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I am new to this stack, and thought my question belongs here.

I am a first-year graduate student currently taking my second course on PDEs (basically covering Evans ch. 5 and onwards). I am planning to do research on the analysis of PDEs (No specific topic or advisor yet).

My question is how to really study the subject and understand it. It might seem a dumb question to ask but I am really struggling to understand how do people approach such a course or a subject. Upon glancing at these chapters, I see a lot of function spaces' definitions and long (sometimes tedious) proofs with well-crafted tricks.

More specifically, do I read every proof and try to reproduce it myself later? Do I write down notes while reading (this turned out very time-consuming for me last semester)? Do I try to prove things myself? Do I...

Maybe it would help if some of the experts in PDEs here shared their experiences when they first came in contact with the subject.

Advice is also welcomed as to focusing on PDEs as a research career.

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    $\begingroup$ You don't "master PDEs": there is no unified theory or even a general program of research. PDEs is an umbrella that contains a lot of individual types of programs each with their idiosyncratic tools. As such, if your goal is to do research in PDEs you probably should prioritize finding a mentor/advisor who will help you set a reasonable path cherry picking those things you do need to study to get started. $\endgroup$
    – Willie Wong
    Commented Jan 26, 2023 at 18:35
  • $\begingroup$ You're right, I didn't think much when I wrote the title, but I think the body of the post explains what I meant $\endgroup$
    – Morcus
    Commented Jan 26, 2023 at 19:10
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    $\begingroup$ @Morcus: Your questions in the paragraph "More specifically, do I read every proof and try to reproduce it myself later? Do I write down notes while reading (this turned out very time-consuming for me last semester)? Do I try to prove things myself?" make me wonder why you believe that the answers to those questions would be different for PDEs than for any other field in mathematics. $\endgroup$
    – Jochen Glueck
    Commented Jan 26, 2023 at 19:21
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    $\begingroup$ @JochenGlueck Why not? At least based on my limited experience I used different ways to approach learning different math subjects as an undergrad for the first time, e.g. abstract algebra vs. real analysis. $\endgroup$
    – Morcus
    Commented Jan 27, 2023 at 7:55
  • $\begingroup$ @Morcus: Thanks for your reply! What different approaches did you use to learn abstract algebra vs. real analysis? $\endgroup$
    – Jochen Glueck
    Commented Jan 27, 2023 at 10:43

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