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Results tagged with ap.analysis-of-pdes
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user 22810
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
1
vote
Invariant definition of the space of symbols on a vector bundle (pseudo-differential operators)
I think the following should work:
Let $M$ be a compact manifold (just to be safe) and $\pi :E \to M$ a vector bundle. Since $E$ carries an action of $\mathbb{R}^{\times}$ there's an invariant notion …
- 7,799
0
votes
1
answer
80
views
Well-posedness for equations of the form $u_t = grad[V(u)]$ and $u_{tt}=grad[V(u)]$?
Let $V \in C^{1}(\mathbb{R}^n, \mathbb{R})$ consider the following PDE:
$$u_t = grad[V(u)]$$
For $u \in C^{1}([0,1]^n\times [0,T),\mathbb{R}^n)$, with boundary conditions specified on the $n$-dimens …
- 7,799
7
votes
2
answers
214
views
Characterizing pseudo-differential operators as a subalgebra of continuous endomorphisms of ...
I'm aware that the following question is at best a refined version of at least 2 questions which are already on this site. I think it is justified however in that it is more precise and has some new c …
- 7,799
2
votes
0
answers
113
views
A global geometric formulation of the fundamental theorem of Picard Vessiot theory?
Let $X$ be a smooth curve over an algebraically closed field of characteristic 0. The category of quasi-coherent $D$-modules $\mathcal{D}_X$-$Mod$ is a symmetric monoidal abelian category. We can ther …
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25
votes
0
answers
749
views
What is a Green's function in the language of $\mathcal{D}$-modules?
Let $P$ be an analytic linear differential operator defined on some open interval $X=(a,b)$ and $\mathcal{M}=\mathcal{D}_X / \mathcal{D}_X \bullet P$ the corresponding $\mathcal{D}$-module. I'm trying …
- 7,799
8
votes
3
answers
837
views
What does the flow of the principal symbol of the differential operator tell us about the PDE?
Disclaimer: Let me apologize in advance for asking this slightly vague question
Let $M$ be a manifold and let $P$ be a partial differential operator acting on $C^{\infty}(M)$. Associated to $P$ there …
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10
votes
0
answers
262
views
Is there a classification of differential equations over the field of fractions of formal po...
Let $k$ be an algebraically closed field of characteristic 0. Consider the field of fractions of formal power series $K:= Frac(k[[T_1,...,T_n]])$. We have the corresponding algebra of differential ope …
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17
votes
0
answers
641
views
Is there an Infinite dimensional sheaf theory for analysis on manifolds?
I apologize if this question is slightly vague but I don't know how to ask it non-vaguely. Moreover, my question is about an ideal situation. If there's a close answer which doesn't satisfy all the co …
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11
votes
1
answer
664
views
Is every continuous endomorphism of the Schwartz space a pseudo-differential operator?
Let $\mathcal{S}:= \mathcal{S}(\mathbb{R}^n)$ be the Schwartz space of smooth functions with rapid decay. The question is pretty simply stated in the title. Pseudo-differential act continuously on the …
- 7,799
18
votes
3
answers
4k
views
What is an "Instanton" in classical gauge theory? (to a mathematician)
There's already a question about the same topic but I think its aim is different.
Classical (non-quantum) gauge theory is a completely rigorous mathematical theory. It can be phrased in completely di …
- 7,799
7
votes
2
answers
498
views
Do pseudo-differential operators form a sheaf of algebras?
Let $M$ be a smooth manifold.
I have been trying to figure out from the literature I know whether (any flavor of) pseudo-differential operators form a sheaf of algebras (w.r.t. the usual topology on …
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23
votes
4
answers
2k
views
What is an "integrable hierarchy"? (to a mathematician)
This is one of those "what is an $X$?" questions so let me apologize in advance.
By now I have already encountered the phrase "integrable hierarchy" in mathematical contexts (in particular the so cal …
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30
votes
8
answers
4k
views
Applications of microlocal analysis?
What examples are there of striking applications of the ideas of microlocal analysis?
Ideally I'm looking for specific results in any of the relevant fields (PDE, algebraic/differential geometry/topol …
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