# Questions tagged [differential-equations]

Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.

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### Is every complex linear algebraic group a differential Galois group?

Let $G$ be a complex linear algebraic group. In other words, $G$ is a subvariety of the space of $n \times n$ complex matrices and $G$ is a group under matrix multiplication. Does there always ...
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### A method for solving certain difference equations

I am faced with difference equations that I need to solve (I need the main term in the asymptotic developpment). I present here the order "2" case and I would like to know if my arguments ...
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### Simply connectedness of leaves of a foliation on an complex manifold

Now I'm searching about leaves of foliation in the following special setting. Let $U,V$ be two holomorphic vector field on $\mathbb{C}^2$ s.t the Lie bracket $[U,V]=UV-VU=0$ and $U$ and $V$ spaned ...
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### When is a solution $P(f'(x)) = Q(f(x))$ periodic or double periodic?

$\newcommand{\cl}{\operatorname{cl}}\newcommand{\sl}{\operatorname{sl}}\newcommand{\cm}{\operatorname{cm}}\newcommand{\sm}{\operatorname{cm}}$Consider the differential equation $$P(f '(x)) = Q(f(x))$$ ...
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### 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 ...
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### Motivation and physical interpretation of the Laplace transform

Concerning the one-sided Laplace transform, $$\mathcal{L}\{f\}(s) = \int_0^\infty f(t)e^{-st} dt$$ what is a motivation to come up with that formula? I am particularly interested in "physical&...
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### Bound on $L^1$ norm of solution of two-point boundary value problem

This has to be known, but I have not been able to find it in the literature (probably due to not being too familiar with two-point boundary value problems). I have a function $u:[0,1]\to\mathbb{R}$ ...
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### Fréchet-valued symbols

Denote by $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n \right)$ the usual space of symbols. Now let $E$ be a Fréchet Space. We can then define $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n; E \right)$...
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### Existence of solution to nonlinear first order PDE with C^1 bounds

I'm looking for general existence of a PDE of the form $$f: U \times [0, \delta) \to \mathbb{R}$$ $$\frac{\partial f}{\partial t}(p,t) = F(f(p,t))$$ where $f(p,0)$ is prescribed and $F$ is non-linear ...
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### Tableau and its first prolongation for linear Pfaffian systems

This question concerns characterization of tableau associated with an exterior differential system (EDS). On the one hand, we have prop 4.2 in the EDS book by Bryant et al.: Given an EDS on a manifold ...
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### Same occupation measure $\Rightarrow$ same trajectory

Let $f$ be a $\mathcal{C}^1$ vector field (VF) on a compact subset $M \subset \mathbb{R}^n$. $M$ inherits the Euclidean metric. We define a dynamical system by $$\dot{x}(t)=f(x(t))$$ The occupation ...
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### Question on the modelling of (viscous) fluid in a bag with holes

Consider some fluid (as nice as possible) in a plastic bag with holes illustrated by the image below (of course no holes have been drawn in this picture) What is the corresponding PDE to model the ...
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### Looking for examples of 3rd-order contact transformations

In the Herglotz Lectures on Contact Transformations and Hamiltonian Systems, after going through contact transformations of the form $X=X(x,y,p)$, $Y=Y(x,y,p)$, $P=P(x,y,p)$ it is stated: As a final ...
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### What is the PDE corresponding to this weak formulation?

Consider a flow $(\mu_t)_{t\ge 0}$ such that every $\mu_t$ is a probability on $\mathbb R_+$; $\mu_0(dx) = \rho(x) \, dx$ with $\rho$ being a probability density (as nice as possible) on $\mathbb R_+$...
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### One solution is known, how to find another one

This question is driven by pure curiosity as for all practical purposes I can generate numerical or series solutions of the given ODE \begin{align} p(a,u) [a'(u)]^2+q(a,u)a'(u)+a(u) &= 0,\\ a(0) &...
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### Relative bounds for vorticity

Write the vorticity equation as \label{Eq20} \begin{split} \dfrac{\partial}{\partial t} v_i & = \biggl[|\textbf{v}|~|\nabla u_i|\cos(\beta_i)- |\textbf{u}|~|\nabla v_i|\cos(\...
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### Upper bound estimation for second-order variable-coefficient ODE

I'm tackling a second-order linear ordinary differential equation with variable coefficients $a(t)$ and seek advice on estimating the upper bound of $y(t)$ s.t $|y(t)|\le M$. The equation in question ...
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