Finite-Time Blow-up of Solutions of an Aggregation Equation in Rn

By Andrea L. Bertozzi, Thomas Laurent. 2007

115 citations recorded by the source corpus.

Subfields: Applied Mathematics, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/2cf7db763c26cb682995b878ce5f7568f312e1c2

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Citations in the corpus

Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion (2010, background): “Some recent work on this problem includes [5, 3, 4] for the inviscid case [26, 25, 24] for the problem with fractional diffusion and [27] for the problem in 1D with nonlinear diffusion.”
Lp theory for the multidimensional aggregation equation (2011, background): “For the inviscid case, much work has been done recently on the question of nite time blowup in equations of this type, from bounded or smooth initial data [10, 6 , 4, 5]. A recent study [16] proves well-posedness of measure solutions for semi-convex kernels.”
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.”
Refined blowup criteria and nonsymmetric blowup of an aggregation equation (2009, methodology): “Remark 3 We point out that the range for q in ( 2 ) is only descriptive in the inviscid case, since due to the smoothing eect of the viscosity one instantly obtains additional regularity for the solution and this yields the biggest ranges, namely 2 q1 in dimension 3 and 2 2 ) and the general case (”
Stationary states of quadratic diffusion equations with long-range attraction (2011, background): “edium equation, which are nicely collected in the book [53], as well as the papers [45, 19]. In the purely nonlocal case, namely when "= 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 com”
Dimensionality of Local Minimizers of the Interaction Energy (2012, background): “ion 5 we will discuss in more detail these questions. Let us nally mention that the gradient ow of the energy E W in the Wasserstein sense d 2 [12, 1, 13] has been extensively studied in recent years [24, 7, 5, 6, 8, 11, 10, 9, 17, 18, 30, 3, 2]. It leads to the nonlocal interaction equation @ @t + ”
Finite-Time Singularities of an Aggregation Equation in $${\mathbb {R}^n}$$ with Fractional Dissipation (2009, methodology): “We state our results starting with an extension of the local existence theorem and continuation result proved by Bertozzi and Laurent [3] in the case ν = 0.”
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 L∞-weak solutions of an aggregation equation (2010, methodology): “For the remainder of this section, we study finite time blow-up for kernels of the form K(x)= e−|x|+g(|x|2), where g ∈ C2(Rn). Our discussion uses results from [ 3 ].”
The derivation of swarming models: Mean-field limit and Wasserstein distances (2013, background): “mmarize the results on the existence and uniqueness of solutions to the aggregation equation (3.1). For the local well-posedness of solutions to equation (3.1), we refer 12 CARRILLO,CHOI,ANDHAURAY to [9, 10, 11, 58]. In particular, unique solutions for the system (3.1) were obtained in [11] with sec”
Swarm dynamics and equilibria for a nonlocal aggregation model (2011, background): “A significant component of this study has dealt with the well-posedness of the initial-value problem for (1a) and (1b) [9–13].”