Annals of Mathematics · 2005 · 124 citations · 9 references
Natural SciencesMathematicsPhysicsDifferential EquationsDynamical SystemsTransport PhenomenaDiffusion ProcessesInitial Value ProblemsPartial Differential EquationsTime IntegrationTime-dependent TransportFinite TimeInitial-value Problems In TransportKinetic Transport EquationsNonlocal TransportSingularity FormationTransport Equations
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.
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Finite time singularities in a 1D model of the quasi-geostrophic equation
Dongho Chae, Antonio Córdoba, Diego Córdoba et al. · Advances in Mathematics · 2004 · 117 citations