Recovering the relativistic kinetic energy
We recover the relativistic kinetic energy as the result of the work of a force.
physics.gen-ph↗
arXiv subjects
Alain Haraux
Publications and source records attributed to Alain Haraux.
A simple toy model for the collapse of the local group with the Shapley attractor
A toy model is proposed for the Cosmic Dipole consisting in the Shapley attractor and the so-called Dipole repeller, whose action is assimilated to an anti-gravitational force. According to this model, the local group will collapse in finite time with the Shapley attractor and, by using the available figures for distances and masses, it is shown that it will happen in less than 100 billion years. To obtain a more precise estimate, more knowledge will be necessary on the equivalent negative mass of the repeller.
math.DS↗
A Newtonian approach to general black holes
We show how it is possible to define and study black holes of arbitrary shapes in the framework of Newtonian mechanics.
math.CA↗
Energy decay estimates for the wave equation with supercritical nonlinear damping
We consider a damped wave equation in a bounded domain. The damping is nonlinear and is homogeneous with degree p -- 1 with p > 2. First, we show that the energy of the strong solution in the supercritical case decays as a negative power of t; the rate of decay is the same as in the subcritical or critical cases, provided that the space dimension does not exceed ten. Next, relying on a new differential inequality, we show that if the initial displacement is further required to lie in L p , then the energy of the corresponding weak solution decays logarithmically in the supercritical case. Those new results complement those in the literature and open an important breach in the unknown land of super-critical damping mechanisms.
math.AP↗
On a linearly damped 2 body problem
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.
math.DS↗
Sharp ultimate velocity bounds for the general solution of some linear second order evolution equation with damping and bounded forcing
We consider a class of linear second order differential equations with damping and external force. We investigate the link between a uniform bound on the forcing term and the corresponding ultimate bound on the velocity of solutions, and we study the dependence of that bound on the damping and on the "elastic force". We prove three results. First of all, in a rather general setting we show that different notions of bound are actually equivalent. Then we compute the optimal constants in the scalar case. Finally, we extend the results of the scalar case to abstract dissipative wave-type equations in Hilbert spaces. In that setting we obtain rather sharp estimates that are quite different from the scalar case, in both finite and infinite dimensional frameworks. The abstract theory applies, in particular, to dissipative wave, plate and beam equations.
math.AP↗
On spatial Gevrey regularity for some strongly dissipative second order evolution equations
Let A be a positive self-adjoint linear operator acting on a real Hilbert space H and $α$, c be positive constants. We show that all solutions of the evolution equation u + Au + cA $α$ u = 0 with u(0) $\in$ D(A 1 2), u (0) $\in$ H belong for all t > 0 to the Gevrey space G(A, $σ$) with $σ$ = min{ 1 $α$ , 1 1--$α$ }. This result is optimal in the sense that $σ$ can not be reduced in general. For the damped wave equation (SDW) $α$ corresponding to the case where A = --$Δ$ with domain D(A) = {w $\in$ H 1 0 ($Ω$), $Δ$w $\in$ L 2 ($Ω$)} with $Ω$ any open subset of R N and (u(0), u (0)) $\in$ H 1 0 ($Ω$)xL 2 ($Ω$), the unique solution u of (SDW) $α$ satisfies $\forall$t > 0, u(t) $\in$ G s ($Ω$) with s = min{ 1 2$α$ , 1 2(1--$α$) }, and this result is also optimal. Mathematics Subject Classification 2010 (MSC2010): 35L10, 35B65, 47A60.
math.AP↗
Universal bounds for a class of second order evolution equations and applications
We consider a class of abstract second order evolution equations with a restoring force that is strictly superlinear at infinity with respect to the position, and a dissipation mechanism that is strictly superlinear at infinity with respect to the velocity. Under the assumption that the growth of the restoring force dominates the growth of the dissipation, we prove a universal bound property, namely that the energy of solutions is bounded for positive times, independently of the initial condition. Under a slightly stronger assumption, we show also a universal decay property, namely that the energy decays (as time goes to infinity) at least as a multiple of a negative power of $t$, again independent of the boundary conditions. We apply the abstract results to solutions of some nonlinear wave, plate and Kirchhoff equations in a bounded domain.
math.AP↗
A sharp stability criterion for single well Duffing and Duffing-like equations
We refine some previous sufficient conditions for exponential stability of the linear ODE $$ u''+ cu' + (b+a(t))u = 0$$ where $b, c>0$ and $a$ is a bounded nonnegative time dependent coefficient. This allows to improve some results on uniqueness and asymptotic stability of periodic or almost periodic solutions of the equation$$ u''+ cu' + g(u)=f(t) $$where $c>0$, $f \in L^\infty (R)$ and $g\in C^1(R)$ satisfies some sign hypotheses. The typical case is $ g(u) = bu + a\vert u\vert^p u $ with $a\ge 0 , b>0.$ Similar properties are valid for evolution equations of the form $$ u''+ cu' + (B+A(t))u = 0$$ where $A(t) $ and $B$ are self-adjoint operators on a real Hilbert space $H$ with $B$ coercive and $A(t)$ bounded in $L(H)$ with a sufficiently small bound of its norm in $L^{\infty}(R+, L(H))$ .
math.DS↗
The universal bound property for a class of second order ODEs
We consider the scalar second order ODE u + |u | $α$ u + |u| $β$ u = 0, where $α$, $β$ are two positive numbers and the non-linear semi-group S(t) generated on IR 2 by the system in (u, u). We prove that S(t)IR 2 is bounded for all t > 0 whenever 0 < $α$ < $β$ and moreover there is a constant C independent of the initial data such that $\forall$t > 0, u (t) 2 + |u(t)| $β$+2 $\le$ C max{t -- 2 $α$ , t -- ($α$+1)($β$+2) $β$--$α$ }.
math.DS↗
An infinite dimensional Duffing-like evolution equation with linear dissipation and an asymptotically small source term
We consider an abstract nonlinear second order evolution equation, inspired by some models for damped oscillations of a beam subject to external loads or magnetic fields, and shaken by a transversal force. When there is no external force, the system has three stationary positions, two stable and one unstable, and all solutions are asymptotic for $t$ large to one of these stationary solutions.We show that this pattern extends to the case where the external force is bounded and small enough, in the sense that solutions can exhibit only three different asymptotic behaviors.
math.AP↗
On the ultimate energy bound of solutions to some forced second order evolution equations with a general nonlinear damping operator
Under suitable growth and coercivity conditions on the nonlinear damping operator $g$ which ensure non-resonance, we estimate the ultimate bound of the energy of the general solution to the equation $\ddot{u}(t) + Au(t) + g(\dot{u}(t))=h(t),\quad t\in\mathbb{R}^+ ,$ where $A$ is a positive selfadjoint operator on a Hilbert space $H$ and $h$ is a bounded forcing term with values in $H$. In general the bound is of the form $ C(1+ ||h||^4)$ where $||h||$ stands for the $L^\infty$ norm of $h$ with values in $H$ and the growth of $g$ does not seem to play any role. If $g$ behaves lie a power for large values of the velocity, the ultimate bound has a quadratic growth with respect to $||h||$ and this result is optimal. If $h$ is anti periodic, we obtain a much lower growth bound and again the result is shown to be optimal even for scalar ODEs.
math.AP↗
The $ε$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors
We prove an estimation of the Kolmogorov $ε$-entropy in H of the unitary ball in the space V, where H is a Hilbert space and V is a Sobolev-like subspace of H. Then, by means of Zelik's result [5], an estimate of the fractal dimension of the attractors of some nonlinear parabolic equations is established.
math.AP↗
Quantization of energy and weakly turbulent profiles of the solutions to some damped second order evolution equations
We consider a second order equation with a linear "elastic" part and a nonlinear damping term depending on a power of the norm of the velocity. We investigate the asymptotic behavior of solutions, after rescaling them suitably in order to take into account the decay rate and bound their energy away from zero.We find a rather unexpected dichotomy phenomenon. Solutions with finitely many Fouriercomponents are asymptotic to solutions of the linearized equationwithout damping, and exhibit some sort of equipartition of theenergy among the components. Solutions with infinitely manyFourier components tend to zero weakly but not strongly. We showalso that the limit of the energy of solutions depends only on thenumber of their Fourier components.The proof of our results is inspired by the analysis of asimplified model which we devise through an averaging procedure,and whose solutions exhibit the same asymptotic properties as thesolutions to the original equation.
math.AP↗
Existence, uniqueness and global behavior of the solutions to some nonlinear vector equations in a finite dimensional Hilbert space
The initial value problem and global properties of solutions are studied for the vector equation: $\Big(\|u'\|^{l}u'\Big)'+\|A^{\frac{1}{2}}u\|^βAu+g(u')=0$ in a finite dimensional Hilbert space under suitable assumptions on $g$.
math.CA↗
A simple characterization of positivity preserving semi-linear parabolic systems
We give a simple and direct proof of the characterization of positivity preserving semi-flows for ordinary differential systems. The same method provides an abstract result on a class of evolution systems containing reaction-diffusion systems in a bounded domain of $ \mathbb{R}^n$ with either Neumann or Dirichlet homogeneous boundary conditions. The conditions are exactly the same with or without diffusion. A similar approach gives the optimal result for invariant rectangles in the case of Neumann conditions.
math.AP↗
A concrete realization of the slow-fast alternative for a semi linear heat equation with homogeneous Neumann boundary conditions
We investigate the asymptotic behavior of solutions to a semilinear heat equation with homogeneous Neumann boundary conditions. It was recently shown that the nontrivial kernel of the linear part leads to the coexistence of fast solutions decaying to 0 exponentially (as time goes to infinity), and slow solutions decaying to 0 as negative powers of t. Here we provide a characterization of slow/fast solutions in terms of their sign, and we show that the set of initial data giving rise to fast solutions is a graph of codimension one in the phase space.
math.AP↗
A Liapunov function approach to the stabilization of second order coupled systems
In 2002, Fatiha Alabau, Piermarco Cannarsa and Vilmos Komornik investigated the extent of asymptotic stability of the null solution for weakly coupled partially damped equations of the second order in time. The main point is that the damping operator acts only on the first component and, whenever it is bounded, the coupling is not strong enough to produce an exponential decay in the energy space associated to the conservative part of the system. As a consequence, for initial data in the energy space, the rate of decay is not exponential. Due to the nature of the result it seems at first sight impossible to obtain the asymptotic stability result by the classical Liapunov method. Surprisingly enough, this turns out to be possible and we exhibit, under some compatibility conditions on the operators, an explicit class of Liapunov functions which allows to do 3 different things: 1) When the problem is reduced to a stable finite dimensional space, we recover the exponential decay by a single differential inequality and we estimate the logarithmic decrement of the solutions with worst (slowest) decay. The estimate is optimal at least for some values of the parameters.
math.AP↗