On the Cylindrically Symmetric Einstein–Vlasov System

In this article we begin to study the question of global existence for the cylindrically symmetric Einstein–Vlasov system with general (in size) data and we show that if a singularity occurs at all, the first one occurs at the axis of symmetry. This is done by a combination of light cone estimates and a careful analysis of the matter terms in the “exterior” region, together with Sobolev methods for t he analysis in the “interior” region.     

Global Solutions for the Two-Component Camassa–Holm System

We prove existence of a global conservative solution of the Cauchy problem for the two-component Camassa–Holm (2CH) system on the line, allowing for nonvanishing and distinct asymptotics at plus and minus infinity. The solution is proven to be smooth as long as the density is bounded away from zero. Furthermore, we show that by taking the limit of vanishing density in the 2CH system, we obtain the global conservative solution of the (scalar) Camassa–Holm equation, which provides a novel way to define and obtain these solutions. Finally, it is shown that while solutions of the 2CH system have infinite speed of propagation, singularities travel with finite speed.     

The Convergence of the Navier–Stokes–Poisson System to the Incompressible Euler Equations

ABSTRACT

The combining quasineutral and inviscid limit of the Navier–Stokes–Poisson system in the torus 𝕋 ^d , d ≥ 1 is studied. The convergence of the Navier–Stokes–Poisson system to the incompressible Euler equations is proven for the global weak solution and for the case of general initial data.     

Quasiunconditional Basis Property of the Faber–Schauder System

Ukrainian Mathematical Journal - We prove that, for any 0 < ?? < 1, there exists a measurable set E?? ? [0, 1], mes (E??) > 1...     

Time-Periodic Solution to the Compressible Navier–Stokes/Allen–Cahn System

In this paper,we investigate the time-periodic solution to a coupled compressible Navier–Stokes/Allen–Cahn system which describes the motion of a mixture of two viscous compressible fluids with a time periodic external force in a periodic domain in R^N.The existence of the time-periodic solution to the system is established by using an approach of parabolic regularization and combining with the topology degree theory,and then the uniqueness of the period solution is obtained under some smallness and symmetry assumptions on the external force.     

Regularity Results for the Spherically Symmetric Einstein–Vlasov System

The spherically symmetric Einstein–Vlasov system is considered in Schwarzschild coordinates and in maximal-isotropic coordinates. An open problem is the issue of global existence for initial data without size restrictions. The main purpose of the present work is to propose a method of approach for general initial data, which improves the regularity of the terms that need to be estimated compared to previous methods. We prove that global existence holds outside the center in both these coordinate systems. In the Schwarzschild case we improve the bound on the momentum support obtained in Rein et al. (Commun Math Phys 168:467–478, 1995) for compact initial data. The improvement implies that we can admit non-compact data with both ingoing and outgoing matter. This extends one of the results in Andréasson and Rein (Math Proc Camb Phil Soc 149:173–188, 2010). In particular our method avoids the difficult task of treating the pointwise matter terms. Furthermore, we show that singularities never form in Schwarzschild time for ingoing matter as long as 3m ≤ r. This removes an additional assumption made in Andréasson (Indiana Univ Math J 56:523–552, 2007). Our result in maximal-isotropic coordinates is analogous to the result in Rendall (Banach Center Publ 41:35–68, 1997), but our method is different and it improves the regularity of the terms that need to be estimated for proving global existence in general.     

Post-Newtonian Approximation of the Vlasov–Nordström System

ABSTRACT

We study the Nordström–Vlasov system, which describes the dynamics of a self-gravitating ensemble of collisionless particles in the framework of the Nordström scalar theory of gravitation. If the speed of light c is considered as a parameter, it is known that in the Newtonian limit c → ∞ the Vlasov–Poisson system is obtained. In this paper we determine a higher approximation and establish a pointwise error estimate of order 𝒪(c ^?4). Such an approximation is usually called a 1.5 post-Newtonian approximation.     

Finite-Energy Global Well-Posedness of the Maxwell–Klein–Gordon System in Lorenz Gauge

It is known that the Maxwell–Klein–Gordon system (M–K–G), when written relative to the Coulomb gauge, is globally well-posed for finite-energy initial data. This result, due to Klainerman and Machedon, relies crucially on the null structure of the main bilinear terms of M–K–G in Coulomb gauge. It appears to have been believed that such a structure is not present in Lorenz gauge, but we prove here that it is, and we use this fact to prove finite-energy global well-posedness in Lorenz gauge. The latter has the advantage, compared to Coulomb gauge, of being Lorentz invariant, hence M–K–G in Lorenz gauge is a system of nonlinear wave equations, whereas in Coulomb gauge the system has a less symmetric form, as it contains also an elliptic equation.     

Some Global Results for the Degn–Harrison System with Diffusion

This paper considers the Degn–Harrison reaction–diffusion system subject to homogeneous Neumann boundary conditions in a smooth and bounded domain. Using the presence of contracting rectangles and the method of Lyapunov, we establish sufficient conditions for the global asymptotic stability of the unique constant steady state.     

Moment Propagation for Weak Solutions to the Vlasov–Poisson System

We consider weak solutions to the Cauchy problem for the three dimensional Vlasov–Poisson system of equations. We obtain a propagation result for any velocity moment of order > 2 as well as a uniqueness statement in ?³. In the periodic case, we show that velocity moments of order > 14/3 are propagated.     

Stability for the Gravitational Vlasov–Poisson System in Dimension Two

We consider the two dimensional gravitational Vlasov–Poisson system. Using variational methods, we prove the existence of stationary solutions of minimal energy under a Casimir type constraint. The method also provides a stability criterion of these solutions for the evolution problem.     

Stable Ground States for the Relativistic Gravitational Vlasov–Poisson System

We consider the three dimensional gravitational Vlasov–Poisson (GVP) system in both classical and relativistic cases. The classical problem is subcritical in the natural energy space and the stability of a large class of ground states has been derived by various authors. The relativistic problem is critical and displays finite time blow up solutions. Using standard concentration compactness techniques, we however show that the breaking of the scaling symmetry allows the existence of stable relativistic ground states. A new feature in our analysis which applies both to the classical and relativistic problem is that the orbital stability of the ground states does not rely as usual on an argument of uniqueness of suitable minimizers—which is mostly unknown—but on strong rigidity properties of the transport flow, and this extends the class of minimizers for which orbital stability is now proved.     

Closed-Loop Control of the Motion of a Disk–Rod System

This work deals with the guidance and control of a system which is composed of a rolling disk and a controlled slender rod that is pivoted, through its center of mass, about the disk center. There are given N points P _i, i=1, ..., N, in the horizontal plane, a set of angles _2i , i=1, ..., N, a finite-time interval [0, t _f], and a sequence of times ₁=0<₂<...< _N =t _f. Using the concept of path controllability, a closed-loop control law is derived to steer the system in such a manner that the disk center and the rod angle of rotation ₂ will pass through (P _i, _2i ) at the times _i , i=1,...,N, respectively. This system serves as a model for the motion of a simple mobile robot.     

Positive Ground States for the Fractional Klein–Gordon–Maxwell System with Critical Exponents

Mediterranean Journal of Mathematics - Let G be a locally compact Abelian (LCA) group which possesses a covering family. We define an atomic Hardy space $$H^1(G)$$ and $$\textrm{BMO}(G)$$ of...     

Blow-up Dynamics of L~2 Solutions for the Davey–Stewartson System

We study the blow-up solutions for the Davey–Stewartson system(D–S system, for short)in L2x(R2). First, we give the nonlinear profile decomposition of solutions for the D–S system. Then, we prove the existence of minimal mass blow-up solutions. Finally, by using the characteristic of minimal mass blow-up solutions, we obtain the limiting profile and a precisely mass concentration of L2 blow-up solutions for the D–S system.     

The Cauchy Problem for the Klein–Gordon–Schrödinger System in Low Dimensions Below the Energy Space

ABSTRACT

We show that the Klein–Gordon–Schrödinger system in one, two, and three dimensions has a global solution below the energy space. The proof uses the I-method recently introduced by Colliander et al. (2001 Colliander , J. , Keel , M. , Staffilani , G. , Takaoka , H. , Tao , T. ( 2001 ). Global well-posedness for Schrödinger equations with derivative . SIAM J. Math. Anal. 33 ( 3 ): 649 – 669 . [CROSSREF]  [Google Scholar]) and mixed type Strichartz estimates for the solutions of Schrödinger and Klein–Gordon equations, respectively.     

Hill Four-Body Problem with Oblate Bodies: An Application to the Sun–Jupiter–Hektor–Skamandrios System

We consider a restricted four-body problem, with a precise hierarchy between the bodies: two larger bodies and a smaller one, all three of oblate shape, and a fourth, infinitesimal body, in the neighborhood of the smaller of the three bodies. The three heavy bodies are assumed to move in a plane under their mutual gravity, and the fourth body to move in the three-dimensional space under the gravitational influence of the three heavy bodies, but without affecting them. We first find that the triangular central configuration of the three heavy oblate bodies is a scalene triangle (rather than an equilateral triangle as in the point mass case). Then, assuming that these three bodies are in such a central configuration, we perform a Hill approximation of the equations of motion describing the dynamics of the infinitesimal body in a neighborhood of the smaller body. Through the use of Hill’s variables and a limiting procedure, this approximation amounts to sending the two larger bodies to infinity. Finally, for the Hill approximation, we find the equilibrium points for the motion of the infinitesimal body and determine their stability. As a motivating example, we identify the three heavy bodies with the Sun, Jupiter, and the Jupiter’s Trojan asteroid Hektor, which are assumed to move in a triangular central configuration. Then, we consider the dynamics of Hektor’s moonlet Skamandrios.

     

Global Existence for the Spherically Symmetric Einstein–Vlasov System with Outgoing Matter

We prove a new global existence result for the asymptotically flat, spherically symmetric Einstein–Vlasov system which describes in the framework of general relativity an ensemble of particles which interact by gravity. The data are such that initially all the particles are moving radially outward and that this property can be bootstrapped. The resulting non-vacuum spacetime is future geodesically complete.