Steady spatial asymptotics for the Vlasov–Poisson system

A collisionless plasma is modelled by the Vlasov–Poisson system in three space dimensions. A fixed background of positive charge, which is independent of time and space, is assumed. The situation in which mobile negative ions balance the positive charge as ∣x∣ tends to infinity is considered. Hence, the total positive charge and the total negative charge are infinite. Smooth solutions with appropriate asymptotic behaviour were shown to exist locally in time in a previous work. This paper studies the time behaviour of the net charge and a natural quantity related to energy, and shows that neither is constant in time in general. Also, neither quantity is positive definite. When the background density is a decreasing function of ∣v∣, a positive definite quantity is constructed which remains bounded. A priori bounds are obtained from this. Copyright © 2003 John Wiley & Sons, Ltd.     

The Vlasov Poisson System with Steady Spatial Asymptotics

Abstract

A collisionless plasma is modeled by the Vlasov-Poisson system in three space dimensions. A fixed background of positive charge, which is independent of time and space, is assumed. The situation in which mobile negative ions balance the positive charge as |x|→∞ is considered. Hence the total positive charge and the total negative charge are infinite. Smooth solutions with appropriate asymptotic behavior are shown to exist locally in time. Conditions for continuation of these solutions are also established.     

Global Existence for the Vlasov-Poisson System with Steady Spatial Asymptotics

ABSTRACT

A collisionless plasma is modelled by the Vlasov-Poisson system in three space dimensions. A fixed background of positive charge—dependant upon only velocity—is assumed. The situation in which mobile negative ions balance the positive charge as | x | → ∞ is considered. Thus, the total positive charge and the total negative charge are both infinite. Smooth solutions with appropriate asymptotic behavior for large | x |, which were previously shown to exist locally in time, are continued globally. This is done by showing that the charge density decays at least as fast as | x |^?6. This article also establishes decay estimates for the electrostatic field and its derivatives.     

Explicit solutions of the one‐dimensional Vlasov–Poisson system with infinite mass

A collisionless plasma is modelled by the Vlasov–Poisson system in one dimension. A fixed background of positive charge, dependent only upon velocity, is assumed and the situation in which the mobile negative ions balance the positive charge as |x| → ∞ is considered. Thus, the total positive charge and the total negative charge are infinite. In this paper, the charge density of the system is shown to be compactly supported. More importantly, both the electric field and the number density are determined explicitly for large values of |x|. Copyright © 2007 John Wiley & Sons, Ltd.     

Blowup in solutions of a quasilinear wave equation with variable‐exponent nonlinearities

We consider, in this paper, the following nonlinear equation with variable exponents: where a,b>0 are constants and the exponents of nonlinearity m,p, and r are given functions. We prove a finite‐time blow‐up result for the solutions with negative initial energy and for certain solutions with positive energy.     

Periodic and solitary-wave solutions of the Degasperis–Procesi equation

Travelling-wave solutions of the Degasperis–Procesi equation are investigated. The solutions are characterized by two parameters. For propagation in the positive x-direction, hump-like, inverted loop-like and coshoidal periodic-wave solutions are found; hump-like, inverted loop-like and peakon solitary-wave solutions are obtained as well. For propagation in the negative x-direction, there are solutions which are just the mirror image in the x-axis of the aforementioned solutions. A transformed version of the Degasperis–Procesi equation, which is a generalization of the Vakhnenko equation, is also considered. For propagation in the positive x-direction, hump-like, loop-like, inverted loop-like, bell-like and coshoidal periodic-wave solutions are found; loop-like, inverted loop-like and kink-like solitary-wave solutions are obtained as well. For propagation in the negative x-direction, well-like and inverted coshoidal periodic-wave solutions are found; well-like and inverted peakon solitary-wave solutions are obtained as well. In an appropriate limit, the previously known solutions of the Vakhnenko equation are recovered.     

Exact solutions with compact and noncompact structures for the one-dimensional generalized Benjamin–Bona–Mahony equation

This paper is devoted to analyzing the physical structures of nonlinear dispersive variants of the Benjamin–Bona–Mahony equation. It is found that these generalized forms give rise to compactons solutions: solitons with the absence of infinite tails, solitons: nonlinear localized waves of infinite support, solitary patterns solutions having infinite slopes or cusps, and plane periodic solutions. It is also found that the qualitative change in the physical structure of solutions depends strongly on whether the exponents of the wave function u(x, t) whether it is positive or negative, and on the speed c of the traveling wave as well.     

On the perron sign change effect for Lyapunov characteristic exponents of solutions of differential systems

The Perron effect is the effect in which the characteristic Lyapunov exponents of solutions of a differential system change sign from negative to positive when passing to a perturbed system. We show that this effect is realized on all nontrivial solutions of two two-dimensional systems: an original linear system with negative characteristic exponents and a perturbed system with small perturbations of arbitrary order m > 1 in a neighborhood of the origin, all of whose nontrivial solutions have positive characteristic exponents. We compute the exact positive value of the characteristic exponents of solutions of the two-dimensional nonlinear Perron system with small second-order perturbations, which realizes only a partial Perron effect.     

On finite time BV blow-up for the p-system

Abstract

The paper studies the possible blowup of the total variation for entropy weak solutions of the p-system, modeling isentropic gas dynamics. It is assumed that the density remains uniformly positive, while the initial data can have arbitrarily large total variation (measured in terms of Riemann invariants). Two main results are proved. (I) If the total variation blows up in finite time, then the solution must contain an infinite number of large shocks in a neighborhood of some point in the t-x plane. (II) Piecewise smooth approximate solutions can be constructed whose total variation blows up in finite time. For these solutions the strength of waves emerging from each interaction is exact, while rarefaction waves satisfy the natural decay estimates stemming from the assumption of genuine nonlinearity.     

Solutions of the Differential Equation u‴+λ2zu+(α−1)λ2u=0

This paper deals with the solutions of the differential equation u?+λ²zu+(α?1)λ²u=0, in which λ is a complex parameter of large absolute value and α is an arbitrary constant, real or complex. After a discussion of the structure of the solutions of the differential equation, an integral representation of the solution is given, from which the series solutions and their asymptotic representations are derived. A third independent solution is needed for the special case when α?1 is a positive integer, and two derivations for this are given. Finally, a comparison is made with the results obtained by R. E. Langer.     

Multiple Nontrivial Solutions for a Class of p-Laplacian Equations

In this paper a class of p-Laplacian equations are considered. The existence of five nontrivial solutions is stated. By the standard supersolution and subsolution method, we prove the existence of minimal positive solution and maximal negative solution in L ^∞(Ω). Furthermore, in virtue of the well-known Mountain-Pass Theorem and Second Deformation Theorem, we can give additional three nontrivial solutions, i.e., one positive solution, one negative solution and one sign-changing solution.     

The <Emphasis Type="Italic">n</Emphasis>-Body Problem in Spaces of Constant Curvature. Part II: Singularities

We analyze the singularities of the equations of motion and several types of singular solutions of the n-body problem in spaces of positive constant curvature. Apart from collisions, the equations encounter noncollision singularities, which occur when two or more bodies are antipodal. This conclusion leads, on the one hand, to hybrid solution singularities for as few as three bodies, whose orbits end up in a collision-antipodal configuration in finite time; on the other hand, it produces nonsingularity collisions, characterized by finite velocities and forces at the collision instant.     

Explicit solutions of the Camassa–Holm equation

Explicit travelling-wave solutions of the Camassa–Holm equation are sought. The solutions are characterized by two parameters. For propagation in the positive x-direction, both periodic and solitary smooth-hump, peakon, cuspon and inverted-cuspon waves are found. For propagation in the negative x-direction, there are solutions which are just the mirror image in the x-axis of the aforementioned solutions. Some composite wave solutions of the Degasperis–Procesi equation are given in an appendix.     

Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation

We consider finite time blow up solutions to the critical nonlinear Schrödinger equation with initial condition u₀ H¹. Existence of such solutions is known, but the complete blow up dynamic is not understood so far. For initial data with negative energy, finite time blow up with a universal sharp upper bound on the blow up rate corresponding to the so-called log-log law has been proved in [10], [11]. We focus in this paper onto the positive energy case where at least two blow up speeds are known to possibly occur. We establish the stability in energy space H¹ of the log-log upper bound exhibited in the negative energy case, and a sharp lower bound on blow up rate in the other regime which corresponds to known explicit blow up solutions.     

A Super Camassa–Holm Equation with N‐Peakon Solutions

A super Camassa–Holm equation with peakon solutions is proposed, which is associated with a 3 × 3 matrix spectral problem with two potentials. With the aid of the zero‐curvature equation, we derive a hierarchy of super Harry Dym type equations and establish their Hamiltonian structures. It is shown that the super Camassa–Holm equation is exactly a negative flow in the hierarchy and admits exact solutions with N peakons. As an example, exact 1‐peakon solutions of the super Camassa–Holm equation are given. Infinitely many conserved quantities of the super Camassa–Holm equation and the super Harry Dym type equation are, respectively, obtained.     

An approximate Gidas–Ni–Nirenberg theorem

We study positive solutions u to Δu + f(u) = 0 in Ω, u = 0 on ?Ω, and we address the following question: If Ω is a small perturbation of a ball, is u a small perturbation of a radially symmetric function? We prove two theorems which give an affirmative answer under different assumptions on the non-linearity f and on the topologies in which perturbations are considered.     

Multiple and sign‐changing solutions for the multivalued p ‐Laplacian equation

We consider a class of elliptic inclusions under Dirichlet boundary conditions involving multifunctions of Clarke's generalized gradient. Under conditions given in terms of the first eigenvalue as well as the Fu?ik spectrum of the p ‐Laplacian we prove the existence of a positive, a negative and a sign‐changing solution. Our approach is based on variational methods for nonsmooth functionals (nonsmooth critical point theory, second deformation lemma), and comparison principles for multivalued elliptic problems. In particular, the existence of extremal constant‐sign solutions plays a key role in the proof of sign‐changing solutions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The interface between regions where u<0 andu > 0in the porous medium equation

We consider the porous medium equation with sign changes. In particular this equation describes the mixing of fresh and salt groundwater due to mechanical dispersion. The unknown function u, which denotes the velocity of the fluids, may take positive as well as negative values. Our main result is the following : under certain monotonicity hypotheses on the initial function, there exists a time T> 0 after which the regions where u < 0 and u > 0 are separated by an interface x = ζ(t) such that ζ is continuously differentiable on [T,∞]. The method of proof is based on a priori estimates for solutions of regularized problems and for their level lines     

Necessary and sufficient conditions on the asymptotic behavior of second‐order neutral delay dynamic equations with positive and negative coefficients

In this paper, we establish necessary and sufficient conditions for the solutions of a second‐order nonlinear neutral delay dynamic equation with positive and negative coefficients to be oscillatory or tend to zero asymptotically. We consider three different ranges of the coefficient associated with the neutral part in one of which it is allowed to be oscillatory. Thus, our results improve and generalize the existing results in the literature to arbitrary time scales. Some examples on nontrivial time scales are also given. Copyright © 2013 John Wiley & Sons, Ltd.