Does there exist the applicability limit of PDE to describe physical phenomena? -- A personal survey of Quantization, QED, Turbulence

What does it mean to study PDE(=Partial Differential Equation)? How and what to do "to claim proudly that I'm studying a certain PDE"? Newton mechanic uses mainly ODE(=Ordinary Differential Equation) and describes nicely movements of Sun, Moon and Earth etc. Now, so-called quantum phenomenum is described by, say Schr\"odinger equation, PDE which explains both wave and particle characters after quantization of ODE. The coupled Maxwell-Dirac equation is also "quantized" and QED(=Quantum Electro-Dynamics) theory is invented by physicists. Though it is said this QED gives very good coincidence between theoretical and experimental observed quantities, but what is the equation corresponding to QED? Or, is it possible to describe QED by "equation" in naive sense?