Questions tagged [differential-equations]
Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.
594
questions with no upvoted or accepted answers
5
votes
0
answers
229
views
Is there a Lie II theorem for monoids?
Lie's second theorem says that if $G$ is a connected simply connected Lie group with Lie algebra $\mathfrak g$, then the functor of "differentiation" from the category $\mathrm{Rep}^f(G)$ of finite-...
- 53.1k
5
votes
0
answers
175
views
Is the space of $C^r$ vector fields inducing locally uniformly bounded trajectories Baire?
Let $\mathcal{V}$ be the space $C^r$ vector fields on a non-compact (smooth) manifold $M$. Being a subspace of $C^r(M, T M)$, it inherits the natural $C^r$ topology (i.e. the strong topology) of that ...
5
votes
0
answers
237
views
Linear ODEs in a locally convex vector space
Let $X$ be a complete, locally convex, Hausdorff topological vector space over $\mathbb{C}$. Let $J \subset \mathbb{R}$ be an open interval. Consider the space $M = C^\infty(J,X)$ of smooth ...
- 245
5
votes
0
answers
198
views
Constructing solutions to matrix equations
Let $k,n$ be integers, $u_1,\dots,u_n \in U(k)$, $d_1,\dots,d_n \in \mathbb Z$ with $\sum_{i=1}^{n} d_i =: d \neq 0$.
Consider the map $w:= U(k) \to U(k)$ with
$$w(v):= u_1 v^{d_1} \cdots u_n v^{d_n} ...
- 25.3k
4
votes
0
answers
132
views
Estimating $p$th moment bound of error between small noise SDE and ODE
For a $d$-dimensional standard Brownian motion $W$, and a locally Lipschitz function $b: \mathbb{R}^d \rightarrow \mathbb{R}^d$, consider an SDE:
$$dX_t^\varepsilon = b(X_t) dt + \varepsilon^t dW_t,\...
- 41
4
votes
0
answers
438
views
A 4th-order linear PDE
I am interested in the following type of $4$-th order linear PDE with $2$ variables (i.e., $x$ and $t$):
$x^3 f_{xxxt}+ f =0$
Does anyone know if this type of PDE already appeared in the literature? ...
- 71
4
votes
0
answers
124
views
Darboux integral for non-polynomial ODEs
Given a polynomial ODE in $n$-dimensions of maximal degree $d$
$$
\dot{x}_j=f_j(x)=\sum_{i_1,\dots,i_n=1}^d a_{i_1,\dots,i_n}^j x_1^{i_1}\dots x_n^{i_n} \quad \forall j=1,\ldots,n
$$
we define ...
- 247
4
votes
0
answers
97
views
Behavior of lapse function at infinity: stability of Minkowski
In the Stability of Minkowski Spacetime, Christodoulou and Klainerman prove a local existence proof for a particular class of quasilinear wave equation for a symmetric, traceless, covariant 2-tensor $...
- 409
4
votes
0
answers
79
views
Reference request, or maybe not really a reference request, on differential algebra
Of differential algebra, Gian-Carlo Rota wrote:
No elementary presentation of this beautiful subject has ever been attempted, to the best of my knowledge; Cohen’s book of the twenties is the closest, ...
- 11.9k
4
votes
0
answers
304
views
Banach's fixed point theorem for quasilinear parabolic PDEs
I have recently started reading into PDE theory, and came across the following question. Consider the PDE $$
\begin{cases}
\partial_t \rho = \Delta (\rho + \rho^2) & \text{ on } (0,T) \times \...
- 213
4
votes
0
answers
231
views
Generalising Bäcklund transform to solve $\omega''(t)=t\sin\omega(t)$
Bäcklund transformations may be used also in ODE to solve non-linear problems; for instance, it's well known that for the equation
$$
\frac{\mathrm{d}^2\omega}{\mathrm{d}t^2}=\sin\omega
\tag{*}\label{...
- 134
4
votes
0
answers
111
views
The semiclassical limit of Virasoro reps $\varphi_{n,1}$ produces certain $\mathfrak{sl}_2$ reps — what is the connection to KdV?
The semiclassical ("light") limit $c\to \infty$ of the irreducible Virasoro representation $\varphi_{n,1}$ with highest weight $h_{n,1}\to -\frac{n-1}{2}$ is $\mathbb{C}[L_{-1},L_{-2},\dotsc]...
- 1,746
4
votes
0
answers
272
views
Integral representation of solution of an elliptic PDE in divergence form
Suppose we have a second order elliptic differential operator
$$
L(v) = -\text{div}(A(x) \nabla v)
$$
$A(x)$ is a bounded and strictly positive definite matrix with Hölder continuous entries. And ...
- 261
4
votes
0
answers
105
views
Poincaré's Lemma in the space of tempered distributions
It is well known that if $f\in \mathcal{D}'(\mathbb{R}^3,\mathbb{R}^3)$ and $\textbf{curl} f= 0$ then there exists a $u\in \mathcal{D}'(\mathbb{R}^3)$ such that $\nabla u = f$.
Question. Does the ...
- 356
4
votes
0
answers
399
views
Intrinsic numerical methods on Riemannian manifolds
I am interested in numerical methods for ordinary differential equations on a Riemannian manifold $M$. The general form of such an equation is $\dot x(t)=V(x(t)), x(0)=x_0 \in M$, where $V$ is a ...
- 345
4
votes
0
answers
123
views
Closed subgroup (Cartan) theorem without transversality nor Lipschitz condition within Banach algebras
Yesterday, I came across the following preliminary theorem.
Theorem Let $\mathcal{B}$ be a Banach algebra (with unit $e$) and $G$ be a closed subgroup
of $\mathcal{B}^{-1}$ (the group of ...
- 3,696
4
votes
0
answers
234
views
Equivalent definitions of differential operator
This puzzles me from some time and is in parts connected to the questions Symmetrized derivatives version and Symmetrized derivatives version II.
For me the linear DO between vector bundles $E$ and $...
- 437
4
votes
0
answers
196
views
Has this functor been studied?
Let $\mathbb{O}$ be the category where the objects are $n$-manifolds, and the morphisms are almost-everywhere smooth cobordisms along with almost-everywhere smooth partial differential equations on ...
- 182
4
votes
0
answers
250
views
An ODE argument of Hitchin
In [1] (p. 183, paragraph of Eq. 7.8), Hitchin makes an argument which I would summarize as follows.
Suppose that $A\in\mathfrak{u}(n)$ (a skew-Hermitian matrix) has distinct eigenvalues and let $S:...
- 1,373
4
votes
0
answers
100
views
Flow lines of a real analytic vector field convergent to a point
Let $X$ be a real analytic vector field defined on an open connected subset $U$ of $\mathbb{R}^n$. Let $p \in \mathbb{R}^n \setminus U$. Let $L$ be union of the flow lines $\ell$ of $X$ such that $p$ ...
- 1,379
4
votes
0
answers
120
views
Ricci flow on locally symmetric noncompact manifold
As it is mentioned by Deane Yang in Ricci flow preserves locally symmetry along the flow, we know the local symmetry is preserved under the Ricci flow on the compact manifold since we have the ...
- 123
4
votes
0
answers
227
views
Orthonormal Basis of Multi-Dimensional Sobolev Space of Different Orders without Reproducing Kernel
Let $\Omega$ be an open subset of $\mathbb{R}^d$. Under regularity conditions, we know that the $s$-th order Sobolev space $H^s(\Omega)$ with $s \geq d/2$ is a reproducing kernel Hilbert space. In ...
- 1,117
4
votes
0
answers
398
views
Spectral Gap of Elliptic Operator
Under what conditions on $a(x)$ and domain $D$, the spectral gap of the elliptic operator $ \nabla \cdot(a(x)\cdot \nabla)$ defined on $D$, can be controlled?
The boundary condition is that the ...
- 325
4
votes
0
answers
142
views
An embedding question: Morrey spaces
Question. If $u\in L^1$ and $Du$ is in the dual of the Holder space $C^\alpha$, then is it possible to say $u$ belongs to some Morrey space $L^{1, \delta}$?
- 41.8k
4
votes
0
answers
173
views
Distributional PDE solutions as topological linear duals of PDE solutions
Let
$$
P \;\colon\; \Gamma_\Sigma(E) \to \Gamma_\Sigma(\tilde E^\ast)
$$
be a formally self-adjoint hyperbolic linear differential operator ($\tilde E^\ast$ denotes the densitized dual of a ...
- 19.6k
4
votes
0
answers
378
views
On modified Bessel solutions to complex ODE's using Kummer's series
I am trying to reduce the following ODE to Bessel's ODE form and hence solve it:
$$x^{2}y''(x)+x(4x^{3}-3)y'(x)+(4x^{8}-5x^{2}+3)y(x)=0\tag{1} \, .$$
I tried to solve it via the standard method, i.e.,...
- 419
4
votes
0
answers
255
views
Basin of attraction of gradient flow
Suppose we have a compact Riemannian manifold $(M,g)$, and a Morse function $f : M \rightarrow \mathbb{R}$. Suppose we consider the gradient flow generated by this function, i.e. $$\dot{x_t} = - \...
- 141
4
votes
0
answers
710
views
When the integral of the product of two matrix exponentials is singular?
Let $A$ and $B$ be two $n \times n$ real matrices. (In my application, $A$ and $B$ are $6\times 6$ traceless singular real matrices) I am interested in finding the smallest $T$ such that the integral $...
- 41
4
votes
0
answers
107
views
Dynamics of pairwise distances in the $n$-body problem
Disclaimer: I have asked this question on Physics SE a week ago, but got no answers. I know that some MO users are interested in the $n$-body problem, so I decided to cross post here as well.
...
4
votes
0
answers
146
views
Connection between cardiac equations and untangling knots?
I was surprised to learn that there is (conjecturally) a connection between a cardiac muscle model known as the FitzHugh-Nagumo equations, and untangling knots:
Maucher, Fabian, and Paul Sutcliffe. ...
- 149k
4
votes
0
answers
580
views
Models used for the Zika virus?
I am currently teaching an ordinary differential equations course, and am thinking about doing a module on infectious disease models, e.g. SIR/SIRS. I thought, if possible, it would be nice to ...
- 5,709
4
votes
0
answers
164
views
Does there exist duality/symmetry between Fuchsian differential equations, like in the case of confluent hypergeometric?
My question is about whether certain dualities or symmetries hold between pairs of Fuchsian differential equations, but I need to give a bit of background to explain what I want to ask. Thank you for ...
- 411
4
votes
0
answers
429
views
Lorenz attractor power spectrum
If considered Lorenz attractor (with classical parameters $\sigma = 10, b = \frac{8}{3},r>25$), it is often noted, that while the spectral density (Fourier transformation of corresponding ...
- 41
4
votes
0
answers
202
views
System of linear ODEs with hypergeometric coefficients
For quite some time I have been trying to solve the following system of differential equations for the two functions $G$ and $H$ defined on the interval $[0,1]$:
$$
\begin{align}x G''(x)=&\mathscr{...
- 656
4
votes
0
answers
429
views
Scattering for rapidly decaying solutions of NLS
Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions
of the Nonlinear Schrödinger Equation" the following property.
Given the problem
\begin{equation}
\left\{
\begin{array}{rl}
...
- 403
4
votes
0
answers
349
views
May a globally bounded G-function have a logarithmic branching? (On a conjecture of Ruzsa)
By a "globally bounded $G$-function," following G. Christol, I will mean a solution to a (minimal) linear differential equation on $\mathbb{P}^1$ (then necessarily of the Fuchsian type and with ...
- 13.7k
3
votes
0
answers
317
views
The local global principle for differential equations
Are there any good reference to tackle the problem below?
Or, are there any know result?
Problem
Let $f_1...f_n\in \mathbb{Z}[x_1,..,x_n]$ and $V:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a vector ...
- 83
3
votes
0
answers
155
views
What d.o. $\sum_i f_i(z)\partial_z^i$ correspond to subalgebras $M$ in polynoms $C[x_i]$ being Langlands dual to motive of $Spec(M) \to X$?
Briefly: The question is about presenting explicit examples of the construction discussed in the recent MO question "Relation between motives and geometric Langlands" and Will Sawin's asnwer ...
- 23.9k
3
votes
0
answers
95
views
Distance between solutions of differential inclusions
Suppose that we have two differential inclusions
$$\frac{dY^1}{dt}(t)\in b_1(Y^1,t)$$
with $Y^1(0)\in Y_0^1$ and
$$\frac{dY^2}{dt}(t)\in b_2(Y^2,t)$$
with $Y^2(0)\in Y_0^2$.
Can we then control $d(Y^1(...
- 1,302
3
votes
0
answers
48
views
Stability of indefinitely damped mechanical system with diagonal stiffness
I'm trying to find conditions for the asymptotic stability of the following linear system,
\begin{equation}
\mathbf{I \ddot{x}} + \mathbf{B \dot{x}} + \mathbf{K x} = 0
\end{equation}
given the ...
3
votes
0
answers
87
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 ...
- 312
3
votes
0
answers
215
views
An attempt to extend polynomial rings
Suppose $\mathbb{K}$ is a field of characteristic $0$. Let $S_n=\mathbb{K}[[x_1,\dots,x_n]]$ be the ring of formal power series in $n$ variables, $L_n$ be its fraction field and $W_n=\mathbb{K}[x_1,\...
- 1,121
3
votes
0
answers
94
views
Mathematical formulation of beam: get stress/strain from forces and momentum
I'm working with static beams with Euler–Bernoulli model which ODE is
$$
\dfrac{d^2}{dx^2} \left(EI \cdot \dfrac{d^2w}{dx^2}\right) = q(x).
$$
With a beam along the $x$ axis, the solution consists of ...
- 253
3
votes
0
answers
240
views
A generalization of Weierstrass transform
As stated in this article, the Weierstrass transform of $f(x)$ is defined as:
\begin{equation}
W[f](x)=\frac{1}{4\pi}\int_{-\infty}^{\infty}f(y)e^{-\frac{(x-y)^{2}}{4}}dy
\end{equation}
which can be ...
- 350
3
votes
0
answers
81
views
Regularity of center manifold
Consider a $C^r$ vector field $f \colon \mathbb{R}^n \to \mathbb{R}^n$ with $r \geq 1$. Let $\bar x$ be a critical point of $f$, that is, $f(\bar x) = 0$.
Suppose that the spectrum of $\mathrm{D}f(\...
- 31
3
votes
0
answers
97
views
Definition clarification: "regular directed distributions"
(I asked this question on math.stackexchange (see here) but didn't receive any reaction, hence I try it here. If it does not fit within here, just let me know in the comments.)
In the definition of ...
- 987
3
votes
0
answers
145
views
Permutahedra Euler characteristic polynomials from cumulant-moment relation, a combinatorial proof?
Given the formal Taylor series, or e.g.f.,
$f(x) = 1 + \sum_{n \geq 1} m_n \; \frac{x^n}{n!}$,
the classical formal cumulants $c_n$ are generated from the formal moments $m_n$ via
$ \sum_{n \geq 1} ...
- 9,931
3
votes
0
answers
44
views
An equality satisfied by the solutions to Kolmogorov forward and backward PDEs
Let $b: \mathbb R_+\times\mathbb R\to \mathbb R$ and $\sigma: \mathbb R_+\times\mathbb R\to (0,\infty)$ be functions as nice as possible (e.g. bounded and of bounded partial derivatives, and $\inf_{(t,...
- 1,230
3
votes
0
answers
74
views
A foliation version of S.Husseini counter example in fixed point theory
In this question we are indirectly inspired by an example by S.Husseini Amer. J.Math 1977
"The Products of Manifolds with the f.p.p. Need Not have the f.p.p"
who gave an example of two ...
- 312
3
votes
0
answers
94
views
L^1 gradient bounds for potentials of weakly closed forms
Context: The Poincaré-lemma is a central statement in differential geometry. It shows that a k-form is closed iff it is exact. A special case is as follows:
Let $\omega\in\Omega^k(U)$ with
$\omega=\...
- 31