Zbl 0809.35057 [3] Córdoba D., Fefferman C., Rodrigo J.: Almost sharp fronts for the surface Quasi-Geostrophic equation. PNAS 101(9), 2487–2491 (2004) · Zbl 1063.76011 [4] Córdoba D., Fontelos M.A., Mancho A.M., Rodrigo J.: Evidence of singularities for a family of contour dynamics equations. PNAS 102(17), 5949–5952 (2005) · Zbl 1135.76315 [5] Fefferman, C., Rodrigo, J.: On the limit of almost sharp fronts for the Surface Quasi-Geostrophic equation. In preparation. · Zbl 1063.76011 [6] Gancedo F.: Existence for the {$$\alpha$$}-patch model and the QG sharp front in Sobolev spaces. Adv. Math. 217(6), 2569–2598 (2008) · Zbl 1148.35099 [7] Majda, A., Bertozzi, A.: Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics 27, Cambridge: Cambridge Univ. Press, 2002 · Zbl 0983.76001 [8] Madja A., Tabak E.: A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow. Physisa D 98(2-4), 515–522 (1996) · Zbl 0899.76105 [9] Rodrigo J.: The vortex patch problem for the Quasi-Geostrophic equation. PNAS 101(9), 2484–2486 (2004) · Zbl 1063.76009 [10] Rodrigo J.: On the evolution of sharp fronts for the quasi-geostrophic equation. Comm. Pure Appl. Math. 58(6), 821–866 (2005) · Zbl 1073.35006 [11] Sammartino M., Caflisch R.E.: Zero Viscosity Limit for Analytic Solutions of the Navier-Stokes Equation on a Half-Space I. Existence for Euler and Prandtl equations. Commun. Math. Phys. 192, 433–461 (1998) · Zbl 0913.35102 [12] Sammartino M., Caflisch R.E.: Zero Viscosity Limit for Analytic Solutions of the Navier-Stokes Equation on a Half-Space II. Construction of Navier-Stokes Solution. Commun. Math. Phys. 192, 463–491 (1998) · Zbl 0913.35103
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