APPROACH TO EQUILIBRIUM FOR A FORCED BURGERS EQUATION

WERNER KIRSCH AND BARRY SIMON^{1}

Abstract. We show that approach to equilibrium in certain forced Burgers equations is implied by a decay estimate on a suitable intrinsic semigroup estimate, and we verify this estimate in a variety of cases including a periodic force.

1. Introduction

This paper is a contribution to the literature [10, 11, 3] on large time asymptotics of the forced Burgers equation

\frac{∂u_{i}}{∂t} + ∑_{j=1}^{ν} u_{j} \frac{∂u_{j}}{∂x_{i}} = \frac{1}{2} Δu_{i} + \frac{∂V (x)}{∂x_{i}}, t ≥ 0, x ∈ ℝ^{ν}, i = 1, …, ν, (1.1)

where u is real valued. We will make two assumptions on the initial data u_{j}(x, t = 0) = u_{j}^{(0)}(x):

(i) \frac{∂u_{i}^{(0)}}{∂x_{j}} = \frac{∂u_{j}^{(0)}}{∂x_{i}}  all i, j  (1.2)

(ii) ψ_{0}(x) ≡ ∫_{0}^{x} u^{(0)}(y) · dy ∈ L^{∞}.  (1.3)

(1.2), which is vacuous in the standard ν = 1 case, implies that the value of ψ_{0} given by (1.3) is independent of the path taken from 0 to x in the line integral. Typical of our results is:

Theorem 1.1. Let V be a C^{1} periodic function on ℝ^{ν}. Then, there is a unique initial condition u_{∞}^{(0)}(x) obeying (1.2), (1.3) for (1.1) whose solution is independent of t. Moreover, if u^{(0)} is any other initial data obeying (1.2)/ (1.3), then

lim_{t→∞} sup_{x}[|u(x, t) − u_{∞}^{(0)}(x)|] = 0.  (1.4)

1Supported in part by NSF Grant No. DMS-9707661.

Date: June 11, 2001.

∗ To appear in Journal of Evolution Equations.

Remark. (1.3) does not imply that u^{(0)} is L^{∞}, but our proof shows that for t > 0, u( · , t) ∈ L^{∞}. Thus, for t > 0, the quantity in the limit in (1.4) is finite.

What is new about our ideas and Theorem 1.1 is that there is no regularity condition on the initial condition u at infinity other than (1.3). Previous approaches require at least that e^{−ψ_{0}} have some kind of average. To understand how we overcome this, we need to begin the proof by reminding the reader of the Cole-Hopf transformation. Define

φ_{0}(x) = exp(−ψ_{0}(x))  (1.5)

so that

u_{i}^{(0)} = −φ_{0}^{−1} \frac{∂φ_{0}}{∂x_{i}} .  (1.6)

V is bounded so

H = −\frac{1}{2}Δ + V  (1.7)

is bounded below. Thus we add a constant to V so that henceforth

inf spec(H) = 0.  (1.8)

Define

φ(x, t) = (e^{−tH}φ_{0})(x).  (1.9)

Then direct manipulation shows that

Proposition 1.2. Let ψ_{0} ∈ L^{∞} be C^{1} and let φ_{0}, φ obey (1.5), (1.9). Then, for t > 0, φ(x, t) > 0 and φ, ∇φ, Δφ are C^{1} and

u(x, t) = −φ(x, t)^{−1}(∇φ)(x, t)  (1.10)

obeys (1.1) for t > 0 and lim_{t↓0} u( · , t) = u^{(0)} ≡ ∇ψ_{0}.

Remarks. 1. It follows from [9] that with V a C^{1} function with bounded derivatives that φ and (−Δ + V)φ are C^{1}. It follows that Δφ is C^{1} which by elliptic regularity means ∇φ is C^{1}. The Laplacian here and in (1.1) may be distributional rather than classical. If ∇V is assumed Hölder continuous, we can replace these by classical derivatives.

2. As we will discuss below, e^{−tH} maps L^{∞} to L^{∞} and (1.9) is intended in the sense of the L^{∞} map.

3. This result does not require that V be periodic; V need only be C^{1} and in L^{∞}. We will use it in this form below.

When V is periodic, it has a periodic ground state Ω, that is, a positive periodic solution of

(−\frac{1}{2}Δ + V)Ω = 0.  (1.11)

Thus

u_{∞}^{(0)}(x) = −Ω(x)^{−1}(∇Ω)(x)  (1.12)

is a stationary solution of (1.1). A natural approach to (1.4) is to prove that

φ → c Ω  (1.13)

and

∇φ → c∇Ω  (1.14)

both in L^{∞}. This is essentially what previous works do.

To understand the limitations of this approach and why one can hope to go beyond them, consider the case V ≡ 0. Then u_{∞} = 0, Ω ≡ 1, and

φ(x, t) = (2πt)^{−ν/2} ∫ exp(−(x − y)^{2}/2t)φ_{0}(y) dy.

If (2R)^{−ν} ∫_{sup_{i}|y_{i}|≤R} φ_{0}(y) d^{ν}y → c as R → ∞, it is not hard to see that φ(x, t) → c as t → ∞ for each fixed x, so (1.13) holds, but this is not true in general.

For example, if R_{n} = e^{e^{n}} and

φ_{0}(y) = 2 + (−1)^{n} if R_{n} < sup_{i}|y_{i}| < R_{n+1},

then for t ∼ R_{n}R_{n+1}, it is not hard to see that φ(0, t) ∼ 2 + (−1)^{n} and thus φ(0, t) does not have a limit. But in this example, one can see that ∇φ does go to zero.

Our key observations are that rather than prove (1.13) and (1.14) separately, it suffices to prove that

∇(φ/Ω) → 0  (1.15)

and that (1.15) is equivalent to some estimates on the intrinsic semigroup associated to H. Specifically, let K_{t}(x, y) be the integral kernel of e^{−tH} and let

L_{t}(x, y) = Ω^{−1}(x)K_{t}(x, y)Ω(y)^{−1}.  (1.16)

L_{t} is the kernel of a semigroup on L^{2}(ℝ^{ν}, Ω^{2}d^{ν}x).

To prove Theorem 1.1, we will prove two estimates:

|∂_{x}L_{t}(x, y)| ≤ C t^{−ν/2}[exp(−D(x − y)^{2}/t) + exp(−E|x − y|)] (1.17)

for suitable C, D and all t > 1, all x, y and

|∂_{x}L_{t}(x, y)| ≤ C t^{−(ν+1)/2}  (1.18)

all t > 1, all x, y.

We will show that if Ω obeys 0 < a ≤ Ω < b, is C^{3} and V ≡ \frac{1}{2}Ω^{−1}(ΔΩ) is bounded and uniformly Hölder continuous, then (even if V is not periodic)

|∂_{x}L_{t}(x, y)| ≤ C t^{−α}[exp(−D(x − y)^{2}/t)]  (1.19)

for some α > ν/2, all t ≥ 1. This will lead to the following generalization of Theorem 1.1:

Theorem 1.3. Let V be a C^{1} function and suppose that V is bounded and uniformly Hölder continuous. Suppose that −\frac{1}{2}Δ+V has a ground state Ω obeying 0 < a ≤ Ω ≤ b for some a, b and all t. Then there is a unique initial condition u_{∞}^{(0)}(x) (= ∇Ω/Ω) obeying (1.2), (1.3) for (1.1) whose solution is independent of t. Moreover, if u^{(0)} is any other initial data obeying (1.2)/ (1.3), then

lim_{t→∞} sup_{x}[|u(x, t) − u_{∞}^{(0)}(x)|] = 0.  (1.20)

Certain quasiperiodic Schrödinger operators have a quasiperiodic ground state [5]. Thus the above theorem applies to this situation as well; see [11].

In Section 2, we will reduce Theorems 1.1 and 1.3 to (1.19). In Section 3, we will derive (1.17) using ideas due to Davies. In Section 4, we will prove (1.18) in the periodic case and (1.19) in general.

We are dedicating this paper to the memory of Tosio Kato, who taught us so much about Schrödinger operators, about semigroups, and about non-linear equations, areas which come together here.

2. Reduction to Intrinsic Heat Kernel Estimates

According to Proposition 1.2, the solution u of (1.1) is related to a solution of (1.9) via (1.16). This leads to

Theorem 2.1. Suppose there exist Ω so

e^{−tH}Ω = Ω  (2.1)

and

0 < a ≤ Ω(x) ≤ b  (2.2)

and that

lim_{t→∞} ∥∇(φ( · , t)/Ω( ·))∥_{∞} = 0.  (2.3)

Then (1.20) holds with u_{∞}^{(0)} = ∇Ω/Ω.

Proof. By (1.3), we have

0 < c_{1} ≤ φ_{0} ≤ c_{2}

so by (2.2)

c_{1}b^{−1}Ω ≤ φ_{0} ≤ c_{2}a^{−1}Ω.

Since e^{−tH} is positivity preserving and (2.1) holds,

c_{1}b^{−1}Ω(x) ≤ φ(x, t) ≤ c_{2}a^{−1}Ω(x).

So by (2.2) again,

c_{1}b^{−1}a ≤ φ(x, t) ≤ c_{2}a^{−1}b.  (2.4)

Now

u( · , t) − u_{∞}^{(0)} = (∇φ)( · , t)/φ( · , t) − (∇Ω)( ·)/Ω(x)
= (Ω∇φ − φ∇Ω)/φΩ
= [∇(φ( · , t)/Ω)][Ω/φ].

Since φ and Ω are uniformly in t and x bounded above and below, we see that (1.20) is equivalent to (2.3).

Now consider the unitary map U : L^{2}(ℝ^{ν}) → L(ℝ^{ν}, Ω^{2} dx) by (Uf)(x) = f(x)Ω(x)^{−1} and let M be the self-adjoint operator UHU^{−1} on L^{2}(ℝ^{ν}, Ω^{2} dx). Then, as is well-known (and a direct calculation),

(f, Mf)_{L^{2}(ℝ^{ν},Ω^{2} dx)} = ∫ (∇f)^{2}Ω^{2} dx

or equivalently,

Mf = −Δf − 2(∇Ω)Ω^{−1} · ∇f.  (2.5)

Now let K_{t}(x, y) be the integral kernel of e^{−tH}, that is,

(e^{−tH}f)(x) = ∫ K_{t}(x, y)f(y) d^{ν}y

and let L_{t}(x, y) be the integral kernel of e^{−tM}, that is,

(e^{−tM}f)(x) = ∫ L_{t}(x, y)f(y)Ω^{2}(y) d^{ν}y.

Since e^{−tM} = Ue^{−tH}U^{−1}, we see that L_{t} and K_{t} are related by (1.16). Now if φ = e^{−tH}φ_{0}, then

[φ( · , t)/Ω( ·)] = Ue^{−tH}φ_{0} = e^{−tM}Uφ_{0}
= ∫ L_{t}( · , y)φ_{0}(y)Ω(y) d^{ν}y.

6 W. KIRSCH AND B. SIMON

Since φ_{0} and Ω are uniformly bounded, we see that

|∇[φ(x, t)/Ω(x)]| ≤ c ∫ |∂_{x}L_{t}(x, y)| d^{ν}y.

Thus: Proposition 2.2. A sufficient condition for (2.3) to hold for any initial φ_{0} (coming from a u_{i}^{(0)} obeying (1.2)/ (1.3)) is that

sup_{x} ∫ |∂_{x}L_{t}(x, y)| d^{ν}y → 0  (2.6)

as t → ∞. Theorem 2.3. If (1.19) holds or if (1.17)/ (1.18) hold, then (2.6) holds. Proof. (1.19) plus scaling implies that

∫ |∂_{x}L_{t}(x, y)| d^{ν}y ≤ C_{1} t^{−α}t^{ν/2}

which goes to zero if t → ∞ since α > ν/2. (1.17)/(1.18) imply

|∂_{x}L_{t}(x, y)| ≤ C t^{−ν/2−1/4}[exp(−D(x − y)^{2}/2t) + exp(−\frac{1}{2}E|x − y|)] which implies

∫ |∂_{x}L_{t}(x, y)| d^{ν}y ≤ C_{1} t^{−1/4} ± C_{2} t^{−ν/2−1/4}

which goes to zero as t → ∞.

3. Exponential-Gaussian Estimates on ∂_{x}L_{t} Our goal in this section is to explain how one can get (1.17) from

ideas of Davies [1, 2]. His ideas immediately imply an estimate

|K_{t}(x, y)| ≤ C_{ε} t^{−ν/2} exp(−(x − y)^{2}/(4 + ε)t)  (3.1)

for any ε > 0. (3.1) implies (even dropping the Gaussian) that

∥e^{−tH}∥_{L^{1}→L^{∞}} ≤ C t^{−ν/2}

so by interpolation with boundedness on L^{∞} (see Simon [9] and references therein),

∥e^{−tH}∥_{L^{1}→L^{2}} = ∥e^{−tH}∥_{L^{2}→L^{∞}} ≤ C t^{−ν/4}.

Thus

∥e^{−(t+is)H}∥_{L^{1}→L^{∞}} ≤ ∥e^{−tH/2}∥_{L^{2}→L^{∞}} ∥e^{−isH}∥_{L^{2}→L^{2}} ∥e^{−tH/2}∥_{L^{1}→L^{2}}
≤ C t^{−ν/2}

and thus for any |θ| < π/2,

∥ exp(−te^{iθ}H)∥_{L^{1}→L^{∞}} ≤ C_{θ} t^{−ν/2}

which yields

|K_{te^{iθ}}(x, y)| ≤ C_{θ} t^{−ν/2}.  (3.2)

By interpolation between (3.1) and (3.2), we see that for complex t in a section S_{θ} = {t | |Arg(t)| ≤ θ} we have if θ < π/2,

|K_{t}(x, y)| ≤ C_{θ} |t|^{−ν/2} exp(−D_{θ}(x − y)^{2}/|t|)  (3.3)

and this implies by a Cauchy estimate that in the same sectors

|∂_{t}K_{t}(x, y)| ≤ C_{θ} |t|^{−ν/2−1} exp(−D_{θ}(x − y)^{2}/|t|)

(where C_{θ}, D_{θ} can change value from one equation to the next). Thus for t ≥ 1 and real,

|(−Δ + V)_{x}K_{t}(x, y)| ≤ C |t|^{−ν/2−1} exp(−D(x − y)^{2}/|t|).  (3.4)

Since V is bounded, (3.3) and (3.4) imply that for t ≥ 1,

|(−Δ_{x} + 1)K_{t}(x, y)| ≤ C |t|^{−ν/2} exp(−D(x − y)^{2}/|t|).

But ∂_{x}(−Δ_{x} + 1)^{−1} has an explicit convolution integral kernel which is L^{1} at short distances and exponentially decaying at large. This implies for t ≥ 1

|(∂_{x}K_{t})(x, y)| ≤ C t^{−ν/2}[exp(−D(x − y)^{2}/t) + exp(−E|x − y|)]

which, with (3.3) and the formula (1.16), implies (1.17). We summarize:

Theorem 3.1. The estimate (1.17) holds for any potential V obeying the hypothesis of Theorem 1.3.

4. Improved Time Decay Estimates on ∂_{t}L_{t}

Here the goal is to show that

|(∂_{x}L_{t})(x, y)| ≤ C t^{−α}  (4.1)

for some α > ν/2. We believe the estimate holds with α = ν/2 + 1/2 and have proven this if V is periodic: One makes a Bloch wave decomposition [8] to write the semigroup as an integral over the Brillouin zone and a Gaussian approximation to control the resulting integral. One uses the fact that the minimum of the bottom band is known to be a unique point with a strictly quadratic minimum [4]. In general, we rely on estimates of Porper-Eidel’man [6] and only get α > ν/2. (In a later paper [7], they get α = ν/2 + 1/2 for the case p = 1 below that does not accommodate our situation.)

Indeed, (4.1) is exactly Corollary 3.4 of their paper which they prove for fundamental solutions of equations of the form

p(x)∂_{t}u = ∇ · (a(t, x)∇u),  (4.2)

where p, a and ∇a are all Hölder continuous.

But (2.5) can be rewritten

Mf = −Ω^{−2}∇ · (Ω^{2}∇f)

so

∂_{t}u = −Mu

is of the form (4.2) where

p = Ω^{2} and a_{ij} = Ω^{2}δ_{ij}.

Thus, their result applies so long as Ω is C^{1} with ∇Ω uniformly Hölder continuous.

Since ΔΩ = V Ω with V ∈ C^{1} and Ω a priori bounded, we see that

∇Ω = ∇(−Δ + 1)^{−1}(−V + 1)Ω.

Since (−V +1)Ω is bounded, the explicit integral kernel for ∇(−Δ+1)^{−1} shows ∇Ω is uniformly Hölder continuous of any order less than 1. Thus, their Corollary 3.4 applies and (4.1) holds.

Acknowledgments. W.K. would like to thank T. Tombrello for the hospitality of Caltech where this research was done. We would like to thank E.B. Davies, T. Coulhon, A. Grigoryan, Y. Pinchover, and D. Robinson for valuable correspondence.

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[8] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV. Analysis of Operators, Academic Press, New York, 1978.

[9] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447–526.

[10] Ya.G. Sinai, Two results concerning asymptotic behavior of solutions of the Burgers equation with force, J. Stat. Phys. 64 (1991), no. 1–2, 1–12.

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W. Kirsch, Institut für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany

E-mail address: werner@mathphys.ruhr-uni-bochum.de

B. Simon, Division of Physics, Mathematics, and Astronomy, 253-37, California Institute of Technology, Pasadena, CA 91125-3700, USA

E-mail address: bsimon@caltech.edu