Self-similar solutions for a transport equation with non-local flux

Abstract

The authors construct self-similar solutions for an N-dimensional transport equation, where the velocity is given by the Riezs 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.

Keywords

Hilbert transform / Riesz transform / Transport equations / Self-similar solutions

References

[1] Baker G. R., Li X., Morlet A. C.. Analytic structure of two 1D-transport equations with nonlocal fluxes. Physica D, 1996, 91: 349-375

[2] Balodis P., Córdoba A.. An inequality for Riesz transforms implying blow-up for some nonlinear and nonlocal transport equations. Adv. Math., 2007, 214: 1-39

[3] Biler, P., Karch, G. and Monneau, R., Nonlinear diffusion of dislocation density and self-similar solutions. arXiv:0812.4979.

[4] Castro A., Córdoba D.. Global existence, singularities and ill-posedness for a nonlocal flux. Adv. Math., 2008, 219: 1916-1936

[5] Castro, A., Córdoba, D. and Gancedo, F., A naive parametrization for the vortex sheet problem. arXiv:0810.0731.

[6] Chae D., Córdoba A., Córdoba D.. Finite time singularities in a 1D model of the quasi-geostrophic equation. Adv. Math., 2005, 194: 203-223

[7] Córdoba A., Córdoba D., Fontelos M. A.. Formation of singularities for a transport equation with non local velocity. Ann. Math., 2005, 162: 1377-1389

[8] Córdoba A., Córdoba D., Fontelos M. A.. Integral inequalities for the Hilbert transform applied to a nonlocal transport equation. J. Math. Pure Appl., 2006, 86: 529-540

[9] Dhanak M. R.. Equation of motion of a diffusing vortex sheet. J. Fluid Mech., 1994, 269: 365-281

[10] Dong H., Li D.. Finite time singularities for a class of generalized surface quasi-geostrophic equations. Proc. Amer. Math. Soc., 2008, 136: 2555-2563

[11] Getoor R. K.. First passage times for symmetric stable processes in space. Trans. Amer. Math. Soc., 1961, 101: 75-90

[12] Morlet A. C.. Further properties of a continuum of model equations with globally defined flux. J. Math. Anal. Appl., 1998, 221: 132-160

[13] Li D., Rodrigo J.. Blow up for the generalized surface quasi-geostrophic equation with supercritical dissipation. Commun. Math. Phys., 2009, 286: 111-124

[14] Okamoto H., Sakajo T., Wunsch M.. On a generalization of the Constantin-Lax-Majda equation. Nonlinearity, 2008, 2110: 2447-2461

[15] Stein E. M.. Singular Integrals and Differentiability Properties of Functions, 1970, Princeton: Princeton University Press