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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 \(f_t-(Hf)f_x=0\).

MSC:
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
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