An engineer's journey in mathematics

In working with my collaborators to justify the diffusion equation resulted from wave turbulence systems dominated by non-local interactions, we need to prove the local well-posedness of some wave kinetic equation. The system under our consideration is a three-dimensional Majda-McLaughlin-Tabak  ( MMT) equation, with its wave kinetic equation given in the form of      $\frac{\partial n_k}{\partial t} = 4\pi {\large{\int}}_{\mathbb{R}^9} |k|^{2\beta} |k_1|^{2\beta} |k_2|^{2\beta} |k_3|^{2\beta} (n_1n_2n_3 + n_2n_3n_k - n_2n_1n_k - n_3n_1n_k)$     $\delta(k+k_1-k_2-k_3) \delta(\omega+\omega_1-\omega_2-\omega_3) dk_1dk_2dk_3$, with $k\in \mathbb{R}^3$ and $\omega=|k|^\alpha$. This equation describes the evolution of wave action spectrum as a result of four-wave interactions. I will not detail more about the physical meaning of the equation and notations used therein, because that will distract readers too much from the main theme of the post. Anyway, one ...

This is Yulin Pan, an assistant professor in the department of Naval Architecture and Marine Engineering at the University of Michigan. Throughout my education I was in engineering departments, receiving all my degrees in engineering. I do have a minor degree in mathematics from MIT, but to achieve that I only needed to take three courses from the math department there. And for me, I took all three courses from the very applied side of mathematics. In summary, my college education of mathematics is highly limited to engineering mathematics, numerical methods and some applied asymptotic analysis to nonlinear problems.  In spite of the lack of training in rigorous mathematics, I do keep a keen interest in mathematics over my research career. During my Ph.D., I got interested in the field of wave turbulence, based on which I later developed my Ph.D. thesis. The field of wave turbulence is quite technical in terms of the derivation, which is usually considered as part of theoretical cl...