1. Introduction The present paper is concerned with the scalar conservation law in one space dimension: u t + (A(u)) x = 0 (1.0.1)

Transcription

1 SINGLE ENTROPY CONDITION FOR BURGERS EQUATION VIA THE RELATIVE ENTROPY METHOD SAM G. KRUPA AND ALEXIS F. VASSEUR Absrac. This paper provides a new proof of uniqueness of soluions o scalar conservaion laws wih convex flux verifying only one enropy condiion. Unlike previous resuls, he proof is no based on he Hamilon Jacobi equaion. I works direcly on he conservaion law, and is based on he mehod of relaive enropy. Conens 1. Inroducion 1 2. Preliminaries 6 3. Consrucion of he shif Lemmas necessary for he proof of Proposiion Proof of Proposiion Main proposiion Lemma 4.2 and Lemma 4.3 imply main proposiion Proof of Lemma Proof of Lemma Main proposiion implies main heorem Appendix Proof of Lemma Proof of Lemma References Inroducion The presen paper is concerned wih he scalar conservaion law in one space dimension: { u + (A(u)) x = 0 (1.0.1) u(x, 0) = u 0 (x). Dae: Sepember 15h, Mahemaics Subjec Classificaion. Primary 35L65; Secondary 35L45, 35L67. Key words and phrases. Conservaion laws, enropy condiions, enropy soluions, relaive enropy, shocks, uniqueness, Oleĭnik, Kruzhkov. The second auhor was parly suppored by NSF Gran DMS

2 2 KRUPA AND VASSEUR In he scalar conservaion law (1.0.1), u(x, ) : R [0, ) R is he unknown, u 0 L (R) is he given iniial daa, and A : R R is he given flux funcion. In he presen paper, we are concerned wih A C 2 (R) sricly convex. A classical soluion of (1.0.1) is a locally Lipschiz funcion u : R [0, ) R which saisfies u + (A(u)) x = 0 almos everywhere and verifies u(x, 0) = u 0 (x) for all x R. We are also ineresed in weak soluions of (1.0.1). A weak soluion o (1.0.1) is a locally bounded measurable funcion u : R [0, ) R which saisfies (1.0.2) 0 [ φu + x φa(u)] dxd + φ(x, 0)u 0 (x) dx = 0 for every Lipschiz coninuous es funcion φ : R [0, ) R, wih compac suppor. In paricular, every weak soluion saisfies (1.0.1) in he sense of disribuions. Noe ha a classical soluion is also a weak soluion. A pair of funcions η, q : R R are called an enropy and enropy flux, respecively, for he scalar conservaion law (1.0.1) if (1.0.3) q (u) = η (u)a (u). We say a weak soluion u of (1.0.1) is enropic for he enropy η if i saisfies he enropy inequaliy (1.0.4) η(u) + x q(u) 0 in a disribuional sense, where q is any corresponding enropy flux. Precisely, (1.0.5) 0 [ φη(u) + x φq(u)] dxd + φ(x, 0)η(u 0 (x)) dx 0 for every nonnegaive Lipschiz coninuous es funcion φ : R [0, ) R, wih compac suppor. Kruzhkov [19] proved exisence and uniqueness for bounded weak soluions o (1.0.1) which are enropic for he large family of enropies {η k } k R, where (1.0.6) η k (u) := u k. For a bounded and measurable soluion o (1.0.1), being enropic for each of he η k is equivalen o being enropic for every convex enropy [26, Proposiion 2.3.4]. See Bolley, Brenier and Loeper [6] for an exension of Kruzhkov s heory, based on he Wassersein disance. Oleĭnik discovered condiion E, and proved exisence and uniqueness for bounded weak soluions o (1.0.1) which saisfy i [24]. A soluion u o (1.0.1) saisfies condiion E if There exiss a consan C > 0 such ha (1.0.7) u(x + z, ) u(x, ) C z for all > 0, almos every z > 0, and almos every x R. I is known ha being enropic for each of he η k is equivalen o Oleĭnik s condiion E when he conservaion law (1.0.1) has a uniformly convex flux funcion A [26, p. 66 and

3 SINGLE ENTROPY CONDITION FOR BURGERS EQUATION 3 p. 57]. When he soluion o (1.0.1) is bounded and A is sricly convex, we can assume A is uniformly convex. In 1989, Kruzhkov [1, p. 164] asked wheher soluions o Burgers equaion (1.0.8) {u + ( u2 2 ) x = 0 u(x, 0) = u 0 (x) which are enropic for he enropy η(u) = u2 2 are unique. Noe ha Burgers equaion is he scalar conservaion law (1.0.1) in he case when A(u) = u2 2. The answer o his quesion is yes; he firs proof was given by Panov in 1994 [25], under he more general siuaion of (1.0.1) wih any A C 2 (R) sricly convex, any enropy η C 1 (R) sricly convex, and u 0 L (R). A second proof was given in 2004 by De Lellis, Oo and Wesdickenberg [12]. Their resul is sronger han Panov s resul: heir proof allows for various righ-hand sides in he enropy inequaliy, and can consider unbounded funcions [12, p. 688]. The wo papers [12, 25] give us a minimal enropy condiion for (1.0.1): we only need o consider one enropy η C 1 (R) sricly convex o ensure uniqueness. The exisence of a minimal enropy condiion is ineresing because in he general case of hyperbolic sysems of conservaion laws in one space dimension, here ofen only exiss one non-rivial enropy for he sysem [28, p. 238]. Boh he proofs of Panov and De Lellis-Oo-Wesdickenberg use a special connecion beween scalar conservaion laws in one space dimension and Hamilon Jacobi equaions: he space derivaive of he soluion o a Hamilon Jacobi equaion is formally a soluion o he associaed scalar conservaion law. However, i is well-known ha he relaion beween scalar conservaion laws in one space dimension and Hamilon Jacobi equaions breaks down in he more general case of hyperbolic sysems of conservaion laws in one space dimension (however, see [15] for a formal connecion beween a general Hamilon Jacobi equaion in n space dimensions and a weakly hyperbolic sysem of conservaion laws). The presen paper gives a new proof of he minimal enropy condiion wihou relying on Hamilon Jacobi heory, under he addiional assumpion of a srong race propery on he soluion. Le us inroduce he srong race propery (originally inroduced in [21]): Definiion 1.1. Le u L (R [0, )). Then u verifies he srong race propery if for any Lipschiz coninuous funcion h : [0, ) R, here exis u +, u L ([0, )) such ha (1.0.9) lim n T 0 for all T > 0. ess sup u(h() + y, ) u + () T d = lim ess sup u(h() + y, ) u () d = 0 y (0, 1 n ) n y ( 1 0 n,0)

4 4 KRUPA AND VASSEUR Noe ha any funcion u L (R [0, )) will saisfy he srong race propery if u has a represenaive such ha for any fixed h, he lef and righ limis (1.0.10) lim y 0 y 0 u(h() + y, ) and lim u(h() + y, ) + exis for almos every. In paricular, any funcion u L (R [0, )) wih a represenaive in BV loc will saisfy he srong race propery. Bu he srong race propery is weaker han BV loc. The resul we show in his paper is Theorem 1.2 (Main heorem). Le u L (R [0, )) be a weak soluion wih iniial daa u 0 L (R) o he scalar conservaion law in one space dimension (1.0.1) wih flux A C 2 (R) sricly convex. Assume u saisfies he enropy inequaliy (1.0.4) for a leas one sricly convex enropy η C 2 (R). Furher, assume u saisfies he srong race propery (Definiion 1.1). Then u is he unique soluion o (1.0.1) verifying (1.0.7) and wih iniial daa u 0. Our mehod is based on he relaive enropy mehod. Given he sysem (1.0.1) and enropy and enropy flux η and q, respecively (or more generally, any hyperbolic sysem of conservaion laws in muliple space dimensions endowed wih any enropy), he mehod of relaive enropy considers he quaniy called he relaive enropy: (1.0.11) η(a b) := η(a) η(b) η (b)(a b) for all a, b R. We have also he associaed relaive enropy-flux, (1.0.12) q(a; b) := q(a) q(b) η (b)(a(a) A(b)) for all a, b R. Boh η(a b) and q(a; b) are locally quadraic in a b. In paricular, for all a and b in a fixed compac se, he sric convexiy of η C 2 (R) gives (1.0.13) c (a b) 2 η(a b) c (a b) 2 for consans c, c > 0. The mehod of relaive enropy was invened by Dafermos [10, 9] and DiPerna [13] o prove weak-srong esimaes: given a weak soluion u o (1.0.1), enropic for η, and a classical soluion ū, he mehod of relaive enropy gives sabiliy esimaes on he growh in ime of u(, ) ū(, ) L 2 by considering he ime derivaive of η(u ū) dx and using he enropy inequaliy (1.0.4) (in paricular, see Lemma 2.2). Due o (1.0.13), he quaniy η(u ū) gives esimaes of L 2 -ype, while being more amenable o sudy han he L 2 norm iself, due o he enropy inequaliy (1.0.4). The relaive enropy mehod is fundamenally an L 2 heory. The weak-srong sabiliy resuls of Dafermos and DiPerna compare a weak soluion o a smooh, classical soluion. The proof of weak-srong sabiliy relies on he he fac ha classical soluions of a hyperbolic sysem of conservaion laws saisfy he enropy inequaliy

5 SINGLE ENTROPY CONDITION FOR BURGERS EQUATION 5 (1.0.4) as an exac equaliy. Weak-srong sabiliy breaks down when a disconinuiy is inroduced ino he classical soluion. In paricular, he relaive enropy mehod is much more involved when considering shocks. For a firs resul, see DiPerna [13] for uniqueness. In he case of sabiliy, here is a growing body of work using he relaive enropy mehod o consider he sabiliy of shock waves in L 2, up o a ime-dependen ranslaion in space of he shock wave soluion. This program of using relaive enropy and sabiliy up o a ranslaion was firs described by he second auhor in [30]. The firs resul in his program was by Leger [20] for he scalar conservaion law (1.0.1) for a sricly convex flux A C 2 (R). We now inroduce he resul of Leger. Le u L, u R R saisfy u L > u R. Le σ saisfy A(u L ) A(u R ) = σ(u L u R ). Define { u L if x < 0 (1.0.14) φ(x) := u R if x > 0. Le u be any Kruzhkov soluion. In his conex, Leger proved he exisence of a Lipschiz coninuous funcion h : [0, ) R such ha u(, ) φ( h() σ) L 2 u(, 0) φ( ) L 2 for all 0. Noe ha by shifing he posiion of he shock wave as a funcion of ime, all L 2 growh in ime is killed. The shif funcion h depends on u. Leger gives conrol on h: h() λ u(, 0) φ( ) L 2 (R), for some consan λ > 0. Leger only considered Kruzhkov soluions, bu his mehods are in fac very general and can be applied whenever a soluion saisfies he srong race propery and is enropic for a leas one sricly convex enropy η C 2 (R). The program of using relaive enropy o consider sabiliy up o a shif has been pushed by he second auhor and coworkers o consider sysems of conservaion laws in one space dimension [16, 29, 31, 27, 21] and scalar viscous conservaion laws in boh one space dimension [17] and muliple space dimensions [18]. For a more general overview of he heory of shifs and he relaive enropy mehod in general, see [28, Secion 3-5]. The heory of sabiliy up o a shif has also been used o sudy asympoic limis when he limi is disconinuous. See [8] for he scalar case, and [32] for he case of sysems. There is a long hisory of using he relaive enropy mehod o sudy he asympoic limi. However, resuls on he asympoic limi which do no use shifs have only been able o consider he case when he limi funcion is Lipschiz coninuous (see [33, 2, 3, 4, 5, 14, 22, 23] and [30] for a survey). There is also an L 1 heory for he sabiliy and uniqueness of soluions for sricly hyperbolic n n sysems of conservaion laws in one space dimension: Bressan, Crasa, and Piccoli [7] show ha in L 1 soluions depend coninuously on heir iniial daa wih a Lipschiz consan which is uniform wih respec o ime. Bressan-Crasa-Piccoli consider soluions wih small oal variaion. Their mehod uses approximae soluions which for each fixed ime are piecewise Lipschiz coninuous. The presen paper, on he oher hand, is anoher sep in he program of using he relaive enropy mehod and shifs o develop an L 2 heory for sabiliy and uniqueness of weak soluions o hyperbolic sysems of conservaion laws.

6 6 KRUPA AND VASSEUR We briefly ouline he presen paper and he proof of he main heorem, Theorem 1.2. Le u be any bounded weak soluion o (1.0.1) wih iniial daa u 0 L (R), enropic for a sricly convex enropy η C 2 (R) and saisfying he srong race propery (Definiion 1.1). To prove u verifies (1.0.7), we use shif funcions very similar o he ones consruced by Leger [20] and in [27] o give, a each fixed ime T, a sequence of piecewise Lipschiz coninuous funcions {ψ ɛ } ɛ>0 defined on a subse of he real line which converge in L 2 o u(, T ) as ɛ 0 +. The ψ ɛ will be consruced by gluing ogeher a ime T various classical soluions o (1.0.1), each of which saisfy (1.0.7). Thus, he ψ ɛ will be shown o verify (1.0.7). By classical measure heory, he ψ ɛ converge (up o a subsequence) o u(, T ) poinwise almos everywhere. Thus, u will saisfy (1.0.7) for ime T, where T is arbirary. This will complee he proof. In Secion 2, we give some preliminary resuls and facs which we will need. Then, in Secion 3 we give a consrucion of a shif funcion as suied o our needs in his paper, based on he consrucion in [27]. Wih he shif consrucion ou of he way, in Secion 4 we sae and prove he main proposiion (Proposiion 4.1) which gives he exisence of he ψ ɛ. The proof of he main proposiion is broken up ino wo lemmas (Lemma 4.2 and Lemma 4.3). To conclude, in Secion 5 we le ɛ 0 + (up o a subsequence) o explain how he main proposiion implies he main heorem (Theorem 1.2). Our mehods are vaguely reminiscen of he Bressan-Crasa-Piccoli L 1 heory; boh heir mehod and ours consider approximae soluions which for each fixed ime are piecewise Lipschiz coninuous. 2. Preliminaries Throughou his paper, we will assume ha here is a consan B > 0 such ha all of he weak soluions w o (1.0.1) ha we consider saisfy (2.0.1) w L (R [0, )) B. For a funcion u L (R [0, )) ha verifies he srong race propery (Definiion 1.1), and a Lipschiz coninuous funcion h : [0, ) R, we use he noaion u(h()±, ) o denoe he values u ± () a a ime when u has a lef and righ race according o Definiion 1.1, where u ± are as in he definiion of srong race. For ime when u has a srong race, he values u ± () are well-defined, and hence u(h()±, ) are also well-defined. Mos of he hings in his preliminary secion are well-known. Bu we include hese facs here and heir proofs so as o make sure we are no using more han we hink. In Theorem 1.2 we do no assume ha he soluion u o he scalar conservaion law is Kruzhkov. Lemma 2.1 (Properies of soluions wih Lipschiz iniial daa). Le vi 0 L (R) be nondecreasing Lipschiz coninuous funcions for i = 1, 2, verifying v1 0(x) v0 2 (x) for all x R. Le v i denoe he unique soluion o (1.0.1) saisfying (1.0.7) and wih iniial daa vi 0. Then he following holds: (a) The v i are classical soluions o (1.0.1) on R [0, ).

7 SINGLE ENTROPY CONDITION FOR BURGERS EQUATION 7 (b) The v i are given by he mehod of characerisics. In oher words, for each (x, ) R [0, ), here exiss x 0 R such ha v 1 (x, ) = v1 0(x0 ) and x = x 0 + A (v1 0(x0 )) (and similarly for v 2 ). (c) The v i saisfy (1.0.7) wih C = 1/ inf A. (d) For i = 1, 2 and 0, v i (, ) is a nondecreasing funcion in he x variable. (e) v 0 L i (R) = v i L (R [0, )) for i = 1, 2. (f) v 1 (x, ) v 2 (x, ) for all (x, ) R [0, ). Proof. From [11, p ], we know ha for each i here exiss a classical soluion v i o (1.0.1) on R [0, ) wih iniial daa vi 0 which is given by he mehod of characerisics. For a fixed i, we now check by direc mehod ha v i saisfies (1.0.7). By uniqueness of soluions saisfying (1.0.7), his will prove pars (a) and (b). Fix, z > 0 and x R. Because v i is given by he mehod of characerisics, here exiss an x 0 and x 0 such ha { v i (x, ) = vi 0(x0 ) (2.0.2) x = x 0 + A (vi 0(x0 )) and (2.0.3) We have (2.0.4) { v i (x + z, ) = v 0 i ( x0 ) x + z = x 0 + A (v 0 i ( x0 )). ( ) x x v i (x, ) = (A ) 1 0 ( ) x + z x v i (x + z, ) = (A ) 1 0 where he funcional inverse (A ) 1 exiss because A is sricly convex. Because he characerisics do no cross, we have (2.0.5) Then from (2.0.5), (2.0.6) Finally, (2.0.7) 0 < x 0 x 0 = z + (A (v 0 i (x 0 )) A (v 0 i ( x 0 ))) < z. 0 < x + z x0 x x0 < z. [ ( ) ( x + z x inf A (A ) 1 0 x x (A ) 1 0 )] ( ( x x A ((A ) 1 0 x + z x )) A ((A ) 1 0 ) ) < z. Lines (2.0.4), (2.0.6) and (2.0.7) imply v i saisfies (1.0.7). In paricular, noe ha we can ake C = 1/ inf A in (1.0.7). This proves pars (a), (b), and (c).

8 8 KRUPA AND VASSEUR Par (d) follows immediaely from par (b). Similarly, par (e) follows immediaely from pars (a) and (b). We now show how par (f) follows from par (b). We argue by conradicion. Assume here exiss (x, ) R [0, ) such ha v 1 (x, ) < v 2 (x, ). Then we have jus proven ha v 1 and v 2 are given by he mehod of characerisics. Thus here exiss x 0 and x 0 such ha (2.0.8) x = x 0 + A (v 0 1(x 0 )) = x 0 + A (v 0 2( x 0 )) and (2.0.9) v 1 (x, ) = v 0 1(x 0 ) < v 0 2( x 0 ) = v 2 (x, ). Then from (2.0.8), (2.0.10) x 0 x 0 = (A (v 0 2( x 0 )) A (v 0 1(x 0 ))). Then he righ-hand side of (2.0.10) is nonnegaive, which means x 0 x 0. Thus, (2.0.11) v 0 1(x 0 ) v 0 1( x 0 ) v 0 2( x 0 ) because v1 0(x) v0 2 (x) for all x R. However, his gives a conradicion wih (2.0.9). This proves par (f). Lemma 2.2 (An exension of he enropy inequaliy). Le ū 0 L (R) be a Lipschiz coninuous and nondecreasing funcion. Le ū be he unique soluion o (1.0.1) wih iniial daa ū 0 and which saisfies (1.0.7). Le u L (R [0, )) be a weak soluion o (1.0.1) wih iniial daa u 0. Assume ha u is enropic for he sricly convex enropy η C 2 (R). Then (2.0.12) η(u ū) + x q(u; ū) 0 in he sense of disribuions. In oher words, he following holds for all nonnegaive, Lipschiz coninuous es funcions φ on R [0, ) wih compac suppor: (2.0.13) [ φη(u ū) + x φq(u; ū)] dxd + φ(x, 0)η(u 0 (x) ū 0 (x)) dx 0. 0 Remark. The inequaliy (2.0.12) exends he enropy inequaliy (1.0.4).

9 SINGLE ENTROPY CONDITION FOR BURGERS EQUATION 9 Proof. By par (a) of Lemma 2.1, ū is a classical soluion. immediaely from he following inequaliy: (2.0.14) 0 [ φη(u ū) + x φq(u; ū)] dxd + 0 Then, Lemma 2.2 follows φ(x, 0)η(u 0 (x) ū 0 (x)) dx φ x ūη (ū)[a(u) A(ū) A (ū)(u ū)] dxd. Dafermos gives his inequaliy as one of he cenral seps in he proof of weak-srong sabiliy. See [11, p. 124, line (5.2.10)]. By par (d) of Lemma 2.1, ū is increasing in x. Furhermore, we have η, A > 0. Thus he righ-hand side of (2.0.14) is conrolled from below by zero. Lemma 2.3. Le [, ) be a bounded inerval. Le h 1 (), h 2 () : [, ) R be Lipschiz coninuous funcions, such ha h 2 () h 1 () > 0 for all [, ). Le ū 0 L (R) be a Lipschiz coninuous and nondecreasing funcion. Le ū be he unique soluion o (1.0.1) wih iniial daa ū 0 and which saisfies (1.0.7). Le u L (R [0, )) be a weak soluion o (1.0.1) wih iniial daa u 0. Assume u is enropic for he sricly convex enropy η C 2 (R). Furhermore, assume ha u verifies he srong race propery (Definiion 1.1). Then for almos every a, b [, ) wih a < b, h 2 (b) η(u(x, b) ū(x, b)) dx h 2 (a) η(u(x, a) ū(x, a)) dx (2.0.15) h 1 (b) b a h 1 (a) [ q(u(h 1 ()+, ); ū(h 1 (), )) q(u(h 2 (), ); ū(h 2 (), ))+ ] ḣ 2 ()η(u(h 2 (), ) ū(h 2 (), )) ḣ1()η(u(h 1 ()+, ) ū(h 1 (), )) d. If = 0, hen (2.0.15) holds for a = 0 and almos every b (, ). Proof. For 0 < ɛ < min{b a, b}, define 0 if < a 1 ɛ ( a) if a < a + ɛ (2.0.16) χ ɛ () := 1 if a + ɛ b 1 ɛ ( (b + ɛ)) if b < b + ɛ 0 if b + ɛ <. Le δ := inf [a,b+ɛ] h 2 () h 1 (). Noe δ > 0. Then for 0 < ɛ < δ 2, define

10 10 KRUPA AND VASSEUR (2.0.17) 0 if x < h 1 () 1 ɛ (x h 1()) if h 1 () < x < h 1 () + ɛ ψ ɛ (x, ) := 1 if h 1 () + ɛ < x < h 2 () ɛ 1 ɛ (x h 2()) if h 2 () ɛ < x < h 2 () 0 if h 2 () < x. The ψ ɛ (x, ) and χ ɛ () are from [20, p. 765]. Use ψ ɛ (x, )χ ɛ () as a es funcion for (2.0.13). The resul is (2.0.18) 1 ɛ b [ h 1()+ɛ a + 1 ɛ h 1 () a+ɛ a q(u; ū) dx h 2 () h 1 () h 2 () h 2 () ɛ η(u ū) dxd 1 ɛ q(u; ū) dx + b+ɛ b h 2 () h 1 () h 2 () h 2 () ɛ ḣ 2 ()η(u ū) dx η(u ū) dxd + O(ɛ) 0. h 1 ()+ɛ h 1 () ] ḣ 1 ()η(u ū) dx d Le ɛ 0 + in (2.0.18). Use he dominaed convergence heorem, he Lebesgue differeniaion heorem, and he fac ha u verifies he srong race propery (Definiion 1.1). This gives (2.0.15) for almos every a, b [, ) wih a < b. When = 0, we wan o show ha (2.0.15) holds for a = = 0. This follows because we include he boundary erm corresponding o = 0 in (1.0.5), and hence he boundary erm corresponding o = 0 appears also in (2.0.13). We now show ha when = 0, (2.0.15) holds for a = = 0. For a = = 0 and 0 < ɛ < b, define (2.0.19) 1 if 0 < b χ 0 ɛ() := 1 ɛ ( (b + ɛ)) if b < b + ɛ 0 if b + ɛ. The χ 0 ɛ is borrowed from [11, p. 124]. Tes (2.0.13) wih ψ ɛ (x, )χ 0 ɛ(). This gives (2.0.20) 1 b ɛ a [ h 1()+ɛ h 1 () q(u; ū) dx + h 2 (0) h 1 (0) h 2 () h 2 () ɛ q(u; ū) dx + η(u 0 (x) ū 0 (x)) dx 1 ɛ h 2 () h 2 () ɛ b+ɛ b ḣ 2 ()η(u ū) dx h 2 () h 1 () h 1 ()+ɛ h 1 () η(u ū) dxd + O(ɛ) 0. ] ḣ 1 ()η(u ū) dx d

11 SINGLE ENTROPY CONDITION FOR BURGERS EQUATION 11 Finally, le ɛ 0 + in (2.0.20). Once again, invoke he dominaed convergence heorem, he Lebesgue differeniaion heorem, and use ha u verifies he srong race propery (Definiion 1.1). We receive (2.0.15) for a = 0 and for almos every b (, ). Lemma 2.4. Le ū 0 L (R) be a Lipschiz coninuous and nondecreasing funcion. Le ū be he unique soluion o (1.0.1) wih iniial daa ū 0 and which saisfies (1.0.7). Le u L (R [0, )) be a weak soluion o (1.0.1) wih iniial daa u 0. Assume ha u is enropic for a leas one sricly convex enropy η C 2 (R). Assume also ha u verifies he srong race propery (Definiion 1.1). Then for all c, d R verifying c < d, he approximae righ- and lef-hand limis (2.0.21) ap exis for all 0 (0, ) and verify (2.0.22) ap d lim η(u(x, ) ū(x, )) dx ± 0 c d d lim η(u(x, ) ū(x, )) dx ap lim η(u(x, ) ū(x, )) dx c c The approximae righ-hand limi also exiss for 0 = 0 and verifies (2.0.23) d c d η(u 0 (x) ū 0 (x)) dx ap lim η(u(x, ) ū(x, )) dx. 0 + c Proof. For some consan C > 0 o be chosen momenarily, define he funcion Γ : [0, ) R, (2.0.24) Γ() := d c η(u(x, ) ū(x, )) dx C. Apply (2.0.15) o he case when h 1 () = c and h 2 () = d for all. The inegrand on he righ-hand side of (2.0.15) is bounded. Thus, here exiss some consan C > 0 such ha Γ() Γ(s) for almos every and s verifying < s. This means ha here exiss a funcion which agrees wih Γ almos everywhere and is non-increasing. This implies ha Γ has approximae lef and righ limis. In paricular, we conclude ha he approximae righ- and lef-hand limis (2.0.21) exis for all 0 (0, ) and verify (2.0.22). Noe he approximae righ-hand limi also exiss for 0 = 0 and because (2.0.15) holds for he ime a = 0, he approximae righ-hand limi verifies (2.0.23) a ime zero. In his secion, we presen a proof of 3. Consrucion of he shif

12 12 KRUPA AND VASSEUR Proposiion 3.1 (Exisence of he shif funcion). Le u L (R [0, )) be a weak soluion o (1.0.1), enropic for a leas one sricly convex enropy η C 2 (R). Assume also ha u verifies he srong race propery (Definiion 1.1). Le ū 0 i L (R) be Lipschiz coninuous and nondecreasing funcions for i = 1, 2, verifying ū 0 1 (x) ū0 2 (x) for all x R. Le ū i L (R [0, )) be soluions o (1.0.1) verifying (1.0.7) and wih iniial daa ū 0 i, for i = 1, 2. Le 0 be some fixed ime, and le x 0 R be some fixed space coordinae. Then for ɛ > 0 here exiss a Lipschiz coninuous funcion h ɛ : [, ) R such ha h ɛ ( (3.0.1) ) = x 0, Lip[h ɛ ] sup A x u L and (3.0.2) q(u + ; ū 2 ) q(u ; ū 1 ) ḣɛ(η(u + ū 2 ) η(u ū 1 )) ɛ for almos every [, ), where u ± = u(h ɛ ()±, ) and ū i = ū i (h ɛ (), ). In [27], he auhors build a general framework for he consrucion of he shif funcions necessary for L 2 sabiliy in he general case of sysems. They also apply heir framework o he scalar case, recovering he resul of Leger [20]. We presen here a consrucion of he shif funcion based on he work [27]. We modify he consrucion slighly. In [27], he daa ū i are consan. In conras, we assume he ū i are Lipschiz coninuous soluions o (1.0.1). The presenaion is also simplified due o our focus only on he scalar case. Lasly, in he general framework for sysems in [27], here is a consrain labeled wih equaion number 7 [27, p. 4]. Alhough his consrain is used in [27] for he scalar case in paricular, we find ha he proof for scalar in [27] as-is does no require i. Lasly, he paper [27] considers only Kruzhkov s soluions [27, p. 9] which are enropic for he family of enropies { u k } k R. We find ha he proofs in [27] hold wihou modificaion for soluions which are no necessarily of his ype Lemmas necessary for he proof of Proposiion 3.1. We need he following srucural lemmas. Lemma 3.2 (Srucural lemma on enropic shocks). Consider he equaion (1.0.1), endowed wih a sricly convex enropy η C 2 (R), and an associaed enropy flux q. Le u L, u R, σ R saisfy (3.1.1) Then (3.1.2) A(u R ) A(u L ) = σ(u R u L ). q(u R ) q(u L ) σ(η(u R ) η(u L )) if and only if u L u R. Which is o say, he shock (u L, u R, σ) is enropic for he enropy η if and only if u L u R. Remark. For he scalar conservaion law (1.0.1) wih flux A sricly convex, we know ha he physical or admissible shock wave soluions will be he ones wih lef- and righhand saes u L and u R, respecively, which saisfy u L > u R. This is he Lax enropy

13 SINGLE ENTROPY CONDITION FOR BURGERS EQUATION 13 condiion for he scalar case. I is immediae ha shock soluions saisfying (1.0.7) verify he Lax enropy condiion. Lemma 3.2 says ha shock soluions o he scalar conservaion law (1.0.1) endowed wih he enropy η, also verify he Lax enropy condiion. Proof. We only consider he case when u L u R. Le (3.1.3) Λ := q(u R ) q(u L ) σ(η(u R ) η(u L )) = We wan o show Λ 0 if and only if u L > u R. We can wrie (3.1.4) Thus (3.1.5) Λ = (3.1.6) (3.1.7) (3.1.8) u R u L u R = u L σ = A(u R) A(u L ) u R u L = ( η (u) A 1 (u) u R u L ( η 1 (u) u R u L u R u L u R u L 1 u R u L ) A (v) dv du ) A (u) A (v) dv du u R u R 1 = η (u)(a (u) A (v)) dvdu u R u L u L u L ( 1 1 = η (u)(a (u) A (v)) dvdu 1 u R u L 2 2 D u R u L u R u L A (u)η (u) ση (u) du. A (v) dv. D ) η (v)(a (u) A (v)) dvdu where D := I(u L, u R ) I(u L, u R ). We use I(a, b) o denoe he inerval wih endpoins a and b. Finally, Λ = 1 1 (3.1.9) (η (u) η (v))(a (u) A (v)) dvdu. 2 u R u L Then, by sric convexiy of A and η, he quaniy (3.1.10) (η (u) η (v))(a (u) A (v)) dvdu D D is always posiive. Thus, he sign of Λ is given by he sign of u R u L. This complees he proof.

14 14 KRUPA AND VASSEUR Lemma 3.3 (Srucural lemma from [27]). Le u L, u R, u, u + R saisfy u L > u R and u > u +. Le η C 2 (R) be a sricly convex enropy, wih associaed enropy flux q. Define (3.1.11) Then, (3.1.12) Moreover, if (3.1.13) for some u R, hen (3.1.14) σ(u, u + ) := A(u +) A(u ) u + u. q(u + ; u R ) q(u ; u L ) σ(u, u + )(η(u + u R ) η(u u L )) 0. η(u u L ) = η(u u R ) q(u; u R ) q(u; u L ) < 0. Lemma 3.3 was proven in [27]. For compleeness, we give he proof of his lemma in he appendix (Secion 6.1). Lemma 3.4 (Srucural lemma). Le η C 2 (R) be a sricly convex enropy for (1.0.1). Le q be he associaed enropy flux. For ɛ > 0, define he funcion V ɛ : {(u, u L, u R ) R 3 u L u R } R: [q(u;u R ) q(u;u L ) ɛ] + V ɛ (u, u L, u R ) := η(u u R ) η(u u L ) if η(u u R ) η(u u L ) (3.1.15) 0 if η(u u R ) = η(u u L ), where [ ] + := max(0, ). Then, V ɛ verifies (3.1.16) V ɛ (u, u L, u R ) A (u). Proof. Le [q(u;u R ) q(u;u L )] + V (u, u L, u R ) := η(u u R ) η(u u L ) if η(u u R ) η(u u L ) 0 if η(u u R ) = η(u u L ). In order o show (3.1.16), he idea is o show (3.1.17) V (u, u L, u R ) A (u) and hen use he basic inequaliy V ɛ (u, u L, u R ) V (u, u L, u R ). The proof of (3.1.17) depends on conrolling he quaniy (3.1.18) q(u; u R ) q(u; u L ) η(u u R ) η(u u L ) for he (u, u L, u R ) values ha make q(u; u R ) q(u; u L ) 0.

15 SINGLE ENTROPY CONDITION FOR BURGERS EQUATION 15 We have hree cases where we ge conrol on he quaniy (3.1.18): u > u L > u R, u L > u R > u, and u R u u L. We begin wih some elemenary facs which will be used repeaedly. Remark ha (3.1.19) b η(a b) = η (b)(b a) and b q(a; b) = η (b)(a(b) A(a)). In paricular, we can wrie for all a, b, c R, (3.1.20) q(a; b) q(a; c) = b c η (v)(a(v) A(a)) dv = We can now begin he casework o prove Lemma 3.4. Case u > u L > u R We noe ha for all u, u L, u R R such ha u > u L > u R, (3.1.21) η(u u R ) η(u u L ) = by sric convexiy of η. We conclude (3.1.22) u R u L v η(u v) dv = η(u u R ) η(u u L ) > 0. b c u R η A(v) A(a) (v)(v a) dv. v a u L η (v)(v u) dv > 0 For all u, u L, u R R such ha u > u L > u R and η(u u R ) η(u u L ) 0, we compue (3.1.23) q(u; u R ) q(u; u L ) η(u u R ) η(u u L ) = u R η (v)(v u) A(v) A(u) v u u L η(u u R ) η(u u L ) By he mean value heorem and sric convexiy of A, A(v) A(u) v u < A (u) for v [u R, u L ]. Thus, recalling also he sric convexiy of η and (3.1.22), we coninue from (3.1.23) o ge A u R (u) η (v)(v u) dv u (3.1.24) < L = A (u). η(u u R ) η(u u L ) In summary, (3.1.25) q(u; u R ) q(u; u L ) η(u u R ) η(u u L ) < A (u). Case u L > u R > u Our mehod is analogous o he case u > u L > u R above. Remark ha for all u, u L, u R R such ha u L > u R > u, (3.1.26) η(u u R ) η(u u L ) = u R u L v η(u v) dv = u R dv u L η (v)(v u) dv < 0

16 16 KRUPA AND VASSEUR by sric convexiy of η. We conclude (3.1.27) η(u u R ) η(u u L ) < 0. For all u, u L, u R R such ha u L > u R > u and η(u u R ) η(u u L ) 0, we calculae (3.1.28) q(u; u R ) q(u; u L ) η(u u R ) η(u u L ) = u R η (v)(v u) A(v) A(u) v u u L η(u u R ) η(u u L ) By he mean value heorem and sric convexiy of A, A(v) A(u) v u > A (u) for v [u R, u L ]. Thus, recalling also he sric convexiy of η and (3.1.27), we coninue from (3.1.28) o ge A u R (u) η (v)(v u) dv u (3.1.29) > L = A (u). η(u u R ) η(u u L ) To summarize, dv (3.1.30) q(u; u R ) q(u; u L ) η(u u R ) η(u u L ) > A (u). Case u R u u L This case is slighly differen han he wo cases above. For all u, u L, u R R such ha u R u u L and η(u u R ) η(u u L ) 0, (3.1.31) (3.1.32) q(u; u R ) q(u; u L ) η(u ur ) η(u u L ) = u R η (v)(v u) A(v) A(u) v u u L η(u ur ) η(u u L ) dv u η (v)(v u) A(v) A(u) u R v u dv + η (v)(v u) A(v) A(u) v u dv u = L u η(u u R ) η(u u L ) By he mean value heorem and sric convexiy of A, A(v) A(u) v u > A (u) for v (u, u L ], and A(v) A(u) v u < A (u) for v [u R, u). Thus, recalling also he sric convexiy of η, u A (u) η (v)(v u) dv + A u R (u) η (v)(v u) dv u < L u (3.1.33) η(u u R ) η(u u L ) (3.1.34) We receive, (3.1.35) = A (u) sgn(η(u u R ) η(u u L )). q(u; u R ) q(u; u L ) η(u u R ) η(u u L ) < A (u) sgn(η(u u R ) η(u u L )).

17 SINGLE ENTROPY CONDITION FOR BURGERS EQUATION 17 We combine all he cases u > u L > u R, u L > u R > u, and u R u u L, and in paricular (3.1.22), (3.1.25), (3.1.27), (3.1.30), and (3.1.35). We keep in mind ha we only consider (u, u L, u R ) values ha make q(u; u R ) q(u; u L ) 0. In conclusion, V ɛ (u, u L, u R ) V (u, u L, u R ) A (u) Proof of Proposiion 3.1. This proof is based on he work [27, p. 7-8]. We modify he proof o consider ū i which are non-consan. Define V ɛ : {(u, u L, u R ) R 3 u L u R } R: [q(u;u R ) q(u;u L ) ɛ] + V ɛ (u, u L, u R ) := η(u u R ) η(u u L ) if η(u u R ) η(u u L ) (3.2.1) 0 if η(u u R ) = η(u u L ), where [ ] + := max(0, ). The funcion V ɛ is coninuous. For (u, u L, u R ) such ha η(u u R ) η(u u L ), V ɛ is clearly coninuous. By (3.1.14), V ɛ = 0 on some neighborhood of {(u, u L, u R ) R 3 η(u u R ) = η(u u L )}. Thus, V ɛ is coninuous. Remark ha by par (a) of Lemma 2.1, he ū i are classical soluions, and by par (f) of Lemma 2.1, ū 1 (x, ) ū 2 (x, ) for all (x, ) R [0, ). We consruc a soluion o {ḣɛ = V ɛ (u(h ɛ (), ), ū 1 (h ɛ (), ), ū 2 (h ɛ (), )) (3.2.2) h ɛ ( ) = x 0 in he Filippov sense. We use he following lemma: Lemma 3.5. There exiss a Lipschiz funcion h ɛ : [, ) R such ha (3.2.3) (3.2.4) h ɛ ( ) = x 0, ḣɛ V ɛ L L, and (3.2.5) ḣ ɛ () I[V ɛ (u, ū 1, ū 2 ), V ɛ (u +, ū 1, ū 2 )], for almos every > 0, where u ± = u(h ɛ ()±, ), and ū i = ū i (h ɛ (), ) for i = 1, 2. Here, I[a, b] denoes he closed inerval wih endpoins a and b. Moreover, for almos every > 0, (3.2.6) (3.2.7) A(u + ) A(u ) = ḣɛ(u + u ), q(u + ) q(u ) ḣɛ(η(u + ) η(u )), which means ha for almos every > 0, eiher he shock (u, u +, ḣɛ) is an enropic disconinuiy for he enropy η or u = u +.

18 18 KRUPA AND VASSEUR The proof of (3.2.3), (3.2.4) and (3.2.5) is nearly idenical o he proof of Proposiion 1 in [21]. For compleeness, we provide a proof of (3.2.3), (3.2.4) and (3.2.5) in he appendix (Secion 6.2). The properies (3.2.6) and (3.2.7) in fac hold for any Lipschiz funcion h : [0, ) R. These properies are well-known in he BV case. When u only saisfies he srong race propery (Definiion 1.1), (3.2.6) and (3.2.7) are given in Lemma 6 in [21]. We do no include a proof of (3.2.6) and (3.2.7) here; a proof is given in he appendix in [21]. We are now ready o prove Proposiion 3.1. Proof of Proposiion 3.1. For almos every such ha u = u +, we borrow noaion from [27] and wrie u ± := u = u +, and hen by Lemma 3.5 we have ḣɛ = V ɛ (u ±, ū 1, ū 2 ). This gives, (3.2.8) q(u ± ; ū 2 ) q(u ± ; ū 1 ) ḣɛ(η(u ± ū 2 ) η(u ± ū 1 )) ɛ. For almos every such ha u u +, Lemma 3.5 says ḣɛ = σ(u, u + ), where σ(u, u + ) = (A(u + ) A(u ))/(u + u ). Then, (3.2.9) q(u + ; ū 2 ) q(u ; ū 1 ) ḣɛ(η(u + ū 2 ) η(u ū 1 )) = (3.2.10) q(u + ; u R ) q(u ; u L ) σ(u, u + )(η(u + u R ) η(u u L )), and hen by (3.1.12), Lemma 3.2, and Lemma 3.5 again, (3.2.11) 0. Thus, for almos every [, ) (3.2.12) q(u + ; ū 2 ) q(u ; ū 1 ) ḣɛ(η(u + ū 2 ) η(u ū 1 )) ɛ. Finally, by Lemma 3.4 and Lemma 3.5, (3.2.13) Lip[h ɛ ] sup A. x u L The proof of Theorem 1.2 will follow from 4. Main proposiion Proposiion 4.1 (Main proposiion). Le u L (R [0, )) be a weak soluion o (1.0.1), wih iniial daa u 0. Le η C 2 (R) be a sricly convex enropy. Assume ha u is enropic for he enropy η and verifies he srong race propery (Definiion 1.1). Then for all R, T, ɛ > 0, here exiss a funcion ψ : [ R, R] R verifying: (a) x R η(u(x, T ) ψ(x)) dx ɛ.

19 SINGLE ENTROPY CONDITION FOR BURGERS EQUATION 19 (b) ψ(x + z) ψ(x) c T z (c) for x [ R, R] and z > 0 wih x + z [ R, R] and where c = 1/ inf A. ψ L ([ R,R]) u L (R [0, )). We decompose he proof of Proposiion 4.1 ino wo lemmas. We uilize funcions of he form (4.0.1) For v L (R), here exiss a finie se of x i wih = x 0 < x 1 < x 2 < < x N < x N+1 = such ha v is nondecreasing and Lipschiz coninuous on (x i, x i+1 ) for i = 0,..., N, and lim v(x) x x i x xi lim v(x) x<x i x>x i for 1 i N. Lemma 4.2. Le R, T > 0. Le u L (R [0, )) be a weak soluion o (1.0.1) wih iniial daa u 0. Assume u is enropic for a leas one sricly convex enropy η C 2 (R). Assume ha u verifies he srong race propery (Definiion 1.1). Choose s > 0 o verify q(a; b) sη(a b) for all a, b [ B, B], where B is defined in (2.0.1). Then for all N N {0}, we have: For any v1 0,..., v0 N+1 L (R) Lipschiz coninuous nondecreasing funcions verifying vi 0(x) v0 i+1 (x) for all 1 i N and x R, and for any [0, T ] and real numbers x 0,..., x N+1 verifying x 0 = R + ( T )s < x 1 < < x N < R ( T )s = x N+1 he following holds: There exis N + 2 real numbers x 0,T,..., x N+1,T such ha x 0,T = R x 1,T x N,T R = x N+1,T and (4.0.2) ap lim T + N x i+1,t η(u(x, ) v i+1 (x, )) dx ap lim x + N x i+1 x i η(u(x, ) v i+1 (x, )) dx where v i is he unique soluion o (1.0.1) wih iniial daa v 0 i and verifying (1.0.7). Lemma 4.3 (Densiy of funcions of form (4.0.1) in L 2 ). Le M, ɛ > 0. Then for all f L 2 ([ M, M]), here is a funcion f ɛ : R R of he form (4.0.1) such ha (4.0.3) f f ɛ L 2 ([ M,M]) < ɛ, (4.0.4) f ɛ L (R) f L ([ M,M]), and all of he disconinuiies of f ɛ are conained in ( M, M).

20 20 KRUPA AND VASSEUR 4.1. Lemma 4.2 and Lemma 4.3 imply main proposiion. As in Lemma 4.2, we choose s > 0 such ha q(a; b) sη(a b) for all a, b [ B, B], where B is defined in (2.0.1). We also choose c > 0 such ha (4.1.1) η(a b) c (a b) 2 for all a, b [ B, B]. By Lemma 4.3, here exiss a funcion v 0 L (R) of he form (4.0.1) such ha u 0 v 0 L ɛ (4.1.2) < 2 ([ R st,r+st ]) c and if here is a leas one disconinuiy in v 0, he disconinuiies are a poins x 1 < x 2 < < x N for some N N, and where x i ( R st, R + st ) for all 1 i N. If v 0 conains a leas one disconinuiy, define he funcions vi 0 : R R for 1 i N +1 as follows: { v1(x) 0 v 0 (x) if x < x 1 := sup x1 <y<x max(v 0 (x 1 ), v 0 (y)) if x 1 < x For 2 i N, inf x<y<xi 1 min(v 0 (x i 1 +), v vi 0 i 1 0 (y)) (x) := v 0 (x) sup xi <y<x max(v 0 (x i ), v 0 (y)) And v 0 N+1(x) := if x < x i 1 if x i 1 < x < x i if x i < x { inf x<y<xn min(v 0 (x N +), vn 0 (y)) if x < x N v 0 (x) if x N < x. If v 0 has no disconinuiies, hen N = 0 and we define v1 0 := v0. By consrucion, he vi 0 are Lipschiz coninuous, nondecreasing in x, and verify v0 i (x) vi+1 0 (x) for all x R and 1 i N. We also have v 0 (4.1.3) L (R) v 0 i L (R) for 1 i N + 1. Le v i denoe he unique soluion o (1.0.1) wih iniial daa vi 0 and which saisfies (1.0.7). From Lemma 4.2, we have N + 2 real numbers x 0,T,..., x N+1,T such ha x 0,T = R x 1,T x N,T R = x N+1,T and (4.1.4) ap lim T + N x i+1,t x i,t η(u(x, ) v i+1 (x, )) dx ap lim 0 + N x i+1 x i η(u(x, ) v i+1 (x, )) dx

21 SINGLE ENTROPY CONDITION FOR BURGERS EQUATION 21 where x 0 := R st and x N+1 := R + st. We now conrol he righ-hand side of (4.1.4). Recalling Lemma 2.4, we have (4.1.5) x N i+1 x N i+1 ap lim η(u(x, ) v i+1 (x, )) dx η(u 0 (x) vi+1(x)) 0 dx 0 + x i Then, from he definiion of he v 0 i, = x i R+sT R st η(u 0 (x) v 0 (x)) dx Using (4.1.1), (4.0.4), and ha u 0 L (R) u L (R [0, )), Then, from (4.1.2) c < ɛ. On he oher hand, by he convexiy of η, (4.1.6) N x i+1,t x i,t R+sT R st η(u(x, T ) v i+1 (x, T )) dx ap lim T + Combining (4.1.4), (4.1.5) and (4.1.6), we find (4.1.7) N We define ψ : [ R, R] R, (4.1.8) x i+1,t (u 0 (x) v 0 (x)) 2 dx N x i+1,t x i,t x i,t η(u(x, T ) v i+1 (x, T )) dx < ɛ. v 1 (x, T ) if R < x < x 1,T v 2 (x, T ) if x 1,T < x < x 2,T ψ(x) :=. v N+1 (x, T ) if x N,T < x < R. η(u(x, ) v i+1 (x, )) dx. By (4.1.7), ψ saisfies par (a) of Proposiion 4.1. We now show ψ saisfies par (b) of Proposiion 4.1: his follows because each of he v i saisfy (1.0.7). In paricular, (4.1.9) v i (x + z, T ) v i (x, T ) C T z

22 22 KRUPA AND VASSEUR for all z > 0 and all x R. From par (c) of Lemma 2.1, we can ake C = 1/ inf A in (4.1.9). Furher, by par (f) of Lemma 2.1, v i (x, T ) v i+1 (x, T ) for all x R and 1 i N. This gives par (b) of Proposiion 4.1. Par (c) of Proposiion 4.1 follows from (4.0.4), (4.1.3), and par (e) of Lemma 2.1. This complees he proof of Proposiion Proof of Lemma 4.2. We prove his lemma by srong inducion on N. Base case For N = 0, le v1 0 L (R) be any Lipschiz coninuous nondecreasing funcion. Le v 1 be he unique soluion o (1.0.1) wih iniial daa v1 0 and verifying (1.0.7). Le h 0 () := R+( T )s and h 1 () := R ( T )s. Then, from Lemma 2.3, Lemma 2.4, and he dominaed convergence heorem, (4.2.1) ap lim T + = + T h 1 (T ) h 0 (T ) η(u(x, ) v 1 (x, )) dx ap lim h 1 ( ) + h 0 ( ) η(u(x, ) v 1 (x, )) dx [ q(u(h 0 ()+, ); v 1 (h 0 (), )) q(u(h 1 (), ); v 1 (h 1 (), )) ] + ḣ1()η(u(h 1 (), ) v 1 (h 1 (), )) ḣ0()η(u(h 0 ()+, ) v 1 (h 0 (), )) d T T 0 by he definiion of h 0, h 2 and s. We ge, (4.2.2) ap lim q(u(h 0 ()+, ); v 1 (h 0 (), )) sη(u(h 0 ()+, ) v 1 (h 0 (), )) d q(u(h 1 (), ); v 1 (h 1 (), )) sη(u(h 1 (), ) v 1 (h 1 (), )) d R T + R η(u(x, ) v 1 (x, )) dx ap lim R ( T )s + R+( T )s η(u(x, ) v 1 (x, )) dx. Thus Lemma 4.2 holds for he base case N = 0. Inducion sep Suppose ha K N {0} is given such ha Lemma 4.2 holds for N = 0, 1,..., K. We now prove ha Lemma 4.2 holds for N = K + 1. Le v1 0, v 2, 0..., vk+2 0 L (R) be any Lipschiz coninuous nondecreasing funcions, saisfying vi 0(x) v0 i+1 (x) for all 1 i K + 1 and x R. For 1 i K + 2, le v i be he unique soluion o (1.0.1)

23 SINGLE ENTROPY CONDITION FOR BURGERS EQUATION 23 wih iniial daa v 0 i and verifying (1.0.7). Le [0, T ] and R + ( T )s < x 1 < < x K+1 < R ( T )s be arbirary. Le ɛ > 0. By Proposiion 3.1 we can consruc Lipschiz coninuous funcions h ɛ,1,..., h ɛ,k+1 on he inerval [, T ] such ha for 1 i K + 1, h ɛ,i ( ) = x i and (4.2.3) q(u i +; v i+1 ) q(u i ; v i ) ḣɛ,i(η(u i + v i+1 ) η(u i v i )) ɛ T (K + 1) for almos every [, T ], where u i ± = u(h ɛ,i ()±, ) and v l = v l (h ɛ,i (), ) for l = i, i + 1. To simplify he exposiion, denoe (4.2.4) (4.2.5) h ɛ,0 () := R + ( T )s, h ɛ,k+2 () := R ( T )s. The Lipschiz consans of he h ɛ,i are uniformly bounded in ɛ by Proposiion 3.1. Thus, by Arzelà Ascoli here exiss a sequence {ɛ j } j N such ha ɛ j 0 + and for each 0 i K + 2, h ɛj,i converges uniformly on [, T ] o a Lipschiz funcion h i. Then, le be he firs ime ha here exis wo of he h i for 0 i K + 2 ha are equal o each oher. If such a ime does no exis, le := T. We now show: Claim. (4.2.6) h K+1 i+1 ( ) ap lim + h i ( ) η(u(x, ) v i+1 (x, )) dx ap h K+1 i+1 ( ) lim η(u(x, ) v i+1 (x, )) dx. + h i ( ) Proof of Claim. Due o he uniform convergence of he h ɛj,i as j, for each τ [, ), here exiss J τ > 0 large enough such ha h ɛj,i+1() h ɛj,i() > 0 for all [, τ], j > J τ and 0 i K + 1. Then, for almos every and τ verifying < τ <, we have from Lemma 2.3 (for j > J τ ): K+1 K+1 h ɛj,i+1(τ) h ɛj,i(τ) τ η(u(x, τ) v i+1 (x, τ)) dx K+1 h ɛj,i+1() h ɛj,i() η(u(x, ) v i+1 (x, )) dx [ q(u(h ɛj,i(r)+, r); v i+1 (h ɛj,i(r), r)) q(u(h ɛj,i+1(r), r); v i+1 (h ɛj,i+1(r), r)) + ḣɛ j,i+1(r)η(u(h ɛj,i+1(r), r) v i+1 (h ɛj,i+1(r), r)) ] ḣɛ j,i(r)η(u(h ɛj,i(r)+, r) v i+1 (h ɛj,i(r), r)) dr

24 24 KRUPA AND VASSEUR Then, we collec he erms corresponding o h ɛj,i ino one sum, and he erms corresponding o h ɛj,i+1 ino anoher sum, = K+1 τ [ ] q(u(h ɛj,i(r)+, r); v i+1 (h ɛj,i(r), r)) ḣɛ j,i(r)η(u(h ɛj,i(r)+, r) v i+1 (h ɛj,i(r), r)) dr + K+1 τ [ q(u(h ɛj,i+1(r), r); v i+1 (h ɛj,i+1(r), r)) ] + ḣɛ j,i+1(r)η(u(h ɛj,i+1(r), r) v i+1 (h ɛj,i+1(r), r)) dr Nex, we peel off he i = 0 erm from he firs sum, and he i = K + 1 erm from he second sum, τ [ ] = q(u(h ɛj,0(r)+, r); v 1 (h ɛj,0(r), r)) ḣɛ j,0(r)η(u(h ɛj,0(r)+, r) v 1 (h ɛj,0(r), r)) dr + K+1 i=1 τ [ ] q(u(h ɛj,i(r)+, r); v i+1 (h ɛj,i(r), r)) ḣɛ j,i(r)η(u(h ɛj,i(r)+, r) v i+1 (h ɛj,i(r), r)) dr + K τ [ q(u(h ɛj,i+1(r), r); v i+1 (h ɛj,i+1(r), r)) ] + ḣɛ j,i+1(r)η(u(h ɛj,i+1(r), r) v i+1 (h ɛj,i+1(r), r)) dr τ + [ q(u(h ɛj,k+2(r), r); v K+2 (h ɛj,k+2(r), r)) ] + ḣɛ j,k+2(r)η(u(h ɛj,k+2(r), r) v K+2 (h ɛj,k+2(r), r)) dr We hen reindex he second sum K firs sum K+1 τ i=1 [ ] dr, = + τ K+1 i=1 τ [ ] dr o sar a i = 1, and combine i wih he [ ] q(u(h ɛj,0(r)+, r); v 1 (h ɛj,0(r), r)) ḣɛ j,0(r)η(u(h ɛj,0(r)+, r) v 1 (h ɛj,0(r), r)) dr τ [ q(u(h ɛj,i(r)+, r); v i+1 (h ɛj,i(r), r)) q(u(h ɛj,i(r), r); v i (h ɛj,i(r), r))

25 SINGLE ENTROPY CONDITION FOR BURGERS EQUATION 25 ] ḣɛ j,i(r)(η(u(h ɛj,i(r)+, r) v i+1 (h ɛj,i(r), r)) η(u(h ɛj,i(r), r) v i (h ɛj,i(r), r))) dr τ + [ q(u(h ɛj,k+2(r), r); v K+2 (h ɛj,k+2(r), r)) ] + ḣɛ j,k+2(r)η(u(h ɛj,k+2(r), r) v K+2 (h ɛj,k+2(r), r)) dr ɛ j T (K + 1) (τ )(K + 1) < ɛ j by (4.2.3), he definiion of s, and noing ha ḣɛ j,0 = s and ḣɛ j,k+2 = s. Thus, (4.2.7) K+1 K+1 h ɛj,i+1(τ) h ɛj,i(τ) h ɛj,i+1() h ɛj,i() η(u(x, τ) v i+1 (x, τ)) dx < η(u(x, ) v i+1 (x, )) dx + ɛ j. Then, le j in (4.2.7) and use he dominaed convergence heorem o ge (4.2.8) K+1 h i+1 (τ) h i (τ) η(u(x, τ) v i+1 (x, τ)) dx K+1 h i+1 () h i () η(u(x, ) v i+1 (x, )) dx for almos every and τ verifying < τ <. From (4.2.8), we ge (4.2.9) (4.2.10) where we have used ap h K+1 i+1 ( ) lim η(u(x, τ) v i+1 (x, τ)) dx τ ap h i ( ) h K+1 i+1 ( ) lim η(u(x, ) v i+1 (x, )) dx + h i ( ) (4.2.11) ap h K+1 i+1 (τ) lim η(u(x, τ) v i+1 (x, τ)) dx = τ h i (τ)

26 26 KRUPA AND VASSEUR (4.2.12) and (4.2.13) h K+1 i+1 () ap lim + h i () ap h K+1 i+1 ( ) lim η(u(x, τ) v i+1 (x, τ)) dx τ h i ( ) η(u(x, ) v i+1 (x, )) dx = ap h K+1 i+1 ( ) lim η(u(x, ) v i+1 (x, )) dx. + h i ( ) The approximae limis exis by Lemma 2.4. Then (4.2.9) and (2.0.22) give he claim (4.2.6). If = T, hen we have proven Lemma 4.2 holds for N = K +1: define x i,t := h i (T ) for 0 i K + 2. By (4.2.6), he x 0,T, x 1,T,..., x K+2,T saisfy he conclusions of Lemma 4.2. Oherwise, < T and we consider he 0 i, j K + 3 such ha he following holds: { h i 1 ( ) < h i ( ) < h j ( ) (4.2.14) for i < k < j, h i ( ) = h k ( ) where h 1 () := and h K+3 () := + for all. Then, le {(i n, j n )} n {1,...,L} for L Z + be he se of i and j pairs which saisfy (4.2.14) (label he i and j pairs such ha i n < i n+1 for all n). Noe ha each i has only one corresponding j. Thus, due o a leas wo of he h i ( ) equalling each oher (for i ranging over 0,..., K + 2), L K + 2. By he inducion hypohesis wih N = L 2: here exis real numbers x 0,T,..., x L 1,T verifying x 0,T = R x 1,T x L 2,T R = x L 1,T and (4.2.15) ap L 2 ap lim T + l=0 L 2 lim + x l+1,t x l,t h il+2 ( ) l=0 h il+1 ( ) η(u(x, ) v il+2 (x, )) dx η(u(x, ) v il+2 (x, )) dx. For each n {1,..., L}, define (4.2.16) x i,t := x n 1,T for all i n i < j n. Then by consrucion x 0,T,..., x K+2,T saisfy he conclusions of Lemma 4.2 wih N = K + 1. In paricular, (4.0.2) follows from (4.2.6) and (4.2.15). Thus, by he principle of srong inducion we have proven Lemma 4.2 for all N N {0} Proof of Lemma 4.3. Sep funcions are dense in L 2 ([ M, M]). Thus, here exiss a sep funcion s L 2 ([ M, M]) such ha f s L 2 ([ M,M]) < ɛ 2 and s L ([ M,M])

27 SINGLE ENTROPY CONDITION FOR BURGERS EQUATION 27 f L ([ M,M]). We can wrie s in he form (4.3.1) ( n + ) ( n s(x) = α i + H(x x+ i ) ) + αi H(x x i ) i=1 for some n +, n N, {α i + }n + i=1 (0, ), {αi }n i=1 (, 0), {x + i }n + i=1 ( M, M) and {x i }n i=1 ( M, M). H is he Heaviside sep funcion { 0 if x < 0 (4.3.2) H(x) := 1 if x > 0. i=1 Define (4.3.3) (4.3.4) s + (x) := s (x) := n + i=1 n i=1 α + i H(x x+ i ), α i H(x x i ). We can hen wrie s = s + + s. Consider he sandard mollifier m : R R, where m is smooh and compacly suppored, m 0, and m(x) dx = 1. Le δ > 0. Define m δ (x) := 1 ( ) x (4.3.5) δ m. δ Define (4.3.6) lim m y M +[(s+ δ )(y) + s (y)] if x M f δ (x) := (s + m δ )(x) + s (x) if M < x < M lim m y M [(s+ δ )(y) + s (y)] if x M. Noe ha f δ is of he form (4.0.1), f δ L (R) f L ([ M,M]) when δ < inf i,j x + i x j, and all of he disconinuiies in f δ are in he inerval ( M, M) because {x i }n i=1 ( M, M). From he Minkowski inequaliy, (4.3.7) f f δ L 2 ([ M,M]) f s L 2 ([ M,M]) + s f δ L 2 ([ M,M]) < ɛ 2 + (s + m δ ) s + L 2 ([ M,M]). Choose δ even smaller such ha (s + m δ ) s + L 2 ([ M,M]) < ɛ 2. This complees he proof.

28 28 KRUPA AND VASSEUR 5. Main proposiion implies main heorem Le u L (R [0, )) be a weak soluion o (1.0.1), wih iniial daa u 0. Le η C 2 (R) be a sricly convex enropy. Assume ha u is enropic for he enropy η and verifies he srong race propery (Definiion 1.1). We will show ha u saisfies (1.0.7). Firs, by sric convexiy of η we can choose a consan c > 0 such ha (5.0.1) c (a b) 2 η(a b) for all a, b [ B, B], where B is defined in (2.0.1). Le R, T > 0. From Proposiion 4.1, for all ɛ > 0, here exiss ψ ɛ : [ R, R] R such ha (5.0.2) η(u(x, T ) ψ ɛ (x)) dx c ɛ 2 and (5.0.3) x R ψ ɛ (x + z) ψ ɛ (x) c T z for x [ R, R] and z > 0 wih x + z [ R, R]. Here c = 1/ inf A. We have also (5.0.4) ψ ɛ L ([ R,R]) u L (R [0, )). By (5.0.4), ψ ɛ L ([ R,R]) B. Likewise, we have u(, T ) L (R) (5.0.1), we ge (5.0.5) c (ψ ɛ (x) u(x, T )) 2 η(u(x, T ) ψ ɛ (x)) for almos every x [ R, R]. Then from (5.0.2) and (5.0.5) we have (5.0.6) ψ ɛ ( ) u(, T ) L 2 ([ R,R]) ɛ. B. Thus, from Thus, here exiss a sequence {ɛ j } j=1 wih ɛ j 0 + such ha (5.0.7) ψ ɛj (x) u(x, T ) as j for almos every x. Addiionally, from (5.0.3): (5.0.8) ψ ɛj (x + z) ψ ɛj (x) c T z for all j N, x [ R, R], and z > 0 wih x + z [ R, R]. We le j in (5.0.8) o ge: (5.0.9) u(x + z, T ) u(x, T ) c T z for almos every x [ R, R], and almos every z > 0 wih x + z [ R, R]. Because R, T > 0 are arbirary, (5.0.9) implies ha u saisfies (1.0.7). This concludes he proof of Theorem 1.2.

29 SINGLE ENTROPY CONDITION FOR BURGERS EQUATION Appendix 6.1. Proof of Lemma 3.3. Le u L, u R, u, u + R. We hen borrow he following noaion from [27]: If F is a funcion of u, hen we define (6.1.1) F R := F (u R ), F L := F (u L ), [F ] := F R F L, F ± := F (u ± ). Proof of (3.1.12) This proof is from [27, p. 9-10]. In [27], for he general sysems case, he auhors develop a condiion which hey label wih he equaion number 7 [27, p. 4]. In paricular, hey claim o use his condiion o show (3.1.12) for he scalar case. In fac, he condiion is no necessary in he scalar case and heir proof goes hrough unchanged wihou he condiion. Denoe (6.1.2) D := q(u + ; u R ) q(u ; u L ) σ(u, u + )(η(u + u R ) η(u u L )). Furher, le σ denoe σ(u, u + ). From Rankine-Hugonio (as noed in [27, p. 5]), (6.1.3) D = [η (u)a(u) q(u)] σ[η (u)u η] + q + q σ(η + η ) [η ](A(u) σu) ±, where (A(u) σu) ± denoes ha (6.1.4) A(u + ) σu + = A(u ) σu because of he Rankine-Hugonio relaion (3.1.11). From he fundamenal heorem of calculus and inegraion by pars, (6.1.5) (6.1.6) We can hen wrie (6.1.7) η (u) du(a(u) σu) ± u u D = η (u)(a(u) σu) du u + u + u L u L η (u)(a(u) σu) du + η (u) du(a(u) σu) ±. u R u R where I, J are disjoin inervals such ha (6.1.8) D = ɛ(i)b(i) + ɛ(j)b(j) I J = ((u +, u ) (u R, u L )) \ ((u +, u ) (u R, u L )). We define he sign ɛ(i) o be +1 if I (u +, u ) and 1 oherwise. Finally, (6.1.9) B(I) := η (u)(a(u) σu) du (A(u) σu) ± η (u) du. I The funcion u A(u) σu is sricly convex and (6.1.4) holds. Thus, B(I) < 0 if I (u +, u ) and posiive oherwise. Thus, for all inervals ɛ(i)b(i) < 0. Thus D < 0. Proof ha (3.1.13) implies (3.1.14) I

30 30 KRUPA AND VASSEUR This proof is from [27, p. 9]. Remark ha he equaliy (6.1.10) is equivalen o [27, p. 4] (6.1.11) Remark also (as noed in [27, p. 4]) (6.1.12) η(u u L ) = η(u u R ) [η ]u = [η (u)u η(u)]. q(u; u R ) q(u; u L ) = [η A q] [η ]A(u). Then, (3.1.14) is equivalen o (as noed in [27, p. 9]) (6.1.13) u L u R η (u)a(u) du u L u R η (u) du > A ( u L u R η (u)u du u L u R η (u) du Finally, (6.1.13) is rue by Jensen s inequaliy because η > 0 so η (u) du is a measure, and A is sricly convex Proof of Lemma 3.5. The following proof of (3.2.3), (3.2.4) and (3.2.5) is based on he proof of Proposiion 1 in [21] and he proof of Lemma 2.2 in [27]. We do no prove (3.2.6) and (3.2.7): hese properies are in Lemma 6 in [21], and heir proofs are in he appendix in [21]. Define (6.2.1) v n (x, ) := 1 0 V ɛ (u(x + y n, ), ū 1(x + y n, ), ū 2(x + y, )) dy. n Le h ɛ,n be he soluion o he ODE: {ḣɛ,n () = v n (h ɛ,n (), ), for > 0 (6.2.2) h ɛ,n ( ) = x 0. Due o V ɛ being coninuous, v n is bounded uniformly in n ( v n L V ɛ L ), Lipschiz coninuous in x, and measurable in. Thus (6.2.2) has a unique soluion in he sense of Carahéodory. Because v n is bounded uniformly in n, he h ɛ,n are Lipschiz coninuous in ime, wih he Lipschiz consan uniform in n. Thus, by Arzelà Ascoli he h ɛ,n converge in C 0 (0, T ) for any fixed T > 0 o a Lipschiz coninuous funcion h ɛ (passing o a subsequence if necessary). Noe ha ḣɛ,n converges in L weak* o ḣɛ. We define (6.2.3) (6.2.4) V max () := max{v ɛ (u, ū 1, ū 2 ), V ɛ (u +, ū 1, ū 2 )}, V min () := min{v ɛ (u, ū 1, ū 2 ), V ɛ (u +, ū 1, ū 2 )}, where u ± = u(h ɛ ()±, ) and ū i = ū i (h ɛ (), ). ).