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.

Memory effects in measure transport equations / Camilli, Fabio; De Maio, Raul. - In: KINETIC AND RELATED MODELS. - ISSN 1937-5093. - 12:6(2019), pp. 1229-1245. [10.3934/krm.2019047]

Memory effects in measure transport equations

Camilli, Fabio;De Maio, Raul
2019

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.
Transport equation; measure-valued solution; fractional derivative; anomalous diffusion
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Memory effects in measure transport equations / Camilli, Fabio; De Maio, Raul. - In: KINETIC AND RELATED MODELS. - ISSN 1937-5093. - 12:6(2019), pp. 1229-1245. [10.3934/krm.2019047]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1359249
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