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Results tagged with ap.analysis-of-pdes
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Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
34
votes
Does Physics need non-analytic smooth functions?
It is worth noting that it is impossible to solve the initial value problem for the standard heat equation in the real analytic category. Here, there are asymptotic expansions available but no Taylor …
- 27.5k
24
votes
Accepted
PDEs and algebraic varieties
What has been studied more extensively is the top order term, which is known as the principal symbol. Since it is a homogeneous polynomial of degree $d$, it defines a real projective algebraic variety …
- 27.5k
21
votes
Does Ricci flow depend continuously on the initial metric?
If you use the right topology on the space of metrics, the answer is yes. Basically, this is always true and a consequence of the proof for any theorem on the existence, uniqueness, and regularity of …
- 27.5k
18
votes
Why can't there be a general theory of nonlinear PDE?
I agree with Craig Evans, but maybe it's too strong to say "never" and "impossible". Still, to date there is nothing even close to a unified approach or theory for nonlinear PDE's. And to me this is n …
17
votes
Motivation for and history of pseudo-differential operators
My vague memories of this:
As others have mentioned, pseudodifferential operators arose from trying to establish existence and regularity theorems for a variable coefficient linear PDE using Fourier …
- 27.5k
16
votes
Epsilon regularity: what does it say and where does it come from?
Below is a rather longwinded description of the special case when the singularity is at worst an isolated point. I suspect you know all this already. The magic comes at the very end (see paragraph tha …
- 27.5k
15
votes
Applications of microlocal analysis?
Although microlocal analysis was developed originally exclusively for linear problems, it has played an increasingly important role in nonlinear PDE via what's known as paradifferential calculus. Ther …
- 27.5k
15
votes
Why can't there be a general theory of nonlinear PDE?
Some more random thoughts:
The closest thing I've ever seen to a "general theory of nonlinear PDE's" is Gromov's book, Partial Differential Relations. He does many things in there that I don't unders …
14
votes
Moduli space of linear partial differential equations
Hormander showed that there is a generic set of scalar linear PDE's that can be studied using general techniques, known as microlocal analysis. This can be linked to algebraic geometry as follows: Any …
- 27.5k
11
votes
PDEs as a tool in other domains in mathematics
The work of Uhlenbeck, Taubes, Donaldson, and others on Yang-Mills connections is a gorgeous application of nonlinear elliptic PDE theory.
10
votes
PDEs as a tool in other domains in mathematics
The Nash isometric embedding theorem
8
votes
Accepted
Inverse of partial differential operator as a smooth tame map
In fact, it's hard to find an example of a PDO which has a right inverse that is not smooth tame. It's certainly true for the standard types: elliptic, hyperbolic, and parabolic.
On the other hand, w …
- 27.5k
8
votes
Book Recommendation - PDE's for geometricians / topologists
I find your question too broad. I would recommend starting with a book that focuses on a particular question or area in differential geometry and presents the PDE theory needed. A very incomplete list …
8
votes
Accepted
Posing Cauchy data for the heat equation: $t=0$ a characteristic surface?
The concept of a non-characteristic surface for a PDE or a system of PDE's is useful primarily for only establishing the existence and uniqueness of real analytic or formal power series solutions to t …
- 27.5k
8
votes
Fundamental solution of an elliptic PDE in divergence form with non-symmetric matrix
If you assume that the coefficients $a^{ij}$ are smooth functions and let $$b^{ij} = \frac{1}{2}(a^{ij} + a^{ji}),$$ then the PDE can be written as
$$
b^{ij}\partial^2_{ij}u + \partial_ia^{ij}\partial …
- 27.5k