DISCRETE AND CONTINUOUS Website: http://AIMsciences.org DYNAMICAL SYSTEMS Volume 00, Number 0, Xxxx XXXX pp. 000–000

REMARKS ON A SMOLUCHOWSKI EQUATION

Peter Constantin^{∗}, Ioannis Kevrekidis^{†}, Edriss S. Titi^{‡}

∗ Department of Mathematics, The University of Chicago, Chicago, Il 60637. †Chemical Engineering, PACM and Mathematics, Princeton University, Princeton, NJ 08544. ‡Department of Mathematics, and Department of Mechanical and Aerospace Engineering,

University of California, Irvine, CA 92697-3875, Department of Computer Science

and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel.

Abstract.

We study the long time dynamics of a Smoluchowski equation arising in the modeling of nematic liquid crystalline polymers. We prove uniform bounds for the long time average of gradients of the distribution function in terms of the nondimensional parameter characterizing the intensity of the potential. In the two dimensional case we obtain lower and upper bounds for the number of steady states. We prove that the system is dissipative and that the potential serves as unique determining mode of the system.

1. Introduction. Certain descriptions of the rheology of non-Newtonian complex fluids containing liquid crystalline polymers combine macroscopic partial differential equations with microscopic stochastic differential equations ([9], [11], [14], [12], [7]). A simple model of nematic liquid crystalline polymers - the rigid rod model - using the Maier-Saupe potential, gives rise to a Smoluchowski equation for the single particle distribution function on the unit sphere ([2], [10], [4],[13], [5]). In spite of its simplicity, this equation exhibits nontrivial nonlinear dynamical features, in contrast with classical Fokker-Planck equations for noninteracting particles ([8]).

We study long time properties of this equation in function of one parameter b > 0 representing the nondimensional potential intensity. We formulate the problem in n dimensions. In the general high dimensional case we obtain bounds for the long time average of gradients of the distribution function which are independent of initial data. In the n = 3 case we obtain long time bounds for the mean square average of the distribution function. The bounds are independent of initial data. In this case we also obtain bounds for the long time average of mean square gradients of the distribution, which again are independent of initial data. We study steady states, and parametrize them. In general, the steady state equations reduce to finitely

1991 Mathematics Subject Classification. 35K35, 76A05, 76A15, 82D60. Key words and phrases. Rheology, non-Newtonian fluids, nematic liquid crystalline polymers,

Smoluchowski equation, long time dynamics, number of steady solutions, gradient bounds, determining modes.

∗ Research partially supported by NSF DMS-0202531. † Research partially supported by AFOSR (Dynamics and Control) and NSF (ITR) . ‡ Research partially supported by NSF DMS-0204794, and by the US Civilian Research and

Development Foundation (CRDF) under grant No. RM1-2343-MO-02. This work was completed while E.S.T. was the Stanislaw M. Ulam Visiting Scholar at the Center for Nonlinear Studies in the Los Alamos National Laboratory supported by the Department of Energy under contract W–7405–ENG–36.

many transcendental equations for a trace-free matrix and a normalization factor. The steady states and corresponding matrix equations have a natural covariance with respect to conjugacy by rotation. This symmetry has to be taken into account when counting distinct steady states. In the n = 2 case we prove that the uniform (constant) distribution is the unique steady state when 0 < b ≤ 4. We prove that, for b > 4 there are at least two distinct steady states. We also prove that, for arbitrary b > 4 there are at most 2 [\frac{b}{4}] distinct steady states ([x] denotes the largest integer not exceeding x). We study in more detail the n = 2 dynamics. A cancellation is used to prove that the system is dissipative in H^{− \frac{1}{2}} (S^{1}): all solutions enter an absorbing ball in finite time, never to leave it again. The system has a finite dimensional global attractor. The long time behavior of solutions can be observed by monitoring one determining mode - the potential itself - and that can be done by measuring the potential at one location alone. Thus, two solutions will approach each other globally if their potentials approach each other at one fixed angle.

2. A Smoluchowski equation. Let S^{n−1} be the unit sphere in R^{n}. We consider the Smoluchowski equation written in local coordinates (ϕ = (ϕ_{1}, … ϕ_{n−1})) as:

∂_{t}ψ = \frac{1}{\sqrt{g}} ∂_{i} (e^{−V} \sqrt{g}g^{ij}∂_{j}(e^{V} ψ))  (1)

The potential V is given by

V (x, t) = −bx_{i}x_{j}S^{ij}  (2)

where x_{i} are Cartesian coordinates in R^{n}, i, j = 1, 2, …, n, and b is a positive constant. The matrix S is determined by

S^{ij}(t) = ∫_{S^{n−1}} x_{i}(ϕ)x_{j}(ϕ)ψ(ϕ, t)σ(dϕ) − \frac{1}{n}δ_{ij}  (3)

with σ(dϕ) = \sqrt{g}dϕ the surface area. Thus, V (x, t) is a homogeneous polynomial of second degree, restricted to the sphere. We will focus on the examples n = 3 and n = 2. When n = 2, the unit circle has local coordinate ϕ ∈ [0, 2π], and one has x_{1}(ϕ) = cos ϕ, x_{2}(ϕ) = sin ϕ, g^{11} = g = 1, ∂_{1} = ∂_{ϕ}. When n = 3, the coordinates on the two dimensional unit sphere are ϕ = (θ, φ), x_{1}(θ, φ) = sin θ cos φ, x_{2}(θ, φ) = sin θ sin φ, x_{3}(θ, φ) = cos θ. Recall also that

g^{11} = 1, g^{22} = (sin θ)^{−2}, g^{ij} = 0, i ≠ j

with ∂_{θ} = ∂_{1} and ∂_{φ} = ∂_{2} and that \sqrt{g} = sin θ.

The equation keeps ψ positive and normalized, if it starts so.

Theorem 2.1. Let ψ_{0} be a nonnegative, continuous function on S^{n−1}. The solutions of (1) with initial data ψ(·, 0) = ψ_{0} exist for all nonnegative time, are smooth, nonnegative and normalized

∫_{S^{n−1}} ψ(ϕ, t)σ(dϕ) = ∫_{S^{n−1}} ψ_{0}(ϕ)σ(dϕ)  (4)

The solutions are real analytic for positive time.

The proof is elementary and will not be given here. From now on we will choose the normalization

∫_{S^{n−1}} ψ(ϕ, t)σ(dϕ) = 1.  (5)

Because of this normalization it follows from the definition (3) that the matrix S is trace-free

T r(S(t)) = 0  (6)

and, consequently, from (2), that the quadratic polynomial V (x, t) is harmonic

Δ_{x}V (x, t) = 0.  (7)

This implies that, when restricted to the unit sphere, V is an eigenfunction of the Laplace-Beltrami operator

Δ_{g} = \frac{1}{\sqrt{g}} ∂_{i} (\sqrt{g}g^{ij}∂_{j})

with eigenvalue equal to −2n

Δ_{g}V = −2nV.  (8)

Because of the way (2) V depends on ψ, the evolution of the quantity

P (t) = − ∫_{S^{n−1}} V (x(ϕ), t)ψ(ϕ, t)σ(dϕ)  (9)

is given by

\frac{d}{2dt}P = − ∫_{S^{n−1}} V (x(ϕ), t)ψ_{t}(ϕ, t)σ(dϕ)

and, using the evolution (1) and one integration by parts, one obtains

\frac{d}{2dt}P (t) = ∫_{S^{n−1}} g^{ij}∂_{i}V (ψ∂_{j}V + ∂_{j}ψ)σ(dϕ).

Integrating by parts and using (8) we deduce

\frac{d}{2dt}P (t) + 2nP = ∫_{S^{n−1}} |∇_{g}V |^{2}ψ σ(dϕ)  (10)

where

|∇_{g}V |^{2} = g^{ij}∂_{i}V ∂_{j}V

On the other hand, considering the (negative) Gibbs-Boltzmann entropy

E(t) = ∫_{S^{n−1}} ψ(ϕ, t) log(ψ(ϕ, t))σ(dϕ)  (11)

one has, using (1) and one integration by parts

\frac{d}{dt}E = − ∫_{S^{n−1}} |∇_{g}ψ|^{2} ψ^{−1}σ(dϕ) − ∫_{S^{n−1}} g^{ij}∂_{j}V ∂_{i}ψ σ(dϕ),

and using one more integration by parts and (8) one obtains

\frac{d}{dt}E = − ∫_{S^{n−1}} |∇_{g}ψ|^{2} ψ^{−1}σ(dϕ) + 2nP (t).  (12)

Note that if the initial data ψ_{0} is strictly positive, then ψ remains strictly positive for all time. Actually, in view of the inequality

−b (1 − \frac{1}{n}) ≤ V (x, t) ≤ \frac{b}{n}  (13)

one can show, using the equations (1), (8) and the maximum principle that

min_{ϕ} ψ(ϕ, t) ≥ e^{−2bt} min_{ϕ} ψ_{0}(ϕ)  (14)

and thus (12) is meaningful. Also, if ψ_{0} is bounded above one has

sup_{ϕ} ψ(ϕ, t) ≤ e^{2b(n−1)t} sup_{ϕ} ψ_{0}(ϕ)  (15)

Theorem 2.2. Let ψ_{0} be a continuous, strictly positive function. Let the minimum and maximum of ψ_{0} be denoted, respectively, m_{0} = min_{ϕ} ψ_{0}(ϕ) and M_{0} = max_{ϕ} ψ_{0}(ϕ). Then the unique solution ψ(ϕ, t) of (1) with initial datum ψ_{0} obeys the inequality

\frac{1}{t} ∫_{0}^{t} (∫_{S^{n−1}} |∇_{g}ψ(ϕ, s)|σ(dϕ))^{2} ds ≤ 2nb + \frac{1}{t} log (\frac{M_{0}}{m_{0}} )  (16)

for all t ≥ 0. Let now t > 0 be fixed and consider a sequence of solutions of (1) coresponding to a sequence of parameters b → ∞. Then

lim_{b→∞} \frac{1}{b^{2}t} ∫_{0}^{t} ∫_{S^{n−1}} |∇_{g}V |^{2} ψ σ(dϕ)ds = 0  (17)

Proof. We start by noting, from (14) and (15) and (5) that

−2bt + log m_{0} ≤ E(t) ≤ 2b(n − 1)t + log M_{0}  (18)

holds for all t ≥ 0. We take the time average of (12):

\frac{1}{t} ∫_{0}^{t} ∫_{S^{n−1}} |∇_{g}ψ|^{2}ψ^{−1}σ(dϕ)ds = 2n\frac{1}{t} ∫_{0}^{t} P (s)ds + \frac{1}{t} (E(0) − E(t)) .

Using (13), (5) and (18) we deduce

\frac{1}{t} ∫_{0}^{t} ∫_{S^{n−1}} |∇_{g}ψ|^{2}ψ^{−1}σ(dϕ)ds ≤ 2nb + \frac{1}{t} log (\frac{M_{0}}{m_{0}} )  (19)

The inequality

∫_{S^{n−1}} |∇_{g}ψ|σ(dϕ) ≤ \sqrt{∫_{S^{n−1}} |∇_{g}ψ|^{2}ψ^{−1}σ(dϕ)}

follows from the normalization (5) and the Schwartz inequality by dividing and multiplying inside the the left hand side integral by \sqrt{ψ}. The above inequality and (19) prove (16). In order to prove (17) we integrate (10) in time from 0 to t and divide by b^{2}t. We use (13) to deduce that

\frac{1}{b^{2}t} ∫_{0}^{t} ∫_{S^{n−1}} |∇_{g}V |^{2} ψ σ(dϕ)ds = O (\frac{1}{b})

and thus prove (17). This ends the proof of the theorem. The relationship (17) implies a strong constraint on moments of the solutions, in the limit of large b (for n = 2 see (43) below).

Let us study the evolution of the L^{2} norm. Using the equation (1), integrating by parts twice and using (8) we deduce

\frac{d}{2dt} ∫_{S^{n−1}} |ψ(ϕ, t)|^{2} σ(dϕ) + ∫_{S^{n−1}} |∇_{g}ψ(ϕ, t)|^{2}σ(dϕ)

= −n ∫_{S^{n−1}} V (x(ϕ), t) |ψ(ϕ, t)|^{2} σ(dϕ)

Now we are going to specialize to the case n = 3. The case n = 2 will be treated separately in the section concerning dynamics. The Gagliardo-Nirenberg-Sobolev inequality ([3]) for W^{1,1} functions in two dimensions, together with straightforward localization implies that there exists a constant C_{3} such that

∫_{S^{2}} |ψ(ϕ)|^{2} σ(dϕ) ≤ C_{3} {1 + (∫_{S^{2}} |∇_{g}ψ(ϕ)|σ(dϕ))^{2}}  (20)

holds for any W^{1,1}(S^{2}) function with unit L^{1} norm. Using this inequality we deduce

Theorem 2.3. Let n = 3, and let ψ_{0} be a continuous, strictly positive function. Let the minimum and maximum of ψ_{0} be denoted as above, respectively, m_{0} = min_{ϕ} ψ_{0}(ϕ) and M_{0} = max_{ϕ} ψ_{0}(ϕ). Then the unique solution ψ(ϕ, t) of (1) with initial datum ψ_{0} obeys the inequalities

\frac{1}{t} ∫_{0}^{t} ∫_{S^{2}} |ψ(ϕ, s)|^{2} σ(dϕ)ds ≤ C_{3} [(6b + 1) + \frac{1}{t} log (\frac{M_{0}}{m_{0}} )]  (21)

and

\frac{1}{t} ∫_{0}^{t} ∫_{S^{2}} |∇_{g}ψ(ϕ, s)|^{2} σ(dϕ)ds ≤ 2C_{3}b(6b + 1)

+\frac{1}{t} [\frac{1}{2} ∫_{S^{2}} |ψ_{0}(ϕ)|^{2} σ(dϕ) + 2bC_{3} log (\frac{M_{0}}{m_{0}} )]  (22)

for all t ≥ 0 with C_{3} the embedding constant of (20).

Proof. The inequality (21) follows directly from (16) and (20). The inequality (13) and the evolution equation for the L^{2} norm imply that

\frac{d}{2dt} ∫_{S^{2}} |ψ(ϕ, t)|^{2} σ(dϕ) + ∫_{S^{2}} |∇_{g}ψ(ϕ, t)|^{2} σ(dϕ)

≤ 2b ∫_{S^{2}} |ψ(ϕ, t)|^{2} σ(dϕ)  (23)

Using (20) in the right hand side of (23), integrating and applying (21) we obtain

\frac{1}{2} ∫_{S^{2}} |ψ(ϕ, t)|^{2} σ(dϕ) + ∫_{0}^{t} ∫_{S^{2}} |∇_{g}ψ(ϕ, s)|^{2} σ(dϕ)ds

≤ \frac{1}{2} ∫_{S^{2}} |ψ_{0}(ϕ)|^{2} σ(dϕ) + 2bC_{3} [(6b + 1)t + log (\frac{M_{0}}{m_{0}} )]  (24)

Dividing by t and gives (22) and ends the proof.

3. Steady States. Let us consider steady states of (1). Because the matrix g^{ij}(ϕ) is nonnegative, diagonal and pointwise invertible, one can prove that any time independent solution of (1) must satisfy

e^{V} ψ = c

for an appropriate constant c. Thus

ψ(ϕ) = Z^{−1}e^{bS^{ij}x_{i}(ϕ)x_{j}(ϕ)}  (25)

We have constraints for the coefficients. Z fixes the normalization ∫ ψdσ = 1. The matrix S is symmetric and traceless. Its eigenvalues must lie between −1/n and (n − 1)/n. Indeed, for any unit vector v, the number

S^{ij}v_{i}v_{j} = ∫ (v · x(ϕ))^{2}ψ(ϕ)dσ(ϕ) − \frac{1}{n}

is in the interval above. The uniform distribution is the special solution for which the matrix S vanishes, Z is the area of S^{n−1} and ψ = Z^{−1}. In order to parametrize all steady solutions let us consider the real valued map

(S, b) ↦ Z(S, b)  (26)

defined for any real, symmetric, traceless matrix S and any positive b by the formula

Z(S, b) = ∫_{S^{n−1}} e^{bS^{ij}x_{i}(ϕ)x_{j}(ϕ)}dσ(ϕ).  (27)

Let us also consider the function

ψ_{S,b}(ϕ) = (Z(S, b))^{−1}e^{b(S^{ij}x_{i}(ϕ)x_{j}(ϕ))}  (28)

associated to any real, traceless, symmetric S and b > 0. Finally, for any real, traceless symmetric S and b > 0, denote

(Ŝ(S, b))^{ij} = ∫_{S^{n−1}} x_{i}(ϕ)x_{j}(ϕ)ψ_{S,b}(ϕ)dσ(ϕ).  (29)

Obviously Ŝ is a function of S and b. Actually, one can check that Z(S, b) depends only on the conjugacy class OSO^{−1}, O ∈ O(n). More specifically, if S_{1} = OSO^{−1} then the rotation invariance of the measure on the unit sphere implies that Z(S, b) = Z(S_{1}, b) and therefore ψ_{S,b}(ϕ) = ψ_{S_{1},b}(T ϕ) where T ϕ is the angle translation associated to the rotation O, Ox(ϕ) = x(T ϕ). The rotation invariance implies then that Ŝ(S_{1}, b) = O (Ŝ(S, b)) O^{−1}.

Theorem 3.1. The steady solutions of (1) are in one-to-one correspondence with the solutions of the implicit trancendental matrix equation

Ŝ(S, b) = S + \frac{1}{n}I  (30)

where Ŝ(S, b) is associated to S and b by the formalism (27), (28), (29) above.

Note that

\frac{∂(log Z(S, b))}{∂b} = T r (SŜ)  (31)

and that

\frac{∂ log Z(S, b)}{∂S} = bŜ.  (32)

Let us consider this problem for n = 2. We look for a symmetric, traceless matrix

S = ( \begin{matrix}A & B \\ B & −A \end{matrix} ) .

To it we associate ψ_{S,b}

ψ_{S,b}(ϕ) = Z^{−1}e^{b(Sx·x)}

with x_{1} = cos ϕ, x_{2} = sin ϕ and (x · y) the usual Euclidean scalar product in R^{2}. Putting

bA = α, bB = β

we get

ψ_{S,b} = Z^{−1}e^{α cos(2ϕ)+β sin(2ϕ)}.

The steady state equation

Ŝ = S + \frac{1}{2}I

together with the normalization

∫_{0}^{2π} ψ_{S,b}(ϕ)dϕ = 1

translate into the four transcendental equations

\frac{α}{b} + \frac{1}{2} = Z^{−1} ∫_{0}^{2π}(cos ϕ)^{2}e^{α cos(2ϕ)+β sin(2ϕ)}dϕ

−\frac{α}{b} + \frac{1}{2} = Z^{−1} ∫_{0}^{2π}(sin ϕ)^{2}e^{α cos(2ϕ)+β sin(2ϕ)}dϕ,

\frac{β}{b} = Z^{−1} ∫_{0}^{2π}(sin ϕ cos ϕ)e^{α cos(2ϕ)+β sin(2ϕ)}dϕ

and

Z = ∫_{0}^{2π} e^{α cos(2ϕ)+β sin(2ϕ)}dϕ

for the unknowns α, β, Z depending on the parameter b. There are only three independent equations, the second equation follows from the first and the last. Elementary trigonometry brings us to

\frac{2α}{b} = Z^{−1} ∫_{0}^{2π} cos ϕ e^{α cos ϕ+β sin ϕ}dϕ,

\frac{2β}{b} = Z^{−1} ∫_{0}^{2π} sin ϕ e^{α cos ϕ+β sin ϕ}dϕ,

and

Z = ∫_{0}^{2π} e^{α cos ϕ+β sin ϕ}dϕ.

Introducing now

α = r cos ϕ_{0}, β = r sin ϕ_{0}

and using the rotation invariance of the measure on the circle (translation invariance of dϕ), we identify ϕ_{0} as the free O(2) conjugacy parameter. The two remaining equations are

Z(r) = ∫_{0}^{2π} e^{r cos ϕ}dϕ  (33)

and

\frac{2r}{b} = (Z(r))^{−1} ∫_{0}^{2π} cos ϕe^{r cos ϕ}dϕ.  (34)

The first of these equations, (33), can be thought of as defining the function Z(r) for all values r ≥ 0, and the second (34) of the equations is then thought of as selecting the appropriate number (or numbers) r corresponding to the parameter b. These, in turn, together with the free parameter ϕ_{0}, determine the matrices S, and steady states ψ. The eigenvalues of the matrix S are

λ_{±} = ±\frac{r}{b} .  (35)

The constraint that the eigenvalues of S lie in the interval [−1/n, 1 − 1/n] implies that

0 ≤ r ≤ \frac{b}{2}  (36)

is the admissible range of solutions r. At r = 0, Z = 2π is the trivial (uniform state) solution. In order to determine whether we have nontrivial solutions we note that Z(r) is a positive function, defined for all r ∈ [0, ∞).

Z(r) = 2π ∑_{k=0}^{∞} r^{2k}2^{−2k} \frac{1}{(k!)^{2}}

and so log(Z(r)) makes sense for r ≥ 0. Now we define, for any continuous periodic function f(ϕ), a transformed function

[f](r) = (Z(r))^{−1} ∫_{0}^{2π} f(ϕ)e^{r cos ϕ}dϕ  (37)

which is a continuous function of r ≥ 0.

Lemma 3.1. For any analytic periodic function f(ϕ) defined for ϕ ∈ [0, 2π], the function [f](r) defined by (37) obeys

lim_{r→∞}[f](r) = f(0).

Proof. We divide both numerator and denominator of

\frac{∫_{0}^{2π} f(ϕ)e^{r cos ϕ}dϕ}{∫_{0}^{2π} e^{r cos ϕ}dϕ}

by e^{r}, and then we use steepest descent: splitting

∫_{−π}^{π} f(ϕ)e^{−r(1−cos ϕ)}dϕ = ∫_{−ϵ}^{ϵ} f(ϕ)e^{−r(1−cos ϕ)}dϕ + O(e^{−r/2})

we get

∫_{−π}^{π} f(ϕ)e^{−r(1−cos ϕ)}dϕ = f(0)\sqrt{\frac{π}{r}} + O(\frac{1}{r})

and dividing by the integral corresponding to f = 1 we finish the proof of the lemma.

The object of our interest is the function

\frac{d}{dr} log Z = [cos](r).

Note that

\frac{d^{2}}{dr^{2}} log Z = [(cos −[cos](r))^{2}](r) > 0,

so r ↦ [cos](r) is increasing. Because [cos](0) = 0 and cos 0 = 1, it follows from the lemma above that

0 ≤ [cos](r) ≤ 1

holds for all r ≥ 0. Introducing

H(r) = \frac{d}{dr} log Z − \frac{2r}{b} = [cos](r) − \frac{2r}{b}  (38)

the equation we wish to study (34) can be written as

Z′(r) − \frac{2r}{b} Z(r) = 0  (39)

or, equivalently, as

H(r) = 0.

Note that, because the range of [cos] is included in the interval [0, 1], it follows that all solutions of (39) are automatically in the required range (36). The explicit formula

Z′(r) − \frac{2r}{b} Z(r) = −\frac{8π}{b} ∑_{k=1}^{∞} (k − \frac{b}{4}) \frac{k}{(k!)^{2}} (\frac{r}{2})^{2k−1}  (40)

shows that there are no zeros other than r = 0 if b ≤ 4. At r = 0 we have

H′(0) = \frac{1}{2} − \frac{2}{b}

so, for b > 4 we have H′(0) > 0. On the other hand, using the lemma

lim_{r→∞} H′(r) = −\frac{2}{b}

so we have at least one nontrivial solution, for b > 4.

Theorem 3.2. Let n = 2. Let N(b) denote the number of distinct steady solutions of (1) modulo the O(2) conjugacy. Then, if b ≤ 4 then N(b) = 1. If b > 4 then N(b) ≥ 2. For b > 4

N(b) ≤ 2 [\frac{b}{4}]

where [x] denotes the largest integer not exceeding x.

Proof. We just proved that there are at least two solutions for b > 4 and only the trivial solution for b ≤ 4. Now we need to estimate the number of zeros of Z′(r) − \frac{2r}{b} Z(r) for b > 4. The expression (40) shows that

Z′(r) − \frac{2r}{b} Z(r) = P (r) − Q(r)

where P (r) is a polynomial of order 2 [\frac{b}{4}] − 1 and

Q(r) = \frac{8π}{b} ∑_{k=[ \frac{b}{4} ]+1}^{∞} (k − \frac{b}{4}) \frac{k}{(k!)^{2}} (\frac{r}{2})^{2k−1}

Differentiating m = 2 [\frac{b}{4}] times we obtain

\frac{d^{m}}{dr^{m}} (Z′(r) − \frac{2r}{b} Z(r)) < 0

for all r > 0. This, and Rolle’s theorem, finish the proof.

4. Dynamics. The two-dimensional (n=2) time dependent problem is

∂_{t}ψ = ∂_{ϕ} (e^{−V} ∂_{ϕ}(e^{V} ψ))

with

V (ϕ, t) = −\frac{b}{2} [cos(2(ϕ − ·))]

and

[f(·)] = ∫_{0}^{2π} f(ϕ)ψ(ϕ, t)dϕ.

Thus,

V (ϕ, t) = −\frac{b}{2} ∫_{0}^{2π} cos(2(ϕ − y))ψ(y, t)dy

and the equation is

∂_{t}ψ = ∂_{ϕ}^{2}ψ + ∂_{ϕ}(V_{ϕ}ψ)

with 2π periodic boundary conditions. Denote

ψ̂(j, t) = ∫_{0}^{2π} e^{−ijϕ}ψ(ϕ, t)dϕ

Clearly V_{ϕ}ψ is a simple quadratic nonlinearity; in Fourier representation

{ \atop ∂_{ϕ}(V_{ϕ}ψ)}(j, t) = \frac{bj}{2} (ψ̂(j − 2, t)ψ̂(2, t) − ψ̂(j + 2, t)ψ̂(−2, t))

Thus, the system becomes

ψ̂(0, t) = 1

and

∂_{t}ψ̂(j, t) = −j^{2}ψ̂(j, t) + \frac{bj}{2} (ψ̂(j − 2, t)ψ̂(2, t) − ψ̂(j + 2, t)ψ̂(−2, t))

Because we are interested in real solutions we have

ψ̂(−j, t) = ψ̂(j, t)^{∗}.

Certain symmetries are preserved. Even functions satisfy

ψ̂(−j, t) = ψ̂(j, t).

This is preserved by the flow. Also, ψ̂(2k + 1, t) = 0 for all k ∈ Z if initially it is zero. Returning to the physical space representation, we have ψ(ϕ, t) > 0 if initially, ψ is probability density,

V_{ϕϕ} = −4V

and V and V_{ϕ} are bounded in absolute value pointwise by \frac{b}{2} and, respectively, by b. No finite time blow up can occur, ψ is bounded pointwise by an exponential function of time. In order to show that the system is dissipative (i.e. the phase space explored is bounded), we denote

y_{k} = y_{k}(t) = ψ̂(2k, t)

From the PDE information we will use

|y_{k}(t)| ≤ 1, y_{0}(t) = 1.

We restrict to the symmetric case (so we take only even Fourier coefficients). This means we are dealing with a cosine series

ψ(ϕ, t) = \frac{1}{2π} + \frac{1}{π} ∑_{k=1}^{∞} y_{k}(t) cos(2kϕ).

The numbers y_{k} are real and obey

y′_{1} = (b − 4 − by_{2})y_{1}  (41)

and

y′_{k} = −4k^{2}y_{k} + bky_{1} (y_{k−1} − y_{k+1})  (42)

for k ≥ 1 where y′ = \frac{dy}{dt}. Note that the sign of y_{1} does not change in the evolution. Note also, with these symmetries in place, the steady state calculation for b > 4 has explicit coefficients y_{k} computed inductively starting from y_{1} = c ≠ 0 and y_{2} = 1 − \frac{4}{b}: y_{k+1} = y_{k−1} − 4k \frac{y_{k}}{by_{1}} . The parameter c is then determined by the condition that the resulting series converge. That becomes a nontrivial calculation in this setting, and the fact that there are at most 2 [\frac{b}{4}] solutions is not easily seen. In the present example the statement (17) is: for every fixed positive t,

lim_{b→∞} \frac{1}{t} ∫_{0}^{t} y_{1}^{2}(s) (1 − y_{2}(s)) ds = 0.  (43)

In order to prove dissipativity of the ODE for y_{k}, we multiply (42) by k^{−1}y_{k} to get

\frac{y_{k}}{k} y′_{k} = −4ky_{k}^{2} + by_{1}(y_{k−1} − y_{k+1})y_{k}, k ≥ 1,

and summing from k = 1 we deduce

\frac{d}{2dt} (∑_{k=1}^{∞} \frac{1}{k} y_{k}^{2}) = −4 ∑_{k=1}^{∞} ky_{k}^{2} + by_{1}^{2}.

Using |y_{1}| ≤ 1 we obtain:

Theorem 4.1. Consider the evolution equation (1) for n=2, with positive continuous initial data

ψ_{0}(ϕ) = \frac{1}{2π} + ∑_{k=1}^{∞} y_{k}(0) cos(2kϕ)

Then

∥ψ(·, t)∥_{H^{− \frac{1}{2}} (S^{1})}^{2} ≤ \frac{b}{4} + e^{−8t}∥ψ_{0}∥_{H^{− \frac{1}{2}} (S^{1})}^{2}  (44)

and

\frac{1}{t} ∫_{0}^{t} ∥ψ(·, s)∥_{H^{\frac{1}{2}} (S^{1})}^{2}ds ≤ \frac{b}{4} + \frac{1}{2t}∥ψ_{0}∥_{H^{− \frac{1}{2}} (S^{1})}^{2}  (45)

hold for all t ≥ 0. Here

∥ψ∥_{H^{s}(S^{1})}^{2} = ∑_{k=1}^{∞} k^{2s}y_{k}^{2}.

The theorem shows that the ball in H^{− \frac{1}{2}} (S^{1}) of radius \sqrt{b} centered at the uniform state ψ = \frac{1}{2π} absorbs all trajectories in finite time.

Let us consider now two solutions ψ^{(1)}(ϕ, t) and ψ^{(2)}(ϕ, t) which are given by cosine series

ψ^{(j)}(ϕ, t) = \frac{1}{2π} + \frac{1}{π} ∑_{k=1}^{∞} y_{k}^{(j)}(t) cos(2kϕ)

Let us consider their difference

ψ(ϕ, t) = ψ^{(1)}(ϕ, t) − ψ^{(2)}(ϕ, t),

and semi-sum

\overline{ψ}(ϕ, t) = \frac{1}{2}(ψ^{(1)}(ϕ, t) + ψ^{(2)}(ϕ, t)).

The Fourier coefficients are denoted accordingly,

y_{k}(t) = y_{k}^{(1)}(t) − y_{k}^{(2)}(t)

and

\overline{y}_{k}(t) = \frac{1}{2} (y_{k}^{(1)}(t) + y_{k}^{(2)}(t))

The equation for the difference is

y′_{k} = −4k^{2}y_{k} + bk\overline{y}_{1}(y_{k−1} − y_{k+1}) + bky_{1}(\overline{y}_{k−1} − \overline{y}_{k+1})  (46)

valid for all k ≥ 1. In the equation for k = 1 it is understood that y_{0} = 0 and \overline{y}_{0} = 1. Multiplying by \frac{1}{k} y_{k} and adding we obtain

\frac{d}{2dt}∥ψ(·, t)∥_{H^{− \frac{1}{2}} (S^{1})}^{2} + 4∥ψ(·, t)∥_{H^{\frac{1}{2}} (S^{1})}^{2} = by_{1} ∑_{k=1}^{∞}(\overline{y}_{k−1} − \overline{y}_{k+1})y_{k}

Using a Schwartz inequality we deduce

\frac{d}{2dt}∥ψ(·, t)∥_{H^{− \frac{1}{2}} (S^{1})}^{2} + 4∥ψ(·, t)∥_{H^{\frac{1}{2}} (S^{1})}^{2}

≤ b|y_{1}|∥ψ(·, t)∥_{H^{\frac{1}{2}} (S^{1})}\sqrt{∑_{k=1}^{∞} \frac{1}{k} (\overline{y}_{k−1} − \overline{y}_{k+1})^{2}}

Therefore, using Young’s inequality

\frac{d}{dt}∥ψ(·, t)∥_{H^{− \frac{1}{2}} (S^{1})}^{2} + ∥ψ(·, t)∥_{H^{\frac{1}{2}} (S^{1})}^{2}

≤ 2b^{2}y_{1}^{2} (1 + ∥ψ^{(1)}(·, t)∥_{H^{− \frac{1}{2}} (S^{1})}^{2} + ∥ψ^{(2)}(·, t)∥_{H^{− \frac{1}{2}} (S^{1})}^{2})

Using the inequality (44) we obtain

\frac{d}{dt}∥ψ(·, t)∥_{H^{− \frac{1}{2}} (S^{1})}^{2} + ∥ψ(·, t)∥_{H^{\frac{1}{2}} (S^{1})}^{2}

≤ 2b^{2}y_{1}^{2} (1 + \frac{b}{2} + e^{−8t}∥ψ_{0}^{(1)}∥_{H^{− \frac{1}{2}} (S^{1})}^{2} + e^{−8t}∥ψ_{0}^{(2)}∥_{H^{− \frac{1}{2}} (S^{1})}^{2}) .  (47)

Theorem 4.2. Let ψ^{(j)}(ϕ, t), j = 1, 2, be two solutions of (1) for n = 2. Assume that the corresponding potentials V^{(1)}(ϕ, t) and V^{(2)}(ϕ, t) become close asymptotically at ϕ = 0:

lim_{t→∞} |V^{(1)}(0, t) − V^{(2)}(0, t)| = 0.

Then the two solutions approach each other:

lim_{t→∞} ∥ψ^{(1)}(·, t) − ψ^{(2)}(·, t)∥_{H^{− \frac{1}{2}} (S^{1})} = 0

Proof. Note that

V^{(j)}(ϕ, t) = −\frac{b}{2}y_{1}^{(j)}(t) cos(2ϕ)

and therefore the asumption implies

lim_{t→∞} |y_{1}(t)| = 0

where y_{1}(t) is the Fourier coefficient of the difference of solutions. The conclusion of the theorem follows then from (47).

This shows that the system has one determining mode. Using standard tools ([1]) one can prove that this system has a finite dimensional global attractor.

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