Global and exploding solutions in a model of self-gravitating systems

Vol. 52 (2003) REPORTS ON MATHEMATICAL PHYSICS No. 2

GLOBAL AND EXPLODING SOLUTIONS IN A MODEL OF SELF-GRAVITATING SYSTEMS

PIOTR BILER

Instytut Matematyki, Uniwersytet Wroclawski, pl. Grunwaldzki 2/4, 5&384 Wroclaw, Poland

(e-mail: [email protected])

and

TADEUSZ NADZIEJA

Instytut Matematyki, Uniwersytet Zielonogbrski, ul. Podg6rna 50, 65-246 Zielona G&a, Poland

(e-mail: [email protected])

(Received November 25, 2002)

We study asymptotic properties of solutions to an extension to arbitrary dimensions of the astrophysical model proposed by Chavanis et al. to explain phenomena of gravitational collapse in clouds of self-gravitating particles. In particular, we show that in the two-dimensional case the solutions can be continued to global ones, while in three space dimensions large data of negative energy blow up in a finite time. Relations between isothermal, Streater’s energy-transport and the present models are also studied.

Keywords: Chavanis-Sommeria-Robert model, mean field equations, drift-diffusion and energy-transport systems, finite time blow-up of solutions, asymptotics of solutions.

2000 Mathematics Subject Classification: 35Q, 35K60, 35B40, 82C21.

1. Introduction and equations

Our aim in this paper is to give some properties of solutions of a system proposed by Chavanis, Sommeria and Robert in [17], and studied recently in [16, 18, 25, 261. The model consists of a nonlinear parabolic equation of drift-diffusion type for the density u = u(x, t) of a cloud of particles, coupled with a Poisson equation for the gravitational potential generated by themselves, and supplemented with the energy relation which determines the temperature 0 = z?(t) uniform in the domain Q c R”, see Eqs. (l)-(3) below.

These equations have been derived from a maximum entropy production princi- ple in the context of statistical mechanics of stellar systems, see [17]. The resulting Eq. (1) can be viewed as a generalization of the classical Smoluchowski equation introduced in the seminal paper [27], describing self-gravitating particles that are sub-

12051

206 P. BILERandT. NADZIELIA

ject to Brownian diffusion and move in a self-consistent gravitational field generated by themselves.

Another motivation to study the Chavanis-Sommeria-Robert model is suggested by its intermediate character between isothermal models of mean field type going back to Nernst, Planck, Debye and Htickel in the context of charged particles in electrolytes theory, see e.g. [30, 2, 8, 11, 141, and general energy-transport models of &eater type which include a nonlinear heat equation for the evolution of the temperature, see [7, 10, 15, 28, 291.

Although these three models are strongly related, their properties differ in a rather important way. Their solutions may exhibit finite time blow-up for large initial data. The simplest isothermal model has blowing up solutions even in two dimensions, while the models involving the temperature have then solutions with uniform a priori bounds on u and 6. The solutions of the latter models can blow up in three (and more) space dimensions. Note that kinetic models with a similar behaviour with respect to blow-up have been considered by G. Wolansky in [31]. The existence, uniqueness and regularity of solutions issues are quite well understood for these simpler models, but those questions for the Streater model are largely open.

The present paper is an extension of recent studies of the Chavanis-Sommeria- Robert model in [16, 18, 251 to the case of solutions without radial symmetry, in arbitrary d-dimensional domains, d > 2.

NOTATION. ]ulp denotes the LP(CJ) norm of the function V; various inessential constants will be denoted by C, even if they may vary from line to line.

The paper is organized as follows: after a discussion of the equations (Section l), the local and global in time existence are proved in the two-dimensional situation (Section 2), blowing up solutions are studied in Section 3 and results on self-similar solutions (of both types: blowing up and forward in time) are in Section 4.

We consider equations for the evolution of the density u = U(X, t) > 0 of a cloud of identical attracting particles, the gravitational potential (9 = (p(x, t) generated by the distribution of mass U(X, t) dx, and the temperature 6 = 0 (t) > 0,

ut=V- (( K VU+ ,

A(P=u. (2)

The energy relation

Ed!d s 2 i-2

ucp dx (3)

with E = const E R, fixed specific heat coefficient c > 0, and the total mass of particles M = Jo u dx, supplements the system of differential equations (l)-(2), thus permitting to determine the current value t9 = 0(t) of the temperature. This CSR model (up to unimportant scalings in (l), (2) and (3)) has been proposed in [17], and has been studied recently in [ 16, 18, 25, 261.

GLOBAL AND EXPLODING SOLUTIONS IN A MODEL OF SELF-GRAVITATING SYSTEMS 207

The coefficient K is a nonnegative function of x, U, 0, q, and K (0) = ~3 was postulated by M. Smoluchowski in his classical equation [27]. We are mainly inter- ested in the cases when either K = 1 or K (6) = 8. The system (l)-(3) is studied either in a bounded connected domain CJ c I@ or in the whole space Rd. In the former case, the equations are supplemented with the boundary condition

;+;&I (4)

expressing that there is no mass flux across the boundary a!G! of the domain. In the latter case, instead of (4), u is supposed to decay at infinity. The potential CP satisfies either the Dirichlet condition

q?=o

on the boundary iXJ or the “free” condition

(5)

Cp=Ed*U (6)

(which is also natural if !YJ = IF?), where Ed is the fundamental solution of the Laplacian in I@: Ed = &lOglzl, Ed(z) = -((d - 2)Cd)-11z]2-d, and fld iS

the area of the unit sphere in R d. The Dirichlet condition (5) leads to a theory which is mathematically somewhat more interesting and sometimes simpler, cf. [7]. The free condition (6) is, however, physically more natural, but sometimes causes additional difficulties, cf. [7, 11, 151. Both of the conditions (5) and (6) lead to the equivalent problems in the case of radially symmetric solutions in balls or in the whole space Rd.

We call Po the system (l)-(4) with (5), and Pr -the system (l)-(4) with (6), both supplemented with the initial conditions

u(x, 0) = uo(x) 2 0, I?0 > 0. (7)

Instead of prescribing the temperature 60, we may pose as the initial condition the density ua and the energy E > i lo ~090 dx, with the potential (~0 determined by uo*

Related but simpler isothermal model consists of the equation

u,=V*(Vu+uVp) (8)

and the Poisson equation (2), see [9, 11, 12, 14, 301. Eqs. (8), (2) with either (5) or (6), the initial condition ua > 0 and no-flux boundary condition g + U% = 0 constitute the model Pi.

A much more complicated energy-transport (Streater) model involves the equations

ut=V. ((

K VU+;@ >>

E-V.j, (9) (c~&)~=V.(hV6)- Vs(c6j)- Vp.j, (10)

208 P. BILER and T. NADZIEJA

and (2), where j = -K (VU + :Vq) denotes the flux and c > 0 is the (generally nonconstant) specific heat coefficient. This system is an extension of the model originally introduced by S-eater (and derived from microscopic considerations) in [28, 291 to the case of mutual interaction of particles, see [lo]. Here h is a strictly positive function of X, U, 6, and (10) represents the balance of heat, including thermal diffusion, convection and heat production effects. Eqs. (9), (lo), (2) with either (5) or (6), and with the no-flux conditions for the heat g = 0, and mass

j . u = 0 on ?JCJ, and for the initial data U(X) 0) = Q(X), z’? (x, 0) = zYa(x), constitute the &eater model which we call further Ps .

One of the major mathematical difficulties in an analysis of the evolution of %-eater’s models is connected with the lack of good a priori estimates for the temperature 0 which may lead to possible degeneracies and/or singularities in the equation (where the temperature enters via l/6 factor). Related energy-transport models in solid state physics considered in e.g. [22, 231 also display such difficulties.

All these models satisfy, thanks to the no-flux conditions (4), the conservation of mass property

Mr s

u(x, t)dx = s

uo(x) dx (11)

and preserve the positivity of uaoand 00: U(X, ,;“z 0 6(x t) > 0 cf. [lo 11, 181. The problem Pi has a natural Lyapunov functional of he (a&entr~&

wi = Wi(f) = SC

1 u logu + -$q

> dx, (12)

n

which is decreasing during the evolution of U, see [ 121. The solutions of the problem Ps obey the first and the second laws of thermo-

dynamics since the energy

(13)

is conserved, while the (neg)entropy

w, = Ws(t) = s

u(logu - clog6)dx (14) L-2

decreases in time according to

s (KU)-lIj12dx + s

hlV6126-2 dx 5 0, Q D

see [lo]. Finally, the problems Po , Pr have their (neg)entropies

(15)

(16)

The conservation of energy is, of course, postulated by the relation (3).


All these models could be also considered with given external potentials, see e.g. an isothermal model of a star surrounded by a cloud of dust in [14] or general Streater’s equations in [7, 10, 1.51.

Recall that replacing the Poisson equation (2) by Alp = --u leads to the change of the gravitational attraction interparticle forces to the electric repulsion of identi- cally charged particles.

One may also consider the drift-diffusion equations analogous to (1) for several species of particles, thus obtaining different versions of either electrolytes or semi- conductors or plasma physics equations of Nernst-Planck-Debye-Htickel, for some mathematical results, see e.g. [5, 6, 81, and [14] for several species of massive particles.

Sometimes, we prefer a version

Ut = v . (OVU + uvqo) (17)

of Eq. (1) with ~(0) = 6. The passage from (1) with K = 1 to (17) can be made by introducing a new time variable t so that t = lo’ S(s) ds. For O(t) E [inf 0, sup rY] c (0, cc), the models with (1) and (17) behave in a completely analogous way with respect to finite time blow-up property of solutions and global in time existence questions. Such a reduction of general Streater systems to the one with a normalized fixed coefficient K is not possible.

The family of &eater models Ps is richer than the isothermal model Pi and the uniform temperature models Pu, Pf. That means, a solution of Pi does not necessarily correspond to a solution of Ps with 0 = const. Similarly, a solution of Ps with 6 = O(t) independent of x satisfies a model with ut + V . j = 0 and &z9 + Vq . j = 0, thus equivalent to neither Pn nor Pr.

However, if we formally pass to the limit of large thermal diffusivity in Ps, we get the model of Chavanis-Sommeria-Robert. This remark, due to Philippe Laurencot, can be viewed as another derivation of that astrophysical model.

Indeed, let (~1, ph, t?~) be solutions of the &eater model with h + cc and A4 = in uk dx. Since

where jA = -K Vuh + ~VP~ , >

under the homogeneous Neumann condition for oh on aQ2, we formally get in the limit h + co

&(x3 t> -+ fico(t), bh, @J + bx~ Pco). (18)

The limit functions satisfy

(k& = -V . .L Acam = u,, 1

CMticcJ + 2 s

um(pm dx = E = const. Q


These are the limit drift-diffusion and Poisson equations (l)-(2), together with the energy relation (3).

The authors of [16, 17, 25, 261 studied the problem Pn with a different Dirichlet condition (p = -aM on INS& c~ > O-a fixed constant, Q-the unit ball in R3, radially symmetric solutions, and the normalized mass A4 = 1. Our approach is more general because we admit arbitrary domains in Jl@, thus not necessarily radial solutions, and a shift of their p translates the boundary condition to that of Pn with the energy adjusted by an additive constant, as was already done in [ 181. In the latter reference, the results on the existence of solutions were proved in (d > 3)-dimensional case, and the radial symmetry was still used in the proof of blow-up of solutions in d = 3.

In an earlier paper [25] certain existence and regularity results for Pn have been considered in the unit ball but for not necessarily radial initial data.

To finish the discussion of the particular properties of the Chavanis-Sommeria- Robert model, let us recall that the authors posed there c = 3/2 since their model had come from a kinetic system under the Maxwellian distribution with respect to velocities. We consider here an arbitrary c > 0 which may lead to a different fraction of the thermal and potential parts in the energy balance (3).

The structure of set of stationary solutions of the Streater, and thus the Chavanis- Sommeria-Robert, model is determined by the Poisson-Boltzmann-Emden equation with 0 = const (following from the entropy production formula (15) with -$ Ws = 0, cf. also (20) below),

(19)

with a suitable boundary condition (either (5) or (6)), and the fixed level of the energy E E R. This problem has been studied in [7, 15, 241. The steady states are basically determined by the solutions of the (unscaled) equation (19) with 0 = 1, so by the stationary solutions of the problem Pi analyzed thoroughly in mathematical and physical literature, see [7, 261 and references therein. Let us recall the most typical results from [7, Theorem 4.2; 15, Theorem 2.81 in this direction.

Given E E R and M > 0 in two-dimensional bounded star-shaped domains or annuli, there is a solution of (19) with (3) and (5). This result is valid for any bounded domain when the free condition (6) is considered.

In higher dimensional star-shaped domains with d >_ 3, the existence of steady states depends on the value of the quotient E/M2. Usually, there are constants JZj = lj (a) E W, j = 0, 1, such that if E/M2 > 4 1, then there is at least one solution of (19), (3), and if E/M2 < .t 0, then there is no stationary solution with given E and M. For certain domains !Z.O = er, e.g. for balls in Rd or, more generally, if the solutions of (19) with 6 = 1 lie on a single branch, and thus they can be parametrized.

The answer to an interesting question, cf. [7, Section 5; 15, Section 31 whether la > 0, depends on Q and, of course, on the actual value of the parameter c in the energy relation (3).


Whenever the initial data (~0, 60) satisfy E/M2 > li, one may expect that solutions of Pu and Pf emanating from (~0, 00) can be continued to global in time solutions, since there is a steady state that could be an accumulation point of the trajectory (u(t), 19(t)). On the other hand, if E/M2 -C CO, it is a natural conjecture that solutions blow up in a finite time because there are no candidates for w-limit points of the trajectories.

We will show results related to this conjecture for star-shaped domains in (d 2 3) dimensions for the both problems Pu, Pr in Theorems 3.1 and 3.2. A similar conjecture for the Streater model Ps, posed in [7, 1.51, seems to be much harder to prove.

2. The two-dimensional case We prove in this section the existence of global in time solutions of the prob-

lems PD and Pf in bounded two-dimensional domains. This will be shown in two separate announcements: Proposition 2.1 (local in time existence) and Theorem 2.1 (continuation of solutions). In fact, we will prove the former result for the problem Pf only. The proof of the local existence for the problem PD is similar, and is contained in [18, Theorem 3.21 (where the three-dimensional case is studied using a slightly different idea than in our Proposition 2.1). However, for the readers convenience, we give here the main ideas of the reasoning for Pf, based on the construction for isothermal problems in [l l] and for the prescribed temperature problem in [18].

We begin with an a priori lower bound estimate for the temperature valid in any bounded d-dimensional domain.

LEMMA 2.1. Each solution (u, 6, ~0) of either the problem PJJ or Pf with 00 > 0 in (7) has its temperature bounded from below by a constant depending on the initial data

f-w> 2 01 = %(uo, 000, a> ’ 0

for each t > 0 whenever the solution exists.

Proof: This was observed already in [25, Proposition 2.2~1 for radially symmetric solutions in the three-dimensional ball. Here we use the entropy W in (16) satisfying the entropy production formula analogous to (15),

-$W + ~(Ku)-~ ~VU + ;Vq12 dx 5 0.

The proof of (20) is formally obvious. Its justification can be made as in [12] and [8] for isothermal models, using regularity of solutions obtained in [18], or as in a simpler situation in [25], i.e. multiplying (1) by log(u + E), integrating by parts using (2), (3), and then passing to the limit I \ 0.

Now, from the estimate W(t) _( W(O), it follows that

cMlog?Y = s ulogudx - W(t) 2 Mlog$ - W(0) > -eK11S21 - W(0). (21) c?


Here the first inequality is a consequence of the Jensen inequality applied to the convex function s H s logs, and the second of s logs >_ --e-l. Therefore, the lower bound in Lemma 2.1 is satisfied with a temperature 61 > 0. 0

The next lemma gives an upper a priori bound for the temperature. This is a specifically two-dimensional result.

LEMMA 2.2. Let (u, 6,~) be a solution of the problem PIJ or Pf in a bounded two-dimensional domain Cl. Then

for each t > 0.

Proof: Since from (3) we have cM6 - E = -i lo UP dx, using the Young inequality with a fixed N > M/(41t) we obtain

2cM0 - 2E = s

u(-v>dx 5 s

Nu log(Nu) dx + s

exp (-p/N - 1) dx cl n n

5 N(W(0) + cM loge) + MN log N + e-‘@(Q).

We used the inequalities sz 5 s logs + eZ-’ for s > 0, z E IR, and

(22)

s expWI)dx I @W2) (23) n

with a constant @(a) independent of +, valid whenever u 3 A+ >_ 0 and m = so u dx < 4~. For the completeness of argument we recall the proof of (23) (due, essentially, to H. B&is and F. Merle): q(x) = IQ K(x, y)v(y) dy, where the kernel K is either K(x, y) = E~(x - y) = & log 1x-y ] in the case of the free condition (6), or K(x, y) = Go(x, y) is the Green function of the domain Q if (5) is assumed. In both cases

I~@? Y)I 5 cw2) - & loi? lx - Y I

(see, e.g., [19, Proposition 1.21]), so that the application of the Jensen inequality leads to

s c.? exp(,~,~dx~Sne-*p(mSnI~(x,Y)lu(Y)m-’dY) dx

< s (s

m-l - v(y)Clx - yl-m’(2k)dy dx >

5 flr,rn-‘v~y) (l Ix - yl-“/““‘dx) dy

= C2n(2 - m/(2n))-1R2-m’(2n) = Q(Q) < 00,


where B is a ball of radius R containing the domain Q (e.g. with R = 2 diam S’J) because m/(27r) -C 2. Since M/N < 4n, the inequality (22) implies

cM(26 - Nlog8) 5 2E + NW(O) + G’(Q) + MNlogN < co,

and this immediately gives

0 < 61 5 G(t) 5 792 = t$(U& Bo, cl) < Ccl,

concluding the proof. 0

A local existence result obtained under the assumption ua E U’(s2), with an exponent p > 1, is in the following Proposition 2.1.

PROPOSITION 2.1. The problem Pf, consisting of Eqs. (1) (4) (6) and the relation of energy (3), in two-dimensional bounded domains with 0 5 ug E LP(Q), p > 1 and IYO > 0, has a unique local solution. This solution belongs to LrC((O, T); LP(st)) for some T > 0, and satisjies u > 0, 0 > 0.

Proof: Let us begin with the case p = 2, the general p > 1 can be treated by a standard approximation procedure as in [8, Theorem 2(ii)]. A solution of the problem Pf on Q x (0, T) is a function u E LDo((O, T); L2(s2)) nL2((0, T); H1(G)) satisfying for each test function 11 E H1(SI x (0, T)) and for a.e. t E (0, T) the identity

s 4x9 t)r(x, t> dx - uyt dxdt + s2

Vu + $Vqj. Vydxdt

= s uo(x>r(x, 0) dx, n with q= E~*u.

We will show that any weak solution u satisfies ut E L2((0, T); (H’(Q))‘), so 6 determined by (3): 6(u)(t) = (CM)-’ (E - i so u(x, t)v(x, t) dx), is a continuous function on [0, T].

Given TO > 0, let X = L4((0, TO); L2(s2)), and let B = BR(O) be the ball of radius R > 0 in the space X.

For a fixed element n E B c X we consider the linear problem

(24)

@(x, t> = s K(x, y)v(y, t) dy with $2

K(x, y) = & log ]x - y],

E=cMB+; s

v$ dx, n

u(x, 0) = z&(x).


If we assume additionally that r?(v) 2 60/2 > 0 on [0, 7’01, then

as well as

Here and below, C are constants which depend on uo and R only. These inequalities imply, as in [ 11, Theorem 11, that (24) has a unique weak

solution U(X, t) on some interval [0, Tt], where Tt > 0 depends on uo and Q. This solution enjoys the following regularity properties:

there exist constants Al, AZ, A3 depending on R, ~0, and Q such that

II~(t)ll~,((O,r);LZ(p)) 5 4t for each t E PI Cl,

IIVUll2L2((O,T,);L2(n,, 5 A2r lbt ll~q(o ~~).(~l(n))‘) 5 A3. , I

In fact, the energy inequality for the problem (24) reads

(25)

;fld: + Ivul; i -&l2~d3lbl6

1 c lbl:‘2(142 + 1) I -II#&,) + -lul;(lvl2 + u4, 2 0:

(26)

where we used the interpolation inequality ]U 13 5 C ]I u ]I if(o, ]U 1:‘” valid for d ( 3. Integrating (26) over the interval [0, t] we get

Hence the two first estimates in (25) are satisfied. To prove the last inequality in (23, note that for each x E H1 (Q) we have

s 1 IWI~ + IW~ dt 5 lwl~ exp

0 ~~ll~ll~l((u,T);L’(p)) + 1) fi4 >

i b&xp(-$R4+1)).

5 Ibll,l(,)lkII~‘(n) + -&lUi31V@~d&P(s2,

I Il~llH’(~)IlxIIHq~) + ~ll~ll~:,,,l~I:‘2cl~12 + UIIXIIH’(Q)~


Hence II~Al&I~n)~~ F I141H~~~2) + ~IIUII~~~n~I~l~‘2~l~12 + 1))2y and so we get the

regularity of uf as stated in (25). Let W be the subset of B c L4((0, T2); L2(s2)), T2 = min{Tr, 00/(2AlA3)},

of V’S which satisfy V(X, 0) = ~0, and inequalities (25) for II instead of U. These conditions guarantee the compactness of W in X. If u E W then 0(v)(t) 2 730/2 on some interval [0, TJ as required in the construction of the solution of (24). We have I$W(t)l = Iso ~$1 5 11~ t (Hl(Q))‘]~]2, and hence s(u)(t) > 60/2 on [O, T3], II and a(t) is a continuous function.

These properties of the auxiliary linear problem (24) imply that if u E W, then the solution u = N(V) of (24) is in W. Hence the nonlinear operator N transforms W into itself and is continuous. The existence of a fixed point of N, which solves the problem (24), is a consequence of the Schauder theorem applied to the compact and convex subset W c X. The positivity property and uniqueness of u can be proved as in [ll]. 0

REMARK 2.1. An analogous local existence result for d-dimensional problems, d > 3, can be proved for ua E U(Q) with some p > d/2. We refer the reader to [18, Theorem 3.81, where the proof used Lemma 2.1 and the constructions in [8, 121. These proofs can be extended also to the case of electric interactions instead of the gravitational ones.

THEOREM 2.1. For every (u0,60) with finite energy (3) and finite entropy (16), local solutions of both problems Pn and Pf in two-dimensional bounded domains can be continued for all t 2 0, and they satisfy the uniform a priori estimates

ulogudsfiluqdxl+fVt)] <m. (27)

Proof: Combining the a priori estimates

sup is

ulogudx + Is I1

uqodx = c = C(u0, a,> < co, (28) f?O c? n

O(t) z 61 > 0, and time regularity of r9 = O(t), following from (16) and (3), respectively, we can apply the argument in the proof of [12, Theorem 2(iv)], cf. also [8, Theorem 31. Indeed, using the inequality

IS,wdx+S, lv.12dx~ 5 CW2, luold

(see [12, Eq. (24.2)]) and the nonlinear embedding inequality

14; 5 ml2 ,q,Jvlog Ml1 + Gl41 (29)

(given E > 0 there exists C, > 0 such that (29) is valid for each 21 E H1(Q), see [8, Eq. (22)]), we get the energy inequality for either Pu or Pr (a counterpart of (26)). That is

216 F! BILER and T. NADZIEJA

12 5 ~I141H’(Q) + ClUll

holds for some C = C (~0, 00). Thus we arrive at

5 w4lH’(n,l41 + CM 5 ;ll~ll2,1,,, + CM(M + l),

and the resulting inequality -$ ]U ]Y$ + ]u]i 5 C leads to the uniform boundedness of u(t) in L2(s2). In fact, even a uniform estimate of SUP,,~ ]u(t)lp with p > 1 would be sufficient.

We refer the reader for a detailed presentation to the original papers since this is a rather lengthy proof, based essentially on the properties of elliptic equations in the Hardy space Xl(Q) in the two-dimensional case, cf. [14, Remark, p. 1931. Once the a priori bound on ]u(t)]2 is obtained, Proposition 2.1 shows that solutions with such a bound can be continued indefinitely in time t. 0

REMARK 2.2. The same a priori estimates (28) have been obtained in [lo, pp. 367-3681 for the Streater model P s. However, since we have no local solvability result for Ps under assumptions like uo E L2(s2), 7%~ > 0 at our disposal, we cannot conclude on the possibility of indefinite continuation of solutions of Ps as was before for the CSR model.

REMARK 2.3. We expect that in the two-dimensional case solutions of the problems PD and Pf converge to corresponding steady states (cf. an analogous situation in [8, 51) which exist for any A4 > 0, E E R, due to the results recalled at the end of Introduction. Recent results in this direction obtained by Ignacio Guerra and Tadeusz Nadzieja will appear in Colloquim Mathematicum.

3. Blow-up of solutions of CSR model in higher dimensions As was remarked in Introduction, solutions of the isothermal problem Pi with

large initial data (in the sense of either large enough mass or a high local concentration) blow up in a finite time. The proofs in the case of the free condition are given in [2, 4, 121, the latter in the case of Q = Ii@. The papers [3, 9, 131 are devoted to an analysis of the radially symmetric blow-up in balls. The authors of [ 181 proved a result on blowing up solutions in d-dimensional ball, d > 3, c > 3 ([ 18, Theorem 5.11) using essentially the same method involving moments of the distribution of mass U(X, t) dx as in [9] and the energy relation (3).


Our aim in this section is to prove the existence of blowing up solutions of the problem Pr (as well as Po) in star-shaped domains in IRd, d > 3. This will be done for particular values of the energy E and the constant c in (3). The method of proving blow-up for Pr is the same as in [12, Theorem 2(v); 2, Theorem l] but for Po the crucial estimate is based on the Pohozaev identity, and it is different from that in [2, Theorem 21 for the problem Pi in the unit ball of 8%’ with not necessarily radial solutions.

THEOREM 3.1. If 52 c Rd, d 2 3, i s a bounded star-shaped domain, then for c > d/(d - 2) the solutions of the problem Pf with the initial data satisfying (3) with E ( 0 cannot be continued to global in time ones.

Proof: First, remark that such initial data do exist for any fixed tYa > 0. It suffices to consider highly concentrated densities like either truncations to Q of Gaussian distributions approaching the point measure or the functions ua = fiI@, with M > 0, w c a, ]w] (< 1.

Since the problem Pr is translation invariant, we may suppose with no loss of generality that CJ is star-shaped with respect to the origin. Define an auxiliary function (the second moment of the distribution of mass U(X, t) dx) w(t) = s u(x, t)lxl”dx. (30)

s-2 The function w measures the concentration of u at the origin. For weak solutions of Pr with K = 1 we get

Vu + $Vcp) . V (1~1~) dx

= -2 s

Vuexdx-2 s

uVq.xdx s-2 fi n

= -2 s

uxevdo+2d s

udx an c2

(31)

Since Q is star-shaped with respect to 0 E s2 c IRd, we have x . u > 0 on XJ, and thus

dw 2 dtWJf-aa,

ss 4x, t)u(y, t) (1~1~ - Y . x) Ix - Y I-d dy dx

c2xn

=ZdM-& u(x, t)u(y, t) lx’* - ’ d

; “_,;; ’ + Iy12 dx dy X

= 2dM - $I, (32)


by the symmetry properties of the integral

I =a,l /I

u(x, t)u(y, t)lx - y12-d dx dy. SlXQ

Clearly, the energy relation (3) can be rewritten as

1 E=cM6-

2(d - 2) I*

Therefore, (32) leads to

so that under the assumptions of Theorem 3.1

dw -c---E<0 dt -

(33)

(34)

(35)

holds for I = 2M((d - 2)c - d) > 0 and E 5 0. The differential inequality (35) implies that w(t) < 0 for t > T with some T I w(O)/& which is absurd. Thus we arrived at a contradiction with the assumption on the global in time existence of nonnegative solutions. This proof easily extends to the case of ~(9) = z9 if c 2 d/(d - 2) and E < 0. 0

The remarks in [2, pp. 233-2361 apply, mututis mutundis, to the present case. In particular, one can weaken the assumption on the geometry of the domain to a condition that s2 resembles a dumbbell, i.e. an anisotropic dilation x = (xi, . . . , xd) H (Wl,..., adxd) with some aj > 0 carries Q onto a star-shaped domain.

Next, we observe that if w(t) -+ 0 for t /” T, then lo u log u dx -+ 00 as t 7 T by the Shannon inequality (see [2, Lemma 11)

s vlogvdx > cl -czlogW P

(36)

valid for all the densities V(X) 2 0, so v(x) dx = M, X = MS’& xv(x) dx-the center of mass of v and the centered moment ZT, = so V(X) Ix - X I2 dx ( w, and with some constants cl E R, c2 > 0, independent of V, cj = cj(Q, M), j = 1, 2.

The nonincreasing property of the entropy (16) shows that 6 + cc as t 7 T, and the energy relation (3) implies so up dx -+ -cc as t 7 T. All these limit relations show that a blow-up of a solution is accompanied by an infinite growth of all the quantities defining the energy (3) and the entropy (16). However, it may happen-and we strongly believe that this is actually the case-that this solution ceases to exist before time T. The reason for such a conjecture is that in higher dimensions the necessary conditions for the existence of regular weak solutions, say


in Lrc, as in [ 18, Theorem 3.8]-based on a bit simpler considerations in [ 121, are connected with a threshold value of the Morrey space Mdi2(Q) norm of u(t),

Il~(thW(~) = x;;;>. R2-d s

BR(X)“n U(Y? t) dY.

Namely, we conjecture that if u is a weak solution, then its Mdj2 norm is controlled locally in the x-space in the sense that

lim sup R2-d s

u(y, t) dy < cd0 (37) n&-2 R-0 h(x)nQ

for an absolute constant Cd. Of course, such a constant Cd should be less than the Md/2(Sl) norm of the singular Chandrasekhar solution 2(d - 2)]xlm2 of Pr in the unit ball in JRd, d > 3. For a thorough discussion of singular solutions of the equation A@ = he- @, h > 0, related to (19) we refer the reader to [ 11.

The proof based on the evolution of the moment w(t) of u(t) involved merely the L’(Q) integrability of the solution, so it could give us only an indirect cause of the loss of u(t) as t 7 T. Finally, we remark that IluljMd/ZcQ2) 2 Cdkf(M/fb)d’2-1,

see [2, (WI, so that ll+)ll&W(n) also grows indefinitely before t = T. Now let us consider the problem P o. We have a result analogous to that in

Theorem 3.1.

THEOREM 3.2. Zf !A c IWd is a bounded star-shaped domain, then for c > d/(d - 2) and E 5 0, the solutions of PD blow up in a finite time.

Proof: Take again the moment w(t) defined in (30) and calculate

---E-Z Vu.xdn:-2 dw dt s s

uV6p. xdx i-2 6 i-2

= -2 s

uxevda+2dM an

(38)

Here I = -(d - 2)jQuqddx = (d - 2)iQ]Vqo]2dx and the integral &,uV~~~.xdx has been transformed using the Pohozaev-Rellich identity

s AqVq.xdx = i L?

2~Qx.u(~)2d~+~jjq12dx (39)

valid for all functions q E C2(s2) vanishing on XL The conclusion of the proof follows the lines of the proof of Theorem 3.1 since

the energy relation is again (34) with Z as above. Noting that x . u > 0 by the

220 P BILER and T. NADZIEJA

star-shapedness of 8, and E ( 0, from (38) we thus obtain

dw -p-E<0

with I = 2M((d - 2)c - d). As before, the assumption that u solves Pn for all t 2 0 leads to a contradiction: w(t) < 0 for large t. 0

REMARK 3.1. Here, we take the opportunity to generalize the results in [2, Theorem 21 for the isothermal problem Pi in balls, extending this to the case of star-shaped domains in I@, d > 3. N amely, we prove that for M = lug 11 large enough the solution of Pi with the Dirichlet condition (5) for (p ceases to exist after a finite time. Indeed, performing calculations analogous to those in (38) we get from (8)

dw dt 5 2dM - I.

Since the entropy (12) decreases: Wi(t) 5 Wi(O) and combining this with (21), we arrive at the estimate

MlogE < s

1

F-4 - n ulogudx I 2(d _ 2)Z + q(O).

Therefore dw dt 52dM-2(d-2) MlogM+Wi(O)

IQI 5-s <0

holds for M >> 1.

REMARK 3.2. It is easy to see that in the case of a strictly star-shaped domain Q, i.e. that with x . v > 6 on afi for some 6 > 0, the proofs of Theorem 3.2 and Remark 3.1 give a little bit better estimates. Namely, the boundary integral term satisfies

which follows from

This gives a supplementary strictly negative term of order M2/6 on the right hand side of h < 2dM - dr - . . . . So, E can be taken positive (but small enough) in assumptions of Theorem 3.2, as well as the bound for M in Remark 3.1 can be refined.


4. Self-similar solutions

The question how do the nonglobal in time solutions explode is usually much more difficult than a proof of their existence. It is natural to exhibit particular blowing up solutions, and frequently the first attempt is to construct self-similar solutions that blow up.

For instance, in the case of semilinear parabolic equation with a power non- linearity, they determine the asymptotics of all blowing up solutions (at least in a generic sense). In the case of nonlocal parabolic problems like Pi and Pr , or the chemotaxis problem of Keller-Segel studied in a series of papers of M. Herrero and J. J. L. Velazquez, the picture is even more complicated. Namely, there are no self-similar blowing up solutions of finite mass for the chemotaxis system and, therefore, for Pi in two dimensions. There exist examples of solutions that blow up in an approximately self-similar way for d = 2, cf. [21]. For exactly self-similar blowing up solutions in d = 3, see [20]. These examples show that blow-up in d = 2 is accompanied by a concentration of a part of mass at a point, while in d = 3 solutions may blow up with mass vanishing at any fixed neighborhood of any point when t approaches the blow-up time T. Interesting results on various self-similar solutions of CRS model are in [16] and [26].

Here we recall some results from [3, Proposition 41 adapting them to the study of the Chavanis-Sommeria-Robert model Pf in the case of fi = I@ and radially symmetric solutions, cf. [16, Section 41 for another approach.

First, observe that if u(x, t) is radially symmetric density, then the integrated density Q(Y, t) = od li pd-‘u(p, t) dp, r = ]X 1, satisfies the second order diflerentiul equation (no more nonlocal terms appear!)

Qr = Qrr - FQr + odti;d-I QQr. (40)

The passage from the formulation Pf to (40) can be done as in [9, Eq. (9)] or [18, Section 2.41.

If U(X, t) = (T - t)-l< (r2(T -t)-‘), r = 1x1, is a Boltzmann-form self-similar solution blowing up as t /” T, and 0 = const (i.e. in the canonical ensemble in the sense of [16, Section 4.2]), then Q is self-similar: Q(r, t) = ad(T-t)d/2-‘<(r2/(T- t)), and the function t satisfies

(41)

Here y = r2(T - t)-’ is the self-similar independent variable, ’ = d/dy, and e(O) = 0. Eq. (41) with r9 = 1 was derived in [3, Eq. (13)].

PROPOSITION 4.1. Zf d >_ 2 then for each nondecreasing solution 0 5 $ f 0 of (41)

J$nmt(Y) = 00,


but the growth of e is controlled

e(y) 5 6 (yd’2 + 2dyd’2-‘).

Moreover, if d = 2 then lim,,, {(y)/y = 812. Proof: The first relation (shown in 1994 by A. Krzywicki with a different

argument, cf. also [20, Section 61) means that there is no self-similar solution of finite mass. All the solutions of (41) must have infinite mass, and therefore, the infinite potential energy Jo UCQ dx.

The Chandrasekhar solution r(y) = 26ydi2-l corresponding to the stationary solution 2(d - 2)6 Jx I-2 of Pr, and another explicit solution e#(y) = 6yd12/d corresponding to the uniform growing-in-time density, satisfy (41). The other solutions, including the explicit one t*(y) = 46ydi2(y + 2(d - 2))-’ discussed for d = 3 in [16, Eq. (77)], will be shown to display (at most) that growth.

Let d = 2 and suppose that e(y) 5 A for some A < co and a nondecreasing solution c of

6” - ;,g + &c’ = 0 (42)

obtained from (41). This assumption leads to <” - be’ + &.$’ > 0 and, after two integrations, to

f3Y) 2 6(Yo) + $f(yo)e-Y0/4yA/(21P) e~/4S-A/(2fi)dS 0

>

for some yo > 0, (‘(~0) > 0, which tends to 00 as y + 03. The case d > 3 is only a bit more difficult. Let z = e’ - y.$, then z satisfies

1 z’- -z- d-2

4 lyl$+&(z+y+=O. (43)

Suppose that e(y) 5 A < 00 so that y (2~~/~@)-‘<(y) - 1 5 0 for large y, say y > ya > 0. Now, Eq. (43) implies z’ + - - i z > 0, and after integration, 20$,2 >

A l-d/2 (d - 2)Oyo >

> o *

Finally, we get

(c(y)~'-~/~)' 3 .,$'/4y'-d/2 and c(y) > ~~/~-'ao s

’ es/4S1-d/2 ds YO

which again tends to 00 as y + co.


The second conclusion of Proposition 4.1 follows from multiplication of (41) by y and integration over [0, Y]:

Yly(Y) - $(Y) - &(Y) + ; s

oyC(Y)dY

+~y1-“/2~2(Y)+(~-1)~~Yy-~~2~~~y~dy=0.

The above relation implies

C2(Y) 5 6 {Yd’2c(Y) + 2dY”‘2-‘c(Y)} )

so the upper bound for c(Y) is obtained. Concerning the asymptotics of c for d = 2, it is routine to check that for

solutions e of (42) with z9 = 1 the function L(y) = 4.$‘(y) -2logc’(y) +y(e(y)/y - 1/2)2 decreases in y since $$ = -O(Y>/Y - 1/212. Thus, UY) 5 UYO) < 00 for an ya > 0 implies the convergence of <(y)/y to l/2. In the general case 6 > 0 we use the scaling c t-+ St. q

In the remainder of this section we adapt to the framework of the Chavanis- Sommeria-Robert model some results from [3] related to the forward in time self- similar solutions enjoying radial symmetry.

Homogeneity properties of the system (I), (2), (6) with a constant 0 > 0 considered in I@ imply that if u solves this system, then the resealed function u~(x, t) = h2u(hx, h2t), h > 0, is also a solution. Thus, it is natural to consider solutions which satisfy the scaling property uk = u, i.e. forward self-similar solutions. By the very definition, they are global in time, and one may expect that they describe large time behaviour of general solutions of the problem Pf in Rd. A discussion of those solutions (existence and singularities) without radial symmetry assumptions can be found in [4].

If uh. G u, then u(x, t) = te1U(x2/t) and the initial condition lim,\a u(x, t) is a function homogeneous of degree -2 at the origin. The associated integrated density Q satisfies Q(r, t) = adtd’2-‘<(r2/t) with a nondecreasing positive function {. It is easy to check that 5 = c(y), y = r2/t, solves the equation (cf. [3, Eq. (ll)])

TT ‘=O, (44)

with ’ = d/dy, and t(O) = 0. The Proposition 3 in [3] leads to the following result.

PROPOSITION 4.2. (i) For d = 2 there exists a solution of (44) such that lim,,, c(y) = Z is finite if and only if Z E [0,48). (ii) Zf d > 3, then for each nondecreasing solution < $0 of (44)

s+(Y) 5 7.9 d - 2 d/2 dy +4(d - l)~~‘~-l

and limy+ca y 1-d’2<(y) E (0, co).


REMARKS 4.1. (i) Means that in two dimensions for each mass M E [0, Snz?) there is a forward self-similar solution of mass M.

(ii) Shows that in d ) 3 dimensions, nontrivial forward self-similar solutions have infinite mass and the asymptotics of c(y) as y--+ cc is that of the stationary Chandrasekhar solution E(X) = 2(d - 2)1Yjnl-~, i.e. c(y) = 273~~‘~~~.

The proof of Proposition 4.2 involves a rather subtle analysis of the asymptotic behaviour of solutions of a nonautonomous system in the plane for the quantities ds) = Y 1-d/2{(y), v(s) = 2~~-~/~<‘(y) with a new independent variable s = i logy, see [3, pp. 183-1851. 0

Acknowledgements The preparation of this paper was partially supported by the KBN grants 5O/PO3/

2000/18, 2/PO3A/Ol l/19, the Austrian-Polish GAD-KBN Technical-Scientific Collab- oration Agreement as well as the Wittgenstein Award 2000 of Peter A. Markowich financed by the Austrian Research Fund FWF. Support by the European Network HYKE, funded by the EC (contract HPRN-CT-2002-00282), is also acknowledged. We are much indebted to Jean Dolbeault for communicating us the CSR model and results in [25], and helpful comments. We thank Carole Rosier, Pierre-Henri Chavanis, Mark A. Peletier and Ignacio A. Guerra for sending us preprints and relevant remarks. The authors are grateful to Peter A. Markowich for hospitality at the Erwin Schrijdinger Institute, Vienna, and interesting conversations. Finally, it is our pleasure to express hearty thanks to Philippe LaurenGot for collaboration on related topics, many pertinent observations and friendly valuable comments.

REFERENCES

[l] M.-F. Bidaut-V&on, L. V&on: Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math. 106 (1991), 489-539; 112 (1993), 447 (errata).

[2] P. Biler: Existence and nonexistence of solutions for a model of gravitational interaction of particles, III, Colloq. Math. 68 (1995), 229-239.

[3] P. Biler: Growth and accretion of mass in an astrophysical model, Appl. Math. (Warsaw) 23 (1995), 179-189.

[4] P. Biler: Local and global solutions of a nonlinear nonlocal parabolic problem, Proc. Banach Center semester “Nonlinear Analysis and Applications, Warsaw 1994”, Gukuto ht. Series Math. Sci. Appl. 7, N. Kenmochi, M. Niezgddka and P. Strzelecki (eds.) 1995, 49-66.

[5] P. Biler and .I. Dolbeault: Long time behaviour of solutions to Nernst-Planck and Debye-Htickel drift- diffusion systems, Ann. Henri Poincare’ 1 (2000), 461472.

[6] P. Biler, .I. Dolbeault, M. .I. Esteban and G. Karch: Stationary solutions, intermediate asymptotics and large time behaviour of type II Streater’s models, Adv. Difl Eq. 6 (2001), 461-480.

[7] P. Biler, J. Dolbeault, M. J. Esteban, P. A. Markowich and T. Nadzieja: Steady states for Streater’s energy-transport models of self-gravitating particles, l-25; Proceedings of a conference held at IMA, Minneapolis, June 2000, Springer IMA Series (2003), to appear.

[8] P. Biler, W. Hebisch and T. Nadzieja: The Debye system: existence and long time behavior of solutions, Nonlinear Analysis 1: M. A. 23 (1994), 1189-1209.

[9] P. Biler, D. Hilhorst and T. Nadzieja: Existence and nonexistence of solutions for a model of gravitational interaction of particles, II, Colloq. Math. 67 (1994), 297-308.

[lo] P. Biler, A. Krzywicki and T. Nadzieja: Self-interaction of Brownian particles coupled with thermodynamic processes, Rep. Math. Phys. 42 (1998), 359-372.


[l l] P. Biler and T. Nadzieja: A class of nonlocal parabolic problems occurring in statistical mechanics, Colloq. Math. 66 (1993), 131-145.

[12] P. Biler and T. Nadzieja: Existence and nonexistence of solutions for a model of gravitational interaction of particles, I, Colloq. Math. 66 (1994), 3 19-334.

[13] P. Biler and T. Nadzieja: Growth and accretion of mass in an astrophysical model, II, Appl. Math. (Warsaw) 23 (1995), 351-361.

[14] P. Biler and T. Nadzieja: A nonlocal singular parabolic problem modelling gravitational interaction of particles, Adv. DiJJ: Eq. 3 (199X), 177-197.

[15] P. Biler and T. Nadzieja: Structure of steady states for Streater’s energy-transport models of gravitating particles, Top. Methods No&in. Analysis 19 (2002) 283-301.

[16] P.-H. Chavanis, C. Rosier and C. Sire: Thermodynamics of self-gravitating systems, (arXiv:cond-mat /0107345 v2), Phys. Rev. E 66 (2002), 036105.

[17] P.-H. Chavanis, J. Sommeria and R. Robert: Statistical mechanics of two-dimensional vortices and colli- sionless stellar systems, Astrophys. J. 471 (1996), 385.

[18] C. .I. van Duijn, I. A. Guerra and M. A. Peletier: Global existence conditions for a non-local problem arising in statistical mechanics, 1-21, to appear.

[19] Q. Han and F. Lin: Elliptic Partial Differential Equations, Courant Lecture Notes in Math., AMS, Providence, RI, 2000.

[20] M. A. Herrero, E. Medina and .I. .I. L. Velazquez: Self-similar blow-up for a reaction-diffusion system, J. Comput. Appl. Math. 97 (1998), 99-119.

[21] M. A. Herrero and .I. J. L. Vel&zquez: Singularity patterns in a chemotaxis model, Math. Ann. 306 (1996), 583-623.

[22] A. Jtingel: Quasi-hydrodynamic Semiconductor Equations, PNLDE 41, Birkhauser, Basel, Boston 2000. [23] Ph. Laurencot: Weak solutions to a Fife-Penrose model with Fourier law for the temperature, J. Math.

Anal. Appl. 219 (1998), 331-343. [24] T. Nadzieja and A. Raczynski: Radially symmetric solutions of the Poisson-Boltzmann equation with a

given energy, AppZ. Math. (Warsaw) 27 (2000), 465473. [25] C. Rosier: Probltme de Cauchy pour une equation parabolique modelisant la relaxation des systemes

stellaires auto-gravitants, C. R. Acad. Sci. Paris, skr. I Math. 332 (2001), 903-908. [26] C. Sire and P.-H. Chavanis: Thermodynamics and collapse of self-gravitating Browniun purticles in D

dimensions (ArXiv: cond-mat/0204303 VI), l-35; Phys. Rev. E 66 (2002), 046133. [27] M. Smoluchowski: Drei Vortrage tiber Diffusion, Brownsche Molekularbewegung und Koagulation von

Kolloidteilchen, Phys. Zeit. 17 (1916), 557-571, 585-599. [28] R. F. Streater: Dynamics of Brownian particles in a potential, J. Math. Phys. 38, no. 9 (1997), 4570-4575. [29] R. F. &eater: The Soret and Dufour effects in statistical dynamics, Proc. R. Sot. London A 456 (ZOOO),

205-2 11. [30] G. Wolansky: On steady distributions of self-attracting clusters under friction and fluctuations, Arch. Rut.

Mech. Anal. 119 (1992), 355-391. [31] G. Wolansky: Comparison between two models of self-gravitating clusters, Nonlinear Analysis 24 (1995),

1119-1129.

Global and exploding solutions in a model of self-gravitating systems

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