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Self-similar solutions for a transport equation with non-local flux. (English) Zbl 1186.35154
Summary: The authors construct self-similar solutions for an \(N\)-dimensional transport equation, where the velocity is given by the Riesz transform. These solutions imply nonuniqueness of weak solution. In addition, self-similar solution for a one-dimensional conservative equation involving the Hilbert transform is obtained.

35Q35 PDEs in connection with fluid mechanics
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35A22 Transform methods (e.g., integral transforms) applied to PDEs
35L67 Shocks and singularities for hyperbolic equations
Full Text: DOI
[1] Baker, G. R., Li, X. and Morlet, A. C., Analytic structure of two 1D-transport equations with nonlocal fluxes, Physica D, 91, 1996, 349–375. · Zbl 0899.76104 · doi:10.1016/0167-2789(95)00271-5
[2] Balodis, P. and Córdoba, A., An inequality for Riesz transforms implying blow-up for some nonlinear and nonlocal transport equations, Adv. Math., 214, 2007, 1–39. · Zbl 1133.35078 · doi:10.1016/j.aim.2006.07.021
[3] Biler, P., Karch, G. and Monneau, R., Nonlinear diffusion of dislocation density and self-similar solutions. arXiv:0812.4979. · Zbl 1207.82049
[4] Castro, A. and Córdoba, D., Global existence, singularities and ill-posedness for a nonlocal flux, Adv. Math., 219, 2008, 1916–1936. · Zbl 1186.35002 · doi:10.1016/j.aim.2008.07.015
[5] Castro, A., Córdoba, D. and Gancedo, F., A naive parametrization for the vortex sheet problem. arXiv:0810.0731. · Zbl 1296.76022
[6] Chae, D., Córdoba, A., Córdoba, D., et al, Finite time singularities in a 1D model of the quasi-geostrophic equation, Adv. Math., 194, 2005, 203–223. · Zbl 1128.76372 · doi:10.1016/j.aim.2004.06.004
[7] Córdoba, A., Córdoba, D. and Fontelos, M. A., Formation of singularities for a transport equation with non local velocity, Ann. Math., 162, 2005, 1377–1389. · Zbl 1101.35052 · doi:10.4007/annals.2005.162.1377
[8] Córdoba, A., Córdoba, D. and Fontelos, M. A., Integral inequalities for the Hilbert transform applied to a nonlocal transport equation, J. Math. Pure Appl., 86, 2006, 529–540. · Zbl 1106.35059 · doi:10.1016/j.matpur.2006.08.002
[9] Dhanak, M. R., Equation of motion of a diffusing vortex sheet, J. Fluid Mech., 269, 1994, 365–281. · Zbl 0821.76013 · doi:10.1017/S0022112094001552
[10] Dong, H. and Li, D., Finite time singularities for a class of generalized surface quasi-geostrophic equations, Proc. Amer. Math. Soc., 136, 2008, 2555–2563. · Zbl 1143.35084 · doi:10.1090/S0002-9939-08-09328-3
[11] Getoor, R. K., First passage times for symmetric stable processes in space, Trans. Amer. Math. Soc., 101, 1961, 75–90. · Zbl 0104.11203 · doi:10.1090/S0002-9947-1961-0137148-5
[12] Morlet, A. C., Further properties of a continuum of model equations with globally defined flux, J. Math. Anal. Appl., 221, 1998, 132–160. · Zbl 0916.35049 · doi:10.1006/jmaa.1997.5801
[13] Li, D. and Rodrigo, J., Blow up for the generalized surface quasi-geostrophic equation with supercritical dissipation, Commun. Math. Phys., 286, 2009, 111–124. · Zbl 1172.86301 · doi:10.1007/s00220-008-0585-3
[14] Okamoto, H., Sakajo, T. and Wunsch, M., On a generalization of the Constantin-Lax-Majda equation, Nonlinearity, 21(10), 2008, 2447–2461. · Zbl 1221.35300 · doi:10.1088/0951-7715/21/10/013
[15] Stein, E. M., Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. · Zbl 0207.13501
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