An integro-differential equation arising as a limit of individual cell-based models

By M. Bodnar, Juan J. López Velázquez. 2006

139 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/ee84cfb3c5f33f230391b0f397a16f79c98b2bd7

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Citations in the corpus

Refined blowup criteria and nonsymmetric blowup of an aggregation equation (2009, background): “): Plugging this estimate into ( 4 ), we obtain”
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].”
On minimizers of interaction functionals with competing attractive and repulsive potentials (2015, background): “2) These functionals are directly connected to a class of self-assembly/aggregation models which recently have received much attention (see for example, [9, 11, 12, 19, 33, 36, 38, 39, 40, 47, 54, 58, 59]).”
Finite-Time Blow-up of Solutions of an Aggregation Equation in Rn (2007, background): “Bodnar and Velazquez consider the one-dimensional problem in [2] and prove finite time blowup by direct comparison with a Burgers-like dynamics.”
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.”
Lp theory for the multidimensional aggregation equation (2011, methodology): “For the inviscid case, much work has been done recently on the question of finite time blowup in equations of this type, from bounded or smooth initial data [10, 6, 4, 5].”
Finite-time blow-up of L∞-weak solutions of an aggregation equation (2010, methodology): “Blow-up for a large class of initial data was proven in one spatial dimension by comparison with a Burgers-like dynamics in [4]; it was also studied, both analytically and computationally, in [17].”
Local and Global Existence for an Aggregation Equation (2007, background): “Bodnar and Velazquez (2006) prove existence and uniqueness in one space dimension for C1 initial data.”
Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations (2011, methodology): “Mathematical modeling of the collective behavior of individuals, such as swarming, has also been treated by continuum models steaming from discrete particle models [33] [14] [41] [34] [42] [15] [36] [13] [21] [23] [16] [17].”
Stationary states of quadratic diffusion equations with long-range attraction (2011, background): “In particular, if nonlocal repulsion acts at a smaller scale with respect to nonlocal attractive forces in the large particle limit, then a nonlocal repulsion term can be replaced by a local term with nonlinear diffusion, we refer to [42, 11, 51, 52, 20, 22, 31, 49, 10, 36] for several examples.”
Blow-up in multidimensional aggregation equations with mildly singular interaction kernels (2009, background): “In both [10, 27] the transformation w = ∫ x −∞ u is used resulting in a non-local scalar conservation law for w: wt − k′′ ∗ wwx = 0.”
Finite-Time Singularities of an Aggregation Equation in $${\mathbb {R}^n}$$ with Fractional Dissipation (2009, background): “In one space dimension with C1 initial data, Bodnar and Vel´azquez [ 6 ] proved global well-posedness for some classes of interaction potentials and finite-time blow-up for others.”