Existence of strong traces for quasisolutions of scalar conservation - - PDF document

Existence of strong traces for quasisolutions of scalar conservation laws E.Yu. Panov (Velikiy Novgorod, Russia) Introduction. Consider a scalar conservation law divxϕ(u) = 0, (1) where u = u(x), x ∈ Ω, Ω ⊂ Rn is a domain with C1-smooth boundary ∂Ω; the flux vector ϕ(u) = (ϕ1(u), . . . , ϕn(u)), ϕi(u) ∈ C(R) ∩ BVloc(R). Definition 1 (Pa,2005). A function u = u(x) ∈ L∞(Ω) is called a quasisolution of (1) if ∀k ∈ F , where F ⊂ R is a dense set (in the sequel we suppose it being countable) there exists a Borel measure γk ∈ ¯ Mloc(Ω) such that divψk(u) = −γk in D′(Ω), ψk(u) = sign (u − k)(ϕ(u) − ϕ(k)). (i) G.e.s. as well as g.e.sub-s., g.e.super-s. are quasis.; (ii) If a, b ∈ F , a < b, and u = u(x) is a quasis. then v = max(a, min(u, b)) is also a quasis. Our main result is the following Theorem 1. There exists a function u0(y) ∈ L∞(∂Ω) (the trace) such that ∀k ∈ R (ψk(u(x)), ν) → (ψk(u0(y)), ν) as x → y ∈ ∂Ω in L1

loc.

Here ν = ν(y) is the normal vector at y ∈ ∂Ω. In particular the normal flux component (ϕ(u(x)), ν) has the strong trace. Corollary. If for a.e. y ∈ ∂Ω functions u → (ϕ(u), ν(y)) are not constant on nondegenerate intervals then u(x) → u0(y) as x → y strongly.

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The above result allows to formulate boundary value problems for (1) in the sense of Bardos, LeRoux & N´ ed´

• elec. For instance the Dirichlet problem

u|∂Ω = ub is understood in the set of the inequality: ∀k ∈ R (sign (u − k) + sign (k − ub))(ϕ(u) − ϕ(k), ν) ≥ 0 on ∂Ω, which is well defined due to existence of strong traces of sign (u − k)(ϕ(u) − ϕ(k), ν) and (ϕ(u), ν). Reformulation of the problem. The assertion of the main Th.1 has the local character. Thus, we can assume that for some f(u) ∈ C1(Rn−1) Ω = { x ∈ Rn | x1 > f(x2, . . . , xn) }. Making the change t = x1 − f(x2, . . . , xn), xi = xi, i = 2, . . . , n we transform (1) to (ϕ0(x, u))t + divx ˜ ϕ(u) = (ϕ0(x, u))t +

n

• i=2 ϕi(u)xi = 0,

(2) u = u(t, x), (t, x) ∈ Π = R+ × Rn−1 ; here ˜ ϕ(u) = (ϕ2(u), . . . , ϕn(u)), ϕ0(x, u) = ϕ1(u) −

n

• i=2 ϕi(u) ∂f

∂xi = (ϕ(u), ν(x)). Clearly u(x) is a quasis. of (1) iff u(t, x) is a quasis. of (2). For equa- tion (2) Th.1 is formulated as follows. Theorem 1. There exists a function u0(y) ∈ L∞(Rn−1) such that ∀k ∈ R ess lim

t→0

ψ0k(x, u(t, x)) = ψ0k(x, u0(x)) in L1

loc(Rn−1).

Here ψ0k(x, u) = sign (u−k)(ϕ0(x, u)−ϕ0(x, k)) = sign (u−k)(ϕ(u)−ϕ(k), ν(x)), u ∈ R, x = (x2, . . . , xn) ∈ Rn−1.

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• Remark. To prove Th. 1 it is sufficient to establish strong convergence:

ess lim

t→0

ψ0k(x, u(t, x)) = vk(x). Indeed, there exists the measure valued trace νx so that ∀p(λ) ∈ C(R) p(u(tm, x)) →

• p(λ)dνx(λ) weakly-∗ in L∞(Rn−1).

Here tm is an arbitrary vanishing sequence, which consists of ”essen- tial points”. By the strong convergence of ψ0k(x, u(tm, x)) we see that ψ0k(x, λ) is constant on supp νx for a.e. x ∈ Rn−1 . This easily implies that supp νx contains in segments where ϕ0(x, λ) is constant. We can define u0(x) =

λdνx(λ). Then for a.e. x ∈ Rn−1

vk(x) =

• ψ0k(x, λ)dνx(λ) = ψ0k(x, u0(x)),

as required. Reduction of the dimension. Suppose one of the component, say ϕn(u) ≡ const. Proposition 1 (Pa,2005). For a.e. xn the function u(t, x′) = u(t, x′, xn) is a quasis. of reduced equations (2) ϕ0(x, u)t +

n−1

• i=2 ϕi(u) = 0, (t, x′) ∈ Π′ = R+ × Rn−2.

Weak traces. From the condition (ψ0k(x, u))t + divx ˜ ψk(u) = −γk ∈ ¯ Mloc(Π), k ∈ F it follows (see Chen & Frid, 1999) existence of the weak traces: ess lim

t→0

ψ0k(x, u(t, x)) = vk(x) weakly-∗ in L∞(Rn−1) for all k ∈ R (we also take into account that ψ0k(x, u) is equicontinuous w.r.t. k). ”Blow-up” procedure. Let u(t, x) be a quasis. of (2), y ∈ Rn−1 . Introduce the sequences uε(t, x; y) = u(εt, y+εx) ∈ L∞(Π), ε = εm → 0; vε

k(x; y) = vk(y + εx) ∈ L∞(Rn−1), 3

and the sequences of measures γε

k(t, x) = εγk(εt, y + εx).

The latter equality is understood in the sense of distributions, i.e. < γε

k, f >= ε1−n

• f(t/ε, (x − y)/ε)dγk(t, x)

∀f ∈ C0(Π). Evidently, (ψ0k(y + εx, uε))t + divxψk(uε) = −γε

k;

(3) ψ0k(y + εx, uε)|t=0 = vε

k(x; y) in the weak sense.

(4) Theorem 2 (Vasseur, 2001; Panov, 2005). (i) There exist a sequence ε = εm → 0 such that for a.e. y ∈ Rn−1 vε

k(x; y) → vk(y) = const in L1 loc(Rn−1); γε k → 0 in ¯

Mloc(Π); (ii) Existence of the strong trace ψ0k(x, u(0, x)) = vk(x) holds iff there exists a sequence ε = εm → 0 such that for a.e. y ∈ Rn−1 the sequence ψ0k(y, uε(t, x; y)) is strictly convergent. Besides, in this case the limit function is vk(y) = const for a.e. y . In order to prove (ii) (the inverse implication) we use some property of isentropic solutions (i.s.) of conservation laws. Choose a sequence ε = εm → 0 and the set of full measure Y ⊂ Rn−1 of values y , for which (i) is satisfied. Suppose that y ∈ Y and ∀k ∈ F ψ0k(y + εx, uε(t, x; y)) → wk(t, x; y) in L1

loc(Π).

(5) We can assume in addition that the sequence uε(t, x; y) weakly converges to some measure valued function νt,x , i.e. ∀p(u) ∈ C(R) p(uε(t, x; y)) →

• p(λ)dνt,x(λ) weakly-∗ in L∞(Π).

From (5) it follows that

• ψ0k(y, λ)dνt,x(λ) = wk(t, x; y), moreover

supp νt,x contains in an interval where ϕ0(y, λ) = const. Passing to the limit as m → ∞ in (3), we obtain that ∀k ∈ F ∂ ∂t

• ψ0k(y, λ)dνt,x(λ) + divx
• ψk(λ)dνt,x(λ) = 0 in D′(Π).

(6)

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Since F is dense and the functions ψ0k , ψk are equicontinuous in k we conclude that (6) holds for all k and νt,x is an isentropic m.v.s. of the equation ϕ0(u)t + divx ˜ ϕ(u) = 0 (7) with some measure valued initial data ν0,x = νx. (8) Here and below we simplify the notations dropping the dependence on y : ϕ0(u) = ϕ(y, u), ψ0k(u) = ψ0k(y, u), wk(t, x) = wk(t, x; y). From (4) in the limit as m → ∞ it follows that also wk(0, x) = vk(y) = const in the weak sense. In particular, taking k sufficiently large, we have w(0, x) = v(y) = const, where w(t, x) =

• ϕ0(λ)dνt,x(λ).

Theorem 3. Let νt,x be an isentropic m.v.s. of problem (7), (8) such that for for a.e. (t, x) ∈ Π supp νt,x contains in a segment where ϕ0(u) is constant. Then (i) ∀k ∈ R the functions wk(t, x) =

ψ0k(λ)dνt,x(λ) have the strong

traces vk(x) =

ψ0k(λ)dνx(λ).

(ii) If vk = const then wk(t, x) ≡ vk . To prove Th.3, observe that νt,x is an isentropic m.v.s. of any equation η(u)t + divxψ(u) = 0, where η′(u) = ϕ′

0(u)f′(u), ψ′(u) = ˜

ϕ′(u)f′(u). Taking f′(u) = α(u), where α is defined by the relation µ(u) = α(u)d|µ|(u), µ = ϕ′

0 we obtain that η′(u) = |µ| ≥ 0, i.e.

η(u) in- creases and it is constant on the same intervals as ϕ0(u). So the problem is reduced to the case when ϕ0(u) increases. This case is treated similarly to the model case ϕ0(u) = u based on the key Kruzhkov relation for two isentropic m.v.s. νt,x , ˜ νt,x |ϕ0(u)−ϕ0(v)|t+divx sign (λ1−λ2)( ˜ ϕ(λ1)− ˜ ϕ(λ2))dνt,x(λ1)d˜ νt,x(λ2) = 0, u =

ϕ0(λ)dνt,x(λ), v = ϕ0(λ)d˜

νt,x(λ). In particular, we apply this relation with ˜ νt,x = νt,x+h .

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H -measures and localization property. Suppose Ω ⊂ Rn and the sequence um = um(x) is bounded in L∞(Ω), um con- verges weakly to a measure valued function νx. We denote Up

m(x) =

δum((p, +∞)) − νx((p, +∞)). Let F(u)(ξ), ξ ∈ Rn be the Fourier trans- form of u(x) ∈ L2(Rn); S = SN−1 = { ξ ∈ Rn | |ξ| = 1 } be the unit sphere; u → ¯ u, u ∈ C be the complex conjugation. Theorem 4 (Pa,1994,1995). (i) There exists a set P ⊂ R with at most countable complement such that Up

m(x) → 0 as m → ∞

weakly-∗ in L∞(Ω); (ii) There exists a family of complex-valued locally finite Borel mea- sures {µpq}p,q∈P on Ω × S and a subsequence Ur(x) = {Up

r (x)}p∈P

such that ∀Φ1(x), Φ2(x) ∈ C0(Ω), ψ(ξ) ∈ C(S) µpq, Φ1(x)Φ2(x)ψ(ξ) = lim

r→∞

• RN F(Φ1Up

r )(ξ)F(Φ2Uq r )(ξ)ψ(ξ/|ξ|)dξ;

(iii) The map (p, q) → µpq is continuous as a map from P ×P into the space Mloc(Ω × S) of locally finite Borel measures on Ω × S . Following L.Tartar, we call the family {µpq}p,q∈P the H -measure cor- responding to the subsequence ur(x). Below some important properties of the H -measure are indicated (see Pa, 1994): 1) ∀p ∈ P µpp ≥ 0; 2) ∀p, q ∈ P µpq = µqp; 3) for p1, . . . , pl ∈ P and for any bounded Borel set C ⊂ Ω × S the matrix aij = µpipj(C), i, j = 1, . . . , l is positively definite; 4) µpq = 0 ∀p, q ∈ P if and only if the sequence ur(x) is strongly convergent as r → ∞. Now, let ϕ(u) ∈ C(R, Rn) be a continuous vector function. Suppose that the sequence ur(x) satisfies the condition: C) ∀k ∈ R the sequence of distributions Lp

r = divx[sign (ur − k)(ϕ(ur) − ϕ(k))] is precompact in H−1 loc (Ω).

Here, as usual, the space H−1

loc (Ω) consists of distributions u(x) such that

for all f(x) ∈ C∞

0 (Ω) the product fu ∈ H−1 2 . The topology in H−1 loc (Ω)

is generated by seminorms fuH−1

2 , f ∈ C∞

0 (Ω). 6

The following theorem plays a key role in the proof of our main Th.1. Theorem 5 (Pa,1995). Suppose that the H -measure {µpq}p,q∈P cor- responding to the sequence ur(x) is not trivial and the condition C) is

• satisfied. Then there exists a nondegenerate segment I = [p0, p0 + δ]

and a vector ξ ∈ S such that (ξ, ϕ(u)) = const on I and µpq = 0 ∀p, q ∈ P ∩ I . In particular from Th.5 it follows that under the nondegeneracy condition ∀ξ ∈ S the function u → (ξ, ϕ(u)) is not constant on nondegenerate intervals µpq ≡ 0 and, therefore, the sequence ur(x) is strongly convergent. Proof of Th.1. We apply the induction in n. If n = 1 (the base) then we have u = u(t), ψ0k(u)t = −γk , i.e. ψ0k(u(t)) are BV-functions in a vicinity of t = 0. Therefore, ψ0,k(u(t)) → vk as t → 0. Clearly vk = ψ0k(u0), where u0 is some limit point of u(t) as t → 0. Now suppose that Th.1 is true for space dimension less than n. In- troduce the countable set I consisting of segments I = [a, b] such that a, b ∈ F and for some ξ ∈ Rn, ξ = 0 (ξ, ϕ(u)) = const on I . We denote uI(t, x) = max(a, min(u(t, x), b)). Recall that uI is also a quasis.

• f (2). Let us show that for I ∈ I

uI satisfies the strong trace property. Let ξ ∈ Rn, ξ = 0 be such that (ξ, ϕ(u)) = const on I . Consider the following two cases: 1) ξ1 = 0. In this case ξ′ = (ξ2, . . . , ξn) = 0 and there exists a linear change z = z(x), x = (x2, . . . , xn) ∈ Rn−1 such that zn = (ξ′, x). After this change (2) reduces to the form ϕ0(x, u)t + divz ¯ ϕ(u) = 0, (9) where ¯ ϕn = (ξ′, ˜ ϕ(u)) = const on I . Thus, uI(t, x′, xn) is a quasisolution

• f reduced equation for a.e. xn. By induction hypothesis, the strong trace

property is satisfied in Π′ = R+ × Rn−1. This easily implies that it is also true in Π.

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2) ξ1 = 0. Let E ⊂ Rn−1 be the set of x such that ξ′ + ξ1∇f(x) = 0. Since ϕ1(u) = ϕ0(x, u) + (∇f(x), ˜ ϕ(u)) we have the relation ξ1ϕ0(x, u) + (ξ′ + ξ1∇f(x), ˜ ϕ(u)) = const ∀u ∈ I. (10) Thus, for x ∈ E ϕ0(x, u) = const on I and therefore for u0 ∈ I ψ0k(x, u(t, x)) → ψ0k(x, u0) as t → 0 in L1

loc(E).

If x / ∈ E , when in a vicinity of this point we can make the change of variables zi = zi(x), i = 2, . . . , n − 1; zn = ξ1(t + f(x)) + (ξ′, x), which reduces (2) to the form (9) with ¯ ϕn = ξ1ϕ0(x, u) + (ξ′ + ξ1∇f(x), ˜ ϕ(u)) = const. As in the case 1), we deduce from this representation the strong trace property in a vicinity of x. The above arguments show that the strong trace property is satisfied on the whole hyperplane t = 0. By Th.2 we see that after extraction of a subsequence ε = εm → 0 the sequence ψ0k(y, (uI)ε(t, x; y)) is strictly convergent for all I ∈ I and y ∈ Y . We assume in addition that for the sequence εm and the set of full measure Y ⊂ Rn−1 the assertion of Th.2(i) holds. Now we fix y ∈ Y and a subsequence of εm such that for sequences uε(t, x; y), ψ0k(y, uε(t, x; y)) H -measures µpq , µpq

k are defined for all k ∈

F . To proof the trace property, we have to show that µpq

k

= 0. If it is not the case then also µpq = 0 for some p, q ∈ P . By Th.5 we conclude that there exists k ∈ F and a segment I ∈ I such that µpq

k =

0 for p, q ∈ P ∩ I . But this contradicts to the strong convergence of ψ0k(y, (uI)ε(t, x; y)). Thus, our subsequences ψ0k(y, uε(t, x; y)) strongly converges to the limit vk(y), which does not depend on the choice of the

• subsequence. Therefore, the strong convergence is true for original sequence

εm and according to Th.2(ii) we conclude that the strong trace property is satisfied.

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Example of a weak solution without a strong trace. Consider the Burgers equation ut + (u2)x = 0. To construct the desired g.s. u(t, x) we introduce the function w(t, x) =

            

, |x| + |t − 6| ≤ 1 −sign x , |x| + |t − 6| > 1, t ∈ (6, 8] sign (1 − x)sign x , |x| + |t − 6| > 1, t ∈ (4, 6) , defined in the square t ∈ (4, 8], −2 ≤ x < 2. We extend this function in the whole layer t ∈ (4, 8], as a 4-periodic function v over the variable x so that for y = x − 4[x/4] − 2 (here [a] denotes the integer part of a) v(t, x) = u(t, y). In the half-space Π we define the piecewise constant function u(t, x) = v(2kt, 2kx) if t ∈ (4 · 2−k, 8 · 2−k], k ∈ Z, see fig. 1. As easily verified, on the discontinuity lines the Rankine-Hugoniot condition is satisfied and therefore u(t, x) is a g.s. of the Burgers equation. As t → 0 u(t, ·) → 0 ∗-weakly in L∞(R) but there is no strong limit of u(tk, x) for any choice of a sequence tk → 0. Remark also that for the constructed g.s. the condition from Def.1 is satisfied with the measures γk ∈ Mloc(Π) ∀k ∈ R, but certainly γk / ∈ ¯ Mloc(Π).

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References. [ChFr99] G.-Q. Chen and H. Frid, Divergence-Measure fields and hyper- bolic conservation laws, Arch. Ration. Mech. Anal. 147 (1999) 89–118. [Pa94] E.Yu. Panov, On sequences of measure-valued solutions of first-

• rder quasilinear equations, Matem. Sbornik 185, No. 2 (1994) 87–106;

English transl. in Russian Acad. Sci. Sb. Math. 81 (1995). [Pa95] E.Yu. Panov, On strong precompactness of bounded sets of mea- sure valued solutions for a first order quasilinear equation, Matemat. Sbornik 186, No. 5 (1995) 103–114; Engl. transl. in Sbornik: Mathe- matics 186 (1995) 729–740. [Pa99] E.Yu. Panov, Property of strong precompactness for bounded sets

• f measure valued solutions of a first-order quasilinear equation, Matemat.

Sbornik 190, No. 3 (1999) 109–128; Engl. transl. in Sbornik: Mathemat- ics 190, No. 3 (1999) 427–446. [Pa05] E.Yu. Panov, Existence of Strong Traces for Generalized Solutions

• f Multidimensional Scalar Conservation Laws, JHDE. V. 2 No. 4 (2005)

885–908. [Ta90] L. Tartar, H -measures, a new approach for studying homogenisa- tion, oscillations and concentration effects in partial differential equations. Proceedings of the Royal Society of Edinburgh 115A, No. 3-4 (1990) 193–230. [Vas01] A. Vasseur, Strong Traces for Solutions of Multidimensional Scalar Conservation Laws. Arch. Ration. Mech. Anal. 160 (2001) 181–193.

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