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2d
comment Kernel of Bianchi operator: Is a (smooth tame) Frechet manifold?
Ok. Now try to do the global result you want in the holder case by following DeTurck's proof. Where does it break down? It has to because otherwise DeTurck would have stated and proved the global result himself.
2d
comment Kernel of Bianchi operator: Is a (smooth tame) Frechet manifold?
Here's my suggestion: First, ignore the subtleties of infinite-dimensional spaces and manifolds. Recall that the standard finite-dimensional implicit function theorem can be used to verify whether an equation defines a submanifold or not.. Work out why your main question can be answered by an analogous infinite-dimensional implicit function theorem. Now figure out what you need in order to use the Banach or tame Frechet (i.e., Nash-Moser) implicit function theorem to get what you want.
2d
comment Kernel of Bianchi operator: Is a (smooth tame) Frechet manifold?
Perhaps you should first make sure you understand the difference between a Banach space and a Frechet space and which function spaces are which. That will then provide a guide to what kind of infinite dimensional manifold the space of metrics is. (The answer is "it depends")
Apr
21
comment Kernel of Bianchi operator: Is a (smooth tame) Frechet manifold?
The space of solutions to a nonlinear elliptic partial differential equation does not need to be a Frechet manifold. You need to know that some kind of constant rank condition holds. That's why DeTurck had to restrict to a sufficiently small ball. My guess is that a sufficient condition for the "kernel" of an underdetermined elliptic operator to be a Banach or tame Frechet manifold is for the linearized operator to be surjective.
Apr
21
comment Kernel of Bianchi operator: Is a (smooth tame) Frechet manifold?
One has to be careful about defining what the "kernel" is of a nonlinear partial differential operator.
Apr
20
comment Moser estimates?
Nice and simple. The more involved Moser iteration or Schauder is needed only if you need weaker regularity assumptions on the coefficients. For example, for a nonlinear PDE.
Apr
19
comment Self-contained book on Ricci Flow/Geometric Analysis
Since the Ricci flow is a PDE, it's not realistic to learn about it without knowing any PDE theory at all. But you don't need much. My advice is to study the books by Chow et al and consult PDE books only as needed.
Apr
19
comment Moser estimates?
Why is right side in $C^\alpha$?
Apr
16
comment Moser estimates?
I suggest googling "Moser iteration PDE". One of the hits is a book by Jurgen Jost on PDE's. That seems like a reasonable place to look.
Apr
16
comment Moser estimates?
Why can't you do Moser iteration applied to the PDE's satisfied by $u$, $(\partial_t, \nabla_x)u$, and $\partial_t^2 + L)u$?
Apr
16
comment Lower order perturbations of 2nd order differential operators
You really should move this question to math.stackexchange.com or figure it out yourself. All you need is integration by parts.
Apr
15
comment Lower order perturbations of 2nd order differential operators
If this is question is about whether the operator is formally self-adjoint, you should be able to do the calculation yourself.
Apr
14
comment Mathematical difference between entropy and energy
I think if you google "Moser iteration parabolic PDE", you'll find lots of presentations on it. They will also cite Moser's original papers.
Apr
14
comment Mathematical difference between entropy and energy
An aside: The $L^p$ inequality is used for Moser iteration, which establishes an a priori estimate for $u$. The basic idea is to rewrite the negative term (up to a constant factor depending on $p$) as $-\int |\nabla u^{p/2}|^2$ and apply the Sobolev inequality. This gives you an estimate for a higher $L^p$ norm of $u$ in terms of a lower one. Repeat and show that the accumulation of constant factors is bounded as $p \rightarrow \infty$. In the limit, you get an $L^\infty$ bound on $u$ in terms of an $L^p$ bound (where $p > 1$). This works for a variable coefficient heat equation.
Apr
13
comment Mathematical difference between entropy and energy
Well, given that entropy is supposed to measure how much randomness or noise there is, it is natural to have it increase in time. So the hyperbolic conservation law people somehow got it wrong.
Apr
13
comment Mathematical difference between entropy and energy
$\int u^2$ decays in time if and only if $\frac{1}{2}\log \int u^2$ does.
Apr
13
revised Mathematical difference between entropy and energy
edited body
Apr
13
revised Mathematical difference between entropy and energy
added 70 characters in body
Apr
13
comment Mathematical difference between entropy and energy
The definition of entropy is usually $-H(t)$.
Apr
13
answered Mathematical difference between entropy and energy