8
votes
0answers
87 views

Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponenent allows for convenient ...
2
votes
0answers
46 views

Why has no body retypeset Ladyzhenskaya et al's “Linear and quasi-linear equations of parabolic type”? [closed]

The book "Linear and quasi-linear equations of parabolic type" is one of the ugliest books I have ever seen in my life. The fonts are awful, the notation is difficult to understand and recall and the ...
1
vote
1answer
54 views

Research Area Choice: PDE vs Optimization

I am on track to starting in applied math PhD this coming Fall. My area of interest is PDE where I have a strong background and have even published a paper. It seems the particular area of PDE I am ...
35
votes
5answers
1k views

Understanding the Laplace operator conceptually

The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it? Any good essays (combining both history and ...
23
votes
6answers
1k views

Should I understand a theorem's proof before using the theorem?

I find myself embarrassed when using results in books. For example, there are so many results in Sobolev spaces that I think I would not be able to understand all of them. Yes, I could try to ...
4
votes
0answers
131 views

How to balance learning and researching as a new PhD student?

As a new PhD student, how to balance learning and researching? I am in Australia and here we don't have any course in PhD period. I know I need to learn something about my programme, but sometimes ...
0
votes
0answers
13 views

what does well posdeness results tells us concerning non linear evolution equations?

Consider a nonlinear Shr\"odinger equation, $$iu_{t}+\bigtriangleup u + f(u)= 0, u(0)= u_{0}$$ where $u(t, x)$ is complex valued function of $(t,x) \in \mathbb R \times \mathbb R^{n}$, $i=\sqrt{-1}, ...
25
votes
2answers
2k views

Why are mathematician so interested to find theory for solving partial differential equations but not for integral equation?

Why are mathematician so interested to find theory for solving partial differential equations (for example Navier-Stokes equation) but not for integral equations?
1
vote
2answers
38 views

PDE course question

What courses do I need for a course in partial differential equations? My university has a prereq of Multivariate Calculus and Ordinary Differential Equations. However, I opened up a book on pdes in ...
22
votes
1answer
389 views

Why is the Laplacian important in Riemannian geometry?

As I've learned more Riemannian geometry, many of my teachers have said that studying the Laplacian (and its eigenvalues) is very important. But I must admit, I've never fully understood why. ...
2
votes
1answer
129 views

How to properly prepare for a graduate level PDE course using the books by Evans and Strauss

For my undergrad background, I have Calculus 1-3, Linear Algebra, one semester of ODE, one semester of real analysis. Never had any PDE before. Thus I know this background is hardly enough to do well ...
1
vote
0answers
32 views

Question pertain to graduate programme in astronomy from mathematical background

I am currently an undergraduate student in mathematics, and wanted to pursue a career in astronomy area. What graduate programme should i take? is applied mathematics sufficient? I am currently ...
0
votes
1answer
60 views

Is it acceptable to use the shorthand “PDEs” in the title of a paper?

If I write a paper, is it acceptable for me to use a title like "analysis of PDEs related to blah blah blah" instead of "analysis of partial differential equations related to blah blah blah"? I ...
2
votes
0answers
61 views

Are Navier-Stokes equations used in molecular biology research?

I am wondering whether Navier-Stokes equations have been used in some molecular biology research papers in the past. A quick Google search revealed that such papers exist but I want to know if there ...
6
votes
1answer
174 views

Is there an integral form of Newton's method?

Warning : This seems like a silly sort of question, not the kind I'd ask out loud. The contraction mapping theorem is a basic tool for proving existence of, and finding solutions to, equations. Given ...
3
votes
4answers
253 views

PDEs in biology

I am student who mostly heard lectures on partial differential equations and homogenization. But I really like the idea of working in biology or with biologists - but (with my lack of overview) it ...
1
vote
3answers
259 views

Thermodynamics for math majors

I'm about to wrap a course in partial differential equations. We've discussed the heat/wave equations and introductory Fourier Analysis. I'd like to do some reading into the field of thermodynamics. ...
1
vote
0answers
168 views

Reference for Finite Difference Schemes

Is there any place that I can find a list of different PDEs and common finite difference schemes used for each? I have seen tables of finite difference coefficients such as the one here ...
2
votes
2answers
230 views

What about some engaging PDE topics for undergraduates?

Does anyone have any suitable PDE topics (research or otherwise) for an undergraduate math student? Consider the student has only completed an introductory class in PDE. The student also has ...
4
votes
2answers
281 views

Homological algebra in PDE

I have been fascinated by the power and wide applicability of homological methods in algebra and topology. Because I am also interested in PDE, there arises a natural question for me. What is ...
3
votes
2answers
322 views

Good reference texts for introduction to partial differential equation?

As the title, are there any good reference texts for introduction to partial differential equation?
1
vote
0answers
150 views

Intuition for PDE Change of Variables

The algebraic manipulations for changing variables in PDE/ODE problems are often very simple once you know the transformation to use (at least at my level it's just applying the chain rule carefully). ...
4
votes
0answers
142 views

Recommendation textbooks on D-module

I am going to take part in a seminar on D-module and applications, the textbooks that will be used are : D-modules, Perverse Sheaves, and Representation Theory, and A Primer of Algebraic D-Modules ...
4
votes
3answers
564 views

What should a PDE/analysis enthusiast know?

What are the cool things someone who likes PDE and functional analysis should know and learn about? What do you think are the fundamentals and the next steps? I was thinking it would be good to know ...
101
votes
5answers
7k views

What do modern-day analysts actually do?

In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about ...
4
votes
1answer
232 views

Which Fourier transform should I use in PDE?

I learned two "different" Fourier transforms in the real analysis course. For any function $f\in C({\bf R/Z;C})$, and any integer $n\in{\bf Z}$, we define the $n$th Fourier coefficient of $f$, ...
25
votes
8answers
1k views

Partial differential equations in “pure mathematics”

One thing I have noticed about PDEs is that they come from Mathematical Physics in general. Almost all the equations I see in Wikipedia follow this pattern. I can't help wondering whether there are ...
5
votes
3answers
629 views

Understanding of the Mean Value Theorem in PDE

I learned the following theorem in Folland's Introduction to Partial Differential Equations(p.69 Chapter 2): Suppose $u$ is harmonic on an open set $\Omega\subset{\mathbb R}^n$. If $x\in\Omega$ ...
9
votes
1answer
199 views

Can anyone recommend some books on PDE in $L^p$ space for me?

I need a book covering $L^p$ theory (is it?) on PDE. Stuff should include: De Giorgi-Nash-Moser’s iteration, Harnack inequalities and Schauder estimates on elliptic/parabolic homogeneous/heterogeneous ...
4
votes
1answer
315 views

How does the boundary property usually work in PDE?

This may be related to the question Is the hypersurface of class $C^k$ a $C^k$-differentiable manifold?. In almost every chapter of the PDE textbook(e.g. Folland's Introduction to Partial Differential ...
8
votes
2answers
1k views

When can one expect a classical solution of a PDE?

When solving a PDE, there may be a classical solution or a weak solution (or distribution solution). But I am wondering that when people talk about "finding a solution" to some PDE, what do they refer ...
9
votes
5answers
2k views

Blow up of a solution

What exactly does blow up mean, when people say, for example, that a solution (to a pde (say)) blows up. Thanks.