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Memory effects in measure transport equations

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Abstract

Transport equations with a nonlocal velocity field have been introduced as a continuum model for interacting particle systems arising in physics, chemistry and biology. Fractional time derivatives, given by convolution integrals of the time-derivative with power-law kernels, are typical for memory effects in complex systems. In this paper we consider a nonlinear transport equation with a fractional time-derivative. We provide a well-posedness theory for weak measure solutions of the problem and an integral formula which generalizes the classical push-forward representation formula to this setting.

Most cited references8

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A well-posedness theory in measures for some kinetic models of collective motion

(2009)
We present existence, uniqueness and continuous dependence results for some kinetic equations motivated by models for the collective behavior of large groups of individuals. Models of this kind have been recently proposed to study the behavior of large groups of animals, such as flocks of birds, swarms, or schools of fish. Our aim is to give a well-posedness theory for general models which possibly include a variety of effects: an interaction through a potential, such as a short-range repulsion and long-range attraction; a velocity-averaging effect where individuals try to adapt their own velocity to that of other individuals in their surroundings; and self-propulsion effects, which take into account effects on one individual that are independent of the others. We develop our theory in a space of measures, using mass transportation distances. As consequences of our theory we show also the convergence of particle systems to their corresponding kinetic equations, and the local-in-time convergence to the hydrodynamic limit for one of the models.
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Fractional diffusion equations and processes with randomly varying time

(2011)
In this paper the solutions $$u_{\nu}=u_{\nu}(x,t)$$ to fractional diffusion equations of order $$0<\nu \leq 2$$ are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order $$\nu =\frac{1}{2^n}$$, $$n\geq 1,$$ we show that the solutions $$u_{{1/2^n}}$$ correspond to the distribution of the $$n$$-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order $$\nu =\frac{2}{3^n}$$, $$n\geq 1,$$ is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions. In the general case we show that $$u_{\nu}$$ coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions $$u_{\nu}$$ and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.
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A Parabolic Problem with a Fractional-Time Derivative

(2015)
We study regularity for a parabolic problem with fractional diffusion in space and a fractional time derivative. Our main result is a De Giorgi-Nash-Moser Holder regularity theorem for solutions in a divergence form equation. We also prove results regarding existence, uniqueness, and higher regularity in time.
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Author and article information

Journal
27 June 2018
1806.10331