VALIDITY OF NONLINEAR GEOMETRIC OPTICS FOR ENTROPY SOLUTIONS OF MULTIDIMENSIONAL SCALAR

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Niveau: Supérieur, Licence, Bac+1
VALIDITY OF NONLINEAR GEOMETRIC OPTICS FOR ENTROPY SOLUTIONS OF MULTIDIMENSIONAL SCALAR CONSERVATION LAWS GUI-QIANG CHEN STEPHANE JUNCA MICHEL RASCLE Abstract. Nonlinear geometric optics with various frequencies for entropy so- lutions only in L∞ of multidimensional scalar conservation laws is analyzed. A new approach to validate nonlinear geometric optics is developed via entropy dissipation through scaling, compactness, homogenization, and L1–stability. New multidimensional features are recognized, especially including nonlinear propagations of oscillations with high frequencies. The validity of nonlinear geometric optics for entropy solutions in L∞ of multidimensional scalar con- servation laws is justified. 1. Introduction We are concerned with nonlinear geometric optics for entropy solutions of mul- tidimensional scalar conservation laws: (1.1) ∂tu+ divxF(u) = 0, u ? R, x ? Rn, where F : R ? Rn is a smooth flux function. Consider the Cauchy problem (1.1) with Cauchy data: (1.2) u|t=0 = u ? 0(x) := u+ ?u1(?1/? ?1 , · · · , ?n/? ?n), where u1 is a periodic function of each of its n arguments whose period is denoted by P = [0, 1]n (without loss of generality), u is a constant ground state, the linear phases ? := (?1, · · · , ?n): (1.3) ?i := n∑ j=1 Jijxj are linearly independent with constant matrix J = (Jij)1≤i,j≤n,

• u?


• geometric optics


• entropy solution


• hunter-majda-rosales


• nonlinear geometric


• dissipation through


• flux function

Sujets

• Geometrical optics

Informations

• Publié par: profil-ontoe-2012
• Nombre de lectures: 43

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VALIDITY OF NONLINEAR GEOMETRIC OPTICS FOR ENTROPY SOLUTIONS OF MULTIDIMENSIONAL SCALAR CONSERVATION LAWS

´ GUI-QIANG CHEN STEPHANE JUNCA MICHEL RASCLE

Abstract.Nonlinear geometric optics with various frequencies for entropy so- lutions only inL ∞  Aof multidimensional scalar conservation laws is analyzed. new approach to validate nonlinear geometric optics is developed via entropy dissipation through scaling, compactness, homogenization, andL 1 –stability. New multidimensional features are recognized, especially including nonlinear propagations of oscillations with high frequencies. The validity of nonlinear geometric optics for entropy solutions inL ∞ of multidimensional scalar con- servation laws is justified.

1.Introduction

We are concerned with nonlinear geometric optics for entropy solutions of mul- tidimensional scalar conservation laws:

(1.1)∂ t u+ div x F(u) = 0, u∈R,x∈R n , whereF:R→R n  the Cauchy problem (1.1) Consideris a smooth flux function. with Cauchy data: (1.2)u| t=0 =u ε 0 (x) :=u+εu 1 (φ 1 /ε α 1 ,∙ ∙ ∙, φ n /ε α n ), whereu 1 is a periodic function of each of itsnarguments whose period is denoted byP= [0,1] n (without loss of generality),uis a constant ground state, the linear phases Φ := (φ 1 ,∙ ∙ ∙, φ n ): n (1.3)φ i := X J ij x j j=1

are linearly independent with constant matrixJ= (J ij ) 1≤i,j≤n , and α= (α 1 ,∙ ∙ ∙, α n )∈[0,∞) n is the magnitude indices of frequency of the initial oscillations. We seek a geometric optics asymptotic expansion: (1.4)u ε (t,x) =u+εv ε (t,x).

Date: July 22, 2004. 1991Mathematics Subject Classification.Primary:35B40,35L65; Secondary:35B35,76N15. Key words and phrases.Nonlinear geometric optics, entropy solutions inL ∞ , multidimensional conservation laws, validity, profile, perturbation, new approach, entropy dissipation, compactness, homogenization, oscillation, scaling, stability, multi-scale,BV. 1

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In this paper, we develop a new approach including several basic frameworks and new tools in Sections 2-5 to validate weakly nonlinear geometric optics via entropy dissipation through compactness, scaling, homogenization, andL 1 –stability and apply this approach first to the one-dimensional case in Section 4 and then to the multidimensional case in Section 5 to recognize new multidimensional features and validate nonlinear geometric optics for multidimensional scalar conservation laws by extending the ideas and techniques. To illustrate multidimensional features clearly in nonlinear geometric optics, we focus now on the two dimensional case. Letu:=u ε be the Krushkov solution of the Cauchy problem:

(1.5)∂ t u+∂ x 1 f 1 (u) +∂ x 1 f 2 (u) = 0, (1.6)u(0, x 1 , x 2 ) =u 0 (x 1 , x 2 )≡u+ε u 1 (φ 1 /ε α , φ 2 /ε β ), where, as for the general setting (1.1)–(1.4), the linear phases (φ 1 , φ 2 ): (1.7)φ 1 :=a 1 x 1 +a 2 x 2 , φ 2 :=b 1 x 1 +b 2 x 2 are linearly independent, and α 1 , α 2 ≥0. The Krushkov solution is anL ∞ functionu=u(t, x 1 , x 2 ) satisfying (1.8)∂ t |u−k|+∂ x 1 (sign(u−k)(f 1 (u)−f 1 (k)))+∂ x 2 (sign(u−k)(f 2 (u)−f 2 (k)))≤0 in the sense of distributions for anyk∈R the Krushkov solution. Foru, we look for an asymptotic expansion: (1.9)u=u ε (t, x 1 , x 2 ) :=u+ε v ε (t, x 1 , x 2 ). After, if necessary, a linear change of coordinates:x 1 →x 1 −a 0 t, x 2 →x 2 −b 0 t, we may assume

f 1 0 (u) =f 0 2 (u) 0. = Sinceφ 1 andφ 2 are linearly independent, we can rewrite equation (1.5) in these co- ordinates, even though they are not necessarily orthonormal, and perform a formal asymptotic expansion to obtain (1.10)∂ t v+ε(a∂ φ 1 +b∂ φ 2 )v 2 +ε 2 (c∂ φ 1 +d∂ φ 2 )v 3 =ε 3 (∂ φ 1 R 1 +∂ φ 2 R 2 ),

where a:= (a 1 f 1 00 (u) +a 2 f 2 00 (u))/2, b:= (b 1 f 0 1 0 (u) +b 2 f 0 2 0 (u))/2, (1.11) c:= (a 1 f 0 1 00 (u) +a 2 f 2 000 (u))/6, d:= (b 1 f 0 1 00 (u) +b 2 f 000 (u))/6, 2 andR j :=R j (v, u, ε), j= 1,2,are Lipschitz functions inv,u, andεwith the form: (1.12)R 1 (v, u, ε) =− 1 6 R 0 1 (1−θ) 3 (a 1 f (4) 1 +a 2 f ) 2 (4 )(u+εθv)dθ v 4 , (1.13)R 2 (v, u, ε) =− 1 6 R 0 1 (1−θ) 3 (b 1 f )4 1 ( +b 2 f )4 2 ( )(u+εθv)dθ v . 4 Define (1.14)M:=  ac bd  =  a b 1 1 a b 2 2   f 0 1 0 (u)/2f 0 1 00 (u)/6  . f 0 2 0 (u)/2f 0 2 00 (u)/6

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We assume that the matrixM=M(u is equivalent to require It) is invertible. that both (1.15)J:=  b a 1 1 a b 2 2  = D D ( ( x φ 1 1 φ, x, 2 2 ) )

and )/2f 0 1 00 (u)/6 (1.16)N:=  f f 0 2 1 00 0 ( ( u u )/2f 0 2 00 (u)/6  be invertible. Note that the invertibility of (1.15) is a corollary of the linear in- dependence of the phases (φ 1 , φ 2 ), and the invertibility of (1.16) is an assumption of genuine nonlinearity and “genuine multidimensionality”, which particularly im- plies that the second derivatives of the fluxesf 1 andf 2 are not proportional on any interval in a neighborhood ofu type of nonlinear assumptions is enough to. This obtain compactness; indeed, this nonlinear assumption is the strongest nonlinear assumption near the constant ground stateu. The compactness of solution oper- ators can be achieved by applying the compensated compactness method and the averaging lemma. In this regard, we refer to Chen-Frid [2, 3], Engquist-E [12], and Lions-Perthame-Tadmor [22].

The main problems concerned in this paper include (i). identification of the formal limitVofv=v ε as a function oftand the fast variables, which turn out to be, for the simplest cases, (1.17)X 1 :=φ/ε min(α 1 ,1) , X 2 :=ψ/ε min(α 2 ,1) ; (ii). justification of this asymptotics, that is, the strong convergence ofv ε toV inL l 1 oc : (1.18)v ε −V→0 strongly inL l 1 oc in the two systems of fast coordinates (t, X 1 , X 2 ) and of slow coordinates (t, x 1 , x 2 ), whereVis the profile.

We emphasize that our purpose here is not necessarily to obtain the sharpest possible results on the convergence rates, say in (1.18). In particular, contrarily to the one-dimensional case, there is even no available result on decayratesof the total variation of the solution to (1.5) with periodic initial data since genuine nonlinearity of the flux function fails, although there are some results of strong convergence to a constant state, see [2, 3, 12]. Precisely, our goal here is to develop a new approach and apply it to prove rigorously the results of strong convergence like (1.18) for entropy solutions, byonlyusing entropy dissipation through compactness, scaling, homogenization, andL 1 –stability, without relying on theBVstructure upon solutions.

There are too many cases to give here a precise statement of the results. Let us just mention for the moment that, in the one-dimensional case with only one phase φand withXdefined as in (1.17), there are three subcases for entropy solutions in L ∞ : (i). Ifα >1, then the initial oscillation is so fast that it is “canceled” by the nonlinearity, that is,v ε converges strongly toσwith (1.19)σ=u 1  value of:= meanu 1 over the period;

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(ii). Ifα= 1 that is the natural situation of weakly nonlinear geometrical optics (WNLGO), thenv ε converges strongly to the profileσin the fast variableX, which is uniquely determined by the Cauchy problem: (1.20)∂ t σ+a∂ X σ 2 = 0, σ(0, X) =u 1 (X); (iii). Ifα <1, then the initial oscillation is so slow that, in the variableX=φ/ε α , v ε converges strongly toσ, which is determined by (1.21)∂ t σ= 0, σ(0, X) =u 1 (X), that is,σ(t, X) =u 1 (X). These results are proved in Section 4. In the multidimensional case, the situ- ation is much more complicated: real new multidimensional features are involved and multidimensional phenomena occur (see Section 5), although some cases are a combination of these three different possibilities. In particular, the links between the linear phases (φ 1 ,∙ ∙ ∙, φ n ) and the fluxesFthrough formulas, such as (1.10) to (1.14) for the two-dimensional case, lead to a number of interesting cases which deserve to be described more precisely. Furthermore, we develop a new approach in Sections 2-5 to validate weakly nonlinear geometric optics , first for the one- dimensional case in Section 4, and then to extend the ideas and techniques from Section 4 to deal with the multidimensional case in Section 5. An important tool to preserve theL l 1 oc –convergence after the triangular change of variables depending onεis introduced in Lemma 3.1 and a “quasi”LUfactorization of matrixMto in- clude all new cases is formulated in Lemma 5.1. We also recognize new phenomena including the blowup of the gradients of the geometric optics asymptotic expan- sions; in contrast to the classical geometric optics expansions with the gradients of order 1 since the amplitude is of orderεand the frequency of orderε −1 . There are essentially two ways to obtain such very high oscillations for the two-dimensional case: (i). A phase gradient is orthogonal to the second flux derivative on the ground state; (ii). There are some precise arithmetic relations between the coefficients of the matrixM. For dimensionn≥can have higher oscillations with combinations of these3, we two ways, or with orthogonality to the second and third flux derivatives, among others. For related results on nonlinear geometric optics, see DiPerna-Majda [11] for one-dimensional 2×hyperbolic systems of conservation laws; also see Cheverry2 [6,7],Gu`es[15],Hunter-Majda-Rosales[16],Junca[18,19],Joly-Metivier-Rauch ´ [17], Majda-Rosales [23], and the references cited therein. 2.Geometric Optics, Compactness, andL 1 –Stability

In this section, we introduce some basic frameworks to validate nonlinear geo- metric optics for entropy solutions only inL ∞ of multidimensional conservation laws.

2.1.Basic Properties of Nonlinear Geometric Optics Expansions.Con- sider the nonlinear geometric optics expansion (1.4) of solutions of the Cauchy ε problem (1.1)–(1.2). Setv ε (t,x) = u( ε t,x)−u  first derive some basic properties. We of the sequencev ε (t,x).

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Lemma2.1.Letu 1 ∈L ∞  that, for each fixed. Assumeε >0,u ε (t,x) is the entropy solution of the Cauchy problem (1.1)–(1.2). Then (2.1)kv ε k L∞ ≤ ku 1 k L∞ for anyε >0. Proof Krushkov’s uniqueness theorem,. Usingu ε (t,x) is the periodic entropy solu- tion of the Cauchy problem (1.1)–(1.2) for any fixedε >0, sinceu 1 is periodic. First, taking the convex entropy (u−u) p of (1.1) for evenp≥2, we obtain from the entropy inequality that ∂ t (u ε −u) p + div x
p Z u ε (ξ−u) p−1 F 0 (ξ)dξ ! ≤0

in the sense of distributions. Equivalently, ∂ t (u ε −u) p + div Φ
p J Z u ε (ξ−u) p−1 F 0 (ξ)dξ ! ≤0

in the sense of distributions, where Φ =Jx.Integrating with respect to Φ and using the periodicity, we obtain Z Pε |u ε (t, J −1 Φ)−u| p dΦ≤ Z |u 0 (φ 1 /ε α 1 ,∙ ∙ ∙, φ n /ε α n )−u| p dφ 1 d∙ ∙ ∙δφ n , Pε whereP ε ={(φ 1 ,∙ ∙ ∙, φ n ) : (φ 1 /ε α 1 ,∙ ∙ ∙, φ n /ε α n )∈P}.This is equivalent to Z P ε |v ε (t, J −1 Φ)| p dΦ Z P ε ∙ ∙, φ n /ε α n )| p dφ 1 ∙ ≤ |u 1 (φ 1 /ε α 1 , ∙∙ ∙dφ n for any evenp≥2. Taking power 1/pin both sides and lettingp→ ∞, we conclude (2.1).

Lemma2.2.For any entropy solutionu ε (t, x 1 , x 2 ) of (1.5)–(1.6),v ε (t, x 1 , x 2 ) is the entropy solution of (1.10) with initial data (2.2)v ε | t=0 =u 1 (φ 1 /ε α 1 , φ 2 /ε α 2 ). sil= k−u with the linear transformation Proof:Notice thatu ε  Choo ngsatisfy (1.8). ε into the (φ 1 , φ 2 )-coordinates, we have (2.3)∂ t |v ε −l|+ε(a∂ φ 1 +b∂ φ 2 )(sign(v ε −l)((v ε ) 2 −l 2 )) +ε 2 (c∂ φ 1 +d∂ φ 2 )(sign(v ε −l)((v ε ) 3 −l 3 )) +ε 3 (∂ φ 1 (sign(v ε −l)(R 1 (v ε , u, ε)−R 1 (l, u, ε))) −∂ φ 2 (sign(v ε −l)(R 2 (v ε , u, ε)−R 2 (l, u, ε))))≤0, which implies thatv ε (t, x 1 , x 2 ) is the entropy solution of (1.10) and (2.2). Remark2.1.The same proof implies that Lemma 2.2 also holds for (1.1)–(1.2) with n≥3, which will be used in Section 5.

2.2.Compactness of Approximate Solutions.We now present several com- pactness lemmas, which can be achieved by compactness arguments and Young measures with the aid of entropy dissipation of the solutions. Lemma2.3.Assume thatu(t,x) is the unique entropy solution inL ∞ of (1.1) with initial data (2.4)u| t=0 =u 0 ∈L ∞ ,

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where the solution is understood in the sense of distributions with initial data included in the integral entropy inequality. Let the Young measureν t,x (λ) be a measure-valued solution to (1.1) and (2.4) withν 0,x =δ u 0 (x) , i.e., (2.5)∂ t hν t,x , η(λ)i+ div x hν t,x ,q(λ)i ≤0, ν 0,x =δ u 0 (x) , for any convex entropy pair (η,q). Then

ν t,x (λ) =δ u(t,x) (λ), a.e.(t,x)∈R + ×R n . That is, ifu ε (t,xsequence of uniformly bounded approximate solutions to) is a (1.1) and (2.4) so that the corresponding Young measureν t,x satisfies (2.5), then u ε (t,x) converges strongly to the unique entropy solutionu(t,x) of (1.1) and (2.4).

This result can be obtained by combining DiPerna’s argument in [10] with the monotonicity argument as in Chen-Rascle [4]; also see Szepessy [27]. In particular, Lemma 2.3 implies that, ifu ε (t,x) are the entropy solutions of ∂ t u ε + div x (F(u ε ) +G ε (u ε )) = 0, u| t=0 =u 0 ∈L ∞ , whereG 0 ε →0 strongly inL l ∞ oc (R) whenε→ ∞, thenu ε (t,x) converges strongly to the unique entropy solutionu(t,x) of (1.1) and (2.4).

On the other hand, the nonlinearity of the flux function can yield the compact- ness of solution operators. We start with the one-dimensional case.

Lemma2.4.Cauchy problem for one-dimensional conservation laws:Consider the

(2.6)∂ t u+∂ x f(u) = 0, u| t=0 =u 0 (x). Assume that there is no interval (α, β) in whichfis affine. Then the entropy solution operatoru(t,∙) =S t u 0 (∙) :L ∞ →L l 1 oc , determined by (2.6), is compact in L l 1 oc (R + ×R). Furthermore, if a uniformly bounded sequenceu ε (t, x) satisfies that (2.7)∂ t η(u ε ) +∂ x q(u ε  compact in) isH l − oc 1 , thenu ε (t, x) strongly converges to anL ∞ functionu(t, x) a.e.

The first proof of this lemma was given in Tartar [28] by using infinite entropy- entropy flux pairs. A simpler proof can be found in Chen-Lu [5] by using only two natural entropy-entropy flux pairs. The corresponding multidimensional version of Lemma 2.4 is the following.

Lemma2.5.Consider the Cauchy problem (1.1)–(1.2). Assume that, for any (τ,k)∈R×R n withτ 2 +|k| 2 = 1,

(2.8)meas{v∈R:τ+F 0 (v)∙k= 0}= 0. Then the entropy solution operatoru(t,∙) =S t u 0 (∙) :L ∞ →L l 1 oc , determined by (1.5), is compact inL l 1 oc (R + ×R n ). In particular, if (2.9) det(F 00 (v),∙ ∙ ∙,F (n+1) (v))6= 0,for anyv∈R, then the entropy solution operator of (1.1) is compact fromL ∞ (R n ) toL l 1 oc (R + × R n ).

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Proof. Thelemma is essentially due to Lions-Perthame-Tadmor first part of this [22]; its complete proof can be found in Chen-Frid [3]. For the second part, it suffices to prove that, for any (τ,k)∈R×R n such that τ 2 +|k| 2 = 1, the setE:={v∈R:τ+F 0 (v)∙k= 0}is countable, someas(E) = 0. Leth(v)≡τ+F 0 (v)∙k. Ifk= (0,∙ ∙ ∙,0), thenh(v) =±1 andv /∈E. Ifk6= (0,∙ ∙ ∙,0), then, for anyv∈R, there existsj∈ {1,∙ ∙ ∙, n}such that d j h(v) 6= 0. dv j Otherwise, k⊥span(F 00 (v)∙ ∙ ∙F (n+1) (v)) =R n , , , i.e.,k= (0,∙ ∙ ∙, the zeros of0). Therefore,h(v) are isolated.

Remark2.2.Lemma 2.5 is also true if the genuine nonlinearity assumption (2.9) is imposed only on the constant stateufor which the compactness, locally nearu, can be achieved.

Remark2.3.also holds if there exist 2Lemma 2.5 ≤i 1 < i 2 <∙ ∙ ∙< i n such that det(F (i 1 ) (v),∙ ∙ ∙,F (i n ) (v))6= 0, followed by a similar proof. Lemma2.6.LetF,G ε ∈C 1 (R;R n ) such thatFsatisfies the nondegeneracy con- dition (2.8) andG 0 ε (u)→0 strongly inL l ∞ oc (R) asε→0. Assume thatu ε 0 (x) is a uniformly bounded sequence converging weak-star tou 0 (x). Letu ε (t,x) andu(t,x) be the entropy solutions of the Cauchy problems: ∂ t u ε + div x (F(u ε ) +G ε (u ε )) = 0, u ε (0,x) =u 0 ε (x), ∂ t u+ div x F(u) = 0, u(0,x) =u 0 (x), respectively. Then the sequenceu ε (t,x) strongly converges tou(t,x) inL l 1 oc (R + × R n ).

Proof:LetF ε :=F+G ε  the kinetic formulation of. Thenu ε (t,x) for (1.1) is ∂ t χ ε +F 0 ε (ξ)∙ r x χ ε =∂ ξ m ε withm ε uniformly bounded inM loc sinceu ε are uniformly bounded inL ∞ .eW can rewrite the kinetic formulation as ∂ t χ ε +F 0 (ξ)∙ r x χ ε =∂ ξ m ε −G 0 ε (ξ)∙ r x χ ε . Then, using Theorem 3.1 of [25] (pp. 124) (also [3, 22]), we get the compactness of the sequenceu ε (t,x) inL l 1 oc  up to a subsequence,. Therefore,u ε →wstrongly inL l 1 oc the limit in the weak formulation of (1.1) and the entropy to . Passing inequality, we have that, for anyϕ∈C 0 ∞ ([0,∞)×R n ), Z 0 ∞ Z Rn ∂ t ϕ+F(w)∙ r x ϕ)(t,x)dtdx+ Z Rn u 0 (x)ϕ(0,x)dx= (2.10) (w0, and

(2.11)∂ t |w−k|+div(sign(w−k)(F(w)−F(k)))≤ any0 fork∈R in the sense of distributions. A priori, we could have lost the initial data in passing to the limit in the entropy inequalities. Now Vasseur’s result in [29] indicates that

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any solution of (2.10)–(2.11) satisfying (2.8) has a strong trace ont Since= 0. the weak formulation of (1.1) implies thatu 0 is a weak trace ofw, we conclude thatu 0 is the strong trace ofwont= 0, which impliesw≡uby the Krushkov’s uniqueness theorem. Since the weak limit is unique, then the whole sequenceu ε (t,x) converges tou(t,x). 2.3.L 1 -Stability with respect to the Flux Functions.We have Lemma2.7.LetF∈C 1 (R;R n ). Letu∈BV loc ∩L ∞ (R n ;R) andv∈L ∞ (R n ;R) be periodic entropy solutions with periodPof the Cauchy problems: ∂ t u+ div x F(u) = 0, u(0,x) =u 0 (x); ∂ t v+ div x G(v) = 0, v(0,x) =v 0 (x), respectively. Then, for any 0< t≤T, Z P |u−v|(t,x)dx≤ Z P |u 0 −v 0 |(x)dx+LT|r x u 0 (∙)| M(P) . whereL:= max 1≤j≤n max{|(F 0 j −G 0 j )(u)|:|u| ≤max(ku 0 k ∞ ,kv 0 k ∞ )}. See [1, 26] for the non-periodic case and [19] for the periodic initial data with respect to one space variable. We can extend the proof of [19] to the case of periodic initial data with respect to each space variable. We will use this lemma with initial datau 0 ε :=u 0 (x 1 /ε α 1 , x 2 /ε α 2 ). Sinceu 0 ∈ BV(P) is periodic with periodP,∂ x 1 u ε 0 is of order 1/ε α and∂ x 2 u 0 ε is of order 1/ε β in the spaceM loc (R 2 ).

3.Scaling,L 1 –Convergence, and Homogenization

In this section, we introduce new tools to validate nonlinear geometric optics for entropy solutions only inL ∞  Theseof multidimensional conservation laws. tools are about the changes of variables that preserve theL l 1 oc –convergence, weak oscillating limits, and uniqueness of the profiles. 3.1.Scaling andL 1 Convergence.first formulate the following useful lemma.We - Lemma3.1 (L 1 -convergence of periodic functions and triangular scaling).Letu ε ∈ L 1 (P;R) be a sequence of periodic functions with periodP= [0,1] n . LetA ε = (a ij ε ) 1≤i,j≤n be a sequence of lower triangularn×nmatrices such that (3.1) 1≤ m i i ≤ n n li ε m → i 0 nf|a ii ε |>0. Setv ε (x) :=u ε (A ε x). Then, whenε→0,u ε (x) converges strongly to 0 inL l 1 oc (R n ) if and only ifv ε (x) converges strongly to 0 inL l 1 oc (R n ). Proof:We first note the following facts: 1. For anya∈R, the translation operatorλ7→ϕ(λ) :=λ+ais obviously one to one from [0,1)≡R/ZtoR/Z. 2. IfA= (a ij ) 1≤i,j≤n is an invertible and lower triangularn×nmatrix, then n (3.2)AΠ 1≤i≤n [0,1/a ii ) = [0,1) n in (R/Z), where [0, a) :={λa:λ∈[0,1)}is independent of the sign ofaandAin (3.2) is the linear map fromR n toR n associated with the matrixA.