7
$\begingroup$

How can one tell which PDE's are, roughly speaking, perhaps more interesting to analyse?

Physical motivation is one reason. For example, the KdV $u_t+u_{xxx} - 6uu_x=0$ for a function $u:\mathbb R\times\mathbb R\to\mathbb R$ is a model for many physical systems, such as the propagation of shallow water waves along water bodies with low height. It is also a completely integrable PDE, which makes its analysis more tempting. But putting aside complete integrability (many studied PDE do not have completely integrable variants) why not study the derivative quasilinear equation $u_t + u_{xxx} - 6uu_{xxx}$ or the another derivative semilinear equation $u_t + u_{xxx} - 6uu_{xx}=0$ (I am only putting the KdV here as an example, and am not really asking this particular question for the KdV)?

Aside from physical motivation, how can one pick PDE to analyse?

Edit: Just to be clear, I am asking this from the perspective of PDE research. When I say "analyse," I mean "do PDE research on."

Improve this question
$\endgroup$
7
  • 2
    $\begingroup$ PDEs can come from other sources than physics: geometry , stochastic processes and economics are examples of such fields. $\endgroup$
    – Piyush Grover
    Commented Mar 28, 2020 at 3:30
  • 1
    $\begingroup$ A PDE is interesting to analyze if mathematicians are interested in its analysis. What kinds of PDEs are other folks in your field thinking about, and why do they find them interesting? $\endgroup$
    – Neal
    Commented Mar 28, 2020 at 3:43
  • 2
    $\begingroup$ Navier–Stokes seems to be OK. $\endgroup$
    – user6976
    Commented Mar 28, 2020 at 4:30
  • 1
    $\begingroup$ “Interesting” PDEs often have more geometrical (e.g. integral) formulations, whose analysis can feel different from “write an arbitrary PDO and crank in the Strichartz estimates”. $\endgroup$
    – Francois Ziegler
    Commented Mar 28, 2020 at 5:02
  • 1
    $\begingroup$ I will echo the answer by Denis Serre. In mathematical, and related theoretical work, one often finds two different situations: "problems in search of solutions" and "solutions in search of problems". If you have some independent motivation for studying a PDE, that is your problem and you need to solve it, which is pretty self-explanatory. By your question, you're probably not in this first situation. On the other hand, if you set yourself the task to study a certain method or technique (a "solution"), then some PDEs naturally manifest as relevant "problems". $\endgroup$
    – Igor Khavkine
    Commented Mar 28, 2020 at 9:41

1 Answer 1

Reset to default
6
$\begingroup$

My impression is that a PDE, or rather a class of PDEs, is interesting to the analysts when it is relevant for some analytical tool. Let me take a few examples. The list is not exhaustive.

  • linear constant coefficient PDEs in the whole space ${\mathbb R}^n$ are treated with Fourier analysis.
  • Elliptic (scalar) PDEs are analyzed with the maximum principle. This is particularly true in the modern treatment.
  • Linear and semi-linear evolution PDEs are the realm of semi-group theory.
  • Dispersive linear PDEs obey to Strichartz inequalities.
  • So-called integrable equations or systems are treated by Lax pairs and spectral theory.
  • Othere examples in homogenization or in hyperbolic conservation laws are solved by using compensated compactness.
Improve this answer
$\endgroup$
1
  • 1
    $\begingroup$ Strichartz inequalities are also useful for semilinear problems (problems where the nonlinearity does not have derivatives), e.g. the nonlinear Schrodinger equation and nonlinear wave equation with power nonlinearities. $\endgroup$
    – David
    Commented Mar 28, 2020 at 17:11

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .