Using some investigations based on information theory, the model proposed by Keller and Segel was extended to the concept of fractional derivative using the derivative with fractional order without singular kernel recently proposed by Caputo and Fabrizio. We present in detail the existence of the coupled-solutions using the fixed-point theorem. A detailed analysis of the uniqueness of the coupled-solutions is also presented. Using an iterative approach, we derive special coupled-solutions of the modified system and we present some numerical simulations to see the effect of the fractional order.

https://www.mdpi.com/1099-4300/17/6/4439/pdf

Analysis of the Keller-Segel Model with a Fractional Derivative without Singular Kernel

We review the progress made in extended irreversible thermodynamics during the ten years that have elapsed since the publication of our first review on the same subject (Rep. Frog. Phys. 1988 51 1105-72). During this decade much effort has been devoted to achieving a better understanding of the fundamentals and a broadening of the domain of applications. The macroscopic formulation of extended irreversible thermodynamics is reviewed and compared with other non-equilibrium thermodynamic theories. The foundations of EIT are discussed on the bases of information theory, kinetic theory, stochastic phenomena and computer simulations. Several significant applications are presented, some of them of considerable practical interest (non-classical heat transport, polymer solutions, non-Fickian diffusion, microelectronic devices, dielectric relaxation), and some others of special theoretical appeal (superfluids, nuclear collisions, cosmology). We also outline some basic problems which are not yet completely solved, such as the definitions of entropy and temperature out of equilibrium, the selection of the relevant variables, and the status to be reserved to the H-theorem and its relation to the second law. In writing this review, we had four objectives in mind: to show (i) that extended irreversible thermodynamics stands at the frontiers of modern thermodynamics; (ii) that it opens the way to new and useful applications; (iii) that much progress has been achieved during the last decade, and (iv) that the subject is far from being exhausted.

https://iopscience.iop.org/article/10.1088/0034-4885/62/7/201/pdf

Extended irreversible thermodynamics revisited (1988-98)

We investigate the relaxation problem for the hydrodynamic isentropic Euler-Poisson system when the momentum relaxation time tends to zero. Very sharp estimates on the solutions, independent of the relaxation time, are obtained and used to establish compactness.

Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation

The bipolar hydrodynamic model for semiconductors and the drift-diffusion equations

In this paper we investigate the Cauchy Problem for coupled NavierStokes-Poisson equation. The global existence of weak solutions in Sobolev framework is proved by using some compactification properties deduced from the Poisson equation.

/pdf/local-and-global-existence-for-the-coupled-navier-stokes-487hqphd13.pdf

Local and global existence for the coupled Navier-Stokes-Poisson problem

Relativistic extended thermodynamics of rarefied polyatomic gas

The Blotekjaer hydrodynamical model for charge carriers transport in semiconductors is reconsidered from the viewpoint of extended thermodynamics In particular the Blotekjaer original closure of the moment equations is shown to be equivalent to that obtained by applying the entropy principle

Extended thermodynamics of the Blotekjaer hydrodynamical model for semiconductors

In extended thermodynamic the entropy principle and the Galilean invariance dictate respectively constraints for the constitutive equations and the velocity dependence. The entropy principle in particular requires the existence of a privileged field, the main field u′, such that the original system becomes symmetric hyperbolic and is generated by four potentials. It is not easy to solve the restrictions of both principles, if we use as field the non convective main field $\widehat{\mathbf{u}}^{\prime}\ $
 and the velocity v. This is due to the fact that $\widehat{\mathbf{u}}^{\prime}$
 are not independent. Rather its components satisfy three scalar constraints. The aim of this paper is to solve the full problem using as new strategy to consider $\widehat{\mathbf{u}}^{\prime}\ $
 as independent variables and requiring an appropriate differential constraint. This new procedure is very efficient and we are able to solve the problem of 13 moments in the full non linear case (far from equilibrium). It turns out that the knowledge of only the equilibrium state function is sufficient to close the system. Keywords: Extended Thermodynamics, Entropy Principle, Galilean invariance, Rarefied Gas, Hyperbolic systems Mathematics Subject Classification (2000): 74A20, 76P05, 35l60

A new method to exploit the Entropy Principle and galilean invariance in the macroscopic approach of Extended Thermodynamics

A New BGK Model for Relativistic Kinetic Theory of Monatomic and Polyatomic Gases

The moment system associated with the Boltzmann equation is largely used in many applications and is the main ingredient of Rational Extended Thermodynamics (RET). The choice of truncation of moments is arbitrary and many possibilities are present in the literature depending in particular on many contracted indexes in each moment tensor considered. Moreover, in a polyatomic gas, we have two hierarchies of moments with different indexes of truncation. As any classical theory can be considered as a limiting case of a relativistic one, we study in this paper the moments associated with the relativistic Boltzmann–Chernikov equation until the index of truncation N. We take the classical limit for both polyatomic and monatomic rarefied gases and prove that there exists only unique possible choice of the moments in the classical case for a given N.

Sebastiano Pennisi

Papers

Relativistic extended thermodynamics of rarefied polyatomic gas

Extended thermodynamics of the Blotekjaer hydrodynamical model for semiconductors

A new method to exploit the Entropy Principle and galilean invariance in the macroscopic approach of Extended Thermodynamics

A New BGK Model for Relativistic Kinetic Theory of Monatomic and Polyatomic Gases

Classical Limit of Relativistic Moments Associated with Boltzmann-Chernikov Equation: Optimal Choice of Moments in Classical Theory