Visiting Graduate Research Student · Feb – Aug 2024 · Baltimore, USA · 📄 Research report (PDF)

1- Overview

From February to August 2024, I was a Visiting Graduate Research Student in the Department of Mathematics of Johns Hopkins University, under the supervision of Professor Yannick Sire. I worked on the long-time behavior of the Fractional Porous Medium Equation: its solutions are known to converge to a stationary profile, and my goal was to prove that this convergence happens at an exponential rate.

  • Role: Visiting Graduate Research Student (6-month research internship)
  • Field: Partial differential equations, functional analysis
  • Methods: entropy methods, weighted Poincaré inequalities, Grönwall’s lemma, linearization, harmonic extension of the fractional Laplacian
  • Output: research report written in LaTeX (English)

2- Context & Motivation

Diffusion of substances, energy or information is central in physics, engineering and even financial markets. Fractional diffusion models long-range interactions and anomalous diffusion that classical models miss. Knowing how fast such systems reach equilibrium, and not only that they do, is what makes these models quantitatively useful.

3- The Problem

We study, on a bounded domain $\Omega \subset \mathbb{R}^d$:

$$\partial_t v = -L\,\varphi(v) + c\,v \quad \text{in } (0,\infty)\times\Omega, \qquad v(0,\cdot) = u_0, \qquad v = 0 \text{ on } \partial\Omega,$$

where $L = (-\Delta)^s$ with $s \in (0,1]$ is a fractional power of the Laplacian, $\varphi$ is a smooth increasing function and $c > 0$. A stationary solution $S$ satisfies $L\varphi(S) = cS$.

Assuming the convergence $v(t) \to S$, the goal is a quantitative estimate of the form

$$\| v(t) - S \|_X \le a\, e^{-bt},$$

where $a, b$ depend only on the data of the problem and $X$ is a suitable Hilbert space.

4- Approach & Results

Entropy method in the classical case

With $\varphi(v) = v^m$ ($m > 1$) and the classical Laplacian, I worked on the relative error $\theta = v/S - 1$ and the weighted entropy

$$E[\theta(t)] = \frac{1}{2}\int_\Omega \big(\theta - \bar\theta\big)^2 S^{1+m}\, dx.$$

Differentiating $E$, controlling the dissipation term with a weighted Poincaré inequality and the reaction term with a Taylor expansion, then applying Grönwall’s lemma, gives exponential decay: for every $\beta < c\,\big(Km + 2(m-1)\big)$, there exists $t^*$ such that

$$E[\theta(t)] \le E[\theta(t^*)]\; e^{-\beta (t - t^*)} \quad \text{for all } t \ge t^*.$$

Toward a general nonlinearity

I then attempted to extend the proof to a general function $\varphi$. My main initiative was the choice of the weight in the entropy integral, adapted to $\varphi$. Some of these directions did not succeed, which will help future work avoid dead ends; others look promising.

A second approach: linearization

I also studied the linearized equation, first with the classical Laplacian, then with the fractional Laplacian using its harmonic extension to $\Omega \times \mathbb{R}_+$, which turns the non-local operator into a local energy term. The remaining step is a functional inequality linking this energy to the entropy, in order to close the Grönwall argument.

5- Takeaways

The project had been paused after earlier work by several researchers. I took ownership of it, synthesized the problem in a structured report, advanced the proofs and proposed new directions. Beyond the mathematics, I learned to read research papers quickly, to communicate abstract ideas in English, and to work autonomously with weekly feedback from my supervisor.

📄 Research report (English, PDF) · 📄 Full internship report (French, PDF)