Formation of singularities for a transport equation with nonlocal velocity

By A. Córdoba, D. C'ordoba, Marco Antonio Fontelos. 2005

191 citations recorded by the source corpus.

We study a 1D transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data. By adding a diffusion term the finite time singularity is prevented and the solutions exist globally in time.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5e74e2ac822d73e42a93acb2319f859f67c34097

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Refined blowup criteria and nonsymmetric blowup of an aggregation equation (2009, background): “Equations similar to (1) with fractional diffusion have been studied in the literature (see (7), (10), (9) and (24)).”
Finite-Time Singularities of an Aggregation Equation in $${\mathbb {R}^n}$$ with Fractional Dissipation (2009, methodology): “We refer the reader to [10],[9] and [27] where this type of dissipation has been used in the context of the surface quasi-geostrophic equation and other one dimensional models, for a more detailed explanation of the 3 regimes.”
On the Finite‐Time Blowup of a One‐Dimensional Model for the Three‐Dimensional Axisymmetric Euler Equations (2017, methodology): “Important modifications of the original model of [8] were proposed by [12, 13] and somewhat differently motivated work of [10].”
The role of self-similarity in singularities of partial differential equations (2008, background): “One-dimensional analogues of this problem, representing the convection of a scalar with a velocity field, which is the Hilbert transform of the scalar itself do have singularities in the form of cusps, as proved in [35], [46].”
Well-posedness for a transport equation with nonlocal velocity (2008, methodology): “To circumvent this difficulty, the authors of [4] considered only positive initial data and thus control the L(2) norm by showing its monotonicity in time.”
Blow-up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation (2008, methodology): “Our proof will follow the approach of Córdoba, Córdoba and Fontelos, (6) with some modifications needed to avoid the problems introduced by the presence of the diffusion term.”