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SELFSIMILAR BLOWUP SOLUTIONS TO AN AGGREGATION EQUATION
"... Abstract. We present numerical simulations of radially symmetric finite time blowup for the aggregation equation ut = ∇ · (u∇K ∗ u), where the kernel K(x) = x. The dynamics of the blowup exhibits selfsimilar behavior in which zero mass concentrates at the core at the blowup time. Computations ..."
Abstract

Cited by 14 (8 self)
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Abstract. We present numerical simulations of radially symmetric finite time blowup for the aggregation equation ut = ∇ · (u∇K ∗ u), where the kernel K(x) = x. The dynamics of the blowup exhibits selfsimilar behavior in which zero mass concentrates at the core at the blowup time. Computations are performed in R n for n between 2 and 10 using a method based on characteristics. In all cases studied, the selfsimilarity exhibits second kind (anomalous) scaling. Key words. aggregation equation, selfsimilarity solution of the second kind, blowup AMS subject classifications. 74H15, 74H35, 76M55, 82B24
Asymptotics of blowup solutions for the aggregation equation
, 2011
"... We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation ut = ∇ · (u∇K ∗ u) in R n, for homogeneous potentials K = x  γ, γ> 0. For γ> 2, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing δring. We develop an a ..."
Abstract

Cited by 5 (2 self)
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We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation ut = ∇ · (u∇K ∗ u) in R n, for homogeneous potentials K = x  γ, γ> 0. For γ> 2, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing δring. We develop an asymptotic theory for the approach to this singular solution. For γ < 2, the solution blows up in finite time and we present careful numerics of second type similarity solutions for all γ in this range, including additional asymptotic behavior in the limits γ → 0 + and γ → 2 −.
THE EVOLUTION OF A CRYSTAL SURFACE: ANALYSIS OF A 1D STEP TRAIN CONNECTING TWO FACETS IN THE ADL REGIME
"... Abstract. We study the evolution of a monotone step train separating two facets of a crystal surface. The model is onedimensional and we consider only the attachmentdetachmentlimited regime. Starting with the wellknown ODE’s for the velocities of the steps, we consider the system of ODE’s giving ..."
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Abstract. We study the evolution of a monotone step train separating two facets of a crystal surface. The model is onedimensional and we consider only the attachmentdetachmentlimited regime. Starting with the wellknown ODE’s for the velocities of the steps, we consider the system of ODE’s giving the evolution of the “discrete slopes. ” It is the l 2steepestdescent of a certain functional. Using this structure, we prove that the solution exists for all time and is asymptotically selfsimilar. We also discuss the continuum limit of the discrete selfsimilar solution, characterizing it variationally, identifying its regularity, and discussing its qualitative behavior. Our approach suggests a PDE for the slope as a function of height and time in the continuum setting. However existence, uniqueness, and asymptotic selfsimilarity remain open for the continuum version of the problem. 1.
pp. X–XX ASYMPTOTICS OF BLOWUP SOLUTIONS FOR THE AGGREGATION EQUATION
"... Abstract. We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation ut = ∇ · (u∇K ∗ u) in R n, for homogeneous potentials K(x) = x  γ, γ> 0. For γ> 2, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing δring. W ..."
Abstract
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Abstract. We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation ut = ∇ · (u∇K ∗ u) in R n, for homogeneous potentials K(x) = x  γ, γ> 0. For γ> 2, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing δring. We develop an asymptotic theory for the approach to this singular solution. For γ < 2, the solution blows up in finite time and we present careful numerics of second type similarity solutions for all γ in this range, including additional asymptotic behaviors in the limits γ → 0 + and γ → 2 −.