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Blow-up phenomena for gradient flows of discrete homogeneous functionals. (English) Zbl 1415.35161

The paper deals with a deterministic particle approximation of the gradient flow of homogeneous functionals in the family \[ \mathcal{G}_m[\rho]=\dfrac{1}{m-1}\int_{\mathbb{R}} \rho(x)^mdx-\dfrac{\chi}{m-1} \iint_{\mathbb{R}\times\mathbb{R}} |x-y|^{1-m} \rho(x)\rho(y)dxdy, \] where \(\chi>0\) is the interaction coefficient, \(m\in(1,2)\) is the non-linear diffusion exponent, and \(\rho\) a probability distribution function, \(\rho\in L^m\cap L^1(|x|^2dx).\) Such functionals arise in the Lagrangian approximation of systems of self-interacting and diffusing particles. The authors focus on the case of negative homogeneity and prove that all solutions become singular in finite time in the case of strong self-interaction (the super critical case). Numerical simulations are also provided that illustrate the striking non-linear dynamics when the initial data have positive energy.

MSC:

35K57 Reaction-diffusion equations
35B44 Blow-up in context of PDEs
35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
65L05 Numerical methods for initial value problems involving ordinary differential equations
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