David Fajman

• PhD
• University of Vienna

We prove the nonlinear stability of homogeneous barotropic perfect fluid solutions in fixed cosmological spacetimes undergoing decelerated expansion. The results hold provided a specific inequality between the speed of sound of the fluid and the expansion rate of spacetime is valid. Numerical studies in our earlier complementary paper provide stron...

We review the status of mathematical research on the dynamical properties of relativistic fluids in cosmological spacetimes–both, in the presence of gravitational backreaction as well as the evolution on fixed cosmological backgrounds. We focus in particular on the phenomenon of fluid stabilization, which describes the taming effect of spacetime ex...

We establish the future nonlinear stability of a large class of FLRW models as solutions to the Einstein-Dust system. We consider the case of a vanishing cosmological constant, which, in particular implies that the expansion rate of the respective models is linear, i.e. has zero acceleration. The resulting spacetimes are future globally regular. Th...

In this paper, we study cosmological solutions to the Einstein–Euler equations. We first establish the future stability of nonlinear perturbations of a class of homogeneous solutions to the relativistic Euler equations on fixed linearly expanding cosmological spacetimes with a linear equation of state $p=K \rho $ for the parameter values $K \in (0,...

We show that any homogeneous initial data set with Λ < 0 on a product 3-manifold of the orthogonal form (F × S ¹ , a 0 ² dz ² + b 0 ² σ, c 0 dz ² + d 0 σ), where (F, σ) is a closed 2-surface of constant curvature and a 0 , . . . , d 0 are suitable constants, recollapses under the Einstein-flow with a negative cosmo- logical constant and forms crush...

In this paper we study cosmological solutions to the Einstein--Euler equations. We first establish the future stability of nonlinear perturbations of a class of homogeneous solutions to the relativistic Euler equations on fixed linearly expanding cosmological spacetimes with a linear equation of state $p=K \rho$ for the parameter values $K \in (0,1...

It is shown that Milne models (a subclass of Friedmann–Lematre–Robertson–Walker (FLRW) spacetimes with negative spatial curvature) are nonlinearly stable in the set of solutions to the Einstein–Vlasov–Maxwell system, describing universes with ensembles of collisionless self-gravitating, charged particles. The system contains various slowly decaying...

Any initial data set on a closed orientable 3-manifold which is close to data for a Friedman-Lema\^itre-Robertson-Walker solution with a negative Einstein metric $\gamma$ as spatial metric and homogenous scalar field matter evolves under the Einstein scalar-field flow in the contracting direction to a past incomplete cosmological spacetime, and in...

We show that any homogeneous initial data set with $\Lambda<0$ on a product 3-manifold of the orthogonal form $(F\times \mathbb S^1,a_0^2dz^2+b_0^2\sigma^2,c_0dz^2+d_0\sigma)$, where $(F,\sigma)$ is a closed 2-surface of constant curvature and $a_0,..., d_0$ are suitable constants, recollapses under the Einstein-flow with a negative cosmological co...

We study the asymptotic behaviour of solutions to the linear wave equation on cosmological spacetimes with Big Bang singularities and show that appropriately rescaled waves converge against a blow-up profile. Our class of spacetimes includes Friedman–Lemaître–Robertson–Walker (FLRW) spacetimes with negative sectional curvature that solve the Einste...

We study the asymptotic behaviour of solutions to the linear wave equation on cosmological spacetimes with Big Bang singularities and show that appropriately rescaled waves converge against a blow-up profile. Our class of spacetimes includes Friedman-Lema\^itre-Robertson-Walker (FLRW) spacetimes with negative sectional curvature that solve the Eins...

We prove the global asymptotic stability of the Minkowski space for the massless Einstein–Vlasov system in wave coordinates. In contrast with previous work on the subject, no compact support assumptions on the initial data of the Vlasov field in space or the momentum variables are required. In fact, the initial decay in v is optimal. The present pr...

We establish the future nonlinear stability of a large class of FLRW models as solutions to the Einstein-Dust system. We consider the case of a vanishing cosmological constant, which in particular implies that the expansion rate of the respective models is linear i.e. has zero acceleration. The resulting spacetimes are future globally regular. Thes...

Motivated by recent problems in mathematical cosmology, in which temporal averaging methods are applied in order to analyse the future asymptotics of models which exhibit oscillatory behaviour, we provide a theorem concerning the large-time behaviour for solutions of a general class of systems. We thus propose our result to be applicable to a wide...

We establish global regularity and stability for the irrotational relativistic Euler equations with equation of state \(\bar{p}{}=K\bar{\rho }{}\), where \(0<K<1/3\), for small initial data in the expanding direction of FLRW spacetimes of the form \((\mathbb R\times \mathbb T^3,-d\bar{t}{}^2+\bar{t}{}^2\delta _{ij} dx^i dx^j\)). This provides the f...

It is shown that Milne models (a subclass of FLRW spacetimes with negative spatial curvature) are nonlinearly stable in the set of solutions to the Einstein-Vlasov-Maxwell system, describing universes with ensembles of collisionless self-gravitating, charged particles. The system contains various slowly decaying borderline terms in the mutually cou...

It is shown that negative Einstein metrics are attractors of the Einstein-Klein-Gordon system. As an essential part of the proof we upgrade a technique that uses the continuity equation complementary to L ²-estimates to control massive matter fields. In contrast to earlier applications of this idea we require a correction to the energy density to o...

Any 2+1 dimensional Einsteinian spacetime with positive cosmological constant and compact hyperbolic spatial topology evolves in the expanding direction to a de Sitter model with homogeneous hyperbolic spatial geometry. In particular, the latter constitutes the uniform future attractor for this class of spacetimes for arbitrary large initial data....

We show that any \(3+1\)-dimensional Milne model is future nonlinearly, asymptotically stable in the set of solutions to the Einstein–Vlasov system. For the analysis of the Einstein equations we use the constant-mean-curvature-spatial-harmonic gauge. For the distribution function the proof makes use of geometric \(L^2\)-estimates based on the Sasak...

We analyse spatially homogenous cosmological models of locally rotationally symmetric Bianchi type III with a massive scalar field as matter model. Our main result concerns the future asymptotics of these spacetimes and gives the dominant time behaviour of the metric and the scalar field for all solutions for late times. This metric is forever expa...

Motivated by recent problems in mathematical cosmology, we present two theorems giving the large-time asymptotics for a general class of systems which we propose to be applicable to a wide range of problems in spatially homogenous cosmology with oscillatory behaviour. Indeed, we improve the standard theory on averaging to one that is applicable to...

We prove the global asymptotic stability of the Minkowski space for the massless Einstein-Vlasov system in wave coordinates. In contrast with previous work on the subject, no compact support assumptions on the initial data of the Vlasov field in space or the momentum variables are required. In fact, the initial decay in $v$ is optimal. The present...

We show that the homogeneous, massless Einstein-Vlasov system with toroidal spatial topology and diagonal Bianchi type I symmetry for initial data close to isotropic data isotropizes towards the future and in particular asymptotes to a radiative Einstein-de Sitter model. We use an energy method to obtain quantitative estimates on the rate of isotro...

We establish global regularity and stability for the irrotational relativistic Euler equations with equation of state $\overline{p}=K\overline{\rho}$, where $0<K<1/3$, for small initial data in the expanding direction of FLRW spacetimes of the form $(\mathbb R\times\mathbb T^3,-d\tb^2+\tb^2\delta_{ij} dx^i dx^j)$. This provides the first case of no...

We analyse spatially homogenous cosmological models of locally rotationally symmetric Bianchi type III with a massive scalar field as matter model. Our main result concerns the future asymptotics of these spacetimes and gives the dominant time behaviour of the metric for all solutions for late times. This metric is forever expanding in all directio...

We analyse the Kantowski–Sachs cosmologies with Vlasov matter of massive and massless particles using dynamical systems analysis. We show that generic solutions are past and future asymptotic to the non-flat locally rotationally symmetric Kasner vacuum solution. Furthermore, we establish that solutions with massive Vlasov matter behave like solutio...

We consider the Einstein-flow on a product manifold with one factor being a compact quotient of 3-dimensional hyperbolic space without boundary and the other factor being a flat torus of fixed arbitrary dimension. We consider initial data symmetric with respect to the toroidal directions. We obtain effective Einsteinian field equations coupled to a...

We show that the homogeneous, massless Einstein-Vlasov system with toroidal spatial topology for initial data close to isotropic data isotropizes towards the future and in particular asymptotes to a radiative Einstein-deSitter model.

We analyse the Kantowski-Sachs cosmologies with Vlasov matter of massive and massless particles using dynamical systems analysis. We show that generic solutions are past and future asymptotic to the non-flat locally rotationally symmetric Kasner vacuum solution. Furthermore, we establish that solutions with massive Vlasov matter behave like solutio...

It is shown that negative Einstein metrics are attractors of the Einstein-Klein-Gordon system. As an essential part of the proof we upgrade a technique that uses the continuity equation complementary to $L^2$-estimates to control massive matter fields. In contrast to earlier applications of this idea we require a correction to the energy density to...

We complement a recent work on the stability of fixed points of the CMC-Einstein-Λ flow. In particular, we modify the utilized gauge for the Einstein equations and remove a restriction on the fixed points whose stability we are able to prove by this method, and thereby generalize the stability result. In addition, we consider the notion of the redu...

We analyze the asymptotic behavior of small perturbations of Minkowski space caused by the presence of ensembles of massive and massless particles in spherical symmetries. The perturbations we consider are compactly supported in space and momenta.

We complement a recent work on the stability of fixed points of the CMC-Einstein-$\Lambda$ flow. In particular, we modify the utilized gauge for the Einstein equations and remove a restriction on the fixed points whose stability we are able to prove by this method, and thereby generalize the stability result. In addition, we consider the notion of...

We consider the Einstein flow on a product manifold with one factor being a compact quotient of 3-dimensional hyperbolic space without boundary and the other factor being a flat torus of fixed arbitrary dimension. We consider initial data symmetric with respect to the toroidal directions. We obtain effective Einsteinian field equations coupled to a...

We prove future nonlinear stability of homogeneous solutions to the Einstein–Vlasov system with massive particles on manifolds with topology M = ℝ×Σ, where Σ is either 𝕊² or 𝕋². For the sphere this implies the existence of an open subset of the initial data manifold with elements of strictly positive scalar curvature, whose developments are future...

We prove the global stability of the Minkowski space viewed as the trivial solution of the Einstein-Vlasov system. To estimate the Vlasov field, we use the vector field and modified vector field techniques developed in [FJS15; FJS17]. In particular, the initial support in the velocity variable does not need to be compact. To control the effect of t...

We prove the global stability of the Minkowski space viewed as the trivial solution of the Einstein-Vlasov system. To estimate the Vlasov field, we use the vector field and modified vector field techniques developed in [FJS15; FJS17]. In particular, the initial support in the velocity variable does not need to be compact. To control the effect of t...

We prove future nonlinear stability for the Einstein–Vlasov system in 2+1 dimensions, for a manifold of the type \({\Sigma \times \mathbb{R}}\), where \({\Sigma}\) is closed with genus > 1, in the expanding direction. This is the first stability result for the Einstein–Vlasov system for the case of vanishing cosmological constant without symmetry a...

We study the long-time behavior of the Einstein flow coupled to matter on 2-dimensional surfaces. We consider massless matter models such as collisionless matter composed of massless particles, massless scalar fields and radiation fluids and show that the maximal globally hyperbolic development of homogeneous and isotropic initial data on the 2-sph...

We find the Courant-sharp Neumann eigenvalues of the Laplacian on some 2-rep-tile domains. In \(\mathbb {R}^{2}\), the domains we consider are the isosceles right triangle and the rectangle with edge ratio \(\sqrt{2}\) (also known as the A4 paper). In \(\mathbb {R}^{n}\), the domains are boxes which generalize the mentioned planar rectangle. The sy...

This paper proves almost-sharp asymptotics for small data solutions of the Vlasov-Nordstr\"om system in dimension three. This system consists of a wave equation coupled to a transport equation and describes an ensemble of relativistic, self-gravitating particles. We derive sharp decay estimates using a variant of the vector-field method introduced...

It is shown that asymptotically flat initial data to the spherically symmetric massless Einstein-Vlasov system with outgoing matter has a future complete maximal globally hyperbolic development. This result complements a previous result of Andréasson, Kunze and Rein for the corresponding system with massive particles.

We consider the vacuum Einstein flow with a positive cosmological constant λ on spatial manifolds of product form M = M1 × M2. In dimensions n = dim M ≥ 4 we show the existence of continuous families of recollapsing models whenever at least one of the factors M1 or M2 admits a Riemannian Einstein metric with positive Einstein constant. We moreover...

We consider the vacuum Einstein flow with a positive cosmological constant on spatial manifolds of product form. In spatial dimension at least four we show the existence of continuous families of recollapsing models whenever at least one of the factors or admits a Riemannian Einstein metric with positive Einstein constant. We moreover show that the...

We consider non-vacuum initial data for the three-dimensional Einstein equations coupled to Vlasov matter composed of massive particles, on an arbitrary compact Cauchy hypersurface without boundary. We show that conservation of the total mass implies future completeness of the corresponding maximal development in the isotropic case, independent of...

We prove a local well-posedness result for the Einstein-Vlasov system in constant mean curvature-spatial harmonic gauge introduced in [L. Andersson and V. Moncrief, Ann. Henri Poincaré, 4 (2003), pp. 1-34], where local well-posedness for the vacuum Einstein equations is established. This work is based on the techniques developed therein. In additio...

We prove existence of spherically symmetric, static, self-gravitating photon shells as solutions to the massless Einstein-Vlasov system. The solutions are highly relativistic in the sense that the ratio $2m(r)/r$ is close to $8/9$, where $m(r)$ is the Hawking mass and $r$ is the area radius. In 1955 Wheeler constructed, by numerical means, so calle...

We prove existence of spherically symmetric, static, self-gravitating photon shells as solutions to the massless Einstein-Vlasov system. The solutions are highly relativistic in the sense that the ratio $2m(r)/r$ is close to $8/9$, where $m(r)$ is the Hawking mass and $r$ is the area radius. In 1955 Wheeler constructed, by numerical means, so calle...

We adapt the vector field method of Klainerman to the study of relativistic transport equations. First, we prove robust decay estimates for velocity averages of solutions to the relativistic massive and massless transport equations, without any compact support requirements (in $x$ or $v$) for the distribution functions. In the second part of this a...

We adapt the vector field method of Klainerman to the study of relativistic transport equations. First, we prove robust decay estimates for velocity averages of solutions to the relativistic massive and massless transport equations, without any compact support requirements (in $x$ or $v$) for the distribution functions. In the second part of this a...

We determine the Courant sharp eigenvalues of the right-angled isosceles triangle with Neumann boundary conditions.

We give a concise proof of nonlinear stability for a large class of solutions to the Einstein equations with a positive cosmological constant and compact spatial topology, where the spatial metric is Einstein with either positive or negative Einstein constant. The proof uses the CMC Einstein flow and stability follows by an energy argument. We prov...

We construct spherically symmetric, static solutions to the Einstein-Vlasov system with non-vanishing cosmological constant $\Lambda$. The results are divided as follows. For small $\Lambda>0$ we show existence of globally regular solutions which coincide with the Schwarzschild-deSitter solution in the exterior of the matter sources. For $\Lambda<0...

A Laplacian eigenfunction on a two-dimensional manifold dictates some natural partitions of the manifold; the most apparent one being the well studied nodal domain partition. An alternative partition is revealed by considering a set of distinguished gradient flow lines of the eigenfunction - those which are connected to saddle points. These give ri...

We prove area inequalities for stable marginally outer trapped surfaces in Einstein-Maxwell-dilaton theory. Our inspiration comes on the one hand from a corresponding upper bound for the area in terms of the charges obtained recently by Dain, Jaramillo and Reiris [1] in the pure Einstein-Maxwell case without symmetries, and on the other hand from Y...

We give an explicit formula for the number of nodal domains of certain eigenfunctions on a flat torus. We apply this to an isospectral but not isometric family of pairs of flat four-dimensional tori constructed by Conway and Sloane, and we show that corresponding eigenfunctions have the same number of nodal domains. This disproves a conjecture by B...

We study the nodal set of eigenfunctions of the Laplace operator on the right angled isosceles triangle. A local analysis of the nodal pattern provides an algorithm for computing the number of nodal domains for any eigenfunction. In addition, an exact recursive formula for the number of nodal domains is found to reproduce all existing data. Eventua...