The Experts below are selected from a list of 1386 Experts worldwide ranked by ideXlab platform
Huicheng Yin - One of the best experts on this subject based on the ideXlab platform.
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on the instability problem of a 3 d transonic oblique shock wave
Advances in Mathematics, 2015Co-Authors: Huicheng YinAbstract:Abstract In this paper, we are concerned with the instability problem of a 3-D transonic oblique shock wave for the steady supersonic flow past an infinitely long sharp wedge. The flow is assumed to be isentropic and irrotational. It was indicated on p. 317 of [7] that if a steady supersonic flow comes from Minus Infinity and hits a sharp symmetric wedge, then it follows from the Rankine–Hugoniot conditions and the physical entropy condition that there possibly appears a weak shock or a strong shock attached at the edge of the sharp wedge, which corresponds to a supersonic shock or a transonic shock, respectively. The question arises which of the two actually occurs. It has frequently been stated that the strong one is unstable and that, therefore, only the weak one could occur. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand such a longstanding open question. We will show that the attached 3-D transonic oblique shock problem is overdetermined with respect to the periodic perturbation, which implies that the 3-D transonic shock is unstable in general.
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On the instability problem of a 3-D transonic oblique shock wave
2014Co-Authors: Li Liang, Xu Gang, Huicheng YinAbstract:In this paper, we are concerned with the instability problem of a 3-D transonic oblique shock wave for the steady supersonic flow past an infinitely long sharp wedge. The flow is assumed to be isentropic and irrotational. It was indicated in pages 317 of [9] that if a steady supersonic flow comes from Minus Infinity and hits a sharp symmetric wedge, then it follows from the Rankine-Hugoniot conditions and the physical entropy condition that there possibly appear a weak shock or a strong shock attached at the edge of the sharp wedge, which corresponds to a supersonic shock or a transonic shock, respectively. The question arises which of the two actually occurs. It has frequently been stated that the strong one is unstable and that, therefore, only the weak one could occur. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand such a longstanding open question. We will show that the attached 3-D transonic oblique shock problem is overdetermined, which implies that the 3-D transonic shock is unstable in general.Comment: 65 page
Gaetano Zampieri - One of the best experts on this subject based on the ideXlab platform.
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Completely integrable Hamiltonian systems with weak Lyapunov instability or isochrony, Commun
2016Co-Authors: Gaetano ZampieriAbstract:Abstract. The aim of this paper is to introduce a class of Hamil-tonian autonomous systems in dimension 4 which are completely integrable and their dynamics is described in all details. They have an equilibrium point which is stable in some rare elements of the class and unstable in most cases. Anyhow, the eigenvalues of the linearization at the equilibrium point are imaginary and no motion is asymptotic in the past, namely no solution has the equilibrium as limit point as time goes to Minus Infinity. In the unstable cases, there is a sequence of initial data which converges to the origin whose corresponding solutions are unbounded and the motion is slow. So instability is quite weak and perhaps no such explicit ex-amples of instability are known in the literature. The stable cases are also interesting since the level sets of the 2 first integrals inde-pendent and in involution keep being non-compact and stability is related to the isochronous periodicity of all orbits and the existence of a further first integral
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Weak instability of Hamiltonian equilibria.
2012Co-Authors: Gaetano ZampieriAbstract:This is an expository paper on Lyapunov stability of equilibria of autonomous Hamiltonian systems. Our aim is to clarify the concept of weak instability, namely instability without non-constant motions which have the equilibrium as limit point as time goes to Minus Infinity. This is done by means of some examples. In particular, we show that a weakly unstable equilibrium point can be stable for the linearized vector field
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Completely Integrable Hamiltonian Systems with Weak Lyapunov Instability or Isochrony
Communications in Mathematical Physics, 2011Co-Authors: Gaetano ZampieriAbstract:The aim of this paper is to introduce a class of Hamiltonian autonomous systems in dimension 4 which are completely integrable and their dynamics is described in all details. They have an equilibrium point which is stable for some rare elements of the class, and unstable in most cases. Anyhow, it is linearly stable (all orbits of the linearized system are bounded) and no motion is asymptotic in the past, namely no non-constant solution has the equilibrium as limit point as time goes to Minus Infinity. In the unstable cases, there is a sequence of initial data which converges to the equilibrium point whose corresponding solutions are unbounded and the motion is slow. So instability is quite weak and perhaps no such explicit examples of instability are known in the literature. The stable cases are also interesting since the level sets of the 2 first integrals independent and in involution keep being non-compact and stability is related to the isochronous periodicity of all orbits near the equilibrium point and the existence of a further first integral. Hopefully, these superintegrable Hamiltonian systems will deserve further research.
Julio Oliva - One of the best experts on this subject based on the ideXlab platform.
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scalar field quasinormal modes on asymptotically locally flat rotating black holes in three dimensions
European Physical Journal C, 2019Co-Authors: Andres Anabalon, Octavio Fierro, Jose Figueroa, Julio OlivaAbstract:The pure quadratic term of New Massive Gravity in three dimensions admits asymptotically locally flat, rotating black holes. These black holes are characterized by their mass and angular momentum, as well as by a hair of gravitational origin. As in the Myers–Perry solution in dimensions greater than five, there is no upper bound on the angular momentum. We show that, remarkably, the equation for a massless scalar field on this background can be solved in an analytic manner and that the quasinormal frequencies can be found in a closed form. The spectrum is obtained requiring ingoing boundary conditions at the horizon and an asymptotic behavior at spatial Infinity that provides a well-defined action principle for the scalar probe. As the angular momentum of the black hole approaches zero, the imaginary part of the quasinormal frequencies tends to Minus Infinity, migrating to the north pole of the Riemann sphere and providing infinitely damped modes of high frequency. We show that this is consistent with the fact that the static black hole within this family does not admit quasinormal modes for a massless scalar probe.
Andres Anabalon - One of the best experts on this subject based on the ideXlab platform.
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scalar field quasinormal modes on asymptotically locally flat rotating black holes in three dimensions
European Physical Journal C, 2019Co-Authors: Andres Anabalon, Octavio Fierro, Jose Figueroa, Julio OlivaAbstract:The pure quadratic term of New Massive Gravity in three dimensions admits asymptotically locally flat, rotating black holes. These black holes are characterized by their mass and angular momentum, as well as by a hair of gravitational origin. As in the Myers–Perry solution in dimensions greater than five, there is no upper bound on the angular momentum. We show that, remarkably, the equation for a massless scalar field on this background can be solved in an analytic manner and that the quasinormal frequencies can be found in a closed form. The spectrum is obtained requiring ingoing boundary conditions at the horizon and an asymptotic behavior at spatial Infinity that provides a well-defined action principle for the scalar probe. As the angular momentum of the black hole approaches zero, the imaginary part of the quasinormal frequencies tends to Minus Infinity, migrating to the north pole of the Riemann sphere and providing infinitely damped modes of high frequency. We show that this is consistent with the fact that the static black hole within this family does not admit quasinormal modes for a massless scalar probe.
Kreuml Andreas - One of the best experts on this subject based on the ideXlab platform.
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Fraktionelle Perimeter und Symmetrisierung
Wien, 2021Co-Authors: Kreuml AndreasAbstract:Arbeit an der Bibliothek noch nicht eingelangt - Daten nicht geprüftAbweichender Titel nach Übersetzung der Verfasserin/des VerfassersIn der vorliegenden Dissertation werden Konvergenz und Symmetrisierung von fraktionellen Perimetern in verschiedenen Räumen untersucht. Zu allererst wird eine Klassifizierung aller Gleichheitsfälle in der anisotropen fraktionellen isoperimetrischen Ungleichung angegeben unter der Annahme, dass die zugrundeliegende Einheitskugel symmetrisch zu jeder Koordinatenhyperebene und strikt konvex ist. Mit deren Hilfe wird gezeigt, dass die anisotrope Symmetrisierung bezüglich dieser Gleichheitsfälle wohldefiniert ist und eine anisotrope fraktionelle Pólya-Szegö-Ungleichung wird für diese Symmetrisierung hergeleitet. Als Nächstes werden fraktionelle Seminormen und Perimeter auf Riemannschen Mannigfaltigkeiten eingeführt und deren Konvergenz zur Sobolev-Seminorm beziehungsweise zum Perimeter für s gegen 1 wird gezeigt. Für fraktionelle Perimeter auf der Sphäre wird ein alternativer Beweis für dieses Resultat mittels sphärischer Integralgeometrie angegeben. In diesem Speziallfall wird die Konvergenz von geeignet renormalisierten fraktionellen Perimetern gegen ein Volumsfunktional für s gegen Minus unendlich gezeigt. Schlussendlich werden isoperimetrische Ungleichungen für sphärische fraktionelle Perimeter mit einer vollständigen Beschreibung aller Gleichheitsfälle hergeleitet. Einige Resultate in dieser Dissertation sind in Zusammenarbeit mit Olaf Mordhorst entstanden.In this thesis, convergence and symmetrization of fractional perimeters in different settings are studied. First, a classification of minimizers of the anisotropic fractional isoperimetric inequality is given whenever the unit ball of the space is unconditional and strictly convex. With its help it is shown that anisotropic symmetrization with respect to these minimizers is well-defined and an anisotropic fractional Pólya-Szegö principle for this symmetrization is established. Next, fractional seminorms and perimeters are introduced on Riemannian manifolds and their convergence to the Sobolev seminorm and the perimeter, respectively, is shown as s tends to 1. For fractional perimeters on the sphere an alternative proof for this result using spherical integral geometry is presented. In this special case, the convergence of suitably normalized fractional perimeters towards a volume functional as s tends to Minus Infinity is shown. Finally, isoperimetric-type inequalities for spherical fractional perimeters with a complete classification of equality cases are derived. Some results of this thesis are joint work together with Olaf Mordhorst.6
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Fraktionelle Perimeter und Symmetrisierung
Wien, 2021Co-Authors: Kreuml AndreasAbstract:Abweichender Titel nach Übersetzung der Verfasserin/des VerfassersIn der vorliegenden Dissertation werden Konvergenz und Symmetrisierung von fraktionellen Perimetern in verschiedenen Räumen untersucht. Zu allererst wird eine Klassifizierung aller Gleichheitsfälle in der anisotropen fraktionellen isoperimetrischen Ungleichung angegeben unter der Annahme, dass die zugrundeliegende Einheitskugel symmetrisch zu jeder Koordinatenhyperebene und strikt konvex ist. Mit deren Hilfe wird gezeigt, dass die anisotrope Symmetrisierung bezüglich dieser Gleichheitsfälle wohldefiniert ist und eine anisotrope fraktionelle Pólya-Szegö-Ungleichung wird für diese Symmetrisierung hergeleitet. Als Nächstes werden fraktionelle Seminormen und Perimeter auf Riemannschen Mannigfaltigkeiten eingeführt und deren Konvergenz zur Sobolev-Seminorm beziehungsweise zum Perimeter für s gegen 1 wird gezeigt. Für fraktionelle Perimeter auf der Sphäre wird ein alternativer Beweis für dieses Resultat mittels sphärischer Integralgeometrie angegeben. In diesem Speziallfall wird die Konvergenz von geeignet renormalisierten fraktionellen Perimetern gegen ein Volumsfunktional für s gegen Minus unendlich gezeigt. Schlussendlich werden isoperimetrische Ungleichungen für sphärische fraktionelle Perimeter mit einer vollständigen Beschreibung aller Gleichheitsfälle hergeleitet. Einige Resultate in dieser Dissertation sind in Zusammenarbeit mit Olaf Mordhorst entstanden.In this thesis, convergence and symmetrization of fractional perimeters in different settings are studied. First, a classification of minimizers of the anisotropic fractional isoperimetric inequality is given whenever the unit ball of the space is unconditional and strictly convex. With its help it is shown that anisotropic symmetrization with respect to these minimizers is well-defined and an anisotropic fractional Pólya-Szegö principle for this symmetrization is established. Next, fractional seminorms and perimeters are introduced on Riemannian manifolds and their convergence to the Sobolev seminorm and the perimeter, respectively, is shown as s tends to 1. For fractional perimeters on the sphere an alternative proof for this result using spherical integral geometry is presented. In this special case, the convergence of suitably normalized fractional perimeters towards a volume functional as s tends to Minus Infinity is shown. Finally, isoperimetric-type inequalities for spherical fractional perimeters with a complete classification of equality cases are derived. Some results of this thesis are joint work together with Olaf Mordhorst.6