Integral inequalities for the Hilbert transform applied to a nonlocal transport equation

DOI10.1016/j.matpur.2006.08.002zbMath1106.35059OpenAlexW2115639921MaRDI QIDQ864195

Antonio Córdoba, Marco Antonio Fontelos, Diego Cordoba

Publication date: 13 February 2007

Published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.matpur.2006.08.002

zbMATH Keywords

finite time blow-upBurgers type equationtransport equation with nonlocal velocityweighted inequalities involving the Hilbert transform

Mathematics Subject Classification ID

Shocks and singularities for hyperbolic equations (35L67) PDEs in connection with fluid mechanics (35Q35) Partial functional-differential equations (35R10) Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type (42A38) Inequalities involving derivatives and differential and integral operators (26D10) Existence, uniqueness, and regularity theory for incompressible inviscid fluids (76B03)

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Cites Work

• Finite time singularities in a 1D model of the quasi-geostrophic equation
• Blow-up in a modified Constantin-Lax-Majda model for the vorticity equation
• Analytic structure of two 1D-transport equations with nonlocal fluxes
• Behavior of solutions of model equations for incompressible fluid flow
• Formation of singularities for a transport equation with nonlocal velocity
• Vorticity and Incompressible Flow
• Explicit solutions of the viscous model vorticity equation
• The spontaneous appearance of a singularity in the shape of an evolving vortex sheet
• On global solutions for the Constantin–Lax–Majda equation with a generalized viscosity term
• A Partial Differential Equation Arising in a 1D Model for the 3D Vorticity Equation
• A simple one‐dimensional model for the three‐dimensional vorticity equation