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Gravitational collapse and the Vlasov-Poisson system

A self-gravitating homogeneous ball of a fluid with pressure zero where the fluid particles are initially at rest collapses to a point in finite time. We prove that this gravitational collapse can be approximated arbitrarily closely by suitable solutions of the Vlasov-Poisson system which are known to exist globally in time.


Introduction
Perhaps the simplest example of a matter distribution which collapses under the influence of its own, self-consistent gravitational field is a homogeneous ball of an ideal, compressible fluid with the equation of state that the pressure is identically zero-this is usually referred to as dust-and with the particles initially being at rest. In suitable units, the radius r(t) of this ball is determined by the initial value problem r = − 1 r 2 , r(0) = 1,ṙ(0) = 0.
The mass density is given by Here 1 S denotes the indicator function of the set S and B r (z) is the ball of radius r > 0 centered at z ∈ R 3 . There exists a time T > 0 such that r(t) > 0 exists on [0, T [ with lim t→T r(t) = 0, i.e., the dust ball collapses to a point in finite time. It is easy to check that the above mass density solves the pressure-less Euler-Poisson system ∂ t ρ + div (ρu) = 0, (1.1) with velocity field u(t, x) =ṙ (t) r(t) x; U = U (t, x) denotes the induced gravitational potential. The fact that such a matter distribution collapses is not surprising since there is no mechanism which opposes gravity, the pressure being put to zero by the choice of the equation of state. This situation changes significantly if dust is replaced by a collisionless gas as matter model in which case smooth, compactly supported initial data launch solutions which exist globally in time and do not undergo a gravitational collapse in the above sense, cf. [3,5]. As in dust, the particles in a collisionless gas interact only by gravity, but the particle ensemble is now given in terms of a density function f = f (t, x, v) ≥ 0 on phase space where t ∈ R, x, v ∈ R 3 stand for time, position, and velocity, and f obeys the Vlasov-Poisson system which consists of the Vlasov equation (1.4) coupled to the Poisson equation (1.3) via the definition ρ(t, x) = f (t, x, v) dv (1.5) of the spatial mass density in terms of the phase space density f ; unless indicated otherwise, integrals always extend over R 3 . In astrophysics, the Vlasov-Poisson system (1.3), (1.4), (1.5) is used as a model for galaxies or globular clusters, cf. [1]. The relation between its solutions and those of the pressure-less Euler-Poisson system (1.1), (1.2), (1.3) was investigated in [2]. If (ρ, u) is a solution of the former system, then formally x)) satisfies the latter system where δ denotes the Dirac distribution. However, when we speak of the Vlasov-Poisson system in the present paper we only consider genuine functions on phase space (which will actually be smooth) as solutions, and we ask the question whether the collapsing solution of the pressure-less Euler-Poisson system presented at the beginning of this introduction can be approximated by suitable smooth solutions of the Vlasov-Poisson system. This is indeed possible, cf. Theorem 2.2 below, and yields the existence of nearly collapsing solutions of the Vlasov-Poisson system. To make the latter more precise we recall the definitions of the kinetic and the potential energy associated with a solution: The total energy E kin (t) + E pot (t) is conserved. We also recall that a solution of the Vlasov-Poisson system is spherically symmetric if f (t, x, v) = f (t, Ax, Av) for any rotation A ∈ SO(3). For a spherically symmetric solution and by abuse of notation, ρ(t, x) = ρ(t, r) and where m(t, r) = 4π r 0 s 2 ρ(t, s) ds is the mass contained in the ball of radius r > 0 centered at the origin. It is well known that spherically symmetric initial data launch spherically symmetric solutions of the Vlasov-Poisson system, cf. [6].
Theorem 1.1. For any constants C 1 , C 2 > 0 there exists a smooth, spherically symmetric solution f of the Vlasov-Poisson system such that initially but for some time t > 0, By choosing C 1 small and C 2 large we see that the matter distribution is initially dilute and the particles are nearly at rest, but at some later time t the matter is very concentrated with large total kinetic and potential energy. In this sense, Vlasov-Poisson solutions can be very close to a gravitational collapse even though-as opposed to the case of the pressure-less Euler-Poisson system-the quantities in the above theorem remain bounded on any bounded time interval. Besides the wish to understand better the relation of the two systems under consideration there is a more specific motivation for the current investigation which also explains why the term m/r is considered in the theorem above. This motivation originates in general relativity.
In 1939, J. R. Oppenheimer and H. Snyder [4] showed how a black hole can develop from regular data. Much as in our introductory example, they considered a spherically symmetric, asymptotically flat spacetime with a dilute homogeneous ball of dust as matter model and showed that a trapped surface and hence a black hole form in the evolution. If r denotes the area radius, then the condition 2m/r > 1 indicates that the sphere of radius r is trapped and the spacetime contains a black hole. Here m is the appropriate general relativistic analogue of the mass function introduced above, the socalled quasilocal ADM mass, and m(t, r = ∞) is a conserved quantity, the ADM mass. The Oppenheimer-Snyder example suffers from the fact that there is no pressure in the matter model, and it remains unclear if a similar calculation is possible with a matter model which is not pressure-less. The analysis in the present note is intended as a blue-print for the analogous analysis in the general relativistic setting which will lead to Oppenheimer-Snyder type solutions which collapse to a black hole, but with the physically more realistic Vlasov equation as a matter model, cf. [8].
Both the Oppenheimer-Snyder solution and its Newtonian analogue can be obtained by taking a spatially homogeneous solution with a big crunch singularity in the future, cutting a suitable, spatially finite piece from it and extending it by vacuum. If the cutting is done along the trajectory of a dust particle, a consistent, asymptotically flat solution of the desired form is obtained. In the present analysis we follow the same recipe. We first introduce a class of spatially homogeneous, cosmological solutions of the Vlasov-Poisson system with a singularity in the future; such solutions, which do of course not satisfy the boundary condition in (1.3), and their perturbations were considered in [7]. From such a spatially homogeneous solution we cut a ball centered at the origin at time t = 0 and extend it smoothly by vacuum. This provides the initial data for the Vlasov-Poisson solution. The latter will for some time have a spatially homogeneous region at the center, the size of which we can control. By choosing the original homogeneous solution sufficiently close to a dust solution in a suitable sense, the time for which the homogeneous core persists can be pushed as close to the collapse time of the homogeneous solution as we wish. This will prove Theorem 2.2 from which Theorem 1.1 will follow.

Solutions with a homogeneous core
We first recall the construction of spatially homogeneous solutions to the Vlasov-Poisson system. To do so we fix a continuously differentiable function For ǫ ∈]0, 1] let whereå > 0 is prescribed. A straight forward computation shows that is a spherically symmetric solution of the Vlasov-Poisson system-where the boundary condition at spatial infinity is dropped-with and i.e., the macroscopic quantities related to this spatially homogeneous solution h ǫ do actually not depend on ǫ. It is well known that (2.2) cannot be solved explicitly. The following information on the behavior of a will be useful.
Proof. As long as the solution exists,ä < 0 and henceȧ(t) <ȧ(0) = 0 for t > 0. We multiply the differential equation in (2.2) byȧ and integrate to find that 1 2ȧ this equation can be integrated once more to yield (2.6). Since lim t→T a(t) = 0, the formula for T is obtained by taking the corresponding limit in (2.6), and the proof is complete. ✷ defines a solution of the pressure-less Euler-Poisson system which coincides with our introductory collapse example ifå = 1.
This solution can be viewed as follows. We start with the spatially homogeneous solution with density ρ h and the given velocity field and cut from it a spherically symmetric piece the boundary of which is given by the curve r = r(t). It should be noted that this curve is precisely the trajectory of the particle which starts at radius r = 1 with zero initial velocity.
An analogous boundary curve for a corresponding cut in the Vlasov case does not exist since there is at each point in space a distribution of particles with different velocities. We therefore proceed as follows. We choose a family of cut-off functions launch a smooth, global, spherically symmetric solution f ǫ of the Vlasov-Poisson system; notice that initially this solution coincides with the homogeneous one on B 1 (0) × R 3 . We aim to show that for ǫ small a homogeneous core at the center persists arbitrarily closely up to the collapse time T of the homogeneous solution from which fact Theorem 1.1 will follow. In order to define the boundary of the homogeneous core, we place a point mass which is slightly larger than the total mass of the initial data at the origin and consider the trajectory of a particle which moves radially inward in the corresponding potential and starts at the cut-off radius 1 with an initial radial velocity which in modulus is larger than the initial radial velocities of the Vlasov particles.
To make this precise we define a strict upper bound for the total mass of f ǫ by We can now state our main result.
Theorem 2.2. Let f ǫ and r ǫ be defined as above for ǫ ∈]0, 1]. Then the following holds.
(b) r ǫ (t) ≤ a(t)/å for t ∈ [0, T ǫ [ and ǫ ∈]0, 1], and r ǫ (t) → a(t)/å for ǫ → 0, uniformly on any time interval [0, The theorem will be proven in a number of steps in the next section. We first indicate how it implies Theorem 1.1.
Proof of Theorem 1.1. For all ǫ ∈]0, 1] the following estimates hold at time t = 0. First of all, (2.8) and the properties of H ǫ and φ ǫ imply that
This in turn implies that Finally, Using parts (a) and (b) of Theorem 2.2 together with the fact that lim t→T a(t) = 0, all these quantities can be made large in the sense of Theorem 1.1 by making ǫ small and choosing t close to T ; when doing this a and hence the estimates at t = 0 remain unchanged. The fact that the kinetic energy behaves in the same way follows from conservation of energy, and the proof is complete. ✷

Proof of Theorem 2.2
We first observe that the parameterå was used only to make sure that the initial estimates in Theorem 1.1 hold. Since it plays no role in the proof of Theorem 2.2 we can for the rest of this paper simplify our notation by choosingå = 1.

Proof of parts (a) and (b) of Theorem 2.2
The initial data for the functions a and r ǫ imply that r ǫ < a on some interval The definition (2.9) together with the properties of φ ǫ imply that We use the lower bound to conclude from the differential equations for a and r ǫ that on ]0, t * [ the estimater ǫ <ä holds, and sinceṙ ǫ (0) = −ǫ <ȧ(0) alsȯ r ǫ <ȧ. This implies that t * = min(T, T ǫ ). Since the maximal existence time T respectively T ǫ is determined by the fact that the function a respectively r ǫ becomes zero there and since the difference a − r ǫ is positive and strictly increasing as long as both functions exist we can conclude that T ǫ < T and r ǫ < a on ]0, T ǫ [. As in the proof of Lemma 2.1, we see that r ǫ is a strictly decreasing function withṙ ǫ < −ǫ, and Henceṙ We note that lim ǫ→0 M ǫ = 1 and 0 < C ǫ < 1 with lim ǫ→0 C ǫ = 1. Let us define a function F : [0, 1] → R by This function is continuous and differentiable on ]0, 1[ with F ′ (r) = ] is strictly decreasing and onto with a continuous inverse. Using F , the above differential equation for r ǫ can be integrated and yields the relation Since lim t→Tǫ r ǫ (t) = 0 and lim r→0 F (r) = π/2, it follows that for ǫ → 0 as claimed. Now fix some t ∈ [0, T [. Then for ǫ sufficiently small, t ∈ [0, T ǫ [, and as ǫ → 0; here we used the limit behavior of M ǫ and C ǫ and the identity (2.6). This proves the desired limit for r ǫ (t), and since 0 ≤ a(s)−r ǫ (s) ≤ a(t)−r ǫ (t) on [0, t] the limit is uniform on compact subintervals of [0, T [. Parts (a) and (b) of Theorem 2.2 are proven. ✷

The behavior of characteristics
To prove part (c) of Theorem 2.2, we use the fact that a smooth function f solves the Vlasov equation (1.4) if and only if it is constant along its characteristics, i.e., along the solutions of the characteristic systeṁ ) denotes the solution of the characteristic system with initial data (X(t, t, x, v), V (t, t, x, v)) = (x, v) with t ≥ 0 and x, v ∈ R 3 prescribed, then a solution of the Vlasov-Poisson system f is related to its initial dataf by the relation To show that the solution f ǫ launched byf ǫ has a homogeneous core bounded by the curve r = r ǫ , we need to control characteristics which cross this curve. This analysis will be facilitated by the spherical symmetry of the solutions in question. In order to exploit this symmetry, we define for x, v ∈ R 3 corresponding spherical variables by If (x(s), v(s)) solves (3.1), then in spherical variables, spherical symmetry of the gravitational field implies that angular momentum and also its square L is constant along particle trajectories. We first establish some control on characteristics under the assumption that the gravitational field is generated by a mass distribution of total mass bounded by M ǫ .  Proof. We first consider part (a). Since by assumption r(0) ≥ 1 = r ǫ (0) andṙ(0) > −ǫ =ṙ ǫ (0), there exists t ∈]0, T ǫ ] such that r(s) > r ǫ (s) for s ∈]0, t[, and we choose t maximal. On the interval ]0, t[, If t < T ǫ we use the assumptions at s = 0 and integrate (3.3) twice to find that r(t) > r ǫ (t) which contradicts the maximality of t. Hence t = T ǫ , and part (a) is proven. As to part (b) we first show that there exists a time t ′ ∈]0, t] such that r(t ′ ) ≥ r ǫ (t ′ ) and w(t ′ ) ≥ṙ ǫ (t ′ ). (3.4) This can be seen as follows. If w(t) ≥ṙ ǫ (t) we choose t ′ = t. If w(t) <ṙ ǫ (t) there exists t * ∈ [0, t[ such that r(s) > r ǫ (s) for s ∈]t * , t[, and we choose t * minimal. Assuming that w(s) <ṙ ǫ (s) on ]t * , t[ it would follow that r(t * ) > r ǫ (t * ). If t * > 0, this contradicts the minimality of t * , and if t * = 0 it contradicts the assumption r(0) ≤ 1 = r ǫ (0) in part (b). Hence there must exist a time t ′ such that (3.4) holds. For any time s ∈ [t ′ , T ǫ [ such that r(s) ≥ r ǫ (s) it follows thatr(s) >r ǫ (s), cf. (3.3). In particular, this holds for s = t ′ . Hence there exists t * ∈]t ′ , T ǫ ] such that r(s) > r ǫ (s) for s ∈]t ′ , t * [, and we choose t * maximal. Assuming t * < T ǫ we integrate the inequalitÿ r(s) >r ǫ (s) twice starting at t ′ and using the properties (3.4) to conclude that r(t * ) > r ǫ (t * ) in contradiction to the maximality of t * . Hence t * = T ǫ , and the proof of part (b) is complete. ✷ Next we consider the characteristics of the homogeneous solution h ǫ . The separating curve is transformed accordingly, i.e., Using the already established part (b) of Theorem 2.2 and recalling that we tookå = 1 we find that on the interval ]0, T ǫ [, Thusw ǫ (s) <w ǫ (0) = −ǫ, and hencė r ǫ (s) < − ǫ a 2 (s) , s ∈]0, T ǫ [. (3.5) To compare this to the given, mass-carrying characteristic of the homogeneous solution, we observe that by the definition (2.1) of H ǫ , Thus for s ∈]0, t[ it follows thatw(s) > −ǫ which by definition ofw and (3.5) implies thatṙ (s) > − ǫ a 2 (s) >ṙ ǫ (s).
Sincer(t) ≤r ǫ (t), it follows thatr(s) <r ǫ (s) and hence r(s) < r ǫ (s) on [0, t[ as desired. So far we assumed that x(s) = 0 on the interval ]0, t[. For times s ∈ [0, t[ where x(s) = 0 the assertion of the lemma holds since r ǫ (s) > 0. If the function x(s) has zeros but is not identically zero, we let [s 1 , s 2 ] ⊂ [0, t] be an interval with non-empty interior and such that x(s) = 0 on ]s 1 , s 2 [; we choose the interval maximal with this property. Then x(s 2 ) = 0 or s 2 = t, and in either case r(s 2 ) ≤ r ǫ (s 2 ) so that the above argument on the interval [0, t] now applies to [s 1 , s 2 ] and implies that r(s) < r ǫ (s) on [s 1 , s 2 [. Since the interval [0, t[ is the union of such subintervals and a set of points where x(s) = 0, the proof is complete. ✷

Proof of part (c) of Theorem 2.2
Using the above information on characteristics, we can prove the remaining assertion of Theorem 2.2. Since in these arguments the parameter ǫ remains fixed, we write f = f ǫ for the solution of the Vlasov-Poisson system launched by the initial dataf =f ǫ specified in (2.8) and h for the homogeneous solution.
We recall that f < M ǫ , the latter follows from the definition (2.3) of the homogeneous solution h and (2.8). We define and we have to show that f | I = h| I . To prove this we first establish the following assertion for these functions on the boundary of I: To prove the assertion for f we consider the characteristic curve (x(s), v(s)) = (X, V )(s, t, x, v) of f which at time s = t passes through a boundary point as specified in (3.6) with w <ṙ ǫ (t). Then there exists some δ > 0 such that |x(s)| < r ǫ (s) for t < s < t + δ and |x(s)| > r ǫ (s) for t − δ < s < t. Hence Lemma 3.1 (b) implies that |x(0)| > 1, and Lemma 3.1 (a) implies that |v(0)| ≥ ǫ and hence f (t, x, v) =f (x(0), v(0)) = 0; by continuity the assertion also holds if w =ṙ ǫ (t). In order to prove (3.6) for h we argue in the same way using a characteristic curve of h, and Lemma 3.2 implies that h(t, x, v) = 0.
The idea now is that there can be at most one solution of the Vlasov-Poisson system on I which has given data at t = 0 and satisfies the boundary condition (3.6). If we take the difference of the Vlasov equations for f and h, we find that which holds for all t ∈ [0, T ǫ [ and x, v ∈ R 3 ; U f and U h denote the potentials induced by f respectively h. We consider a characteristic curve (x(s), v(s)) = (X, V )(s, t, x, v) of f with |x| < r ǫ (t) and define s * := sup {s ∈ [0, t] | |x(τ )| < r ǫ (τ ), s ≤ τ ≤ t} so that |x(s)| < r ǫ (s) on ]s * , t]. Integrating (3.7) along the characteristic curve implies that , v(s)) ds; (3.8) notice that either s * = 0 in which case the first term on the right hand side vanishes because both functions have the same initial data for |x| ≤ 1, or s * > 0 in which case |x(s * )| = r ǫ (s * ) and w(s * ) ≤ṙ ǫ (s * ), and the first term on the right hand side vanishes due to (3.6). For t ∈ [0, T ǫ [ we define D(t) := sup{|f − h|(t, x, v) | |x| ≤ r ǫ (t), v ∈ R 3 }.
Spherical symmetry and standard estimates imply that notice that the velocity supports of both f and h are bounded, uniformly on [0, T ǫ [, and hence |ρ f (t, x) − ρ h (t, x)| ≤ CD(t) for |x| ≤ r ǫ (t). Hence (3.8) implies that so that D(t) = 0 for t ∈ [0, T ǫ [, and the proof of Theorem 2.2 is complete.