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Results tagged with ap.analysis-of-pdes
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user 36688
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
votes
1
answer
825
views
Does $P_xP_y+Q_xQ_y=0 \implies$ "non-existence of limit cycle" for $P\partial_x+Q\partial_y$...
Let $X=P\partial_x+Q\partial_y$ be a vector field on the plane $\mathbb{R}^2$. Assume that we have :$$P_xP_y+Q_xQ_y=0$$ Does this imply that the vector field $X$ is a divergence-free vector field w …
CommunityBot
- 1
4
votes
0
answers
243
views
Dynamical obstruction for a vector field to have a Harmonic divergence
Let $(M,g)$ be an analytic Riemannian manifold and $X$ be an analytic vector field on $M$. Can we always have a volume form $\Omega$ such that $\operatorname{Div}_{\Omega} X$ is a harmonic …
- 356
3
votes
0
answers
89
views
Cycloid on manifolds
Inspired by differential equation
$$y(1+y'^2)=c$$
which generates the cycloid we consider the following differential equation on a Riemannian manifold:
$$f(1+|\nabla f|^2)=c$$
On the other hand th …
- 356
2
votes
0
answers
463
views
A Fourier elliptic vector field on a Riemannian manifold
Motivation for this question:
Let $X$ be a vector field on a manifold $M$. Obviously the differential operator $D:C^{\infty}(M)\to C^{\infty}(M)$ with $D(f)=X.f$ is not an elliptic opetator when $\di …
- 356
6
votes
0
answers
266
views
Elliptic foliations of the plane
A $1$ dimensional foliation of the plane $\mathbb{R}^2$is called elliptic if it admits a non vanishing smooth tangent vector field $X$ with the following properties:
The differential operator …
- 356
1
vote
0
answers
97
views
The module generated by kernel of an elliptic differential operator
Let $D$ be an elliptic differential operator defined on $\Gamma(E)$ where $\Gamma (E)$ is the space of smooth sections of vector bundle $E$ over a smooth manifold $M$. So $\Gamma (E)$ is a $C^\i …
- 356
6
votes
1
answer
340
views
Fredholm theory of non elliptic operators
In this question we search for a big list of non elliptic operators whose Fredholm index is finite or whose Fredholm theory is extensively discussed. The main motovation is the conference linked in th …
- 2,821
3
votes
1
answer
272
views
Harmonic interpolation with analytic initial condition
Let $n>1$ and $M\subset \mathbb{R}^n$ be a (sufficiently low dimensional) compact analytic submanifold.
Assume that $f:\mathbb{R}^n\to \mathbb{R}$ is an analytic function.
Is there a Harmonic funct …
- 6,367
4
votes
1
answer
373
views
Differential inequalities under which a flat function must be identically zero
Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function which is flat at $0\in \mathbb{R}$. That is $f^{(k)}(0)=0,\; k=0,1,2,\ldots $.
Assume that $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ …
- 4,899
5
votes
2
answers
459
views
A question on certain elliptic PDE
Consider the elliptic PDE
$$(CR)\;\;\;\;\;\;\begin{cases} U_{xx}=V_{yy}\\U_{yy}=-V_{xx} \end{cases}$$
And its consequence $$(LAP)\;\;\;\;\;\;U_{xxxx}+U_{yyyy}=0$$.
Somehow, these equations are si …
- 356
2
votes
1
answer
235
views
Elliptic operators and Leibniz rule
Let $M$ be a manifold. Does it necessarily admit an elliptic operator on $C^{\infty}(M)$ which satisfy Leibniz rule?
Let $M$ be a symplectic manifold with the standard Poisson structure on $C^{\inf …
- 6,825
0
votes
1
answer
147
views
Is it a sufficient condition for linearity?
Edit: According to the comment by LSpice we come back to the initial version of this question
Let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a smooth function such that for every $x\in \mathbb{R}^n$ the der …
- 356
6
votes
0
answers
283
views
A concept weaker than geodesibility of flows which is possibly useful in limit cycle theory
The main objective of this post is to apply the Gauss Bonnet Theorem to count the number of limit cycles of a polynomial vector field as described in this MO post and its linked MO posts But in thi …
- 356
2
votes
1
answer
131
views
Global first integral for certain $3$ dimensional system
A physicist colleague asks me the following question. I have no idea to answer him. Your answer is very appreciated.
Is there a global first integral on $\mathbb{R}^3$ for the following vector field? …
- 6,696
6
votes
2
answers
613
views
On equation $\Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemann...
Assume that $M$ is a compact Riemannian manifold whose Laplacian is denoted by $\Delta$. Assume that the Euler characteristic of $M$ is zero. Does $M$ admit a non vanishing vector field …
- 39k