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Mathematical analysis of a nonlinear parabolic equation arising in the modelling of non-newtonian flows

The mathematical properties of a nonlinear parabolic equation arising in the modelling of non-newtonian flows are investigated. The peculiarity of this equation is that it may degenerate into a hyperbolic equation (in fact a linear advection equation). Depending on the initial data, at least two situations can be encountered: the equation may have a unique solution in a convenient class, or it may have infinitely many solutions.


Introduction
Modelling the flow of complex fluids is a very intricate problem which is far from being solved up to now. Besides studies which aim at improving phenomenological rheological models (purely macroscopic constitutive laws), only a few attempts are made to recover the rheological behavior of a complex fluid from elementary physical processes arising in its microstructure.
The mesoscopic model which has been proposed by Hébraud and Lequeux in [3] deals with simple shear flows of concentrated suspensions. It is obtained by dividing the material in a large number of mesoscopic elements ("blocks") with a given shear stress σ (σ is a real number; it is in fact an extra-diagonal term of the stress tensor in convenient coordinates) and by considering the evolution of the probability density p(t, σ) which represents the distribution of stress in the assembly of blocks. Under various assumptions on the evolution of the stresses of the blocks which will be described below, the equation for the probability density p(t, σ) for a block to be under stress σ at time t may be written as: where for f ∈ L 1 (IR), we denote by In equation (1.1a), χ I R\[−σc,σc] denotes the characteristic function of the open set IR \ [−σ c , σ c ] and δ 0 the Dirac delta function on IR. Each term arising in the above equation (1.1a) (HL equation in short) has a clear physical interpretation. When a block is sheared, the stress of this block evolves with a variation rate b(t) = G 0γ (t) proportional to the shear rateγ(t) (G 0 is an elasticity constant); in this study, the shear rateγ(t), and therefore the function b(t), are assumed to be in L 2 loc (IR + ). When the modulus of the stress overcomes a critical value σ c , the block becomes unstable and may relax into a state with zero stress after a characteristic relaxation time T 0 . This phenomenon induces a rearrangement of the blocks and is modelled through the diffusion term D(p(t)) ∂ 2 σσ p. The diffusion coefficient D(p(t)) is assumed to be proportional to the amount of stress which has to be redistributed by time unit and the positive parameter α is supposed to represent the "mechanical fragility" of the material.
In all that follows, the parameters α, T 0 and σ c are positive, and the initial data p 0 in (1.1c) is a given probability density; that is (1.2) We will be looking for solutions p = p (t, σ) in C 0 t (L 1 σ ∩L 2 σ ) such that σ p belongs to L ∞ t (L 1 σ ) to the nonlinear parabolic partial differential equation (1.1). The subscript σ refers to integration over IR with respect to σ, whereas the subscript t refers to time integration on [0, T ] for any T > 0. Note that the average stress in the material is given by and therefore the above condition on σp ensures that the average stress is an essentially bounded function of time.
Actually in practice, the shear rate is not uniform in the flow and in order to better describe the coupling between the macroscopic flow and the evolution of the microstructure we introduce and study in a second paper [1] a micro-macro model. In this model p is also a function of the macroscopic space variables and the average stress defined by (1.3) is inserted into the macroscopic equation governing the velocity of the macroscopic flow.
In order to lighten the notation and without loss of generality we assume from now on that σ c = 1 and T 0 = 1. This amounts to changing the time and stress scales.
The main difficulties one encounters in the mathematical analysis come from the nonlinearity in the diffusion term and also and even more from the fact that the parabolic equation may degenerate when the viscosity coefficient D(p) vanishes, and this will be shown to may appear only when D(p 0 ) = 0. This difficulty is illustrated on a simplified example just below and also in Section 5 where we discuss the existence of stationary solutions in the case when the shear rate b is a constant.
Let us first of all look at the following simplified model which already includes the difficulties we are going to face to in the study of equation (1.1). We consider the equation : where χ ]−1,1[ is the characteristic function of the interval ] − 1, 1[. The initial condition is on purpose chosen in such a way that D(u(t = 0)) = 0. The function u = 1 2 χ ]−1,1[ (σ) is a stationary solution to this equation and for this solution D(u(t)) is identically zero. But it is not the unique solution to (1.4) . It is indeed possible to construct a so-called vanishing viscosity solution for which D(u(t)) > 0 for all t > 0, and there are actually infinitely many solutions to this equation. (This statement is obtained as a corollary of Lemma 4.2 in Section 4 below.) As far as equation (1.1) is concerned, we show that, in the case when D(p 0 ) = 0 and b ≡ 0, we may have either a unique or infinitely many solutions, depending on the initial data (see Proposition 4.1 in Section 4).
On the other hand, we are able to prove the following existence and uniqueness result in the non-degenerate case when D(p 0 ) > 0 : Theorem 1.1 Let the initial data p 0 satisfy the conditions and assume that D(p 0 ) > 0 .
Then, for every T > 0, there exists a unique solution p to the system (1 , I R p(t, σ) dσ = 1 for all t > 0, D(p) ∈ C 0 t and for every T > 0 there exists a positive constant ν(T ) such that min 0≤t≤T D(p(t)) ≥ ν(T ) .
The first step toward the existence proof of solutions to (1.1) will consist in the study of so-called vanishing viscosity approximations, which are the unique solutions to the family of equations (recall that we have rescaled the time and stress units to get T 0 = 1 and σ c = 1). Section 2 below is devoted to the proof of the following Proposition 1.1 (Existence and uniqueness of vanishing viscosity approximations) Let T > 0 be given. We assume that the initial data satisfies the same conditions (1.5) as in the statement of the theorem. Then, for every T > 0 and 0 < ε ≤ 1, there exists a unique solution and for every T > 0, there exist positive constants C 1 (T, p 0 ), C 2 (T, p 0 ) and C 3 (T, p 0 ) which are independent of ε such that Theorem (1.1) is then proved in Section 3 while the degenerate case is investigated in Section 4. Lastly, the description of stationary solutions in the constant shear rate case is carried out in Section 5.

The vanishing viscosity approximation
This section is devoted to the proof of Proposition 1.1.
We begin with the following : Lemma 2.1 (Uniqueness) Let p 0 satisfy (1.2). Then for every T > 0 and 0 < ε, there exists at most one solution p ε to (1.6) in . Moreover, p ε ∈ C 0 t (L 2 σ ) (thus, the initial condition makes sense) and Proof of Lemma 2.1: We begin with proving that every solution to (1.6 t (H 1 σ ) satisfies (2.1). We fix R > 1 and we consider a cut-off C 2 function φ R = φ R (σ) with compact support which is equal to 1 when 0 ≤ |σ| ≤ R and to 0 when |σ| ≥ 2R and such that where here and below C denotes a positive constant that is independent of R.
We bound from above the terms on the right-hand side as follows. First, we have thanks to (2.2) and using that p ε ∈ L ∞ t (L 1 σ ) and b ∈ L 1 t . Next, thanks again to (2.2), Cauchy-Schwarz' inequality and since ∂ σ p ε is in L 2 t,σ . Finally, and the right-hand side goes to 0 as R goes to infinity since p ε is in L ∞ t (L 1 σ ). All this together yields for almost every t in [0, T ]. In particular, this implies that D(p ε ) ≤ α .
Let us now argue by contradiction by assuming that there exist two solutions p 1 and p 2 to (1.6a) corresponding to the same initial data p 0 . By subtracting the equations satisfied by p 1 and p 2 respectively, we obtain We multiply (2.3) by q and integrate over IR with respect to σ to obtain, after integrations by parts, We first remark that since I R p 1 = I R p 2 = 1 thanks to (2.1), we get with the help of Cauchy-Schwarz' inequality. Next, using the Sobolev embedding of H 1 (IR) into L ∞ (IR), we bound from above the terms on the right-hand side in the following way : Finally, by applying the Gronwall lemma, we prove that q 2 We now turn to the existence part in the statement of Proposition 1.1. From now on we fix a positive constant ε ≤ 1. The proof of Proposition 1.1 will be carried out by the Schauder fixed point theorem. For given positive constants M (≥ ε) and R, we introduce D ε,M and Y R two closed convex subsets of respectively L 2 t and L 2 t,σ as follows: To simplify notation we denote We first prove the following and Proof of Proposition 2.1: Let us first observe that for every q in for almost every t in [0, T ]. Therefore the source term D(q(t)) δ 0 (σ) in (2.5a) is in L ∞ t (H −1 σ ) and the existence and the uniqueness of a solution p ∈ C 0 t (L 2 σ ) ∩ L 2 t (H 1 σ ) to the system (2.5) is well-known (see for example [2]). In particular, the initial condition makes sense. Owing to the fact that the source term is non -negative, the proof that p ≥ 0 is also standard (see again [2]).
We now check the pointwise inequality (2.6). This is ensured by the maximum principle with observing that p − and p + given respectively by (2.7) and (2.8) are the unique solutions to the systems respectively. We now turn to the proof of statement i. and assume that p 0 belongs to L ∞ (IR). Then, using the two facts that for every ν > 0, I R ϕ ν = 1 and ϕ ν ≤ 1 √ 2π ν , (2.9) is easily deduced from p ≤ p + with the help of (2.11) and since a ≥ ε.
Suppose now that I R |σ| p 0 < +∞. This together with the assumption p 0 ∈ L 2 (IR), guarantees that p 0 ∈ L 1 (IR) (see also below). Using (2.6) again, we now have With the help of (2.11) and observing that , we then deduce (2.10).
We now use this bound to check that p ∈ C 0 t (L 1 σ ) and D(p) ∈ C 0 t . Indeed, for any t, any sequence t n in [0, T ] which converges to t and A > 1, we have For any fixed A the first term in the right-hand side goes to 0 as n goes to infinity since p ∈ C 0 t (L 2 σ ) and then the second term is arbitrarily small as A goes to infinity. The same argument yields the continuity of D(p(t)) with respect to t. ♦ The following proposition aims at checking the required assumptions to apply the Schauder fixed point theorem.
2 Let T f > 0 be given. We assume that Let 0 < ε ≤ 1, R = 1 + I R |σ| p 0 and M = 1 + 2 α. We define (2.17) Then, for every T ≤ min 1 R ; T c , the function T : (a; q) → D(p) + ε ; p , with p being the solution to the system (2.5), maps D ε,M × Y R into itself. Moreover T is continuous and Proof of Proposition 2.2: Step 1: T is well-defined.
According to Proposition 2.1, p is in C 0 t (L 1 σ ) and D(p) ∈ C 0 t . We now prove that with our choice for M ( which ensures that ε+D(p 0 ) ≤ 1+α ≤ M ), D(p)+ε ∈ D ε,M . For this, we again use the inequality p ≤ p + , the definition (2.8) of p + , the rough estimate |σ|>1 ϕ ν ≤ I R ϕ ν = 1 and (2.11) to obtain and observe that this condition holds provided Since we already have demanded that t ≤ T ≤ 1 R a sufficient condition is then which reduces to T ≤ T c with T c given by (2.17).
Our next step will consist in establishing a priori bounds on p in Step 2: A priori bounds.
If we multiply equation (2.5a) by p and integrate by parts over IR with respect to σ we easily obtain Since from the Sobolev embedding of H 1 (IR) into L ∞ (IR) and the bound (2.11) on D(q) we get we may write We recall that a ≥ ε and we apply the Gronwall lemma to obtain sup 0≤t≤T p 2 We now return to (2.18) and integrate it over [0; T ] to obtain (2.20) Step 3: The function T is continuous.
We consider a sequence (a n ; q n ) in D ε,M × Y R such that a n converges to a strongly in L 2 t and q n converges to q strongly in L 2 t,σ , and we denote T (a n ; q n ) = (D(p n ) + ε; p n ). We have to prove that p n converges strongly to p in L 2 t,σ and D(p n ) converges to D(p) strongly in L 2 t , with (D(p) + ε ; p) = T (a; q).
In virtue of (2.19) and (2.20), the sequence p n is bounded in . This together with the fact that p n is bounded in L 2 t (H 1 σ ) implies that, up to a subsequence, p n converges strongly towards p in L 2 t (L 2 loc,σ ) (the convergence being weak in L 2 t (H 1 σ )) thanks to a well-known compactness result [4]. In particular, p n converges to p almost everywhere. Thus p ≥ 0 and by the Fatou's lemma, I R |σ| p ≤ R almost everywhere on [0; T ]. Hence p belongs to Y R . We are going to show that the convergence is actually strong in L 2 t,σ . In virtue of (2.9) in Proposition 2.1, we dispose of a uniform a priori bound on p n in L ∞ t,σ (hence also on p). For the strong convergence in L 2 t,σ we then argue as follows. For any fixed positive real number K, we have owing to the fact that p n and p belong to a bounded subset of Y R ∩ L ∞ t,σ . We then conclude by letting n next K go to infinity.
We now prove that D(p n ) converges to D(p) strongly in L 2 t . We shall actually prove that D(p n ) converges to D(p) strongly in L 1 t and then use the fact that D(p n ) is bounded in L ∞ t , in virtue of (2.11) and because p n lies in Y R . Let us fix K > 1. Then, we have because p and p n belong to Y R . Since p n converges to p strongly in L 1 t (L 1 loc,σ ), we conclude that D(p n ) converges to D(p) in L 1 t by letting n next K go to infinity in (2.21).
In order to pass to the limit in the equation satisfied by p n (thereby proving that (D(p) + ε ; p) = T (a; q)), we now observe that the strong convergence of q n to q in L 2 t,σ , together with the argument in (2.21) above shows that D(q n ) converges to D(q) strongly in L 2 t . It is then easily proved that p is a weak solution to (1.6a) and since p is in L 2 t (H 1 σ ) it is the unique solution to (2.5) corresponding to a and q. In particular, the whole sequence p n converges and not only a subsequence.
Let (D(p n ) + ε ; p n ) = T (a n ; , q n ) be a sequence in T (D ε,M × Y R ). We have to prove that we may extract a subsequence which converges strongly in L 2 t × L 2 t ,σ . Exactly as for the proof of the continuity, the a priori estimates (2.19) and (2.20) ensure that the sequence p n is bounded , we can mimic the argument in Step 3 above to deduce that up to a subsequence the sequence p n converges to some p in Y R strongly in L 2 t ,σ and that D(p n ) converges to D(p) strongly in L 2 t . ♦ We are now in position to conclude the proof of Proposition 1.1.
Let T f > 0 and 0 < ε ≤ 1 being given. We are going to prove the existence of a unique solution on [0; T f ].
Being given an initial data p 0 which satisfies (1.5), existence of a solution p ε is ensured from Proposition 2.2 by applying the Schauder fixed point theorem on "short" time interval [0; T 1 ] with T 1 = min( α R 1 , T c ) and where R 1 = 1+ I R |σ|p 0 . This solution is uniquely defined in virtue of Lemma 2.1 and we know from (2.1) that I R p ε (T 1 ) = 1. Moreover from Proposition 2.1 p ε (T 1 ) ∈ L ∞ σ and by construction I R |σ| p ε (T 1 ) ≤ R 1 . Therefore p ε (T 1 ) satisfies the same conditions (2.16) as p 0 . Then, repeating the same argument we may build a solution to (1.6) with initial data p ε (T 1 ) on [T 1 ; T 2 ] with T 2 = min 1 R 2 , T c , where R 2 = R 1 + 1 = I R |σ| p 0 + 2. Thanks to the uniqueness result (Lemma 2.1), if we now glue this solution to p ε at t = T 1 we obtain the unique solution to (1.6) on [0; T 1 + T 2 ]. It is now clearly seen that for any integer n ≥ 1 we may build a solution to (1.6) on [0; 1≤k≤n T k ] with T k = min k+ I R |σ| p 0 −1 ; T c .
Since 1≤k≤n T k obviously goes to +∞ together with n, existence (and uniqueness) of the solution p ε to (1.6) is obtained on every time interval.
For the proof of (1.8) we argue as for the proof of (2.9) in Proposition 2.1. Defining p + ε as in (2.8) with a replaced by D(p ε ) + ε and D(q) by D(p ε ) we obtain from which (1.10) follows gathering together (2.1) and (1.8) and, with the notation of the proposition, The proof of (1.9) follows the same lines as the proof of (2.14). Indeed, we again use the pointwise inequality p ε ≤ p + ε and replace D(q) by D(p ε ) (≤ α) and a by D(p ε ) + ε (≤ α + 1) in (2.14) and use (2.16) to deduce whence (1.9) with C 1 (T, p 0 ) being the quantity in the right-hand side of (2.22).
In order to prove (1.11), we multiply (1.6a) by p ε , and we integrate by parts over IR with respect to σ to obtain We use the L ∞ bound (1.8) to bound the right-hand side and we integrate (2.23) with respect to t over [0; T ] to deduce (1.11) with The non degenerate case: D(p 0 ) > 0 The main result of this section corresponds to the statement of Theorem 1.1 and fully describes the issue of existence and uniqueness of solutions to the HL equation (1.1) in the non-degenerate case. It is summarized in the following : Proposition 3.1 Let p 0 satisfy (1.5). We assume that D(p 0 ) > 0. Then, the HL equation (1.1) has a unique solution p in C 0 t (L 2 σ ) ∩ L 2 t (H 1 σ ) and p is the limit (in L 2 t,loc (L 2 σ ) ∩ C 0 t,loc (L 2 σ )) of (p ǫ ) when ǫ goes to 0 where p ε is the vanishing viscosity solution whose existence and uniqueness is ensured by Proposition 1.1. Moreover, p ∈ L ∞ t,σ ∩ C 0 t (L 1 σ ), σ p ∈ L ∞ t (L 1 σ ) and I R p = 1. Furthermore, D(p) ∈ C 0 t and for every T > 0 there exists a positive constant ν(T ) such that min We begin with proving the following : for every 0 < ε ≤ 1.
Remark 3.1 Note that this bound from below is independent of ε, but it comes out from the proof that it depends on p 0 and on the shear b.
Proof of Lemma 3.1: The proof relies on the bound from below in (2.6) that we integrate over |σ| > 1 to obtain is a Gaussian probability density with mean σ ′ + χ(t) and squared width 2 t 0 (D(p ε ) + ε). Therefore, for every σ ′ ∈ IR \ K χ , we have In the zero shear case (b ≡ 0, thus χ ≡ 0) the proof is over and In the general case, a strictly positive bound from below is available as long as the support of p 0 is not contained in K χ . We thus define Then 0 < t * (t * possibly even infinite), the support of p 0 is contained in [−1− χ(t * ), 1− χ(t * )[, and for every T < t * 2 , (3.2) holds for some positive constant ν 1 (T ) defined by It is worth emphasizing that this quantity is independent of ε. If t * = +∞, the proof is over and ν(T ) = ν 1 (T ) fits. Let us now examine the case when t * < +∞ and T ≥ t * 2 .
We go back to (3.3), take t in [ t * 2 ; T ] and denote x = t 0 (D(p ε ) + ε) for shortness. Then −1−χ(t * ) p 0 = 1 and x ≥ t * /2 0 D(p ε ) ≥ t * ν 1 (t * /2)/2 thanks to (3.5). The proof of Lemma 3.1 then follows by defining ν(T ) = min(ν 1 (T ); ν 2 (T )) , with ν 1 (T ) given by (3.5) and ν 2 (T ) being the positive quantity in the right-hand side of (3.6), that is ♦ Proof of Proposition 3.1: We first go back to the proof of the bound (1.11) on ∂ σ p ε and more precisely we look at (2.23) and observe that in virtue of (3.1) Let now ε n denote any sequence in [0, 1] which goes to 0 as n goes to infinity. To shorten the notation we denote by p n instead of p εn the corresponding sequence of solutions to (1.6). With the above bound (3.7) on p n and (1.10), we know that p n is bounded in L 2 t (H 1 σ ) independently of n. Moreover thanks to (2.1) and (1.8) p n is bounded in L ∞ t (L 1 σ ∩ L ∞ σ ) and we also dispose of a uniform bound on I R |σ|p n in virtue of (1.9). Therefore arguing exactly as in the proof of Proposition 2.2 (Step 4) where we have proved that the mapping T is relatively compact in L 2 t × L 2 t,σ we show that p n converges to some p strongly in L 2 t,σ and D(p n ) converges to D(p) in L 2 t . Then p is a solution to the initial problem (1.1) in This non-degeneracy condition on the viscosity coefficient ensures that there is at most one solution to (1.1) in L 2 t (H 1 σ ) ∩ L ∞ t (L 2 σ ) (this follows by an obvious adaptation of the proof of Lemma 2.1 to this case). Therefore the limiting function p is uniquely defined and does not depend on the sequence ε n . Moreover the whole sequence p n converges to this unique limit and not only a subsequence. ♦ As a conclusion of this subsection let us make the following comment which is a byproduct of Proposition 3.1. Let p be a solution to (1.1) in C 0 t (L 1 σ ∩ L 2 σ ), then as soon as D(p(t)) is positive for some time t it remains so afterwards since the solution can be continued in a unique way beginning from time t. 4 The degenerate case : D(p 0 ) = 0 Throughout this section we assume that D(p 0 ) = 0 and therefore the support of p 0 is included in [−1; +1]. Assume that we dispose of a solution to (1.1) in C 0 t (L 1 σ ∩ L 2 σ ). We may define t * ∈ IR + ∪ {+∞} by t * = max t > 0 ; The above system reduces to p(t, σ) = p 0 σ − χ(t) ; D(p(t)) = 0 , for all t in [0; t * ] . (4. 2) The second equation in (4.2) is compatible with the first one as long as Therefore there exists a maximal time interval [0; T c ] on which the HL equation may reduce to a mere transport equation and this is for an intrinsic time T c (possibly infinite) defined by Note that T c is completely determined by the data p 0 and b. If T c = +∞, the steady state p(t, σ) = p 0 (σ − χ(t)) is a solution of the HL equation for all time. We shall now exhibit circumstances under which it is not the unique solution. For convenience, we restrict ourselves to the case when b ≡ 0 (we then have obviously T c = +∞).
For p 0 ∈ L 1 (IR) ∩ L ∞ (IR) such that p 0 ≥ 0, let us denote by F p 0 the function from IR + to IR + defined by F p 0 (0) = D(p 0 ) and by ii. Otherwise, (1.1) has an infinite number of solutions in C 0 t (L 2 σ ). The set of solutions to (1.1) is made of the steady state p(t, σ) = p 0 (σ) and of the functions (q t 0 ) t 0 ≥0 defined by It follows that for any x > 0, Let us consider the problem ∂ t w = D(w(t))∂ 2 σσ w − γw w(0, σ) = p 0 (σ). (4.7) i. If F p 0 satisfies (4.4) then p(t, σ) = p 0 (σ) is the unique solution to (4.7) in C 0 t (L 2 σ ); ii. Otherwise, (4.7) has an infinite number of solutions in C 0 t (L 2 σ ). The set of solutions to (4.7) is made of the steady state w(t, σ) = p 0 (σ) and of the functions Proof of Corollary 4.1: The only point to be checked is that 1 0 dx Fp 0 (x) < +∞. With the standard notation erfc(z) ≡ +∞ z e −t 2 dt, and by using (4.6) and symmetry considerations, simple calculations yield Since erfc(z) ∼ 1 2 e −z 2 /z for z going to +∞, F p 0 (x) ∼ α √ π √ x near 0 and the integrability of Let us consider a non-negative function D ∈ C 0 ([0, +∞[, IR + ). The unique solution in C 0 is given by where t * = inf t > 0, t 0 D > 0 . Any solution to (4.7) thus satisfies w = w D(w) and therefore D(w(t)) = D w D(w) (t) It follows that the function D(w) is solution in C 0 ([0, +∞[) to the nonlinear integral equation On the other hand, if D ∈ C 0 ([0, +∞[) is solution to (4.10) it is easy to check that the function w D defined by (4.9) is solution to (4.8).
If condition (4.4) is fulfilled, equation (4.10) has a unique solution in C 0 ([0, +∞[) (the constant function equal to zero) and the steady state w(t, ·) = p 0 thus is the unique solution to (4.7) in C 0 t (L 2 σ ); otherwise, the set of solutions to (4.10) is made of the steady state w(t, ·) = p 0 and of the family (y t 0 ) t 0 ≥0 with where the function z is defined on [0, +∞[ by otherwise .
Statement ii. is obtained by denoting by v the solution to (4.8) associated with the function z(t). ♦

Proof of Proposition 4.1:
The solution p ǫ to equation (1.6) satisfies the inequalities Therefore on the one hand and on the other hand If (4.4) is not fulfilled, using (4.11) and the property that F p 0 is strictly increasing on [0, +∞[, we obtain that where z(t) is the function defined in the proof of Lemma 4.2. As for any 0 < t 0 ≤ T , there exists η > 0 such that z(t) ≥ η on [t 0 , T ] the same reasoning as in the non-degenerate case leads to the conclusion that (p ǫ ) converges up to an extraction to p in D ′ (]0, +∞[×IR) and in L 2 ([t 0 , T ], L 2 (IR)) for any 0 < t 0 < T < +∞, p being a solution to (1.1) in C 0 (]0, +∞[, L 2 σ ) such that D(p) > 0 on ]0, +∞[. ♦

Steady states
Throughout this section the shear rate b is assumed to be a given constant and we are looking for solutions in L 1 (IR) to the following system : Our main results are summarized in the following : Proposition 5.1 .
i If b ≡ 0, any probability density which is compactly supported in [−1; +1] is a solution to (5.1) which satisfies D(p) = 0. If α ≤ 1 2 , these are the only stationary solutions (and there are infinitely many), whereas when α > 1 2 there exists a unique stationary solution corresponding to a positive value of D, which is explicitly given by (5.4) and (5.6) below. This solution is even and with exponential decay at infinity.
ii If b ≡ 0, for any α > 0, there exists a unique stationary solution to (5.1), and it corresponds to a positive value for D, which is implicitly given by (5.7) and (5.8) below. This solution has exponential decay at infinity.
Remark 5.1 The statement in the above proposition is already pointed out by Hébraud and Lequeux [3].
Proof of Proposition 5.1: The case when b ≡ 0 We first observe that any non-negative function p which is normalized in L 1 (IR) and with support in [−1; +1] is a solution to the system (5.1) since in that case all terms in equation (5.1) cancel. We now examine the issue of existence of solutions of (5.1) such that D(p) > 0. For simplicity we denote D = D(p). For given constant D > 0, it is very easy to calculate explicitly the solutions of (5.1) on each of the three regions σ < −1, σ ∈ [−1; +1] and σ > 1.
Using compatibility conditions on IR and the fact that p has to be in L 1 (IR) one obtains :  The compatibility condition D = D(p) happens to be then automatically satisfied and the normalization constraint I R p = 1 imposes that D solves (5.5) Since D ≥ 0, we immediately reach a contradiction when α < 1 2 , whereas when α > 1 2 equation (5.5) admits a unique positive solution; namely The case when b ≡ 0 First of all, we observe that if D = 0 every term in equation (5.1) but b ∂ σ p vanish. Thus p has to be a non-zero constant which is in contradiction with p ∈ L 1 (IR). So necessarily D > 0. For given positive constant D, we then solve (5.1) as above and obtain Dβ + e b/2D αb β + e b/2D − β − e −b/2D . It is tedious but easy to check that this function always fulfills the self-consistency condition D = D(p) and that the normalization condition I R p = 1 reads D b (1 + β + ) + (β − − 1) e −b/D β + − β − e −b/D + D = α . (5.8) For any b > 0 (the negative values of b are dealt with by replacing σ by −σ), the left-hand side of (5.8) is a continuous function which goes to +∞ when D goes to infinity and goes to zero when D goes to 0. This already ensures the existence of at least one steady state for any α > 0. Moreover, setting z = b 2 D (for example) we may rewrite the left-hand side of (5.8) as f (z) = b 2 z + 2 b 2 z 1 + 1 2b z coth(z/2b) + 1 2b (z 2 + 4z) 1/2 z + (z 2 + 4z) 1/2 coth(z/2b) .
Next we check that the function f is monotone decreasing (thus, the left-hand side of (5.8) is increasing with respect to D), whence the uniqueness result. ♦