Memory effects in measure transport equations

Fabio Camilli; Raul De Maio

doi:10.3934/krm.2019047

Article Contents

2019, Volume 12, Issue 6: 1229-1245. Doi: 10.3934/krm.2019047

This issue Previous Article Sonic-supersonic solutions for the two-dimensional pseudo-steady full Euler equations Next Article Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation

Memory effects in measure transport equations

Fabio Camilli and
Raul De Maio

Dip. di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, via Scarpa 16, 00161 Roma, Italy

Received: June 2018

Revised: March 2019

Published online: December 2019

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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.

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Mathematics Subject Classification: Primary: 35R11, 26A33; Secondary: 35Q35.

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