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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 ..."
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Cited by 12 (6 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
A generalized BirkhoffRott Equation for 2D Active Scalar Problems
, 2011
"... In this paper we derive evolution equations for the 2D active scalar problem when the solution is supported on 1D curve(s). These equations are a generalization of the BirkhoffRott equation when vorticity is the active scalar. The formulation is Lagrangian and they are valid for nonlocal kernels K ..."
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Cited by 5 (4 self)
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In this paper we derive evolution equations for the 2D active scalar problem when the solution is supported on 1D curve(s). These equations are a generalization of the BirkhoffRott equation when vorticity is the active scalar. The formulation is Lagrangian and they are valid for nonlocal kernels K that may include both a gradient and an incompressible term. We develop a numerical method for implementing the model which achieves second order convergence in space and fourth order in time. We verify the model by simulating classic active scalar problems such as the vortex sheet problem (in the case of inviscid, incompressible flow) and the collapse of delta ring solutions (in the case of pure aggregation), finding excellent agreement. We then study two examples with kernels of mixed type i.e., kernels that contain both incompressible and gradient flows. The first example is a vortex density model which arises in superfluids. We analyze the effect of the added gradient component on the KelvinHelmholtz instability. In the second example, we examine a nonlocal biological swarming model and study the dynamics of density rings which exhibit complicated milling behavior.
The McKean –Vlasov Equation in Finite Volume
, 2009
"... Abstract: We study the McKean–Vlasov equation on the finite tori of length scale L in d–dimensions. We (re)derive the necessary and sufficient conditions for the existence of a phase transition – first uncovered in [13] and [20]. Therein and in subsequent works, one finds indications pointing to cri ..."
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Cited by 4 (1 self)
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Abstract: We study the McKean–Vlasov equation on the finite tori of length scale L in d–dimensions. We (re)derive the necessary and sufficient conditions for the existence of a phase transition – first uncovered in [13] and [20]. Therein and in subsequent works, one finds indications pointing to critical transitions at a particular model dependent value, θ ♯ of the interaction parameter. We show (there is a basin of) dynamical stability for θ < θ ♯ and prove, abstractly, that a critical transition must occur at θ = θ ♯. However for this system we show that under generic conditions – L large, d ≥ 2 and isotropic interactions – the phase transition is in fact discontinuous and occurs at some θT < θ ♯. Finally, for H–stable, bounded interactions with discontinuous transitions we show that, with suitable scaling, the θT(L) tend to a definitive non–trivial limit as L → ∞. 1
Stability and clustering of selfsimilar solutions of aggregation equations
, 2012
"... In this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation ρt = ∇ · (ρ∇K ∗ ρ) in Rd, d ≥ 2, where K(r) = r γ /γ with γ> 2. It was previously observed 1 that radially symmetric solutions are attracted to a selfsimilar collapsing ..."
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Cited by 3 (3 self)
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In this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation ρt = ∇ · (ρ∇K ∗ ρ) in Rd, d ≥ 2, where K(r) = r γ /γ with γ> 2. It was previously observed 1 that radially symmetric solutions are attracted to a selfsimilar collapsing shell profile in infinite time for γ> 2. In this paper we compute the stability of the similarity solution and show that the collapsing shell solution is stable for 2 < γ < 4. For γ> 4, we show that the shell solution is always unstable and destabilizes into clusters that form a simplex which we observe to be the long time attractor. We then classify the stability of these simplex solutions and prove that two dimensional (in)stability implies n dimensional (in)stability.
Nonlinear Diffusion with Fractional Laplacian Operators
"... Abstract We describe two models of flow in porous media including nonlocal (longrange) diffusion effects. The first model is based on Darcy’s law and the pressure is related to the density by an inverse fractional Laplacian operator. We prove existence of solutions that propagate with finite speed. ..."
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Abstract We describe two models of flow in porous media including nonlocal (longrange) diffusion effects. The first model is based on Darcy’s law and the pressure is related to the density by an inverse fractional Laplacian operator. We prove existence of solutions that propagate with finite speed. The model has the very interesting property that mass preserving selfsimilar solutions can be found by solving an elliptic obstacle problem with fractional Laplacian for the pair pressuredensity. We use entropy methods to show that they describe the asymptotic behaviour of a wide class of solutions. The second model is more in the spirit of fractional Laplacian flows, but nonlinear. Contrary to usual Porous Medium flows (PME in the sequel), it has infinite speed of propagation. Similarly to them, an L 1contraction semigroup is constructed and it depends continuously on the exponent of fractional derivation and the exponent of the nonlinearity. 1 Nonlinear diffusion and fractional diffusion Since the work by Einstein [39] and Smoluchowski [62] at the beginning of the last century (cf. also Bachelier [9]), we possess an explantation of diffusion and Brownian motion in terms of the heat equation, and in particular of the Laplace operator. This explanation has had an enormous success both in Mathematics and Physics. In the decades that followed, the Laplace operator has been often replaced by more general types of socalled elliptic operators with variable coefficients, and later by nonlinear differential operators; a huge body of theory is now available, both for the evolution equations [49] and for the stationary states, described by elliptic equations of different kinds [50, 42].
DOI 10.1007/s109550099913z The McKean–Vlasov Equation in Finite Volume
"... Abstract We study the McKean–Vlasov equation on the finite tori of length scale L in ddimensions. We derive the necessary and sufficient conditions for the existence of a phase transition, which are based on the criteria first uncovered in Gates and Penrose (Commun. Math. Phys. 17:194–209, 1970) an ..."
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Abstract We study the McKean–Vlasov equation on the finite tori of length scale L in ddimensions. We derive the necessary and sufficient conditions for the existence of a phase transition, which are based on the criteria first uncovered in Gates and Penrose (Commun. Math. Phys. 17:194–209, 1970) and Kirkwood and Monroe (J. Chem. Phys. 9:514–526, 1941). Therein and in subsequent works, one finds indications pointing to critical transitions at a particular model dependent value, θ ♯ of the interaction parameter. We show that the uniform density (which may be interpreted as the liquid phase) is dynamically stable for θ<θ ♯ and prove, abstractly, that a critical transition must occur at θ = θ ♯.Howeverforthis system we show that under generic conditions—L large, d ≥ 2 and isotropic interactions— the phase transition is in fact discontinuous and occurs at some θT <θ ♯. Finally, for Hstable, bounded interactions with discontinuous transitions we show that, with suitable scaling, the θT(L) tend to a definitive nontrivial limit as L →∞.
EQUIVALENCE OF GRADIENT FLOWS AND ENTROPY SOLUTIONS FOR SINGULAR NONLOCAL INTERACTION EQUATIONS IN 1D
"... Abstract. We prove the equivalence between the notion of Wasserstein gradient flow for a onedimensional nonlocal transport PDE with attractive/repulsive Newtonian potential on one side, and the notion of entropy solution of a Burgerstype scalar conservation law on the other. The solution of the for ..."
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Abstract. We prove the equivalence between the notion of Wasserstein gradient flow for a onedimensional nonlocal transport PDE with attractive/repulsive Newtonian potential on one side, and the notion of entropy solution of a Burgerstype scalar conservation law on the other. The solution of the former is obtained by spatially differentiating the solution of the latter. The proof uses an intermediate step, namely the L 2 gradient flow of the pseudoinverse distribution function of the gradient flow solution. We use this equivalence to provide a rigorous particlesystem approximation to the Wasserstein gradient flow, avoiding the regularization effect due to the singularity in the repulsive kernel. The abstract particle method relies on the socalled wavefronttracking algorithm for scalar conservation laws. Finally, we provide a characterization of the subdifferential of the functional involved in the Wasserstein gradient flow. 1.