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Jack Schaeffer - One of the best experts on this subject based on the ideXlab platform.
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The Nonrelativistic Limit of Relativistic Vlasov-Maxwell System
Mathematical Methods in the Applied Sciences, 2016Co-Authors: Jack SchaefferAbstract:We consider the one and one-half dimensional multi-species relativistic Vlasov-Maxwell System with non-decaying(in space) initial data. We prove its well-posedness and nonrelativistic limit as the speed of light $c\rightarrow\infty$. These results mainly rely on a delicate analysis of energy structure and application of estimates along the characteristic lines.
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Large time behavior of the relativistic Vlasov Maxwell System in low space dimension
Differential and Integral Equations, 2010Co-Authors: Robert Glassey, Stephen Pankavich, Jack SchaefferAbstract:When particle speeds are large the motion of a collisionless plasma is modeled by the relativistic Vlasov Maxwell System. Large time behavior of solutions which depend on one position variable and two momentum variables is considered. In the case of a single species of charge it is shown that there are solutions for which the charge density $(\rho = \int f dv)$ does not decay in time. This is in marked contrast to results for the non-relativistic Vlasov Poisson System in one space dimension. The case when two oppositely charged species are present and the net total charge is zero is also considered. In this case, it is shown that the support in the first component of momentum can grow at most as $t^{\frac{3}{4}}$.
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Large Time Behavior of the Relativistic Vlasov Maxwell System in Low Space Dimension
arXiv: Analysis of PDEs, 2009Co-Authors: Robert Glassey, Stephen Pankavich, Jack SchaefferAbstract:When particle speeds are large the motion of a collisionless plasma is modeled by the relativistic Vlasov Maxwell System. Large time behavior of solutions which depend on one position variable and two momentum variables is considered. In the case of a single species of charge it is shown that there are solutions for which the charge density does not decay in time. This is in marked contrast to results for the non-relativistic Vlasov Poisson System in one space dimension. The case when two oppositely charged species are present and the net total charge is zero is also considered. In this case, it is shown that the support in the first component of momentum can grow at most like t to the three-fourths power.
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Steady states of the Vlasov-Maxwell System
Quarterly of Applied Mathematics, 2005Co-Authors: Jack SchaefferAbstract:The Vlasov-Maxwell System models collisionless plasma. Solutions are considered that depend on one spatial variable, x, and two velocity variables, v 1 and v 2 . As x → -∞ it is required that the phase space densities of particles approach a prescribed function, F (v 1 , v 2 ), and all field components approach zero. It is assumed that F (v 1 , v 2 ) = 0 if v 1 < W 1 , where W 1 is a positive constant. An external magnetic field is prescribed and taken small enough so that no particle is reflected (v 1 remains positive). The main issue is to identify the large-time behavior; is a steady state approached and, if so, can it be identified from the time independent Vlasov-Maxwell System? The time-dependent problem is solved numerically using a particle method, and it is observed that a steady state is approached (on a bounded x interval) for large time. For this steady state, one component of the electric field is zero at all points, the other oscillates without decay for x large; in contrast the magnetic field tends to zero for large x. Then it is proven analytically that if the external magnetic field is sufficiently small, then (a reformulation of) the steady problem has a unique solution with B → 0 as x → +∞. Thus the downstream condition, B → 0 as x → +∞, is used to identify the large time limit of the System.
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The Relativistic Vlasov‐Maxwell System in Two Space Dimensions: Part II
Archive for Rational Mechanics and Analysis, 1998Co-Authors: Robert Glassey, Jack SchaefferAbstract:The motion of a collisionless plasma is modeled by the Vlasov‐Maxwell System. For the relativistic Vlasov‐Maxwell System in the plane with smooth initial data, bounds on particle speeds are derived. In an earlier work [10] it was shown that solutions remain smooth as long as particle speeds do not approach the speed of light. When these results are combined, it follows that solutions remain smooth for all time.
Gerhard Rein - One of the best experts on this subject based on the ideXlab platform.
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a numerical investigation of the steady states of the spherically symmetric einstein Vlasov Maxwell System
Classical and Quantum Gravity, 2009Co-Authors: Mikael Eklund, Hakan Andreasson, Gerhard ReinAbstract:We construct, by numerical means, static solutions of the spherically symmetric Einstein-Vlasov-Maxwell System and investigate various features of the solutions. This extends a previous investigation (Andreasson and Rein 2007 Class. Quantum Grav. 24 1809) of the chargeless case. We study the possible shapes of the energy density profile as a function of the area radius when the electric charge of an individual particle is varied as a parameter. We find profiles which are multi-peaked, where the peaks are separated either by vacuum or a thin atmosphere, and we find that for a sufficiently large charge parameter the solutions break down at a finite radius. Furthermore, we investigate the inequality root M <= root R/3 + root R/9 + Q(2)/3R, which is derived in Andreasson (2009 Commun. Math. Phys. 288 715) for general matter models, and we find that it is sharp for the Einstein-Vlasov-Maxwell System. Here M is the ADM mass, Q is the charge and R is the area radius of the boundary of the static object. We find two classes of solutions with this property, while there is only one in the chargeless case. In particular we find numerical evidence for the existence of arbitrarily thin shell solutions to the Einstein-Vlasov-Maxwell System. Finally, we consider one-parameter families of steady states, and we find spirals in the mass-radius diagram for all examples of the microscopic equation of state which we consider.
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a numerical investigation of the steady states of the spherically symmetric einstein Vlasov Maxwell System
arXiv: General Relativity and Quantum Cosmology, 2009Co-Authors: Mikael Eklund, Hakan Andreasson, Gerhard ReinAbstract:We construct, by numerical means, static solutions of the spherically symmetric Einstein-Vlasov-Maxwell System and investigate various features of the solutions. This extends a previous investigation \cite{AR1} of the chargeless case. We study the possible shapes of the energy density profile as a function of the area radius when the electric charge of an individual particle is varied as a parameter. We find profiles which are multi-peaked, where the peaks are separated either by vacuum or a thin atmosphere, and we find that for a sufficiently large charge parameter there are no physically meaningful solutions. Furthermore, we investigate if the inequality \sqrt{M}\leq \frac{\sqrt{R}}{3}+\sqrt{\frac{R}{9}+\frac{Q^2}{3R}}, derived in \cite{An2}, is sharp within the class of solutions to the Einstein-Vlasov-Maxwell System. Here M is the ADM mass, Q the charge, and R the area radius of the boundary of the static object. We find two classes of solutions with this property, while there is only one in the chargeless case. In particular we find numerical evidence for the existence of arbitrarily thin shell solutions to the Einstein-Vlasov-Maxwell System. Finally, we consider one parameter families of steady states, and we find spirals in the mass-radius diagram for all examples of the microscopic equation of state which we consider.
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Global weak solutions to the relativistic Vlasov-Maxwell System revisited
arXiv: Mathematical Physics, 2004Co-Authors: Gerhard ReinAbstract:In 1989, R. DiPerna and P.-L. Lions established the existence of global weak solutions to the Vlasov-Maxwell System. In the present notes we give a somewhat simplified proof of this result for the relativistic version of this System, the main purpose being to make this important result of kinetic theory more easily accessible to newcomers in the field. We show that the weak solutions preserve the total charge.
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Global Weak Solutions to the Relativistic Vlasov-Maxwell System Revisited
Communications in Mathematical Sciences, 2004Co-Authors: Gerhard ReinAbstract:In their seminal work (3), R. DiPerna and P.-L. Lions established the existence of global weak solutions to the Vlasov-Maxwell System. In the present notes we give a somewhat simplified proof of this result for the relativistic version of this System, the main purpose being to make this important result of kinetic theory more easily accessible to newcomers in the field. We show that the weak solutions preserve the total charge.
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A stability result for the relativistic Vlasov-Maxwell System
Archive for Rational Mechanics and Analysis, 1992Co-Authors: Kai -olaf Kruse, Gerhard ReinAbstract:We consider a space-periodic version of the relativistic Vlasov-Maxwell System describing a collisionless plasma consisting of electrons and positively charged ions. As our main result, we prove that certain spacially homogeneous stationary solutions are nonlinearly stable. To this end we also establish global existence of weak solutions to the corresponding initial value problem. Our investigation is motivated by a corresponding result for the Vlasov-Poisson System, cf. [1, 14].
Robert Glassey - One of the best experts on this subject based on the ideXlab platform.
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Large time behavior of the relativistic Vlasov Maxwell System in low space dimension
Differential and Integral Equations, 2010Co-Authors: Robert Glassey, Stephen Pankavich, Jack SchaefferAbstract:When particle speeds are large the motion of a collisionless plasma is modeled by the relativistic Vlasov Maxwell System. Large time behavior of solutions which depend on one position variable and two momentum variables is considered. In the case of a single species of charge it is shown that there are solutions for which the charge density $(\rho = \int f dv)$ does not decay in time. This is in marked contrast to results for the non-relativistic Vlasov Poisson System in one space dimension. The case when two oppositely charged species are present and the net total charge is zero is also considered. In this case, it is shown that the support in the first component of momentum can grow at most as $t^{\frac{3}{4}}$.
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Large Time Behavior of the Relativistic Vlasov Maxwell System in Low Space Dimension
arXiv: Analysis of PDEs, 2009Co-Authors: Robert Glassey, Stephen Pankavich, Jack SchaefferAbstract:When particle speeds are large the motion of a collisionless plasma is modeled by the relativistic Vlasov Maxwell System. Large time behavior of solutions which depend on one position variable and two momentum variables is considered. In the case of a single species of charge it is shown that there are solutions for which the charge density does not decay in time. This is in marked contrast to results for the non-relativistic Vlasov Poisson System in one space dimension. The case when two oppositely charged species are present and the net total charge is zero is also considered. In this case, it is shown that the support in the first component of momentum can grow at most like t to the three-fourths power.
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The Relativistic Vlasov‐Maxwell System in Two Space Dimensions: Part II
Archive for Rational Mechanics and Analysis, 1998Co-Authors: Robert Glassey, Jack SchaefferAbstract:The motion of a collisionless plasma is modeled by the Vlasov‐Maxwell System. For the relativistic Vlasov‐Maxwell System in the plane with smooth initial data, bounds on particle speeds are derived. In an earlier work [10] it was shown that solutions remain smooth as long as particle speeds do not approach the speed of light. When these results are combined, it follows that solutions remain smooth for all time.
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the relativistic Vlasov Maxwell System in two space dimensions part ii
Archive for Rational Mechanics and Analysis, 1998Co-Authors: Robert Glassey, Jack SchaefferAbstract:The motion of a collisionless plasma is modeled by the Vlasov‐Maxwell System. For the relativistic Vlasov‐Maxwell System in the plane with smooth initial data, bounds on particle speeds are derived. In an earlier work [10] it was shown that solutions remain smooth as long as particle speeds do not approach the speed of light. When these results are combined, it follows that solutions remain smooth for all time.
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The “Two and One–Half Dimensional” Relativistic Vlasov Maxwell System
Communications in Mathematical Physics, 1997Co-Authors: Robert Glassey, Jack SchaefferAbstract:The motion of a collisionless plasma is modeled by solutions to the Vlasov–Maxwell System. The Cauchy problem for the relativistic Vlasov–Maxwell System is studied in the case when the phase space distribution function f = f(t,x,v) depends on the time t, \(\) and \(\). Global existence of classical solutions is obtained for smooth data of unrestricted size. A sufficient condition for global smooth solvability is known from [12]: smooth solutions can break down only if particles of the plasma approach the speed of light. An a priori bound is obtained on the velocity support of the distribution function, from which the result follows.
Walter A. Strauss - One of the best experts on this subject based on the ideXlab platform.
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A sharp stability criterion for the Vlasov–Maxwell System
Inventiones mathematicae, 2008Co-Authors: Zhiwu Lin, Walter A. StraussAbstract:We consider the linear stability problem for a 3D cylindrically symmetric equilibrium of the relativistic Vlasov–Maxwell System that describes a collisionless plasma. For an equilibrium whose distribution function decreases monotonically with the particle energy, we obtained a linear stability criterion in our previous paper [24]. Here we prove that this criterion is sharp; that is, there would otherwise be an exponentially growing solution to the linearized System. We also treat the considerably simpler periodic $1\frac{1}{2}$ D case. The new formulation introduced here is applicable as well to the non-relativistic case, to other symmetries, and to general equilibria.
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A SHARP STABILITY CRITERION FOR THE Vlasov-Maxwell System
Inventiones mathematicae, 2008Co-Authors: Zhiwu Lin, Walter A. StraussAbstract:We consider the linear stability problem for a 3D cylindrically symmetric equilibrium of the relativistic Vlasov–Maxwell System that describes a collisionless plasma. For an equilibrium whose distribution function decreases monotonically with the particle energy, we obtained a linear stability criterion in our previous paper [24]. Here we prove that this criterion is sharp; that is, there would otherwise be an exponentially growing solution to the linearized System. We also treat the considerably simpler periodic \(1\frac{1}{2}\)D case. The new formulation introduced here is applicable as well to the non-relativistic case, to other symmetries, and to general equilibria.
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a sharp stability criterion for the Vlasov Maxwell System
arXiv: Plasma Physics, 2007Co-Authors: Zhiwu Lin, Walter A. StraussAbstract:We consider the linear stability problem for a 3D cylindrically symmetric equilibrium of the relativistic Vlasov-Maxwell System that describes a collisionless plasma. For an equilibrium whose distribution function decreases monotonically with the particle energy, we obtained a linear stability criterion in our previous paper. Here we prove that this criterion is sharp; that is, there would otherwise be an exponentially growing solution to the linearized System. Therefore for the class of symmetric Vlasov-Maxwell equilibria, we establish an energy principle for linear stability. We also treat the considerably simpler periodic 1.5D case. The new formulation introduced here is applicable as well to the nonrelativistic case, to other symmetries, and to general equilibria.
Hakan Andreasson - One of the best experts on this subject based on the ideXlab platform.
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a numerical investigation of the steady states of the spherically symmetric einstein Vlasov Maxwell System
Classical and Quantum Gravity, 2009Co-Authors: Mikael Eklund, Hakan Andreasson, Gerhard ReinAbstract:We construct, by numerical means, static solutions of the spherically symmetric Einstein-Vlasov-Maxwell System and investigate various features of the solutions. This extends a previous investigation (Andreasson and Rein 2007 Class. Quantum Grav. 24 1809) of the chargeless case. We study the possible shapes of the energy density profile as a function of the area radius when the electric charge of an individual particle is varied as a parameter. We find profiles which are multi-peaked, where the peaks are separated either by vacuum or a thin atmosphere, and we find that for a sufficiently large charge parameter the solutions break down at a finite radius. Furthermore, we investigate the inequality root M <= root R/3 + root R/9 + Q(2)/3R, which is derived in Andreasson (2009 Commun. Math. Phys. 288 715) for general matter models, and we find that it is sharp for the Einstein-Vlasov-Maxwell System. Here M is the ADM mass, Q is the charge and R is the area radius of the boundary of the static object. We find two classes of solutions with this property, while there is only one in the chargeless case. In particular we find numerical evidence for the existence of arbitrarily thin shell solutions to the Einstein-Vlasov-Maxwell System. Finally, we consider one-parameter families of steady states, and we find spirals in the mass-radius diagram for all examples of the microscopic equation of state which we consider.
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a numerical investigation of the steady states of the spherically symmetric einstein Vlasov Maxwell System
arXiv: General Relativity and Quantum Cosmology, 2009Co-Authors: Mikael Eklund, Hakan Andreasson, Gerhard ReinAbstract:We construct, by numerical means, static solutions of the spherically symmetric Einstein-Vlasov-Maxwell System and investigate various features of the solutions. This extends a previous investigation \cite{AR1} of the chargeless case. We study the possible shapes of the energy density profile as a function of the area radius when the electric charge of an individual particle is varied as a parameter. We find profiles which are multi-peaked, where the peaks are separated either by vacuum or a thin atmosphere, and we find that for a sufficiently large charge parameter there are no physically meaningful solutions. Furthermore, we investigate if the inequality \sqrt{M}\leq \frac{\sqrt{R}}{3}+\sqrt{\frac{R}{9}+\frac{Q^2}{3R}}, derived in \cite{An2}, is sharp within the class of solutions to the Einstein-Vlasov-Maxwell System. Here M is the ADM mass, Q the charge, and R the area radius of the boundary of the static object. We find two classes of solutions with this property, while there is only one in the chargeless case. In particular we find numerical evidence for the existence of arbitrarily thin shell solutions to the Einstein-Vlasov-Maxwell System. Finally, we consider one parameter families of steady states, and we find spirals in the mass-radius diagram for all examples of the microscopic equation of state which we consider.
