A Singular Problem in Electrolytes Theory

Mathematical Methods in the Applied Sciences, Vol. 20, 767—782 (1997)MOS subject classification: 35B 40, 35K 60, 76W05, 82D, 92E

A Singular Problem in Electrolytes Theory

Piotr Biler* and Tadeusz Nadzieja

Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Communicated by B. Brosowski

Existence and nonexistence of solutions (both stationary and evolution) for a parabolic—elliptic systemdescribing the electrodiffusion of ions are studied. The answers to some questions concerning a generalizationof a problem posed by I. Rubinstein are given. ( 1997 by B.G. Teubner Stuttgart—John Wiley & Sons Ltd.

Math. Meth. Appl. Sci., Vol. 20, 767—782 (1997)(No. of Figures: 0 No. of Tables: 0 No. of Refs: 24)

1. Introduction

This paper is devoted to a study of a particular model problem of electrodiffusion ofions. The intriguing phenomenon of electrodiffusion appears to be of importance invarious areas of science: electrolysis, semiconductor theory, biological systems (poly-electrolytes, membrane transport). The book [21] is a very good comprehensiveintroduction to mathematical aspects of these problems.

The history of basic PDE’s in electrolytes theory is quite long. W. Nernst andM. Planck formulated at the end of the nineteenth century a system of para-bolic—elliptic equations whose simplified form reads

ut"+ · (+u!u+u), (1)

*(u#» )"u. (2)

Here u, u : )]R`PR, » :)PR, and ) is a bounded smooth domain in Rn. Thisnon-local (since +u depends in a non-local way on u) system governs the temporalevolution of the spatial density u"u(x, t) of charged particles (ions of the same sign)moving in an electrolyte confined to the container ). The ions create the electricpotential u through collective effects, and move directed by the gradient of u anda given potential ». In this description the diffusion and frictional forces proportionalto the ions velocity are taken into account. For a more detailed physical backgroundof this model we refer the reader to [13, III Abstract, Kap. 14.2]; [21]; [22; chapter 2,section 2]. We assume one of the conditions (3.1)—(3.3) for u and u on the boundary )

u

l"0,

ul

"0 on ), (3.1)

*Correspondence to: P. Biler.Contract grant sponsor: KBN; contract grant number: 0472/P3/94/07

CCC 0170—4214/97/090767—16$17.50 Received 11 June 1995( 1997 by B. G. Teubner Stuttgart—John Wiley & Sons Ltd.

u

l!u

ul

"0, u"0 on ), (3.2)

u

l!u

ul

"0 on ), u"En *

u!», (3.3)

where l denotes the exterior unit vector normal to ). Either (3.1) or the first (no flux)condition in (3.2)—(3.3) guarantees the conservation of the total charge :) u,q

0. The

condition imposed on u means that either the wall of the container is grounded(equation (3.2)), or isolated (equation (3.3)). Equation (3.1) is relevant in the radialsymmetry case, i.e. if ) is a ball, when the global electroneutrality condition issatisfied: q

0":) u":) +» · l,q* .

To determine completely the evolution, the system (1)— (3) is supplemented with theinitial condition

u(x, 0)"u0(x)*0. (4)

The problem (1)— (4) has been studied with »,0 in [2], [15] and, in a greatergenerality, with sufficiently regular » ’s in [8]. The full Nernst—Planck—Debye—Huckelsystem for several ions species has been treated in [6].

The system (1)— (3) with a singular potential »"q*En, q*'0, E

nbeing the

fundamental solution of the Laplacian on Rn, is of our interest here. Recall thatE2(z)"(2n)~1 log Dz D, E

n(z)"!((n!2)p

n)~1 Dz D2~n for n*3, p

n!the area of the

unit sphere in Rn . This problem generalizes the equations proposed by I. Rubinstein in[21, section 2.4, Problem 4] to describe the situation when a charge q* is fixed at theorigin x"03) and surrounded by ions of opposite sign moving in a solute. Theoriginal formulation of the Rubinstein problem 4 in [21, section 2.4]

u

t"

1

r2

r Ar2A

u

r!u

urBB ,

1

r2

rAr2

urB"u!q*

d (r)

r2, (5)

ur K

r/1

"

u

r Kr/1

"0,

u(r, 0)"u0(r)*0,

concerns the radially symmetric situation in the unit ball )"B(0, 1)LR3, and hereq*"q

0":B(0, 1) u

0expresses the global electroneutrality assumption.

I. Rubinstein conjectured that in the above problem blow-up phenomena mayoccur when steady states are lacking. As far as we know, this problem has been leftopen in the literature. We will study in the sequel various generalizations of (5) inarbitrary dimensions. The three-dimensional system (1)— (4) with cylindrical symmetrycorresponds to a charged thin wire in an electrolyte. This case is of interest in thetheory of polyelectrolytes, and this leads to the two-dimensional version of (5). As wewill see later, it seems to be the most interesting case from the mathematical point ofview. We will refer to the problem (1)—(4) with the singular potential »"q* E

nas to

the Rubinstein problem whose particular version is (5).

768 P. Biler and T. Nadzieja

( 1997 by B. G. Teubner Stuttgart—John Wiley & Sons Ltd. Math. Meth. Appl. Sci., Vol. 20, 767—782 (1997)

In the case when ) is the ball B(0, R)LRn and u is radially symmetric, thenon-local parabolic—elliptic problem (1)— (4) can be reformulated as a parabolicequation

Qt"Q

rr!(n!1)r~1Q

r!p~1

nr1~n(Q!q*)Q

r, (6)

with the boundary conditions (the same for each of the conditions (3.1)—(3.3))

Q(0, t)"0, Q(R, t)"q0. (7)

Here Q(r, t)":B (0,r)

u (x, t) dx is the integrated density, (/r) Q(r, t)"pnrn~1u (r, t),

compare [7, (6)— (7)]. The initial condition

Q(r, 0)"Q0(r), 0)r)R. (8)

is a positive non-decreasing function, and the obvious compatibility condition is

Q0(0)"0, Q

0(R)"q

0. (9)

Such a formulation allows us to consider some non-smooth densities u.The scaling properties of (6) permit us to assume, without loss of generality, that

R"1. Indeed, together with Q(r, t) and q*, the function R2~nQ(Rr, R2t) and R2~nq*is a solution of (6). The problem (6)— (8) can be transformed, using a new independentvariable y"rn (cf. [7, (12)]; [5, (15)]), into

Qt"n2y2~2@nQ

yy!np~1

n(Q!q*)Q

y, (10)

Q(0, t)"0, Q(1, t)"q0, Q(y, 0)"Q

0(y). (11)

The contents of this paper is as follows. Stationary solutions are studied in section 2:the existence in the radial planar case in Proposition 1, non-existence in higherdimensions in Proposition 2, existence in arbitrary domains in the plane in Theorem 1.The evolution problem is considered in section 3: the construction of global in timesolutions in the radial case in Theorem 2, non-existence of local in time radialsolutions in Proposition 3, non-existence of global solutions in star-shaped domainsin Proposition 4, global existence in planar domains in Theorem 3, and existence ofradial self-similar solutions in Proposition 5.

Besides standard ODE methods used in the analysis of radial stationary solutions,we apply comparison techniques (classical and probabilistic) for parabolic PDE in theproofs of Proposition 3 and Theorem 2, Schauder fixed point arguments in Theorem1, and more subtle approximation tools for weak solutions of non-linear PDE,including various a priori estimates, Lyapunov techniques and compactness resultsvia non-linear imbedding inequalities of Sobolev type in Theorem 3.

We note that the Rubinstein problem, although related to the Debye systemconsidered in particular in [6], and to the astrophysical model in [9], [7], [4], is moredifficult because of singular terms containing the derivatives of the potential ».Therefore, we will make stress on these difficulties while some parts of standardarguments, which can be found in the above cited items, will be merely sketched.However, we will try to keep this paper self-contained.

We use standard notation D · Dp, E · E

m,rfor the ¸p()), resp. ¼m,r ())

(¼m,2())"Hm())) norms of functions on ). Inessential constants will be denoted bythe same letter C, even if they may vary from line to line.

A Singular Problem in Electrolytes Theory 769


2. Stationary solutions

We collect in this section some results on the existence and nonexistence of steadystates of the system (1)— (3). We begin with the radial case in the balls of Rn in theformulation (6)— (7) or (10)—(11).

Proposition 1. For n"2 and each q*3[0, 4n], q0*0, there exists a unique stationary

solution Q of the system (10)—(11), i.e. the steady state of the Rubinstein problem in theunit disk B(0, 1)LR2.

For q*'4n and q0'0, there is no solution of (10)— (11). Similarly, the stationary

equation (10) does not admit any nontrivial positive solution Q, Q (0)"0, in the wholeplane R2.

Proof. For n"2 the stationary version of (10) becomes

4yQyy!n~1(Q!q*)Q

y"0 (12)

which is an integrable equation. Indeed, (12) leads to

4yQy!Q2/(2n)#(q*/n!4)Q"c

0.

After an integration we can easily verify that the constant of integration c0"0, since

the boundary condition Q(0)"0 must hold, and therefore either

Q(Q#8n!2q*)~1"cyc

with c"1!q*/(4n)'0 and an arbitrary c'0, or

Q~1"c!(8n)~1 log y

when q*"4n (i.e. c"0) with c3R. These equalities correspond to either

Q(y)"(8n!2q*)cyc(1!cyc )~1, (13)

with q*(4n, q0"Q(1)"(8n!2q*)c(1!c)~1, c3 (0, 1), or

Q(y)"(q~10

!(8n)~1 log y)~1 (14)

with q*"4n.The non-existence part of Proposition 1 is a simple consequence of the analysis of

the formulas (13), (14).The next result is the non-existence of steady states of the radially symmetric

Rubinstein problem in dimensions n*3.

Proposition 2. For n*3 and q*'0, q0'0, there is no stationary solution Q of the

problem (10)—(11) in the n-dimensional ball, or the equation (10) with Q(0)"0 in thewhole space Rn.

Remark. It is well-known that for q*"0 and each q0*0 there exists a unique

stationary solution of (10)—(11), see e.g. the discussion in [2]; [6, section 4]; [8, section 3].This can be proved using either the phase plane method (similarly to [7, section 2]), ofmonotonicity methods alluded to in [2], or variational methods (cf. the approach in[23, section 3]; [24, section 3]), or in a more general context, the compactness methodfor non-linear integral equations, cf. [8, Theorem 2] and Theorem 1 below.

Moreover, the first part of Proposition 2 can be deduced from non-existence resultsfor the evolution problem in Proposition 3.



Proof. Suppose that QI 0 is a nontrivial positive solution of (10) satisfying Q(0)"0.Since 0IQ (y))q*/2 for some y

0'0 and all y3[0, y

0], it follows from (10) that

n2y2~2@nQyy#np~1

nq*Q

y/2)0

on [0, y0]. Therefore, the function log Q

y(y)!c

1y2@n~1 is non-increasing for some

c1'0. In other words, Q

y(y)*c

2exp(c

1y2@n~1) for some c

2'0 on [0, y

1],

0(y1)y

0is such that Q

y(y

1)'0, so Q

yis not integrable on [0, y

1], hence Q cannot

satisfy the condition Q(0)"0.Now we turn to the stationary problem (1)—(3) in arbitrary bounded domains

)LR2. Observe that time independent solutions º, ' of (1)— (3) satisfy the system

+ · (exp(')+ (exp(!')º) )"0, (15)

*('#»)"º, (16)

and the no-flux boundary condition (3.2) or (3.3) becomes

l(exp(!')º)"0.

Multiplying (15) by exp(!')º and integrating over ), we arrive at

P)exp(') D+ (exp(!')º) D2"0,

which gives º(x)"q0k~1 exp('(x) ) a.e. in x3), with the normalization constant

k"P)exp(' (x)) dx. (17)

Together with (16) this implies the Poisson—Boltzmann equation for the potential

*('#»)"q0k~1 exp('). (18)

The calculations above leading to the Boltzmann form of the density º are somewhatformal but they can easily be justified for, e.g., º3H1()) and '#»3H1 ())W¸=())(see [6, section 4]), so for » bounded and smooth, cf. [8, section 3]. In our case, for»(x)"(2n)~1 q* log Dx D, q*(4n, we look for the steady states of (1)— (3) also in theabove form (18).

The second part of the boundary conditions (3.1)— (3.3) permits us to write (18) inthe form

("!(q0k~1J (exp((!»))#!), (19)

with ("'#» and an integral operator J defined by a kernel K

J(º) (x)"P)K (x, y)º (y) dy.

Here!K is either the Neumann, or the Green function of ) for (3.1), (3.2) resp., or thefundamental solution of the Laplacian for (3.3), and ! is a bounded harmonic functioncorrecting the boundary conditions imposed on '#» (not on ' solely!) in (3). Inparticular, for (3.3) we have !"0. Anyway, K is a function on )]) smooth off thediagonal, whose singularities do not exceed those of DE

nD. Of course, the regularity of

K near the boundary depends on the regularity of ).



As was remarked after the formulation of Proposition 2, the existence of a uniquesolution of the nonlinear integral equation (19) with the normalization (17) for n"2,each q

0'0 and each sufficiently small q**0 can be studied by monotone operators

techniques. For n"2 the variational approach is also possible—this consists inlooking for global minimizers of the convex (and lower semicontinuous in suitablefunction spaces) functional

¼"P)º logº#

1

2 P)J (º )º#P)

º (»#!). (20)

The functional (20), which can be interpreted physically as the free energy, will play animportant role in the proof of Theorem 3. This will serve as a Lyapunov functioncontrolling the size and compactness of families of solutions to the evolution problem.However, we prefer here an approach adapted from the proof of Theorem 2(i), (iii) in[8] (cf. [9, Theorem 1(i)]) and based on compactness properties of the operator J. Thiswill allow us to obtain existence and uniqueness of solutions to (19), (17) for smallq0'0 and q*3[0, 4n) in a more general setting, i.e. irrespective of the sign of the

kernel K defining J, while for the electrostatic repulsion of self-interacting ions of thedensity º, the kernel K is bounded from below. Caution: because of the notationalconvenience, another form of (19) has been used in [8] where »3¸=, while we shoulddeal with a singular potential ».

Theorem 1. ¸et n"2 and ) is a bounded domain in R2. For each q*3[0, 4n) thereexists a constant qN '0 such that for each q

03[0, qN ) there is a solution (3¸= ()) of (19),

(17). ¹hese solutions are unique for sufficiently small q0’s.

Proof. We briefly recall the scheme of the proof in [8, Theorem 2].First, the normalization factor k is well-defined for (3¸= ()) and q*(4n. In fact,

because q*/(2n)(2

0(exp(!D(D=

) P)Dx D!q*/(2n) dx)k)exp( D(D

=) P)

Dx D!q*/(2n) dx(R,

so q0k~1 is finite.

Next, observe that J (exp((!» )) is a bounded function on ) whenever ( is.Indeed, J (exp(() Dx D!q*/(2n) ) is well-defined because exp(() is bounded, Dx D!q*/(2n)

3

¸1`d ()) for some d'0 (recall that q*/(2n)(2), and the kernel K is in ¸r for everyr(R. Note that since DK (x, y) D)k

1( DE

2(x!y) D#1) with some k

1'0, cf. Example

III after Theorem 2 in [8], the singularities of K are at most logarithmic in Dx!y D.Finally, the operator J is compact from ¸=()) into itself, which is also a conse-

quence of the integrability properties of K. Thus, the non-linear operator

T(()"!(q0k~1 J(exp((!» ) )#!)

is defined for (3¸= ()), T is continuous in the D · D=

norm, and the closure of theimage of each ball in ¸= ()) is compact. The estimate

DT(() D=)q

0eR AP) Dx D!q*/(2n) dxB

~1

]Ck1eR AP) (log Dy D#1) Dy D!q*/(2n) dyB#D!D

=



shows that for sufficiently large R'0 and small q0, q

03[0, qN ), the ball

B(0, R)L¸=()) isT-invariant. Hence the Schauder fixed point theorem is applicableand ("T (() solves the problem (19), (17).

Concerning the uniqueness of solutions, note that the argument in [8, Theorem2(iii)] applies when we have at our disposal an a priori estimate on ( in ¸=()).Actually, such an estimate can be proved exactly as in [8, Theorem 2(iii)], so theuniqueness of steady states for sufficiently small q

0'0 follows.

3. Evolution problem

We study in this section the local and global in time solvability of the Rubinsteinproblem (1)—(4) and its radial version (10)—(11). The results proved for parabolic-elliptic systems describing electrolytes in [2], [6] would suggest that one can expectthe existence of local in time solutions in arbitrary dimensions, and global solutions inthe two-dimensional case, as well as the convergence to the unique steady states whentP#R.

In our situation when a highly singular distribution of a fixed charge q* describedby the potential » is present, the solvability questions are much more complicated.Namely, if n"2 and q* is large enough (q*'4n), one cannot define even local in timesolutions to the evolution problem. For n*3 any q*'0 prohibits the local existenceof solutions. This is shown for the radial case in Proposition 3. Such a non-existenceresult demonstrates that quite a natural idea to approximate a point charge bycharges supported on small balls shrinking to the point, will lead to divergence ofsolutions to approximating problems. Therefore, it is of interest to study the Rubin-stein problem in the two-dimensional case with moderate charges q* only. The globalin time solutions are constructed in this setting in Theorem 2: using comparisontechniques for parabolic equations in the radially symmetric case—when q*)4n, andin Theorem 3: using compactness properties of families of approximating solutions inthe general case—under the assumption q*(4n. We note that even a reasonabledefinition of a (very) weak solution to (1)—(4) in the spirit of [16], [17], is notimmediate since the term + · (u+u) as it stands in (1) may contain products ofdistributions (remember that *»"q*d).

Let us begin with the radial case, which has been studied for related parabolic—elliptic systems in [6, Appendix]; [5], [7], [10]. The essential difficulty is the singulardiffusion coefficient 4y in (10) for n"2. This does not allow to apply directly standardexistence results from [12], [16]. To overcome this difficulty we will consider approx-imations of the original problem (10)— (11) by non-singular parabolic problems like in[7, Theorem 2] and [10, Theorem 1(i), (iii), Theorem 3(i)]

Qt"4 (y#e)Q

yy!n~1 (Q!q*)Q

y, e'0,

(Pe) Q(0, t)"0, Q(1, t)"q0,

Q(y, 0)"Q0(y).

Theorem 2. If n"2 and q*3[0, 4n], then the problem (10)— (11) has a global in timesolution for any Holder continuous initial condition Q

0.



Proof. The regularized problem (Pe ), with the initial condition Q03Ca ([0, 1]) for

some a'0, has a unique solution Q"Qe3C2à,1à@2([0, 1]][0, ¹]) for each¹'0. This follows from the standard theory of uniformly parabolic problems, see[16, chapter VI, Theorem 5.2]. Moreover, the estimate of the Holder norm

EQeECa,a@2([d, 1]][0, ¹]))C(d) (21)

for each d'0 with C(d ) independent of e is a consequence of the result cited above.The comparison principle holds for solutions of the problem (Pe), hence suitable

supersolutions QM e can supply us with bounds on Qe showing the compactness of thefamily of approximating solutions MQe : e'0N in C([0, 1]][0,¹]). To construct suchQM e we make the crucial observation that the function, whose form is reminiscent of thesolution (14) of (12) with q*"4n,

QM (y)"A(c!log y)~1 (22)

is a supersolution of the equation (10) for all A*8n and c'0. Clearly, if thenon-linear parabolic operator L is defined by Lq"4yq

yy!n~1 (q!q*) q

y!q

t,

then LQM "Ay~1 (c!log y)~3 M (8!A/n)!(4!q*/n)(c!log y)N)0.Moreover, under the Holder continuity assumption on Q

0lim sup

y?0Q

0(y) D log y D(R, so there exists a sufficiently large A and small c such that

Q0(y))QM (y).Now, by similar calculations, a counterpart of (22) QM e(y)"A (c!log(y#e))~1 is

a supersolution of (Pe ), and therefore Qe (y, t))QM e (y). This, combined with (21),implies the existence of a common modulus of continuity for MQe : e'0N. Thus, there isa sequence e

kP0 and a function Q3C ([0, 1]][0, ¹]) such that Qe

k

PQ as ekP0,

uniformly in (y, t). Again from the bound Qe)QM e we infer that Q satisfies theboundary condition (11).

Next let (y, t)3 (0, 1)](0, ¹ ). From standard results in [16, chapter III, Theorem10.1], [12, chapter 3] it follows that MQe : e'0N is bounded in Cmà, (mà)@2([y/2, 1]][t/2, ¹]) for all m3N, a3 (0, 1). Then, since Qe solves (Pe ), Q satisfies (10)in [y/2, 1]][t/2, ¹]. The point (y, t) was chosen arbitrarily in (0, 1)](0, ¹ ), so Q isa solution of (10) in (0, 1)](0, ¹ ) for each ¹'0.

The assumption on q* in Theorem 2 guaranteeing the existence of global solutionsis sharp, see the result below.

Proposition 3. If either n"2 and q*'4n, or n*3 and q*'0, then there is no positivenon-decreasing solution QI0 of (10) satisfying the boundary condition Q(0, t)"0.

Proof. This can be shown in a completely elementary, although somewhat tediousmanner, similar to that used in the proof of Proposition 2. However, we prefer a quickargument involving the Feller test for the existence of evolution solutions to one-dimensional singular diffusion equations, see e.g. [19, chapter II, Theorem 2.6] or [14,chapter 15, section 6]. Observe that a hypothetical solution Q of (10) with theboundary condition Q (0, t)"0 would satisfy locally in t and y, i.e. for all t3[0, t

0],

y3[0, y0], with some t

0'0, y

0'0

Q(y, t))q*/2!2n if n"2, or Q(y, t))q*/2 if n*3.

Hence, Q would be a solution of a linear singular parabolic equationn2y2~2@n q

yy#np~1

n(q*!Q)q

y!q

t"0 whose diffusion coefficient n2y2~2@n

(i.e. 4y for n"2) and the drift coefficient np~1n

(q*!Q)*np~1n

q*/2'0



(n~1 (q*!Q)*q*/(2n)#2'4 for n"2 resp.) correspond to the case of an entrancepoint in the Feller test, i.e. Q cannot preserve the boundary condition Q(0, t)"0.

The next result shows that for q* large compared to q0

any reasonable weaksolution u (irrespective of actual difficulties of its construction) in a star-shapeddomain )LR2 of the Rubinstein problem with the boundary condition (3.3) ceases toexist after a finite time.

Proposition 4. If ) is a star-shaped domain of R2, 03) and 2q*!q0'8n, then

solutions to (1), (2), (3.3), (4) cannot be defined globally in time.

Proof. The proof is based on virial calculations, i.e. concerning the evolution ofmoments of the density u, similar to those in [7, Theorem 3], [9, Theorem 2(v)], [4,Theorem 1], [10, Theorem 3(ii)], [18, Theorem 4]. Suppose, with no loss of generality,that ) is star-shaped with respect to the origin. Define an auxiliary function

w(t)"P)u (x, t) Dx D2 dx*0. (23)

Then, we have

dw

dt"!2 P)

+u ·x#2P)u+u · x

"!2 P )ux · l#4 P)

u#2 PP)])u (x, t) (+

xE2(x!y)) · xu (y, t) dydx

!q*n~1 P)u (x, t) Dx D~2x · x dx.

Since, x · l*0 holds for star-shaped domains ), after dropping the first term on theright-hand side, and after the symmetrization of the third integral leading to

(2n)~1PP)])u(x, t) u (y, t) Dx!y D~2( (x!y) ·x#(y!x) · y) dydx

"(2n)~1 PP)])u (x, t) u (y, t) dxdy"(2n)~1 q2

0,

we arrive at

dw

dt)4q

0#

q20

2n!

q0q*

n"q

0 A4#q0

2n!

q*

n B .

Hence if 2q*!q0'8n, then w (t) becomes negative in a finite time, a contradiction.

Our main result in this paper is the existence of global in time solutions to theRubinstein problem (1)—(4) if n"2 and q*(4n.

Theorem 3. Suppose that )LR2 is a bounded domain, 03), )3C1`e for some e'0,q*(4n, and 0)u

03¸p()) for some p'1. ¹hen there exists a solution u"u (t) of the

problem (1)— (4) defined for all t*0, and u (x, t)*0 a.e. in x, t. ¹his solution, obtained asa limit of solutions to suitably regularized problems, has an accumulation point in ¸1())as tP#R.



Proof. Since » is quite singular (remember that D2» is a measure), it is not at all clearhow to define directly a weak solution to (1)—(4), cf. the remark at the beginning of thissection. A natural idea is to approximate (1)— (2) by similar parabolic—elliptic systemswith less singular potentials, and then define u as a limit in a certain sense of solutionsto approximating problems. Such an approach is by now standard for elliptic andparabolic problems involving measures as data, see e.g. [1], [11], [20]. We adopt thispoint of view, leading to a priorily very weak solutions (for instance: the renormalizedones), nicknamed by Boccardo and Murat: SOLA,solution obtained as a limit ofapproximations.

We begin with recalling the definition of the weak solution of the problem (1)—(4)when 0)u

03¸2()) and the potential » is assumed to be smooth and bounded, see

[6, Theorem 1]; [8, section 2, (D)]. We will apply it to »"»k"q*E

2sk(q*E

2),

where sk(x)"s (x/k), k3N, and s is a smooth function with compact support,

s(x),1 for x3[!1, 1], 0)s)1. The definition is based on a standard one in [16]and reads u3¸= ((0, ¹ ); ¸2()) )W¸2(0, ¹ ); H1())) is a weak solution to (1)—(4) on(0, ¹ ) if the identity

P)u(x, t) g (x, t) dx!P

1

0P)

ugt#P

t

0P)

(+u!u+u) ·+g"P)u0(x)g (x, 0) dx

holds for each g3H1()](0, ¹ )), a.e. t3 (0, ¹ ), and u"u(t)3H1()) is a weaksolution of (2)—(3), i.e.

P)+u ·+m#P)

(u!*» )m"0

for each m3H1()) and a.e. t3 (0, ¹ ); for (3.2) we assume u (t)3H10()), for (3.3) we

simply put u"E2 * u!».

We divide the proof of Theorem 3 into several steps.

Lemma 1. If » is bounded and smooth, then there exists ¹'0 and a unique weaksolution u of (1)—(4) on (0, ¹ ) in the sense of the definition above. Moreover,:)

u (x, t) dx":)

u0(x) dx and u(x, t)*0 a.e.

The idea of the construction is the following, cf. [2, Theorem 1]; [6, Theorem 1]with modifications in [8, Theorem 1]: given uN 3¸4 ((0, ¹ ); ¸2())) we define uN satisfy-ing *(uN #» )"uN a.e. in t3 (0, ¹ ) and the appropriate boundary conditions. Then weconstruct the solution of the linear parabolic problem u

t"+ · (+u!u+uN ), with either

u/l"0 or u/l!u uN /l"0 on ). Finally, we look for a fixed point of thetransformation uN >u. The crucial estimate to establish for ¹'0 small enoughsuitable boundedness and compactness properties (via an Aubin—Lions-type lemma,[17]) of this transformation is

1

2

d

dtDu D2

2#D+u D2

2"P)

u+u ·+u)D+u D2Du D

3D+u D

6

)C D+u D2EuE1@3

1Du D2@3

2( Du D

2#C(» ) ). (24)

Here u"uN and u"uN in the last factors on the right-hand side. For a proof of thisestimate, see e.g. [2, (9)] with a simplification due to the assumption n"2.



The well-posedness of the problem (1)—(4), i.e. the continuity of u with respect to u0,

the regularity property ut3¸2( (0, ¹ ); H~1())), and the positivity and integral

preserving properties are discussed in more details in [2, Proposition 1, 2].

Lemma 2. ¸ocal in time solutions to (1)— (4) with bounded » can be defined wheneveru03¸p()), p'1. ¹hey enjoy some regularity properties including u3¸=

-0#( (0, ¹ );

¸=())).

The proof of this lemma is based on [6, Theorem 2] where »,0, p'n/2 andn*2. This uses the Moser iteration technique and an approximation of u

0, and it can

be generalized in a straightforward way to the case of a bounded potential » withD2»3¸q()) and q large enough, see [8, Theorem 1].

We note that, because of the non-linear no-flux condition on u in (3.2), (3.3), wecannot expect the parabolic maximum principle for u be useful; no a priori control onu is available on the parabolic boundary of )](0, ¹ ). This makes the question of thecontinuation of solutions in time quite delicate, even for smooth ». The global in timeexistence for the full Debye system and n"2 has been proved in [6, Theorem 3] (seealso [9, Theorem 1(iv)] for a similar construction in the gravitational case) usinga supplementary a priori estimate suggested by the physical origin of the problem.

Lemma 3. ¹he function

¼(t)"¼(u(t) )"P)u(x, t) log u (x, t) dx

#

1

2 P)D+(u(x, t)#»(x)#!(x)) D2dx

#P)u(x, t)(»(x)#! (x)) dx (25)

is a ¸yapunov function for the problem (1)— (4) with a bounded potential ».

Clearly, for (3.1) or (3.2) ¼ is the free energy introduced in (20) for stationarysolutions since :) J (u)u":)D+ (u#»#! ) D2. Multiplying formally (1) by (log u!u),integrating by parts and integrating on [0, t], we arrive at the inequality

¼(t)#Pt

0P)

u D+ (log u!u) D2)¼(0), (26)

i.e. ¼ (t) is a non-increasing function on the trajectories of (1)— (4):¼(t))¼(0)"¼ (u

0). This computation is made rigorous by approximating u by

u#d (d'0), using solvability results in Lemmas 1, 2, and passing to the limit dP0,cf. [2, Lemma 3]. The case of (3.3) is slightly different. The inequality:) D+ (u#») D2):) J (u)u#C(), q

0) holds (compare [9, (24.2)]), and this implies

¼(t))¼(u0)#C(), u

0).

Lemma 4. For n"2 and bounded smooth » the solutions of (1)— (4) can be continuedglobally in time and satisfy the estimate supt*0 Du (t) D

2(R.

Clearly, by Lemma 1 it is sufficient to show an a priori bound on the ¸2())-normof u. This is done along the lines of the proof of [6, Theorem 3] combining (24),



a non-linear imbedding inequality in [6, (22)]

Du D33)eEuE2

1Du log DuE

1#Ce Du D

1(27)

valid for every e'0, some Ce and each u3H1 ()), )LR2, and the uniform in timebound on :) u log u and :) D+u D2)2 :) D+ (u#»#!) D2#2 :) D+ (»#! ) D2 from theLyapunov function ¼ in Lemma 3. These ingredients lead to

1

2

d

dtDu D2

2#D+u D2

2)D+u D

2Du D

3D+u D

6)C D+u D

2Du D3@2

3( D+u D1@2

2#C(» ))

)C D+u D2((eEuE2

1Du log u D

1#Ce Du D

1)1@2#C(» ) )

)

1

2EuE2

1#C(u

0, e, » )

for e'0 small enough. Finally, we obtain

d

dtDu D2

2#2 Du D2

2#D+u D2

2)3 Du D2

2#C)EuE2

1#C Du D2

1#C,

so the differential inequality dw/dt#w)C for the function w (t)"Du (t) D22, which

implies the uniform boundedness of w(t) for all t*0.

Lemma 5. Suppose that 0)u03¸p()) for some p'1, »

k"q*E

2sk(q*E

2) is

a bounded smooth truncation of the potential »"q*E2

with q*(4n, and uk

arethe solutions of (1)— (4) corresponding to »"»

k. ¹hen ¼

k(t)"¼

k(u

k(t)))

supk¼

k(u

0)(R for each k3N and all t*0, and

supk, t P)

uk(x, t) log u

k(x, t) dx(R. (28)

For the proof it is sufficient to establish a lower bound for the last term in theLyapunov function ¼"¼

kin (25). Using the Young inequality, or simply the

inequality st)s log s#exp(t!1) valid for all s*0, t3R, we estimate

K P) uk(x, t)»

k(x) dx K)P)

uk(x, t) q*(2n)~1 D log Dx D Ddx

)(1!d ) P)Du

kD log Du

kD#P)

Dx D!q*/(2n(1!d) ) dx.

Here d'0 is chosen small enough so that q*(4n(1!d). The integral :) uk!

kposes

no problem since !k’s are uniformly bounded. Finally, we have

d P)uklog u

k!C(), q

0))¼

k(t)

)supk

¼k(u

0))2 P)

u0log u

0#

1

2 P)J (u

0)u

0#C (), q

0)(R,

as ¸p ())L¸ log¸ ()).



Lemma 6. ºnder the assumptions of ¸emma 5 there exists a function u"u(t) which isfor every t*0 the weak-¸1()) limit of a subsequence of u

k"u

k(t) solving (1)— (4) with

the regularized potentials »k. ¹he potentials u

k(t)#»

kcorresponding to the approxi-

mating solutions uk(t) converge in ¸q()) for each q(R and t*0. ¹he trajectory u (t)

has an accumulation point as tP#R.

The uniform in k3N bound (28) on Duk(t) log u

k(t) D

1proved in Lemma 5 enables us

to pass to the weak-¸1()) limit for each t*0 thanks to the compactness criterion ofde la Vallee-Poussin and the Dunford-Pettis theorem. It is natural to call suchu a SOLA. Concerning the behaviour of potentials u

k#»

k, the uniform boundedness

of the Lyapunov functions ¼kgives

supk, t

D+ (uk(t)#»

k#!

k) D2(R,

so by the Rellich lemma a subsequence of uk(t)#»

kconverges in ¸2()) for each t*0.

Moreover, for any solution of (2)—(3) if uk3¸ log¸()), then (u

k#»

k)3¸=()).

Indeed, e.g. for (3.3) by the Young inequality

DE2 * u

kD=)P)

exp(!(2n)~1 log Dx D ) dx#P)uklog u

k(R.

Finally, the convergence of potentials in ¸2()) together with their uniform bounded-ness implies for each q(R the ¸q()) convergence.

The bound (28) is also uniform in t*0, hence by a diagonal argument u(t) is weaklyconvergent in ¸1()) along a sequence of t’s, tP#R, to a limit, which is expected tobe a steady state of (1)—(3).

Remark. Further uniform boundedness properties of uk’s include

supkP

T

0P)

D+(log(1#uk) ) D2(R

for every ¹'0. This can be proved using 1!(1#uk)~1 as a test function in the

definition of the weak solution to (1)— (4), cf. [20, Lemma 1].

Our last result is the existence of special global in time solutions of the Rubinsteinproblem in the whole plane, namely self-similar solutions enjoying some scalingproperties. As it is well-known, self-similar solutions are expected to describe asymp-totics of global ones, see e.g. [3]. Their properties are of interest even for the localbehaviour of solutions to the boundary value problem (1)—(4) near x"0 when tPR.

We are looking for self-similar solutions of the equation (6) for n"2 in (0, R), i.e.with radially symmetric u (x, t)"t~1º(xt~1@2) of the Boltzmann invariance form in(1)—(2), hence Q(r, t)"p

ntn@2~1f(r2/t) in (6), see [3, (21), (31)—(34)]. For n"2 this

leads to the problem

fA#1

4f@!

1

2y Af!q*

2nB f@"0, y"r2/t , @"d

dy, (29)

for non-decreasing f such that f(0)"0.We remark that functional analytic methods developed in [3, Proposition 2(i)] for

the gravitational case can be applied to the problem of the existence of self-similar



solutions when n"2, q*"0 and u0"q

0d with small q

0'0; the !sign in (1) does

not matter compared to the #sign in [3, (1)]. We prove below, using ODE methods,a counterpart of Proposition 3(i) in [3] for the equation (29) with q*(4n.

Proposition 5. For n"2 and q*3[0, 4n) there exist self-similar solutions of theRubinstein problem with a finite total charge q

0"f(R)/(2n), i.e. solutions of (29) with

f(R)3(0, R).

Proof. Concerning the local existence of positive non-decreasing solutions f of thesingular equation (29) satisfying the condition f (0)"0, note that f

1(y)"ay with

0(a)1/2 is a subsolution, and f1 (y)"byc with c"1!q*/(4n) and b*1/2 isa supersolution of (29) on [0, 1]. Using approximation techniques like for the evolu-tion case in the proof of Theorem 2, we prove that there exist non-trivial solutions of(29) with f (0)"0. Namely, the regularized problems

(Re) fA#1

4f@!

1

2(y#e) Af!q*

2nB f@"0, e'0,

f (0)"e/2, f(1)3 (16, 13),

have non-decreasing (see (32) below) solutions f"fe satisfying the inequalities

(y#e)/6)fe(y))(y#e)c/2, y3[0, 1], 0(e(1. (30)

Indeed, a (y#e), b(y#e)c with 0(a)1/2)b are sub-, resp. supersolutions of (Re).Next, observe that f@e(1))1/2. Indeed, for some ye3[0, 1] f@e (ye )(1/2 holds. Since

fe(1))1/2, fe cannot quit the set Mf)(y#e)/2#q*/(2n)N and

4(y#e)fAe"f@e (2fe!(y#e)!q*/n))0, (31)

so fe is concave and f@e decreases. Moreover, the derivative

f@e(y)"f@e(1) expA!P1

y

( (2(y#e) )~1 (fe!q*/(2n))!1/4) dyB (32)

can be estimated uniformly on each interval [d, 1] with d'0

sup0(e(1, d)y)1

f@e (y))C(d)(R. (33)

The inequality (30) for y3[0, d] brought together with (33) give the existence ofa common modulus of continuity for the family of approximating solutions Mfe :e'0N. Hence there exists a continuous function fI 0 such that fe

k

Pf for somesequence e

kP0. A similar argument applies to all the derivatives of fe , since

fAe , f@@@e ,2 can be calculated from the equation (Re ). Therefore, the limit functionf satisfies (29) for every y3 (0, 1]. Obviously, the derivative f@ does not change the sign,which is seen from an analogue of (32) for f with e"0. Now, if f(z))z/2 for somez'0 (this holds, e.g., for z"1), then the counterpart of (31) with e"0

4yfA"f@(2f!y!q*/n))0 (34)

shows that f remains in the set Mf)y/2N for all y*z. Thus, solutions of (29) which aresmall on [0, 1] can be continued to the global non-decreasing ones. Therefore, (34)leads to an estimate for f@ of the form f@ (y))ayc~1, so f(y))Ayc for some a, A'0 (if



c"1, A can be chosen so that A(1/2). Using this bound in (29), we arrive at 4fA)f@(2Ayc~1!1!q*/(ny)), and then f@(y))B exp (A (2c)~1 yc!y/4)yc~1 . Since eitherc(1, or c"1 and A(1/2, so f@ is integrable on (1, R), and sup

yf(y)"

f(R)"q0/(2n)(R follows.

Remark. A related problem when a cloud of particles surrounding a mass fixed ata point in ) interacts with itself via the gravitational attraction has been mentioned in[24, Lemma 4.5]. Equation (1) is replaced then by u

t"+ · (+u#u+u). Radially

symmetric version of this, the so-called ‘star problem’, has been considered in [10,section 4]. The general case is studied using methods developed in [3—5], [9], [7], ina paper in preparation. In particular, steady states, local and global solutions of theevolution problem, self-similar solutions, gravitational collapse (i.e. blow up) of largesolutions are discussed.

Acknowledgements

We are deeply grateful to Andrzej Krzywicki for interesting discussions and, in particular, for suggestionsconcerning Proposition 5. The first named author benefited also from discussions with Saıd Benachour andMichel Pierre when he enjoyed a warm hospitality at Institut Elie Cartan (Nancy) in May 1995.

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