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Results tagged with differential-equations
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user 5371
Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.
2
votes
Accepted
helmholtz zero in R^3
If $u$ is a solution to the equation $\triangle u +k^2 u=0$ in a 3D domain $\Omega$, then
for any $x\in\Omega$ and any $r>0$ such that $\{y\in\mathbb R^3:\ |x-y|\leq r \}\subset\Omega$, we have
$$u(x …
- 22.3k
22
votes
Accepted
Existence/Uniqueness of solutions to quasi-Lipschitz ODEs
Yes. This follows from the classical uniqueness theorem due to Osgood (the original paper appeared in 1898).
Osgood's Criterion. Let $\omega(t,u)=\phi(t)\psi(u)$ where $\phi(t)\geq 0$ is continuous
o …
- 22.3k
24
votes
Why are differential forms called closed and exact?
According to Hans Samelson's historical note "Differential Forms, the Early Days", both notions were introduced in Les Méthodes nouvelles de la Mécanique Céleste by Poincaré (vol. 3, Gauthier-Villar …
- 22.3k
29
votes
Accepted
On the non-rigorous calculations of the trajectories in the moon landings
The existence of the Arenstorf Orbits was discovered in 1963 on the basis of numerical computations. The Arenstorf orbits appear as periodic solutions to the equations for the plane restricted three b …
- 22.3k
2
votes
Newton equations, second order equation and (im)possible motions
Newton's second law implies basically that the evolution of a mechanical system is completely determined as soon as the particles' initial positions $x(0)$ and velocities $\dot x(0)$ are specified. S …
- 22.3k
4
votes
Accepted
A formula for the Jacobian of a flow
This is the Liouville formula. It is explained nicely in Ordinary Differential Equations by Arnold.
- 22.3k
13
votes
Where was/is Compensated Compactness used?
Compensated compactness helps when one needs to find the limit of $u_n \cdot v_n$, where the sequences of vector fields $u_n$ and $v_n$ converge weakly in $L^2$: $u_n\rightharpoonup u$, $v_n\righth …
- 22.3k