scholarly journals On Some Damped 2 Body Problems

2020 ◽  
Author(s):  
Alain Haraux
Keyword(s):  
Singular Potential ◽  
Blow Up ◽  
Central Charge ◽  
The Sun ◽  
Bounded Solutions ◽  
High Scale ◽  
Electrical Devices ◽  
Moving Particle ◽  
New Phenomena ◽  
Dissipative Model

The usual equation for both motions of a single planet around the sun and electrons in the deterministic Rutherford-Bohr atomic model is conservative with a singular potential at the origin. When a dissipation is added, new phenomena appear. It is shown that whenever the momentum is not zero, the moving particle does not reach the center in finite time and its displacement does not blow-up either, even in the classical context where arbitrarily large velocities are allowed. Moreover we prove that all bounded solutions tend to $0$ for $t$ large, and some formal calculations suggest the existence of special orbits with an asymptotically spiraling exponentially fast convergence to the center. A related model with exponentially damped central charge or mass gives some explicit exponentially decaying solutions which might help future investigations. An atomic contraction hypothesis related to the asymptotic dying off of solutions proven for the dissipative model might give a solution to some intriguing phenomena observed in paleontology, familiar electrical devices and high scale cosmology.

scholarly journals On Some Damped 2 Body Problems

2020 ◽  
Author(s):  
Alain Haraux
Keyword(s):  
Singular Potential ◽  
Blow Up ◽  
Central Charge ◽  
The Sun ◽  
Bounded Solutions ◽  
High Scale ◽  
Electrical Devices ◽  
Moving Particle ◽  
New Phenomena ◽  
Dissipative Model

The usual equation for both motions of a single planet around the sun and electrons in the deterministic Rutherford-Bohr atomic model is conservative with a singular potential at the origin. When a dissipation is added, new phenomena appear. It is shown that whenever the momentum is not zero, the moving particle does not reach the center in finite time and its displacement does not blow-up either, even in the classical context where arbitrarily large velocities are allowed. Moreover we prove that all bounded solutions tend to $0$ for $t$ large, and some formal calculations suggest the existence of special orbits with an asymptotically spiraling exponentially fast convergence to the center. A related model with exponentially damped central charge or mass gives some explicit exponentially decaying solutions which might help future investigations. An atomic contraction hypothesis related to the asymptotic dying off of solutions proven for the dissipative model might give a solution to some intriguing phenomena observed in paleontology, familiar electrical devices and high scale cosmology.

scholarly journals On Some Damped 2 Body Problems

2021 ◽  
Author(s):  
Alain Haraux
Keyword(s):  
Singular Potential ◽  
Central Charge ◽  
The Sun ◽  
Initial State ◽  
High Scale ◽  
All Solutions ◽  
Electrical Devices ◽  
New Phenomena ◽  
Formal Calculation ◽  
Dissipative Model

The usual equation for both motions of a single planet around the sun and electrons in the deterministic Rutherford-Bohr atomic model is conservative with a singular potential at the origin. When a dissipation is added, new phenomena appear which were investigated thoroughly by R. Ortega and his co-authors between 2014 and 2017, in particular all solutions are bounded and tend to $0$ for $t$ large, some of them with asymptotically spiraling exponentially fast convergence to the center. We provide explicit estimates for the bounds in the general case that we refine under specific restrictions on the initial state, and we give a formal calculation which could be used to determine practically some special asymptotically spiraling orbits. Besides, a related model with exponentially damped central charge or mass gives some explicit exponentially decaying solutions which might help future investigations. An atomic contraction hypothesis related to the asymptotic dying off of solutions proven for the dissipative model might give a solution to some intriguing phenomena observed in paleontology, familiar electrical devices and high scale cosmology

scholarly journals On a Linearly Damped 2 Body Problem

2020 ◽  
Author(s):  
Alain Haraux
Keyword(s):  
Finite Time ◽  
Body Problem ◽  
Singular Potential ◽  
Blow Up ◽  
Fast Convergence ◽  
Atomic Model ◽  
The Sun ◽  
Bounded Solutions ◽  
Moving Particle ◽  
New Phenomena

The usual equation for both motions of a single planet around the sun and electrons in the deterministic Rutherford-Bohr atomic model is conservative with a singular potential at the origin. When a dissipation is added, new phenomena appear. It is shown that whenever the momentum is not zero, the moving particle does not reach the center in finite time and its displacement does not blow-up either, even in the classical context where arbitrarily large velocities are allowed. Moreover we prove that all bounded solutions tend to $0$ for $t$ large, and some formal calculations suggest the existence of special orbits with an asymptotically spiraling exponentially fast convergence to the center.

scholarly journals On a Linearly Damped 2 Body Problem

2020 ◽  
Author(s):  
Alain Haraux
Keyword(s):  
Body Problem ◽  
Singular Potential ◽  
Blow Up ◽  
Atomic Model ◽  
The Sun ◽  
Moving Particle ◽  
New Phenomena

The usual equation for both motions of a single planet around the sun and electrons in the deterministic Rutherford-Bohr atomic model is conservative with a singular potential at the origin. When a dissipation is added, new phenomena appear. It is shown that whenever the momentum is not zero, the moving particle does not reach the center in nite time and its displacement does not blow-up either, even in the classical context where arbitrarily large velocities are allowed. Moreover some formal calculations suggest the existence of special orbits with an asymptotically spiraling convergence to the center.

Coercive elliptic systems with gradient terms

2017 ◽  
Vol 6 (2) ◽  
pp. 165-182 ◽  
Author(s):  
Roberta Filippucci ◽  
Federico Vinti
Keyword(s):  
Blow Up ◽  
Elliptic Systems ◽  
Continuous Functions ◽  
Necessary Condition ◽  
Bounded Solutions ◽  
Radial Solutions ◽  
Existence Of A Solution ◽  
Content Type ◽  
Positive Radial Solutions

AbstractIn this paper we give a classification of positive radial solutions of the following system:$\Delta u=v^{m},\quad\Delta v=h(|x|)g(u)f(|\nabla u|),$in the open ball ${B_{R}}$, with ${m>0}$, and f, g, h nonnegative nondecreasing continuous functions. In particular, we deal with both explosive and bounded solutions. Our results involve, as in [27], a generalization of the well-known Keller–Osserman condition, namely, ${\int_{1}^{\infty}(\int_{0}^{s}F(t)\,dt)^{-m/(2m+1)}\,ds<\infty}$, where ${F(t)=\int_{0}^{t}f(s)\,ds}$. Moreover, in the second part of the paper, the p-Laplacian version, given by ${\Delta_{p}u=v^{m}}$, ${\Delta_{p}v=f(|\nabla u|)}$, is treated. When ${p\geq 2}$, we prove a necessary condition for the existence of a solution with at least a blow up component at the boundary, precisely ${\int_{1}^{\infty}(\int_{0}^{s}F(t)\,dt)^{-m/(mp+p-1)}s^{(p-2)(p-1)/(mp+p-1)}% \,ds<\infty}$.

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