Why are so many inequalities and estimate in PDEs? I want to study PDE, but when I read some PDE books, I notice that there are so many inequalities there and always lots of estimates. I really want to know why  before I continue my study. I mean, 


*

*what is motivation of estimates and the main ideas of PDEs? 

*How do the main ideas involve estimate and 

*why should we estimate solutions? 
I have read some books, but I am still confused and I am afraid if I can't understand this I may not continue my study. Thanks! I hope someone could answer me.
Actually, I just found the answers from the introduction of Gilbarg and Trudinger's book now. But I still want to hear some point of view of others. So if anyone has some opinions , I still hope you could reply me. Many thanks. 
 A: In a pure study of Partial Differential Equations, somehow you have to come up with the existence of a function to solve some kind of equation. How do you establish the existence of a function? There are a few techniques. One is to use the Riesz representation theorem on $L^{2}$ spaces (Lax-Milgram is just an extension of Riesz representation.) Another is to use fixed point theory, where you have a function $F : X \rightarrow X$ on a function space $X$ with some continuity property, and you come up with a function $f$ such that $F(f)=f$. Another method is to use distributions where you can more easily get solutions, but not so easily show that a distributional solution is related to an actual function. Such methods require estimates to establish the continuity of some map.
PDE theory is one of the most difficult areas of analysis, and non-linear PDE theory is practically a show-stopper, which is why vonNeumann invented the CPU in hopes of solving equations with machines. I thinking computing methods should be a strong emphasis for someone interested in PDEs. Some of the best theoretical tools to deal with PDEs have come from analyzing numerical solution methods using Statistics. The real advances in non-linear theory seem to come from computations. Best of all, studying computations helps break the compulsive mode of looking at one inequality after another after another ...
Here is an interesting interview with Lax (of Lax-Milgram fame) from about 10 years ago. It's fascinating reading from a technical and historical perspective.
http://www-personal.umich.edu/~yryamada/Lax_interview.pdf
"You remember that – or you may not remember that – already during the war von Neumann realized that to do bomb calculations analytical methods were useless and one needed massive computing. He also realized that computing is good not only for bomb making but for solving any large-scale scientific or engineering problem. And he also realized that it’s not only used for schemes for solving concrete problems but rather to explore which way science should be developed."
One of Lax's final remarks: "Well, computational science, computational mathematics, has a very bright future, and I warmly recommend them to young people to go into that field."
