Weak solutions to the time-fractional g-B\'enard equations

TL;DR

This paper investigates the existence and uniqueness of weak solutions for time-fractional g-Bénard equations modeling heat conduction in fractal media, using techniques from Navier-Stokes and fractional calculus.

Contribution

It introduces the g-Bénard equations with fractional derivatives and establishes foundational results on weak solutions in this novel context.

Findings

Proved existence of weak solutions

Established uniqueness under certain conditions

Extended classical models to fractional, fractal media context

Abstract

In this paper, we introduce the g-B\'enard equations with time-fractional derivative of order $\alpha \in (0, 1)$ in domains of $\mathbb R^2$. This equations model, the memory-dependent heat conduction of liquids in fractal media considered in g-framework. We aim to study the existence and uniqueness of weak solutions by means of standard techniques from Navier-Stokes equations theory and fractional calculus theory.