Functional Analysis and Its Applications, Vol. 35, No. 1, pp. 1–12, 2001
Spectral Properties of Solutions of the Burgers Equation with Small Dissipation ` Biryuk A. E.
UDC 517.9
We study the asymptotic behavior as δ → 0 of the Sobolev norm um of the solution to the Cauchy problem for the one-dimensional quasilinear Burgers type equation ut + f (u)x = δuxx (It is assumed that the problem is C ∞ , the boundary conditions are periodic, and f σ > 0.) We show that the locally time-averaged Sobolev norms satisfy the estimate cm δ −m+1/2 < u2m 1/2 < Cm δ −m+1/2 (m 1). The estimates obtained as a consequence for the Fourier coefficients justify Kolmogorov’s spectral theory of turbulence for the case of the Burgers equation.
1. Introduction We consider the Cauchy problem for the one-dimensional (dim x = 1) Burgers type parabolic equation ∂ ∂ (1.1) u+ f (u) = δuxx ∂t ∂x with the initial data (1.2) u(0, x) = u0 (x) and the periodic boundary conditions u(t, x ± ) = u(t, x).
(1.3)
Here δ < 1 is a small positive parameter and the initial function u0 ( · ) is C ∞ and -periodic. The function f : R → R is C ∞ and uniformly convex (f σ > 0). The classical theory of parabolic equations (e.g., see [1]) states that for each δ > 0 problem (1.1)–(1.3) has a unique solution u = uδ (t, x), which is C ∞ . It is also known that the solution of problem (1.1)–(1.3) converges as δ → +0 to the so-called (weak) entropy solution of the quasilinear equation ∂ u ∂ f (u) + = 0. (1.1 ) ∂t ∂x (See [2], where the issue was studied under √ the most general assumptions.) Moreover, the convergence rate in L1 (R) is at least Const · δ [3]. See the appendix for the proof of this fact in our situation. We note that Eq. (1.1 ) is not solvable in general even in the class of continuous functions. For this reason, one considers weak solutions in the sense of the corresponding integral identity. However, this results in the loss of uniqueness. The entropy condition selects a unique solution, which is most natural from the physical point of view. We shall study the behavior of the Sobolev H m -seminorms m 2 ∂ u 2 um = m dx ∂x as δ → 0. Throughout the paper, we write . . . dx (omitting the integration limits) for 0 . . . dx = x0 + . . . dx, i.e., an integral over the period or, equivalently, an integral over the circle S = R/Z. x0 From now on we omit the prefix semi- from the word “seminorm.” Moscow State University; Heriot-Watt University, Edinburgh. Translated from Funktsional nyi Analiz i Ego Prilozheniya, Vol. 35, No. 1, pp. 1–15, January–March, 2001. Original article submitted September 15, 1999. 0016–2663/01/3501–0001 $25.00
c 2001 Plenum Publishing Corporation
1
The Sobolev H m -norm is a function of time t. We average this function over [0, T ]: 1 T ∂ m u 2 2 um [0,T ] = ∂xm dx dt. T 0 We claim that um
m−1/2 1 Cm , δ
(1.4)
for any m 1, sufficiently small δ, and all t, whereas for some T , which depends on the initial data of problem (1.1)–(1.3) (see Eq. (3.2) below), one has the opposite inequality 1/2 u2m [0,T ]
m−1/2 1 cm . δ
(1.5)
Here the constants cm and Cm are independent of δ and can be explicitly expressed via the C k -norms (k m + 1) of the initial data of problem (1.1)–(1.3). The Sobolev H m -norm can be expressed via the Fourier coefficients as u2m =
2π
2m +∞
s2m |ˆ u s |2 ,
s=−∞
1 where u ˆs = √
u(x)e−2πisx/ dx .
Together with (1.4) and (1.5), this relation provides some information about the behavior of the Fourier coefficients of the solution as δ → +0. Namely, let θ > 0 be a given number. By Aˆ2s,θ we denote the mean value of |ˆ uk |2 (t) over t ∈ [0; T ] and over k such that δ −s+θ < |k| δ −s−θ , i.e., T uk |2 (t) 1 δ −s+θ <|k|δ −s−θ |ˆ 2 dt, Aˆs,θ = T 0 δ −s+θ <|k|δ −s−θ 1 where s > 0. Then Aˆ2s,θ satisfies the following estimates for sufficiently small δ: du0 2 2s 2 ˆ δ As,θ < 2 π dx L1 2m 2 2 2+2m(s−1−θ) ˆ Cm δ As,θ < 2π 2 2 c1 2+2θ 2 ˆ δ A1,θ > 2π 4
∀m > 0
for s > 0,
(1.6)
for s > 1 + θ,
(1.7)
for s = 1.
(1.8)
Our main goal is to prove the estimates (1.4)–(1.8) for problem (1.1)–(1.3). We note that relations (1.6)–(1.8) are closely related to the so-called Kolmogorov asymptotics for freely decaying turbulence (i.e., unforced turbulence) described by the Burgers equation (1.1). In particular, it follows from (1.6)–(1.8) that the Kolmogorov dissipation scale of solutions of the Burgers equation (1.1) is equal to δ and the exponent in the spectral law is equal to 2 (see [8]; cf. [9, 10]). The following interpolation inequalities for Sobolev norms and C k norms will be used in the sequel: 1−k/m
uk u0 |u|k 2
uk/m m
1−k/m Cmk |u|0 |u|k/m m
(0 k m),
(1.9)
(0 k m).
(1.10)
Here |u|k = maxx∈S |dk u/dxk |. The proof of inequality (1.9) is so easy that we give it here: 2k u2k = (|ˆ us |2 )(m−k)/m (|ˆ us |2 s2m )k/m 2π k/m (m−k)/m k/m 2m 1−k/m 2 2 2m 2 |ˆ us | |ˆ us | s = u0 um . (2π) The proof of inequality (1.10) is nontrivial and can be found in [4], where the following sharp values of the constants Cmk are also given: Km−k 4 1 (−1)i+1 (−1)i+1 Cmk = 1−k/m , where Ki = + + + · · · . 1+ π 3i+1 5i+1 7i+1 Km We also need the easy-to-verify inequality (Cm1 )m 2. Furthermore, we shall use the Gagliardo–Nirenberg inequality for periodic functions: m α/m α ∂ u 1−α/m ∂ u α(m−α) 4 |u|L∞ m , ∂xα ∂x L2 L2m/α
(1.11)
(1.12)
where α is an integer between 0 and m (see [5, pp. 106–107]). We denote ∂ m u/∂xm by u(m) ; likewise, ux = ∂u/∂x, uxx = ∂ 2 u/∂x2 , and ut = ∂u/∂t. Unless otherwise specified, the symbol Lp stands for Lp (S). The author is grateful to Professor S. B. Kuksin, who posed the problem and paid much attention to the research. 2. Upper Bounds Let u+ and u− be the maximum and minimum value, respectively, of the initial function u0 = u0 (x). The oscillation of the solution at time t is defined as oscx u(t, x) = maxx u(t, x)−minx u(t, x). We introduce the number 1 u+ − u − D = osc u = . (2.1) 2 x 2 t=0 By the maximum principle, u− u(t, x) u+ and 1 osc u(t, x) D for all t. 2 x Multiplying Eq. (1.1) by u and integrating over the period, we obtain uut dx = −δ u2x dx, or
(2.2)
(2.3)
oscx u(t, x) |ut |L1 Dut |t=0 L1 . 2 In the last inequality, we have used inequality (2.2) and the fact that |ut |L1 = |(∂u(t, x)/∂t)| dx is a nonincreasing function of t (see Lemma A in the appendix). We set
2 df (u0 ) d u0 C1 = D · (2.4) dx dx + dx2 dx . δ
u2x dx = −
uut dx
Expressing ut |t=0 from Eq. (1.1), we find that uδ 1 C1 δ −1/2
(2.5)
for all t 0 and 0 < δ < 1. 3
Differentiating equation (1.1) m−1 times with respect to x, multiplying by u(m+1) , and integrating over the period, we obtain m+1 2 m+1 m 2 ∂ ∂ u ∂ u u ∂ m f (u) 1 d dx = −δ dx + dx. (2.6) 2 dt ∂xm ∂xm+1 ∂xm+1 ∂xm To prove the main estimates of this section, we need the following auxiliary technical inequality. Lemma. For all m 1, one has m+1 m ∂ ∂ f (u) u (2.7) ∂xm+1 ∂xm dx Bm um um+1 , where Bm = Qm |f ( · )|C m [u− ,u+ ] max(1, |u0 |m−1 L∞ )
(2.8)
and Qm depends on m alone. Proof. By induction on m, one can readily prove that m ∂ m f (u) dj f (u) (α1 ) (α2 ) = Aj,m u · · · u(αj ) , α1 ,...,αj u ∂xm duj α +···+α =m j=1
1
j
1α1 ···αj
where the Aj,m α1 ,...,αj are nonnegative integers. Consequently, j m+1 m m ∂ d f (u) u ∂ f (u) j,m |u(m+1) | |u(α1 ) | · · · |u(αj ) | dx Aα1 ,...,αj ∂xm+1 ∂xm dx j du α +···+α =m j=1
1
j
1α1 ...αj
(we use the H¨older inequality with regard for the fact that 1/2 + α1 /(2m) + · · · + αj /(2m)=1) j m d f (u) j,m |u(m+1) |L |u(α1 ) |L Aα1 ,...,αj · · · |u(αj ) |L2m/αj 2 2m/α1 j du L ∞ α +···+α =m j=1
1
j
1α1 ···αj
2 (we use inequality (1.12) with regard for the fact that ji=1 4αi (m−αi ) 4m ) j m d f (u) j,m |u(m+1) |L 4m2 |u|j−1 |u(m) |L Aα1 ,...,αj 2 2 L∞ j du L∞ α +···+α =m j=1
=
1
j
1α1 ···αj
m
m2
j=1 α1 +···+αj =m 1α1 ···αj
Aj,m α1 ,...,αj 4
j d f (u) duj
L∞
j−1 um um+1 . |u|L ∞
Since |u|L∞ |u0 |L∞ by the maximum principle, the proof of the lemma is complete. Combining (2.6) with (2.7), we obtain the estimate 1 d u2m −δu2m+1 + Bm um um+1 = um+1 (−δum+1 + Bm um ). 2 dt Theorem. If m ∈ N and δ
δ0
= Bm
C1 √ Bm u0 m
(2.9)
1/(m−1/2) ,
(2.10)
then m−1 C1 δ −m+1/2 u(t, · )m Bm
for all t 0. Here C1 and Bm are defined in (2.4) and (2.8), respectively. 4
(2.11)
Proof. Substituting ux for u into (1.9), we obtain 1/m
um u1
(m−1)/m
um+1
.
(2.12)
We rewrite this as um+1 um (um /u1 )1/(m−1) . Combining (2.9) with the last inequality, we have um 1/(m−1) 1 d 2 um um+1 um − δ + Bm . 2 dt u1 Hence if um > (Bm /δ)m−1 u1 , then um is decreasing. With regard for (2.5), we conclude that m−1 if u(t, · )m > Bm C1 δ −m+1/2 ,
then um is decreasing.
(2.13)
Condition (2.10) implies that inequality (2.11) holds for t = 0. Combining this with (2.13), we conclude that um never exceeds the right-hand side of (2.11). 3. Lower Bounds In this section we shall use Lax’s explicit formula [6] for the weak entropy solution u0 (t, x) of the quasilinear equation (1.1 ). Let g be the Legendre transform of the convex smooth function f . Let ymin = ymin (t, x) be the y point of global minimum of the auxiliary function ψt,x (y) = 0 u0 (η) dη + t · g((x − y)/t). The point for almost all t and x. Then u0 (t, x) = g ((x − ymin (t, x))/t). ymin is uniquely determined Let M = −1 u0 dx be the mean value of the initial function. In our (-periodic) case the explicit formula for the solution of Eq. (1.1 ) implies the estimate . (3.1) sup |u0 (t, x) − M | tσ x Indeed, let y1 be the point of minimum of the function t·g((x−y)/t)+M y, i.e., g ((x−y1 )/t) = M . Since the functions f and g areinverses of each other, we have (x − y1 )/t = f (M ). It follows from y the -periodicity of the function 0 u0 (η) dη−M y that |ymin −y1 | < . Thus (x−ymin )/t = f (M )+κ, where |κ| /t. Now we can write 1 , |u0 (t, x) − M | = |g (f (M ) + κ) − g (f (M ))| sup |g | · |κ| σ t 0 just as claimed in (3.1). As a consequence, we find that (u (t, x) − M )2 dx 3/(t2 σ 2 ). Hence at time 23/2 (3.2) T = σ|u0 ( · ) − M |L2 we have the estimate 1 (u0 (T, x) − M )2 dx |u0 ( · ) − M |2L2 . (3.3) 4 To simplify the formulas, in what follows we assume that M = 0. This assumption results in no loss of generality, since we can make the substitution u → u − M , f ( · ) → f ( · + M ). Theorem. Suppose that m ∈ N and δ < δ0 = Then
T 0
|u0 |4L2 1 . 3072 T |u0 |2L∞ ( |du0 /dx| dx)2
u(t, · )2m dt
1/2
cm
m−1/2 1 , δ
(3.4)
where T is defined in (3.2) and cm is independent of δ. Proof. We rewrite (2.3) as 1 d δ 2 δ (ux (t, x)) dx = − (uδ (t, x))2 dx 2 dt 5
and integrate over [0, T ] with respect to t: T 1 1 δ 2 δ 2 δu 1 dt = (u (0, x)) dx − (uδ (T, x))2 dx. 2 2 0
(3.5)
Inequality (3.4) with m = 1 will be proved once we show that the last term in (3.5) is small. Suppose that δ < δ0 ; then, using inequality (A.6), we have δ 2 0 2 (u (T, x)) dx − (u (T, x)) dx |uδ (T, x) − u0 (T, x)| |uδ (T, x) + u0 (T, x)| dx du0 4 √ dx · 2|u0 |L∞ 1 |u0 |2L . Tδ √ 2 dx 12 3 Combining this with (3.5) and (3.3), we obtain T 1 1 1 1 1 δ 2 2 2 2 δu 1 dt |u0 |L2 − |u0 |L2 + |u0 |L2 = |u0 |2L2 , 2 2 4 12 3 0 or |u0 |2L2 1 T δ 2 . u 1 dt T 0 3T δ √ Thus inequality (3.4) holds for m = 1 with the constant c1 = |u0 |L2 / 3T . It follows from (2.2) and the interpolation inequality (1.10) with k = 1 that
(3.6)
1/m
|ux |L∞ Cm1 D(m−1)/m |u(m) |L∞ , where D is defined in (2.1). Using (1.11), we obtain |u(m) |L∞ (|ux |L∞ )m/(2Dm−1 ). Using inequality (A.4), we obtain |∂uδ /∂x|L∞ uδ 21 /|du0 /dx|L1 . Thus we have |u(m) |L∞
uδ 2m 1 . 2Dm−1 |du0 /dx|m L1
(3.7)
Lemma. Let v be a periodic function vanishing at some point. Then |v|2L∞ |v|L2 |vx |L2 . Proof. For each y, we have y d 2 1 2 v (x) dx v (y) = 2 v −1 (0) dx
d 2 v (x) dx = |v(x)| |v (x)| dx |v|L |vx |L . 2 2 dx
We apply this lemma to v = ∂ m u/∂xm and, using inequality (2.12), obtain 1/2
1/(2m)
|u(m) |L∞ u1/2 m um+1 u1 Hence
um+1
|u(m) |2m L∞ u1
1/(2m−1)
(3.7)
(2m−1)/(2m)
um+1
.
u2m+1 1 . 2m/(2m−1) (2Dm−1 |du0 /dx|m L1 )
Now we square this inequality, use the inequality T 2m+1 1 1 T 2 u4m+2 dt u dt , 1 1 T 0 T 0 and average over t ∈ [0, T ]: T (3.6) (|u0 |2L2 /(3T δ))2m+1 ( T1 0 u21 dt)2m+1 1 T 2 um+1 dt . 4m/(2m−1) 4m/(2m−1) T 0 (2Dm−1 |du0 /dx|m (2Dm−1 |du0 /dx|m L1 ) L1 ) This proves inequality (3.4) for all m ∈ N. 6
It is of interest to compare inequality (3.4) for the case m = 1 with the upper bound for the averaged L1 -norm (which follows from inequality (A.4)): 1/2 2 −1/2 c1 δ , |u (x)| dx Const . u (x) dx Roughly speaking, this means that the graph of the gradient u (x) is a curve with rare high peaks. This behavior is closely related to the intermittency phenomenon in turbulence theory (see [8]). 4. The Behavior of the Fourier Coefficients In this section, we prove the estimates (1.6)–(1.8). The Fourier coefficients of a function defined on the circle S are the coefficients of its expansion in the orthonormal basis formed by the exponentials ek ( · ), where 1 2πikx , k ∈ Z. ek (x) = √ exp The expansion of u(x) in this basis has the form u(x) =
+∞ k=−∞
The Sobolev
H m -norm
is given by u2m
=
2π
Let a ˆ2k then we have 1 T
0
T
1 = T
u2m dt
1 where u ˆk = √
u ˆk ek (x),
u(x) exp
2m +∞
−2πikx dx.
k 2m |ˆ u k |2 .
k=−∞
T 0
=
|ˆ uk (t)|2 dt;
2π
2m +∞
k 2m a ˆ2k .
k=−∞
Combining this formula with the results of Secs. 2 and 3, we see that for every m 1 and for sufficiently small δ (namely, δ < δ0 = min(1, δ0 , δ0 )), the estimates c˜m δ −2m+1 <
+∞
m δ −2m+1 k 2m a ˆ2k < C
(∗)
k=−∞ 2 . We set m = (/(2π))2m Cm and C du0 2 . V = 2 4π dx L1
hold, where c˜m =
(/(2π))2m c2m
Then
√ √ i −2πikx V 1/2 ∂ u ∂u |ˆ uk | = exp dx . 2πk ∂x 2π|k| ∂x L1 |k|
In the last estimate we have used inequality (A.4). Thus, a ˆ2k V /k 2 . Let s 0 and θ > 0. By Aˆ2s,θ we denote the mean value of a ˆ2k over |k| ∈ (δ −s+θ ; δ −s−θ ]. Suppose that δ θ < 1/2; this condition ensures that the averaging is carried out over a nonempty k+1/2 set of indices k. Then, using the obvious inequality k −2 < k−1/2 x−2 dx, we have [δ−s−θ ]+1/2 V [δ−s+θ ]+1/2 x−2 dx V 2 = < 4V δ 2s , Aˆs,θ < [δ −s−θ ] − [δ −s+θ ] ([δ −s−θ ] + 1/2)([δ −s+θ ] + 1/2) 7
and the estimate (1.6) is thereby proved. Let us proceed to the estimate (1.7). Suppose that 1 < s − θ < s < s + θ, δ θ < 1/2, and condition (∗) holds. Then
a ˆ2k
(δ
s−θ 2m
)
|k|>δ −s+θ
k 2m a ˆ2k
(δ
)
Aˆ2s,θ
ˆ2k |k|>δ −s+θ a 2([δ −s−θ ] − [δ −s+θ ])
m δ 2m(s−θ−1)+1 k 2m a ˆ2k C
k=−∞
|k|>δ −s+θ
and
+∞
s−θ 2m
m δ 2m(s−θ−1)+1 C m δ 2+2m(s−θ−1) ,
which proves (1.7). Let us obtain a lower bound for the mean value of a ˆ2k over |k| ∈ (δ θ−1 , δ −θ−1 ]. We denote this 2 )δ θ/˜ mean value by Aˆ21,θ . Suppose that (2V + C c1 < 1/2 and (∗) holds. Since 2 δ −1+θ k2 a ˆ2k δ 2+θ k4 a ˆ2k C |k|>δ −1−θ/2
and
|k|>δ −1−θ/2
|k|δ −1+θ
we have
k2 a ˆ2k 2
V 2V δ −1+θ ,
0
a ˆ2k δ 2+θ
δ −1+θ <|k|δ −1−θ
k2 a ˆ2k
δ −1+θ <|k|δ −1−θ/2
=δ
2+θ
+∞
k2 a ˆ2k
−
k=−∞
k2 a ˆ2k
−
|k|δ −1+θ
k2 a ˆ2k
|k|>δ −1−θ/2
2 c˜1 2V + C 1+θ 1− c˜1 δ δ 1+θ . δθ c˜1 2 Thus
c˜1 δ 1+θ/2 c˜1 2+2θ δ = , Aˆ21,θ −1−θ 2δ 4 and the estimate (1.8) is thereby proved. Appendix Let a function u =
uδ (t, x)
satisfy the equation
∂ f (u) ∂ u+ = δuxx ∂t ∂x in the cylinder [0, +∞) × S and the initial condition
(A.1)
u(0, x) = u0 (x) ∈ C ∞ (S).
(A.2)
Here f : R → R is an arbitrary C ∞ function (not necessarily convex) and δ > 0. Then u is C ∞ for all t > 0 (see [1]). We recall that if a function v(t, x) satisfies the equation vt + ψ(t, x)vx = δvxx , where ψ(t, x) is a
C∞
function, then the following maximum principle holds:
inf v(t1 , · ) inf v(t2 , · ) sup v(t2 , · ) sup v(t1 , · )
for t1 < t2 .
Let the symbol u˙ stand for either ∂u/∂t or ∂u/∂x (the following lemma holds in both cases). ˙ L1 (t) = |u(t, ˙ x)| dx. We consider the L1 -norm of u˙ on the circle S as a function of t, |u| Lemma A. The function t → |u| ˙ L1 (t) is nonincreasing. 8
Proof. The function u(t, ˙ x) satisfies the equation ˙ x = δ u˙ xx . u˙ t + (f (u)u) We multiply this equation by a smooth function ϕ(t, x) on the cylinder [T1 , T2 ] × S , T1 < T2 , and integrate by parts: T2 u(ϕ ˙ t + f (u)ϕx + δϕxx ) dx dt. uϕ| ˙ t=T2 dx = uϕ| ˙ t=T1 dx + (A.3) T1
Now let ϕ(t, x) be a solution of the Cauchy problem for the equation ϕt + f (u)ϕx + δϕxx = 0 in the cylinder [T1 , T2 ] × S . The initial data must be given at the top of the cylinder. Set ˙ 2 , x)), ϕ(T2 , x) = signε (u(T
where the function signε (u(T ˙ 2 , · )) is a smooth approximation to the function sign u(T ˙ 2 , · ) such ˙ 2 , x))| 1 for all x and the distance between these two functions in the L1 (S)-norm that | signε (u(T is less then ε. Now, by the maximum principle, we have |ϕ(t, x)| 1 for all (t, x) ∈ [T1 , T2 ] × S . Substituting ϕ(t, x) into (A.3), we obtain ε ˙ 2 , x)) dx = uϕ| ˙ t=T1 dx |u(T ˙ 1 , x)| dx. u(T ˙ 2 , x) sign (u(T Letting ε → 0, we have
|u| ˙
t=T2
dx
|u| ˙ t=T1 dx.
It follows from this lemma that δ ∂u ∂x (t, x) dx
du0 dx dx
for any t 0 and δ > 0. For δ ∈ (0, 1), we have |uδ (t2 , x) − uδ (t1 , x)| dx C|t2 − t1 |
(A.4)
(A.5)
where the constant C is independent of δ. For example, one can take 2 df (u0 ) dx + d u0 dx. C = 2 dx dx We shall now study the behavior of the solutions of problem (A.1), (A.2) as δ → 0. We are interested in the rate of convergence to the limit function. Theorem. For any T 0 and δ 0, we have du0 4 √ δ 0 (x) dx. (A.6) Tδ |u (T, x) − u (T, x)| dx √ dx 3 In what follows, we omit the superscript δ and denote the limit function u0 by v. Prior to proving the theorem, we establish a number of technical inequalities. Lemma 1. For every real number a and every nonnegative C 2 function ψ(t, x), one has |u(T, x) − a|ψ(T, x) dx |u(0, x) − a|ψ(0, x) dx T F (u(t, x), a)ψx (t, x) + |u(t, x) − a|(ψt (t, x) + δψxx (t, x)) dx dt, (A.7) + 0
where F (u, a) = sign(u − a)(f (u) − f (a)). 9
Proof. We multiply Eq. (A.1) by Φn (u)ψ(t, x), where Φn ( · ) is a convex C 2 function (Φn ( · ) 0). Then we integrate over the cylinder [0, T ] × S : T T T Φn (u)ut ψ dx dt + Φn (u)f (u)ux ψ dx dt = δ uxx Φn (u)ψ dx dt. 0
Since Φn (u)f (u)ux = (∂/∂x)
u 0
0
0
Φn (ξ)f (ξ) dξ, we can integrate by parts, thus obtaining
Φn (u)ψ|t=T dx−
Φn (u)ψ|t=0 dx−
T
0
= −δ
Φn (u)ψt dx dt− 0
T
T
0
u 0
u2x Φn (u)ψ dx dt
−δ
0
Φn (ξ)f (ξ) dξ
T
ψx dx dt
Φn (u)ux ψx dx dt,
or Φn (u)ψ|t=T dx Φn (u)ψ|t=0 dx +
0
T
u
0
Φn (ξ)f (ξ) dξ
ψx dx dt +
T 0
Φn (u)(ψt + δψxx ) dx dt.
We now suppose that the Φn ( · ) approximate the function | · − a| in the supremum norm, and moreover, 1 for u > a, d dΦn (u) → |u − a| = du du −1 for u < a, in L1 (u− , u+ ), where u− = min u0 and u+ = max u0 . Thus we can pass to the limit and obtain inequality (A.7). It follows from the maximum principle for uδ and inequality (A.4) that the family {uδ (t, · )}δ∈(0,1); t∈[0,T ]
(/)
is uniformly bounded and has a uniformly bounded variation. By Helly’s selection principle (see [7]), every infinite subset of this family contains a pointwise convergent sequence. By the dominated convergence theorem, this sequence converges in L1 (S). This means that the family (/) is precoma theorem that the family pact in L1 (S). It follows from this, inequality (A.5), and the Ascoli–Arzel` δ {u (t, x)}δ∈(0,1) is precompact in C([0, T ]; L1 (S)). Thus there exists an element v ∈ C([0, T ]; L1 (S)) and a sequence δk → 0 (k → ∞) such that uδk → v in C([0, T ]; L1 (S)). We note that this convergence and the maximum principle for uδk imply that u− ess inf v(t, · ) ess sup v(t, · ) u+ for all t; otherwise |uδk (t, · ) − v(t, · )|L1 would not tend to 0. For the next step, we need the inequalities |u(t, x) − a| − |v(t, x) − a| |u(t, x) − v(t, x)| and |F (u(t, x), a) − F (v(t, x), a)| max |f ( · )| |u(t, x) − v(t, x)| for almost all (t, x). [u− ,u+ ]
(The first is just the triangle inequality. To prove the second, one must consider two cases: a) a is between u and v; b) both u and v are either greater or less than a. For each t, the second inequality holds for almost all x.) 10
We pass to the limit in inequality (A.7) along the sequence δk → 0 (the function ψ(t, x) and the initial state u0 are given): ˜ x |v(T, x ˜) − b|ψ(T, ˜) d˜ x T ˜ |v(0, x ˜) − b|ψ(0, x ˜) d˜ x+ (F (v(t˜, x ˜), b)ψ˜x˜ (t˜, x ˜) + |v(t˜, x ˜) − b|ψ˜t˜(t˜, x ˜)) d˜ x dt˜. (A.8) 0
˜ Here we have intentionally replaced a with b and t, x, and ψ with t˜, x ˜ , and ψ. 2 Lemma 2. Suppose that Ω : S → R is an arbitrary nonnegative C function on the circle S . Then |u(T, x) − v(T, x ˜)|Ω(x−˜ x) dx d˜ x T |u(t, x) − v(t, x ˜)|Ω (x − x |u(0, x) − v(0, x ˜)|Ω(x − x ˜) d˜ x dx + δ ˜) dx d˜ x dt. 0
Proof. Suppose that ωε : R → R is an even nonnegative C 2 function such that ωε (z) ≡ 0 for |z| ε and +∞ ωε (z) dz = 1. −∞
ˆ t˜, x, x ˆ ˆ t˜ and ∂ ψ/∂x ˆ We note that the function ψ(t, ˜) = ωε (t − t˜)Ω(x − x ˜) satisfies ∂ ψ/∂t = −∂ ψ/∂ = ˆ ˆ −∂ ψ/∂ x ˜ . We substitute a = v(t˜, x ˜) and ψ(t, x) = ψ(t, t˜, x, x ˜) into (A.7) and integrate with resect ˜ t˜, x ˆ t˜, x, x to (t˜, x ˜) over [0, T ] × S . Next, we substitute b = u(t, x) and ψ( ˜) = ψ(t, ˜) into inequality (A.8) and integrate with respect to (t, x) over [0, T ]×S . Then we add the resulting two inequalities: T |u(T, x) − v(t˜, x ˜)| ωε (T − t˜)Ω(x − x ˜) dx d˜ x dt˜ 0
T
|v(T, x ˜) − u(t, x)| ωε (t − T )Ω(x − x ˜) d˜ x dx dt
+
T
0
0
|u(0, x) − v(t˜, x ˜)| ωε (−t˜)Ω(x − x ˜) dx d˜ x dt˜
T
|v(0, x ˜) − u(t, x)| ωε (t)Ω(x − x ˜) d˜ x dx dt
+ 0
T
+δ 0
0
T
|u(t, x) − v(t˜, x ˜)| ωε (t − t˜)
∂2 Ω(x − x ˜) dx d˜ x dt dt˜. ∂x2
Letting ε → 0, we arrive at the assertion of the lemma. Lemma 3. Suppose that Ψ : R → R is a nonnegative C 2 function with compact support. Then +∞ +∞ |u(T, x + y) − v(T, x)|Ψ(y) dx dy |u(0, x + y) − v(0, x)|Ψ(y) dx dy −∞
T
−∞ +∞
+δ 0
−∞
|u(t, x + y) − v(t, x)|Ψ (y) dx dy dt. (A.9)
˜ by y Proof. We apply the previous lemma with Ω( · ) = +∞ k=−∞ Ψ( · + k), replacing x − x and x ˜ by x. To obtain the desired result from (A.9), we apply the triangle inequality in the form |u(t, x + y) − v(t, x)| = |u(t, x) − v(t, x)| + w(t, x, y), 11
where |w(t, x, y)| |u(t, x + y) − u(t, x)|. Since u(0, x) ≡ v(0, x) and +∞ Ψ(y) dy |u(T, x) − v(T, x)| dx
+∞
−∞
−∞
|u(T, x) − u(T, x + y)| dx Ψ(y) dy + T
+∞
+δ 0
−∞
+∞ −∞
+∞ −∞
Ψ (y) dy = 0, we have
|u(0, x) − u(0, x + y)| dx Ψ(y) dy
|u(t, x) − u(t, x + y)| dx |Ψ (y)| dy dt.
(A.10)
+∞ In the following, we assume that −∞ Ψ(y) dy = 1. Using the inequalities y y ∂u ∂u (t, x + ξ) dξ dx (t, x + ξ) |u(t, x) − u(t, x + y)| dx = dx dξ ∂x 0 ∂x 0 (A.4) du0 ∂u dx, = |y| (t, x) dx |y| ∂x dx we rewrite (A.10) as follows: |u(T, x) − v(T, x)| dx 2
+∞
−∞
|y|Ψ(y) dy + T δ
+∞
−∞
du0 |y||Ψ (y)| dy dx dx.
(A.11)
Now let the functions Ψ(y) approximate the function max((1 − |y|/ε)/ε, 0). Then expression in parentheses in (A.11) tends to ε |y| 1 2 3T δ 1 dy + 2δT = ε+ . |y| 1 − 2 ε ε ε 3 ε −ε Minimizing with respect to ε, we arrive at the estimate (A.6). The proof of the theorem is complete. References 1. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural tseva, Linear and Quasi-Linear Equations of Parabolic Type, Amer. Math. Soc., Providence, R.I., 1968. 2. S. N. Kruzhkov, “First order quasilinear equations in several independent variables,” Mat. Sb., 81 (123), No. 2, 228–255 (1970); English transl. Math USSR-Sb., 10, 217–243 (1970). 3. N. N. Kuznecov, “The accuracy of some approximate methods for computing weak solutions of a first order quasilinear equation,” Zh. Vychisl. Mat. i Mat. Fiz., 16, No. 6, 1489–1502 (1976); English transl. USSR Comp. Math. and Math. Phys., 16, No. 6, 105–119 (1976). 4. A. N. Kolmogorov, “On inequalities for supremums of successive derivatives of a function on an infinite interval,” In: Selected works of A. N. Kolmogorov, Vol. 1, Kluwer, 1991. 5. L. H¨ ormander, Lectures on Nonlinear Hyperbolic Differential Equations, Springer-Verlag, Berlin, 1997. 6. P. D. Lax, “Hyperbolic systems of conservation laws II,” Comm. Pure Appl. Math., 10, 537–566 (1957). 7. A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Dover Publ., New York, 1975. 8. U. Frish, Turbulence. The Legacy of A. N. Kolmogorov, Cambridge Univ. Press, 1995. 9. S. Kuksin, “Spectral Properties of Solutions for Nonlinear PDE’s in the Turbulent Regime, Geom. Funct. Anal., 9, 141–184 (1999). 10. Weinan E, K. Khanin, A. Mazel, and Ya. Sinai, “Invariant measures for Burgers equation with stochastic forcing,” Ann. of Math., 151, No. 3, 877–960 (2000). ` Biryuk Translated by A. E.
12