I am a math major and I am studying PDE. When I read papers in this field, I sometimes find some formal calculations, for example, they differentiate as if the function is smooth or they interchange integral and differentiation. Furthermore, they even write in their paper that regularity is not an issue in the paper.
However, when I look at the basic graduate level of PDE books, such as Evans' PDE, the theory of PDE has been rigorously developed and the calculation is delicately done. For example, there are concepts of weak solutions and distributions. Also, there is functional analysis which is highly rigorous. I feel like this kind of rigorous theory is really mathematics and also this is one of the aspects of mathematics I have been drawn by in my life.
Going back to the original issue, there are papers that deal with formal calculation even written by famous mathematicians, such as Cedric Villani. (I will attach the part of one of his papers below.) My question is what if they face some issues in the end when it comes to regularity. It seems to me that formal calculation is literally 'Formal' calculation, not a proper calculation in some rigorous sense. What if the formal calculation doesn't make any sense..? It is not physics. (I am not disparaging physics and physicists please. Only to get across my point.)
How do I understand this situation? I am really desperate. Please give me advice, great PDEists!
Evans' PDE book is designed to be a first look at treating the subject of PDEs rigorously. So he assumes nothing - if there exist some regularity conditions on $V$ such that interchanging limits, derivatives or integrals is justified, he must prove it carefully. However, once he manages to do this, it is assumed common knowledge (again, since the text is considered introductory). So why would a paper bother going through the justification, especially if it does not shed much light on the subject matter?
Let's look at a specific example. Suppose we are interested in the behavior of solutions $u=u_\epsilon$ to the PDE
$$\frac{\epsilon}2\Delta u + b(x)\cdot\nabla u = 0$$
in some domain $\Omega$, with zero (Dirichlet) boundary conditions, as $\epsilon\to0$. One can use the theory developed in Evans' book to show that, given certain conditions on $b$, solutions to the above PDE always exist and have nice regularity properties. Moreover, since we are interested in the behavior of said solutions as $\epsilon\to0$, the mere existence or regularity of solutions is not really our primary interest. So why would we bother spending time in our paper re-proving existence and regularity?
We might not even care to state explicit conditions on $b$; we may have a specific $b$ in mind which is $C^\infty$, but certainly we do not require $b\in C^\infty$ for existence and regularity of the solutions. Are we really going to spend time tracking down the optimal assumptions on $b$? Or does it make more sense to just say "for $b$ assumed regular enough"? This isn't rhetorical, by the way. It's a matter of personal preference, and there's not necessarily a right or wrong answer.