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Integral inequalities for the Hilbert transform applied to a nonlocal transport equation. (English) Zbl 1106.35059
We prove several weighted inequalities involving the Hilbert transform of a function f(x) and its derivative. One of those inequalities,
\[ -\int\frac{f_x(x)[Hf(x)-Hf(0)]}{|x|^\alpha}\,dx\geq C_\alpha\int\frac{f(x)-f(0))^2}{|x|^{1+\alpha}}\, dx, \] is used to show finite time blow-up for a transport equation with nonlocal velocity \(f_t-(Hf)f_x=0\).

35Q35 PDEs in connection with fluid mechanics
26D10 Inequalities involving derivatives and differential and integral operators
35R10 Functional partial differential equations
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
35L67 Shocks and singularities for hyperbolic equations
Full Text: DOI
[1] Baker, G.R.; Li, X.; Morlet, A.C., Analytic structure of 1D-transport equations with nonlocal fluxes, Physica D, 91, 349-375, (1996) · Zbl 0899.76104
[2] Bertozzi, A.L.; Majda, A.J., Vorticity and the mathematical theory of incompressible fluid flow, (2002), Cambridge Univ. Press Cambridge, UK
[3] Constantin, P.; Lax, P.; Majda, A., A simple one-dimensional model for the three-dimensional vorticity, Comm. pure appl. math., 38, 715-724, (1985) · Zbl 0615.76029
[4] Córdoba, A.; Córdoba, D.; Fontelos, M.A., Formation of singularities for a transport equation with nonlocal velocity, Ann. of math., 162, 3, 1375-1387, (2005) · Zbl 1101.35052
[5] Chae, D.; Córdoba, A.; Córdoba, D.; Fontelos, M.A., Finite time singularities in a 1D model of the quasi-geostrophic equation, Adv. math., 194, 203-223, (2005) · Zbl 1128.76372
[6] De Gregorio, S., A partial differential equation arising in a 1D model for the 3D vorticity equation, Math. methods appl. sci., 19, 1233-1255, (1996) · Zbl 0860.35101
[7] Moore, D.W., The spontaneous appearance of a singularity in the shape of an evolving vortex sheet, Proc. R. soc. London A, 365, 1720, 105-119, (1979) · Zbl 0404.76040
[8] Sakajo, T., On global solutions for the constantin – lax – majda equation with a generalized viscosity term, Nonlinearity, 16, 1319-1328, (2003) · Zbl 1140.76332
[9] Schochet, S., Explicit solutions of the viscous model vorticity equation, Comm. pure appl. math., 41, 531-537, (1986) · Zbl 0623.76012
[10] Yang, Y., Behavior of solutions of model equations for incompressible fluid flow, J. differential equations, 125, 133-153, (1996) · Zbl 0851.76012
[11] Vasudeva, M.; Wegert, E., Blow-up in a modified constantin – lax – majda model for the vorticity equation, Z. anal. anwend., 18, 183-191, (1999) · Zbl 0943.76023
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