Local and Global Existence for an Aggregation Equation

By Thomas Laurent. 2007

118 citations recorded by the source corpus.

Subfields: Modeling and Simulation, Molecular Biology, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/68b40f2072e72ddb96e9d49dd5404960e6f6f123

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Citations in the corpus

Finite-time blow-up of L∞-weak solutions of an aggregation equation (2010, methodology): “It has been shown that if the kernel K has bounded second derivatives, then solutions exist for all time [16, 22]; again, we see that the degree of singularity of K controls blow-up.”
Stationary states of quadratic diffusion equations with long-range attraction (2011, background): “In the purely nonlocal case, namely when ε = 0, this issue has been studied extensively in many papers [37, 13, 35, 5, 4, 16, 6, 7, 29, 25, 24], combined with the study of the regularity of solutions compared to the attractive singularity of the interaction kernel.”
Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations (2011, background): “Therefore, we can easily check that the previous constructed solutions in the series of papers [27, 7, 5, 20] for a family of more restrictive potentials W than the ones presented in this work, are indeed weak measure solutions up to their maximal time of existence.”
Local and global well-posedness for aggregation equations and Patlak–Keller–Segel models with degenerate diffusion (2010, background): “For the inviscid problem, see the recent works of [41, 5, 4, 6, 22].”
Swarm dynamics and equilibria for a nonlocal aggregation model (2011, background): “A significant component of this research has dealt with the well-posedness of the initial-value problem for (1) [9, 10, 11, 12, 13].”
Dimensionality of Local Minimizers of the Interaction Energy (2012, background): “Well-posedness theories for these repulsiveattractive potentials in various functional settings have been provided in [24, 1, 8, 11, 9, 3].”
Ring patterns and their bifurcations in a nonlocal model of biological swarms (2015, unknown): “2) known as the aggregation equation [17, 18, 3, 4] which takes the form ρt(x,t)+∇·(ρ(x,t)u(x,t)) = 0, x∈R(2), t≥0 u(x,t) = ∫ R2 f (|x−y|)(x−y)ρ(y,t) dy.”
Lp theory for the multidimensional aggregation equation (2011, background): “3) arises in a number of models for biological aggregation [11, 14, 15, 23, 28, 37, 36, 38, 39] as well as problems in materials science [24, 25] and granular media [3, 17, 18, 31, 40].”
Blow-up in multidimensional aggregation equations with mildly singular interaction kernels (2009, methodology): “In this paper, we focus on the case involving only attractive forces [5, 6, 23, 35] and no diffusion.”
Finite-Time Singularities of an Aggregation Equation in $${\mathbb {R}^n}$$ with Fractional Dissipation (2009, methodology): “1 (Persistence of positivity and L norm [24]).”
Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion (2010, methodology): “Related model, without the diffusion, has been studied by Topaz and Bertozzi [33], Bodnar and Velazquez [6], Laurent [21], Bertozzi and Brandman [3], Li and Zhang [25], Carrillo, Di Francesco, Figalli, Laurent, Slepčev [12] and others.”
The derivation of swarming models: Mean-field limit and Wasserstein distances (2013, background): “For the local well-posedness of solutions to equation (3.1), we refer to [9, 10, 11, 58].”
Refined blowup criteria and nonsymmetric blowup of an aggregation equation (2009, background): “LHS of (21) = ∫ Rd×Rd×Rd ( x− y |x− y| · x− z |x− z| ) g(x)g(y)g(z) = dxdydz”