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Results tagged with ap.analysis-of-pdes
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user 9652
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
25
votes
Applications of microlocal analysis?
Microlocal analysis is used in computed tomography and other tomographic imaging techniques e.g. in medicine .
Specifically, it is used to describe which wavefront sets (here: boundaries of objects, e …
- 1,286
21
votes
How to generalize the various vector calculus theorems to distributions?
This may be borderline for this site, but here goes: What you are looking for may be the notion of weak derivative. A function $f$ defined on some $d$-dimensional set $\Omega$ has a weak partial deriv …
- 12.7k
0
votes
Practical applications of Sobolev spaces
What about the Eikonal equation
$$
|\nabla u| = 1?
$$
As far as I understand, one can construct solutions directly, but still, the solution does not have the desired classical regularity in general bu …
- 12.7k
8
votes
Making the Fourier transform quantitative
There are various mathematical formulations of this phenomenon and some of them are quite quantitative.
First, there is the classical Heisenberg uncertainty principle which roughly states that the pr …
- 12.7k
2
votes
Accepted
Iterative method for $p$-Laplacian
I know this method under the name lagged diffusivity. I learned it from the paper
Vogel, Curtis R., and Mary E. Oman. "Iterative methods for total variation denoising." SIAM Journal on Scientific …
- 12.7k
5
votes
Mathematical difference between entropy and energy
It is probably worth mentioning that the the heat equation is inherently linked to energy and entropy in two ways:
The heat equation is the gradient flow of the energy in the Hilbert space $L^2$ (cl …
- 12.7k
5
votes
Concentration compactness. Can this concept be stated in a theorem?
Well, I think you have to accept that concentration compactness is concept rather than a result. The intro of the mentioned book starts with
The subject of this book, concentration compactness, i …
- 12.7k
5
votes
Relativistic Control Theory
There is also
Inverse problems in spacetime I: Inverse problems for Einstein equations - Extended preprint version, by Yaroslav Kurylev, Matti Lassas, Gunther Uhlmann
and related papers.
Sound …
- 12.7k
12
votes
0
answers
7k
views
Otelbayev's approach to Navier-Stokes [closed]
Recent news post that Mukhtarbai Otelbayev from Eurasian National University has shown existence of strong solutions of the Navier-Stokes equation in the article
"Existence of a strong solution of …
CommunityBot
- 1
1
vote
Parametrised Hilbert spaces; can we put a norm on the following space of Hilbert spaces?
If you want to put a norm on $H$, you should first think about the vector space structure that you want to impose on $H$ (I don't see a canonical one). If you are satisfied with a metric on $H$ (sound …
- 12.7k
1
vote
Accepted
What are the basis functions for a product space?
Edit: Note that I understood the question for $L^1([0,1]^3)$, c.f. Robert Israels comment.
First, the term "basis" for general Banach spaces, especially ugly spaces such as $L^1$, can be complicated. …
- 12.7k
5
votes
When to use more exciting function spaces than ordinary Sobolev spaces?
I've heard that regularity theory for $p$-Laplacian equations, i.e. equations of the form
$$
\mathrm{div}(c_1|\nabla u|^{p-2}\nabla u) + c_2 |u|^{p-2} = f
$$
with $c_1,c_2 > 0$, $p>2$ need Besov spac …
- 12.7k
5
votes
"Wild" solutions of the heat equation: how to graph them?
I made a MATLAB plot of a partial sum of the first seven terms of the series Carlo referred to. The series is
$$
u(x,t) = \sum_{n=0}^\infty f^{(n)}(t)\frac{x^{2n}}{(2n)!}
$$
where
$$
f(t) = \begin{cas …
CommunityBot
- 1