arXiv subjects

Geremy Loachamin

Publications and source records attributed to Geremy Loachamin.

3 recordsLinked to original sources

arXiv2024

On the fractional approach to quadratic nonlinear parabolic systems

We introduce a general coupled system of parabolic equations with quadratic nonlinear terms and diffusion terms defined by fractional powers of the Laplacian operator. We develop a method to establish the rigorous convergence of the fractional diffusion case to the classical diffusion case in the strong topology of Sobolev spaces, with explicit convergence rates that reveal some unexpected phenomena. These results apply to several relevant real-world models included in the general system, such as the Navier-Stokes equations, the Magneto-hydrodynamics equations, the Boussinesq system, and the Keller-Segel system. For these specific models, this fractional approach is further motivated by previous numerical and experimental studies.

math.AP↗

arXiv2023

From non-local to local Navier-Stokes equations

Inspired by some experimental (numerical) works on fractional diffusion PDEs, we develop a rigorous framework to prove that solutions to the fractional Navier-Stokes equations, which involve the fractional Laplacian operator $(-Δ)^{\fracα{2}}$ with $α<2$, converge to a solution of the classical case, with $-Δ$, when $α$ goes to $2$. Precisely, in the setting of mild solutions, we prove uniform convergence in both the time and spatial variables and derive a precise convergence rate, revealing some phenomenological effects. Finally, our results are also generalized to the coupled setting of the Magnetic-hydrodynamic (MHD) system.

math.AP↗

arXiv2023

From anomalous to classical diffusion in a non-linear heat equation

In this paper, we consider the heat equation with the natural polynomial non-linear term; and with two different cases in the diffusion term. The first case (anomalous diffusion) concerns the fractional Laplacian operator with parameter $1<α<2$, while, the second case (classical diffusion) involves the classical Laplacian operator. When $α\to 2$, we prove the uniform convergence of the solutions of the anomalous diffusion case to a solution of the classical diffusion case. Moreover, we rigorous derive a convergence rate, which was experimentally exhibit in previous related works.

math.AP↗