Questions tagged [ap.analysis-of-pdes]
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
4,467 questions
9
votes
3
answers
1k
views
Green's formula for nonorientable manifolds
Usually in differential geometry one proves the Stokes theorem and then obtains divergence theorem and Green's formulas as corollaries. However, divergence theorem is also valid for nonorientable ...
20
votes
2
answers
3k
views
Why don't existence and uniqueness for the Boltzmann equation imply the same for Navier-Stokes?
As I understand it, Lions and DiPerna demonstrated existence and uniqueness for the Boltzmann equation. Moreover, this paper claims that
Appropriately scaled families of
DiPerna–Lions ...
- 1
3
votes
1
answer
452
views
Identifying the nonlinear parabolic PDE $u_t = (u^2)_{xx}$.
A friend of mine in the department needs to know if the following PDE has been extensively studied
$$ u_t = (u^2)_{xx}$$
Or more generally, replacing the square by any function of $u$. One would like ...
- 15.4k
5
votes
0
answers
159
views
Regularity of reflection coefficients (or more generally the scattering transform)
Consider the Schrodinger operator $L(q) = -\partial_x^2 + q(x)$ where the potential $q$ is a real-valued function of a real variable which decays sufficiently rapidly at $\pm \infty$.
We define the ...
- 471
6
votes
2
answers
600
views
What are the interesting cases of the generalized Korteweg-de Vries equation?
The generalized Korteweg-de Vries equation is
$u_t + u_{xxx} + (u^p)_x=0$
for integer $p$. (The original Korteweg-de Vries equation is the case $p=2$.) I need to understand solutions for $p=1$, ...
- 5,293
5
votes
1
answer
470
views
Sign-Gordon Equation
What can be said and done about the "SIGN-Gordon equation"?
$$\varphi_{tt}- \varphi_{xx} + \text{sgn}(\varphi) = 0.$$
It came up here.
CommunityBot
- 1
1
vote
1
answer
311
views
References for weak ellipticity
There are good books (like Evans) for strongly elliptic second order linear PDE. I want to learn about weakly elliptic PDE (of any order). Are there any good books for the same? I am very curious as ...
- 25.8k
9
votes
3
answers
1k
views
Newlander-Nirenberg for surfaces
Quite a long ago, I tried to work out explicitly the content of the Newlander-Nirenberg theorem. My aim was trying to understand wether a direct proof could work in the simplest possible case, namely ...
- 932
2
votes
1
answer
580
views
Entropy of Markov processes
Consider a Markov process $X_t$ with generator $L$ and invariant distribution $\pi$, whose distribution at time $t$ is given by $\pi(t,dx)=\phi(t,x) \pi(dx)$ - in other word, $\phi(t,x)$ is the ...
- 15.4k
19
votes
5
answers
2k
views
"Physical" construction of nonconstant meromorphic functions on compact Riemann surfaces?
Miranda's book on Riemann surfaces ignores the analytical details of proving that compact Riemann surfaces admit nonconstant meromorphic functions, preferring instead to work out the algebraic ...
CommunityBot
- 1
-1
votes
2
answers
1k
views
Rellich-Necas identity
I am looking for a book/paper which has the proof of the Rellich-Nicas identity.
[EDIT by Yemon Choi] It seems that what was meant is "the Rellich-Necas identity", although the original poster hasn't ...
- 25.8k
4
votes
2
answers
3k
views
Solutions to the diffusion equation
When it comes to solving the heat diffusion equation u_t=u_xx the two most important solutions are
a) a combination (sum) of sin-terms to resemble the function of the initial condition (that is ...
CommunityBot
- 1
3
votes
3
answers
2k
views
Error analysis of implicit functions
I'm trying to do propagation of error using the linearized variance method (assuming independent variables, thus no need for the covariance terms):
$$\sigma^2_f = \sum^n_{k=0} \left(\frac{\partial f}{...
- 15.4k
0
votes
1
answer
166
views
Transformation from domains to half-spaces
In a paper I read, an elliptic boundary value problem
on a bounded domain D x (0,T) is solved by first transforming
it in a set of equations on half-spaces R^n and then applying
partial Fourier ...
- 15.4k
6
votes
1
answer
779
views
primitive of an exact differential form with special properties
We were working on a smoothing problem and ran across the apparently simple following question:
X is a triangulated smooth manifold of dimension $n$, and $\alpha$ is an exact differential form of ...
CommunityBot
- 1
28
votes
4
answers
2k
views
What is the relationship between various things called holonomic?
The following things are all called holonomic or holonomy:
A holonomic constraint on a physical system is one where the constraint gives a relationship between coordinates that doesn't involve the ...
- 243
12
votes
3
answers
1k
views
Equations for Integrable Systems
So, let's say we have a symplectic variety over $\mathbb{C}$, $M$, of dimension $2n$, and $f_1,\ldots,f_n$ Poisson commuting functions with $df_1\wedge\ldots\wedge df_n$ generically nonzero. Further ...
- 5,293