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

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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 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, and ? = (?1, · · · ,

  • geometric optics

  • entropy solution

  • hunter-majda-rosales

  • solution oper- ators

  • nonlinear geometric

  • dissipation through


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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, andL1–stability. New multidimensional features are recognized, especially including nonlinear propagations of oscillations with high frequencies. The validity of nonlinear geometric optics for entropy solutions inLof 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 uRxRnwhereF:RRn the Cauchy problem (1.1) Consideris a smooth flux function. with Cauchy data: (1.2)u|t=0=uε0(x) := +εu1(φ1εα1   φnεαn)whereu1a periodic function of each of itsis narguments whose period is denoted byP= [01]n(without loss of generality), is a constant ground state, the linear phases Φ := (φ1   φn): n (1.3)φi:=XJijxj j=1
are linearly independent with constant matrixJ= (Jij)1ijn, and α= (α1   αn)[0)n is the magnitude indices of frequency of the initial oscillations. We seek a geometric optics asymptotic expansion: (1.4)uε(tx) = +εvε(tx)
Date: July 22, 2004. 1991Mathematics Subject Classification.Primary:35B40,35L65; Secondary:35B35. 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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MICHEL RASCLE
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, andL1–stability, and we 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 in Section 4. To illustrate multidimensional features clearly in nonlinear geometric optics, we focus now on the two dimensional case. Letu:=uεbe the Krushkov solution of the Cauchy problem: (1.5)tu+x1f1(u) +x1f2(u) = 0(1.6)u(0 x1 x2) =u0(x1 x2)+ε u1(φ1εα φ2εβ)where, as for the general setting (1.1)–(1.4), the linear phases (φ1 φ2): (1.7)φ1:=a1x1+a2x2 φ2:=b1x1+b2x2 are linearly independent, and α1 α20The Krushkov solution is anLfunctionu=u(t x1 x2) satisfying (1.8)t|uk|+x1(sign(uk)(f1(u)f1(k)))+x2(sign(uk)(f2(u)f2(k)))0 in the sense of distributions for anykR the Krushkov solution. Foru, we look for an asymptotic expansion: (1.9)u=uε(t x1 x2 +) :=ε vε(t x1 x2)After, if necessary, a linear change of coordinates:x1x1a0t x2x2b0t we may assume
f1) =f2) = 0Sinceφ1andφ2independent, we can rewrite equation (1.5) in these co-are linearly ordinates, even though they are not necessarily orthonormal, and perform a formal asymptotic expansion to obtain (1.10)tv+ε(a∂φ1+b∂φ2)v2+ε2(c∂φ1+d∂φ2)v3=ε3(φ1R1+φ2R2)
where a:= (a1f1′′) +a2f2′′))2 b:= (b1f1) +b2f2))2(1.11) c:= (a1f1′′) +a2f2′′′))6 d:= (b1f1′′) +b2f2′′′))6andRj:=Rj( εv ) j= 12are Lipschitz functions inv and, ,εwith the form: (4 (1.12)R1( εv ) =6R01(1θ)3(a1f)1+a2f(4)2) +εθv)dθ v4(1.13)R2( εv ) =6R10(1θ)3(b1f4)1(+b2f)42() +εθv)dθ v4Define (1.14)M:=bcad=ab11ab22 ff12))22ff12))66
NONLINEAR GEOMETRIC OPTICS FOR MULTI-D CONSERVATION LAWS
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We assume that the matrixM=M) is invertible. It is equivalent to require that both φ1 φ2) (1.15)J:=ab11ab22=D(x1 x2) D(
and (1.16)N:=ff12))22ff21))66be invertible. Note that the invertibility of (1.15) is a corollary of the linear in-dependence of the phases (φ1 φ2), 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 fluxesf1andf2are not proportional on any interval in a neighborhood of . This type of nonlinear assumptions is enough to obtain compactness; indeed, this nonlinear assumption is the strongest nonlinear assumption near the constant ground state . 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=vεas a function oftand the fast variables, which turn out to be, for the simplest cases, (1.17)X1:=φεmin(α11) X2:=ψεmin(α21); (ii). justification of this asymptotics, that is, the strong convergence ofvεtoV inLl1oc: (1.18)vεV0 strongly inLl1oc in the two systems of fast coordinates (t X1 X2) and of slow coordinates (t x1 x2), 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, andL1–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 withXin (1.17), there are three subcases for entropy solutions indefined as L: (i). Ifα >1, then the initial oscillation is so fast that it is “canceled” by the nonlinearity, that is,vεconverges strongly toσwith (1.19)σ=u1 value of:= meanu1over the period;
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(ii). Ifαthe natural situation of weakly nonlinear geometrical optics= 1 that is (WNLGO), thenvεconverges strongly to the profileσin the fast variableX, which is uniquely determined by the Cauchy problem: (1.20)tσ+a∂Xσ2= 0 σ(0 X) =u1(X); (iii). Ifα <1, then the initial oscillation is so slow that, in the variableX=φεα, vεconverges strongly toσ, which is determined by
(1.21)tσ= 0 σ(0 X) =u1(X)that is,σ(t X) =u1(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, especially the examples in the two-dimensional case), although some cases are a combination of these three different possibilities. In particular, the links between the linear phases (φ1   φn) and the fluxesFformulas, such as (1.10) to (1.14) for the two-dimensionalthrough case, lead to a number of interesting cases which deserve to be described more pre-cisely. 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 multidimen-sional case in Section 5. An important tool to preserve theLl1oc–convergence after the triangular change of variables depending onεis introduced in Lemma 3.1 and a “quasi”LUfactorization of matrixMto include all new cases is formulated in Lemma 5.1. We also recognize new phenomena including the blowup of the gra-dients of the geometric optics asymptotic expansions; in contrast to the classical geometric optics expansions with the gradients of order 1 since the amplitude is of orderεand the frequency of orderε1 are essentially two ways to obtain. There 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 dimensionn3, we can have higher oscillations with combinations of these 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],Gue`s[15],Hunter-Majda-Rosales[16],Junca[18,19],Joly-M´etivier-Rauch [17], Majda-Rosales [23], and the references cited therein. 2.Geometric Optics, Compactness, andL1–Stability
In this section, we introduce some basic frameworks to validate nonlinear geo-metric optics for entropy solutions only inLof 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 problem (1.1)–(1.2). Setvε(tx) =uε(εtx). We first derive some basic properties of the sequencevε(tx).
NONLINEAR GEOMETRIC OPTICS FOR MULTI-D CONSERVATION LAWS 5 Lemma2.1.Letu1L. Assume that, for each fixedε >0,uε(tx) is the entropy solution of the Cauchy problem (1.1)–(1.2). Then (2.1)kvεkL≤ ku1kLfor anyε >0Proof Krushkov’s uniqueness theorem,. Usinguε(tx) is the periodic entropy solu-tion of the Cauchy problem (1.1)–(1.2) for any fixedε >0, sinceu1is periodic. First, taking the convex entropy (u)pof (1.1) for evenp2, we obtain from the entropy inequality that t(uε)p+ divxpZuε(ξ)p1F(ξ)!0
in the sense of distributions. Equivalently, t(uε)p+ divΦp JZuε(ξ)p1F(ξ)!0
in the sense of distributions, where Φ =JxIntegrating with respect to Φ and using the periodicity, we obtain Z|uε(t J1Φ)− |pdΦZPε|u0(φ1εα1   φnεαn)− |p1  nPε wherePε={(φ1   φn) : (φ1εα1   φnεαn)P}This is equivalent to ZPε|vε(t J1Φ)|pdΦZPε|u1(φ1εα1   φnεαn)|p1  n for any evenp2. Taking power 1pin both sides and lettingp→ ∞, we conclude (2.1).
Lemma2.2.Ifuε(t x1 x2) is the entropy solution of (1.5)–(1.6), thenvε(t x1 x2) is the entropy solution of (1.10) with initial data (2.2)vε|t=0=u1(φ1εα1 φ2εα2)Proof:Notice thatuε osing 8)satisfy (l=εk 1. . Cho with the linear transformation into the (φ1 φ2)-coordinates, we have (2.3)t|vεl|+ε(a∂φ1+b∂φ2)(sign(vεl)((vε)2l2)) +ε2(c∂φ1+d∂φ2)(sign(vεl)((vε)3l3)) +ε3(φ1(sign(vεl)(R1(vε ε )R1(l  ε))) φ2(sign(vεl)(R2(vε  ε)R2(l  ε))))0which implies thatvε(t x1 x2) 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 n3, 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 inLof (1.1) with initial data (2.4)u|t=0=u0L
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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 measureνtx(λ) be a measure-valued solution to (1.1) and (2.4) withν0x=δu0(x), i.e., (2.5)thνtx η(λ)i+ divxhνtxq(λ)i ≤0 ν0x=δu0(x)for any convex entropy pair (ηq). Then
νtx(λ) =δu(tx)(λ) ae(tx)R+×RnThat is, ifuε(txsequence of uniformly bounded approximate solutions to) is a (1.1) and (2.4) so that the corresponding Young measureνtxsatisfies (2.5), then uε(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, ifuε(tx) are the entropy solutions of tuε+ divx(F(uε) +Gε(uε)) = 0 u|t=0=u0LwhereGε0 strongly inLloc(R) whenε→ ∞, thenuε(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.Consider the Cauchy problem for one-dimensional conservation laws:
(2.6)tu+xf(u) = 0 u|t=0=u0(x)Assume that there is no interval (α β) in whichfis affine. Then the entropy solution operatoru(t) =Stu0() :LLl1oc, determined by (2.6), is compact in Ll1oc(R+×R if a uniformly bounded sequence). Furthermore,uε(t x) satisfies that 1 (2.7)tη(uε) +xq(uε) is compact inHlocthenuε(t x) strongly converges to anLfunctionu(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. that, for any AssumeConsider the Cauchy problem (1.1)–(1.2). (τk)R×Rnwithτ2+|k|2= 1,
(2.8)meas{vR:τ+F(v)k= 0}= 0Then the entropy solution operatoru(t) =Stu0() :LLl1oc, determined by (1.5), is compact inLl1oc(R+×Rn). In particular, if (2.9) det(F′′(v)  F(n+1)(v))6= 0 for anyvRthen the entropy solution operator of (1.1) is compact fromL(Rn) toLl1oc(R+× Rn).
NONLINEAR GEOMETRIC OPTICS FOR MULTI-D CONSERVATION LAWS
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Proof. The first part of this lemma is essentially due to Lions-Perthame-Tadmor [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×Rnsuch that τ2+|k|2= 1, the setE:={vR:τ+F(v)k= 0}is countable, someas(E) = 0. Leth(v)τ+F(v)k. Ifk= (0  0), thenh(v) =±1 andv E Ifk6= (0  0), then, for anyvR, there existsj∈ {1   n}such that djhdv(jv)60= Otherwise, kspan(F′′(v)  F(n+1)(v)) =Rni.e.,k= (0  0). Therefore, the zeros ofh(v) are isolated.
Remark2.2.Lemma 2.5 is also true if the genuine nonlinearity assumption (2.9) is imposed only on the constant state for which the compactness, locally near , can be achieved.
Remark2.3.Lemma 2.5 also holds if there exist 2i1< i2<  < insuch that det(F(i1)(v)  F(in)(v))6= 0followed by a similar proof. Lemma2.6.LetFGεC1(R;Rn) such thatFsatisfies the nondegeneracy con-dition (2.8) andGε(u)0 strongly inLloc(R) asε0. Assume thatuε0(x) is a uniformly bounded sequence converging weak-star tou0(x). Letuε(tx) andu(tx) be the entropy solutions of the Cauchy problems: tuε+ divx(F(uε) +Gε(uε)) = 0 uε(0x) =uε0(x)tu+ divxF(u) = 0 u(0x) =u0(x)respectively. Then the sequenceuε(tx) strongly converges tou(tx) inLl1oc(R+× Rn).
Proof:LetFε:=F+Gε. Then the kinetic formulation ofuε(tx) for (1.1) is tχε+Fε(ξ) ∇xχε=ξmε withmεuniformly bounded inMlocsinceuεare uniformly bounded inL. We can rewrite the kinetic formulation as tχε+F(ξ) ∇xχε=ξmεGε(ξ) ∇xχεThen, using Theorem 3.1 of [25] (pp. 124) (also [3, 22]), we get the compactness of the sequenceuε(tx) inLl1oc. Therefore, up to a subsequence,uεwstrongly inLl1oc to the limit in the weak formulation of (1.1) and the entropy. Passing inequality, we find that, for anyϕC0([0)×Rn), (2.10)Z0ZRn(w∂tϕ+F(w) ∇xϕ)(tx)dtdx+ZRnu0(x)ϕ(0x)dx= 0and
(2.11)t|wk|+div(sign(wk)(F(w)F(k)))0 for anykR 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 the Since= 0. weak formulation of (1.1) implies thatu0is a weak trace ofw, we conclude thatu0 is the strong trace ofwont= 0, which implieswuby the Krushkov’s uniqueness theorem. Since the weak limit is unique, then the whole sequenceuε(tx) converges tou(tx). 2.3.L1-Stability with respect to the Flux Functions.We have Lemma2.7.LetFC1(R;Rn). LetuBVlocL(Rn;R) andvL(Rn;R) be periodic entropy solutions with periodPof the Cauchy problems: tu+ divxF(u) = 0 u(0x) =u0(x); tv+ divxG(v) = 0 v(0x) =v0(x)respectively. Then, for any 0< tT, ZP|uv|(tx)dxZP|u0v0|(x)dx+LT|∇xu0()|M(P)whereL:= max1jn(max{|(FjGj)(u)|:|u| ≤max(ku0kkv0k)}). 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 datau0ε:=u0(x1εα1 x2εα2). Ifu0BV(P) is periodic with periodP, thenx1u0εis of order 1εα1andx2uε0is of order 1εα2in the spaceMloc(R2).
3.Scaling,L1–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 theLl1oc–convergence, weak oscillating limits, and uniqueness of the profiles. 3.1.Scaling andL1-Convergence.We first formulate the following useful lemma. Lemma3.1 (L1-convergence of periodic functions and triangular scaling).LetuεL1(P;R) be a sequence of periodic functions with periodP= [01]n. LetAε= (aεji)1ijnbe a sequence of lower triangularn×nmatrices such that (3.1)1miinnliεmi0nf|aiiε|>0Setvε(x) :=uε(Aεx). Then, whenε0,uε(x) converges strongly to 0 inLl1oc(Rn) if and only ifvε(x) converges strongly to 0 inLl1oc(Rn). Proof:We first note the following facts: 1. For anyaR, the translation operatorλ7→ϕ(λ) :=λ+ais obviously one to one from [01)RZtoRZ. 2. IfA= (aij)1ijnis an invertible and lower triangularn×nmatrix, then (3.2)AΠ1in[01aii) = [01)nin (RZ)nwhere [0 a) :={λa:λ[01)}is independent of the sign ofa, andAin (3.2) is the linear map fromRntoRnassociated with the matrixA.
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