A Semigroup Approach to an Integro-Differential Equation Modeling Slow Erosion

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1 A Semigrop Approch to n Integro-Differentil Eqtion Modeling Slow Erosion Alberto Bressn nd Wen Shen Deprtment of Mthemtics, Penn Stte University, University Prk, PA 16802, USA E-mils: bressn@mthpsed, shen w@mthpsed Jnry 19, 2014 Abstrct The pper is concerned with sclr conservtion lw with nonlocl flx, providing model for grnlr flow with slow erosion nd deposition While the soltion t, x cn hve jmps, the inverse fnction x xt, is lwys Lipschitz continos; its derivtive hs bonded vrition nd stisfies blnce lw with mesre-vled sorces Using bckwrd Eler pproximtion scheme combined with nonliner projection opertor, we constrct continos semigrop whose trjectories re the niqe entropy wek soltions to this blnce lw Going bck to the originl vribles, this yields the globl well-posedness of the Cchy problem for the grnlr flow model 1 Introdction In this pper we stdy the sclr conservtion lw with nonlocl flx t t, x exp f x t, y dy 0, 0, x ūx 11 x Here x IR is the spce vrible, nd one cn think of t, the height of stnding profile of snd or some other grnlr mteril We ssme tht x t, x is strictly incresing, with x 1 s x ± In this model, the vrible t shold not be thoght s the sl time on the clock Rther, t mesres the totl mont of snd pored from the top, ie t x + As it slides downwrd, this thin moving lyer of snd will pt frther snd into motion, t points where the slope is x > 1 On the other hnd, if the slope is x < 1, prt of the moving lyer will be deposited nd become prt of the stnding profile To nderstnd the mening of the flx in 11, consider nit mont of snd pored down t x + Let σt, x be the mont of snd which crosses the point x, from right to left For fixed t, this is determined by solving the liner ODE x d dx σt, x f xt, x σt, x, σ

2 Here f is clled the erosion fnction, since it describes the mont of erosion s fnction of the slope, per nit distnce trvelled in spce nd per nit mss pssing throgh We shll lwys ssme tht f is n incresing fnction with f1 0 Solving 12 one obtins σt, x exp x f x t, y dy The rte t which snd is deposited inside ny given intervl, b is ths compted by d dt b t, x dx σt, b σt, b exp x f x t, y dy dx x Since < b re rbitrry, this yields the conservtion lw 11 Eqtion 11 ws first derived in 1 s the slow erosion limit for the two-lyer model of grnlr flow by Hdeler nd Kttler 13, with the specific erosion fnction fp p 1/p In this pper, more generl incresing fnctions f will be considered Differentiting 11 wrt x, nd denoting by p x > 0 the slope, one obtins the dditionl conservtion lw p t t, x + fpt, x exp If the fnction f stisfies x fpt, y dy x 0, p0, x px 13 f1 0, f > 0, f < 0, lim fp, lim p 0+ fp 0, 14 p + p then one cn show tht soltions pt, x of 13 remin bonded for ll t 0 In prticlr, this is the cse when fp p 1/p, s for the limit of Hdeler-Kttler model, stdied in 1 Under sitble ssmptions on the initil dt, the existence nd niqeness of BV soltions for 13 hs been estblished in 2, 3, sing front trcking nd opertor splitting techniqes If the erosion fnction f is llowed to hve symptoticlly liner growth, then it is known tht the slope p x cn blow p in finite time Throghot this pper, insted of 14 we shll se the following ssmptions on the erosion fnction: A1 The fnction f : IR + IR is twice continosly differentible nd stisfies f1 0, f < 0, η lim p + f p > 0, lim fp, lim fp pf p < p 0+ p + 15 These conditions imply tht, s p +, the grph of f pproches liner symptote with slope η > 0 When the slope p x becomes infinite nd the fnction becomes discontinos, the eqtion 13 is no longer pproprite nd one mst stdy the originl eqtion 11 As shown in 15, soltions cn hve three types of singlrities These re kinks where x hs jmps bt is continos, shocks where hs jmps, nd hyperkinks where is continos bt x pproches + With the presence of the jmps in, the distribtionl derivtive x contins point msses, csing technicl difficlties in the 2

3 nlysis For sitble fmily of initil dt, the globl existence of entropy dmissible soltions ws proved in 15, by mens of piecewise ffine pproximtions generted by n dpted front trcking lgorithm However, the niqeness of these soltions hs remined n open problem We observe tht, s long s x t, x c 0 > 0, the inverse fnction x Xt, is lwys well-defined nd globlly Lipschitz continos Whenever t, x hs jmp, with left nd right sttes < +, the mp Xt, remins constnt over the intervl, + If t, x is smooth soltion of 11, strightforwrd compttion shows tht X Xt, stisfies the conservtion lw X t t, + exp + g X t, v dv 0, X0, X 16 Here the fnction g is recovered from f ccording to 1 gz z f 17 z A strightforwrd compttion yields 1 g z f z 1 z f 1 z From the ssmptions 15 on f it ths follows, g z 1 1 z 3 f 18 z g1 0, g < 0, lim z + gz, g0 > 0, g 0 < 19 Differentiting 16 wrt, nd writing zt, X t,, one obtins z t t, gzt, exp + gzt, v dv 0, z0, z 110 The dvntge of this lterntive formltion is tht, while in 11 cn be discontinos nd p in 13 cn become distribtion with point msses, the vrible X in 16 is lwys Lipschitz continos nd z in 110 remins globlly bonded fnction However, this comes t price, becse soltion of 110 my well become negtive In this cse, the mp Xt, is no longer invertible nd the connection with the originl eqtion 13 is lost To preserve its physicl mening, the eqtion 110 mst be spplemented by the pointwise constrint z 0 This leds to + z t t, gzt, exp gzt, v dv µ t, z0, z, 111 where, for ech t 0, µ t is sitble mesre spported on the set where z 0 Throghot this pper we shll consider soltions of 111 which re nonnegtive, lower semicontinos, nd sch tht zt, BV for every t 0 In this cse, precise set of conditions on the mesres µ t cn be stted s follows 3

4 C There exists jointly mesrble fnction Θ Θt, sch tht Θt, BV nd µ t Θt, is the derivtive in distribtionl sense, for e t 0 Moreover zt, 0 µ t, 0, 112 zt, 0, zt, b 0 b µ t, d A semigrop of soltions to 111 ws first constrcted in 9 The nlysis in 9 shows tht the limits of front trcking pproximtions yield entropy wek soltions which depend continosly on the initil dt s well s on the erosion fnction g The prpose of the present pper is three-fold First, we provide n entirely different constrction of the flow generted by 111 Soltions re here obtined by flx-splitting method, lternting bckwrd Eler steps for 110 with nonliner projection opertor on the cone of positive fnctions This pproch is mch in the spirit of nonliner semigrop theory, s in 10 We then prove the niqeness of entropy wek soltions of 111 by clssicl Krzhkov-type rgment Finlly, we prove the eqivlence between entropy soltions of 111 nd entropy soltions of the originl eqtion 11 As conseqence, this yields the globl existence nd niqeness of entropy dmissible soltions to 11, nd their continos dependence on the initil dt The reminder of the pper is orgnized s follows In Section 2 we define bckwrd Eler step for 110 nd estblish severl estimtes In Section 3 we stdy nonliner projection opertor from sbset of L 1 loc into the cone of non-negtive fnctions By combining these two steps, pproximte soltions to 111 re constrcted in Section 4 Letting the time step pproch zero, compctness rgment derived from Helly s theorem yields continos semigrop of entropy wek soltions See Definition 51 nd Theorem 54 in Section 5 for precise reslt The niqeness of entropy wek soltions to 111 is proved in Section 6, by dpting the clssicl vrible dobling techniqe 14 Finlly, Theorem 74 in Section 7 shows tht the entropy wek soltions to 111 correspond to entropy dmissible soltions for the originl problem 11 This eqivlence hevily relies on the fct tht or soltions re BV fnctions nd the flx fnction is convex In this cse, the Krzhkov entropy conditions re stisfied if nd only if the Lx dmissibility conditions hold t every point of pproximte jmp From the existence nd niqeness of soltions to 111, thnks to this eqivlence reslt we eventlly obtin the well-posedness of the Cchy problem for 11 For the bsic theory of conservtion lws we refer to 5, 16, 17 The dmissibility conditions nd the vrible-dobling techniqe to estblish niqeness of entropy wek soltions were introdced in the clssicl ppers 18 nd 14 The semigrop pproch to sclr conservtion lw, bsed on bckwrd Eler pproximtions, is originlly de to Crndll 10 It is interesting to compre the eqtion 11 with similr conservtion lws with nonlocl flx In the models considered in 6, 7, 8, the strctre of the eqtion provides niform priori bonds on the integrl 2 x dx As conseqence, soltions remin niformly Hölder continos nd no shock is ever formed On the other hnd, soltions to 11 cn become discontinos in finite time, nd disply vrios types of singlrities 4

5 As relted reslt, we mention tht the existence nd locl stbility of trveling wve soltions for 111 hve been recently estblished in 12 2 Bckwrd Eler pproximtions In this section we stdy the bckwrd Eler step for the slow erosion model withot the constrint z 0 Note tht in this cse the soltion of 110 cold become negtive We ths need to extend the definition of the erosion fnction gz lso for negtive vles of z For convenience, we extend the domin of g by setting gz g0 + g 0z, z 1, 0 21 nd frther extend g in smooth wy for z 1 Reclling 19, we cn ssme tht this extended fnction g stisfies the following ssmptions: A2 The fnction g : IR IR is continosly differentible, vnishes for s 2, is ffine for s 1, 0, nd is twice continosly differentible for s > 0 Moreover it stisfies g0 0, g1 0, g s < 0 for ll s > 0 22 fw gz 0 1 w z Figre 1: A fnction f nd the corresponding fnction g in 17, extended to negtive vles ccording to A2 In the rest of the pper, we denote TV{ } the totl vrition of fnction Or pproximte soltions will tke vles inside the domin D 0 {z : IR IR ; z is bsoltely continos nd there exist constnts M, U 0 > 0 sch tht z 1 L 1 M, TV{z } M, z 1 for ll U 0 } 23 The set of nonnegtive fnctions in D 0 will be written s D + 0 { } z D 0 ; z 0 for ll IR 24 5

6 For z D 0, we define G; z exp Using the ssmptions A2 one obtins gzy dy 25 G ; z gz G; z, lim G; z 1, 26 + nd { 0 < G; z exp gzy dy exp z 2,1 mx 2 z 1 g z z 1 L 1 } 27 We shll constrct pproximte soltions sing bckwrd Eler scheme For convenience, insted of 110 we consider the eqivlent eqtion z t t, gz G; z λz 0, 28 where the constnt λ > 0 is chosen lrge enogh depending on the initil condition sch tht ll the chrcteristic speeds for 28 become 1 By 26, this is the cse provided tht sp t, g zt, sp t, G; zt, + λ 1 29 It is cler tht these two problems re entirely eqivlent: zt, is soltion of 28 if nd only if zt, λt is soltion of 110 Definition 21 Bckwrd Eler opertor Consider fnction z D 0 + nd let ε > 0 be given We define the bckwrd Eler opertor Eε : D 0 + D 0 by setting E ε z w, 210 where w D 0 is the niqe fnction stisfying the implicit ODE w z + ε gwg; w + ελw 211 Notice tht the condition w D 0 singles ot the niqe soltion of 211 sch tht w 1 for ll sfficiently lrge 212 The next lemm shows tht the bckwrd Eler opertor is well defined, nd estblishes some of its properties Lemm 22 Let g stisfy the ssmptions A1 Let z D 0 nd let M, U 0 be the corresponding constnts in 23 We introdce the constnts { κ M g L 2, M+1, λ 1 + e κ g L 2, M+1, { C0 e κ sp s 0 g 2 s, C 1 g 2 L 2,M+1 2M eκ 213 Then for every ε > 0 the problem dmits niqe soltion w E ε z D 0 Moreover, the following properties hold 6

7 i sp w sp z M + 1 ii w 1 L 1 z 1 L 1 M iii e κ G; w e κ vi inf w C 0 ε v TV{w} 1 εc 1 1 TV{z}, provided tht ε < 1/C 1 vi w z L 1 ε 4λ + 2κe κ TV{z}, provided tht ε 1/2C 1 vii If z D 0 nd w E ε z is the corresponding soltion to , then 1 εc w w L 1 z z L 1, 214 for some constnt C depending only on the fnction g nd on M viii For ny constnt c > 0 nd ny positive test fnction ψ, one hs w c z c gw ψ ψ d + gcg; w + λw c d ε gc signw cgwg; wψ d 215 Proof The implicit ODE 211 cn be rewritten s λ + g w G; w w w z ε + g 2 wg; w 216 By ssmption z D 0, hence there exists U 0 sch tht z 1 for ll U 0 Since g1 0, it is cler tht w E ε z is the niqe bsoltely continos fnction tht solves the ODE 216 for U 0 nd sch tht w 1 for ll U Becse of the reglrity of the coefficients, this ODE hs niqe locl soltion It ths remins to check tht this soltion cn be prolonged bckwrds for, U 0 This reqires to prove priori estimtes showing tht w remins niformly bonded, while the coefficient λ + g wg; w in 216 remins niformly positive i - Upper nd lower bonds on w We begin by showing tht, on ny domin 0, where the soltion of 211 is defined, one hs the priori bonds 2 w sp z 218 Indeed, consider ny w > sp z 1 If w w for some, contrdiction is obtined s follows Define sp { IR ; ws > w } 7

8 We then hve w w, w w for ll >, so w 0 However, this is impossible becse w λ + g w 1 w G z ; w + g 2 w G ; w > 0 ε Next, if w < 2 for some, define sp { ; w < 2} We then hve w 2, w 2 for ll >, so w 0 However, this is impossible becse by sing g 2 0 nd g 2 0 we hve w 1 2 z λ + g 2 G ; w + g 2 2G ; w < 0 ε ii-iii Bonds on w 1 L 1 nd on G Let the soltion of 211 be defined on 0, We rewrite 211 s w 1 z 1 + ε gwg; w + λw Mltiplying by signw 1 nd integrting in, for ny 0 one obtins w 1 d z 1 d + ε signw 1 gwg; w + λw 1 d z 1 L 1 ε signw 1 g w G ; w + λ w 1, 220 becse w is bsoltely continos nd gwg; w + λw 1 0 whenever w 1 We clim tht the lst term on the right hnd side of 220 is non-positive Indeed, by 218 it follows sp g w sp g s g L 2 s M+1 2, M+1 Observing tht gw g L 2, M+1 w 1, to prove tht sign w 1 g w G ; w + λ w 1 0, 221 8

9 by 213 it sffices to show tht G ; w e κ If this ineqlity fils, contrdiction is obtined s follows Define sp { ; G; w > e κ } By continity, G ; w e κ Hence by 213 there exists δ > 0 sch tht g L 2, M+1 G ; w < λ for ll δ, 222 Using 222 in 220, for every > δ we obtin w 1 d z 1 L 1, so gwy dy g L 2, M+1 w 1 L 1 g L 2, M+1 z 1 L 1 κ Hence G ; w e κ for ll > δ, ginst the ssmption The previos nlysis shows tht, if soltion of is defined on 0, for some 0, then the bonds 218 hold, together with λ + g wg, w 1 for ll We ths conclde tht the soltion w of cn be extended bckwrds to the entire rel line, nd stisfies i iii iv Lower bond on w We now refine the lower bond in 218, deriving n ε- dependent estimte Reclling 213, consider ny vle w < C 0 ε If w < w for some, contrdiction is obtined s follows Define sp { IR ; w < w } We then hve w w, w w for ll > However, this is impossible becse the ineqlities G ; w e κ nd z 0 yield w λ + g w 1 w G z ; w + g 2 w G ; w ε λ + g w 1 w G ; w ε + C 0 < 0 v Bond on the totl vrition Fix ny h > 0 Then w z + ε gwg; w + λw, w h z h + ε gw hg h; w + λw h 9

10 Writing σ signw w h, we obtin { w w h z z h } d 1 TV{w} TV{z} lim h 0+ h 1 ε lim σ gwg; w gw hg h; w h 0+ h +λw w h d 1 ε lim h 0+ h + ε lim h 0+ 1 h σ gw gw hg; w + λw w h d σ gw hg; w G h; w d Here the first term vnishes becse w is bsoltely continos nd signw w h sign gw gw hg; w + λw w h Ths, we hve 1 TV{w} TV{z} ε lim h 0+ h To simplify nottion, cll G; w G; w G h; w h σ gw hg; w G h; w d h h G s; w ds J h G, where the right hnd side denotes the convoltion of the derivtive G ; w with the step fnction { h 1 if s 0, h, J h s 0 otherwise By i nd iv, nd by choosing ε > 0 sfficiently smll we cn ssme tht w 1, M + 1, so gw M g L 2,M By stndrd properties of convoltions one obtins TV{G ; w} TV {G ; w} TV{gw G ; w} We now hve the following estimtes: TV{gw} G ; w L + gw L TV{G ; w} 226 gw L g L 2,M+1 w 1 L g L TV{w}, 227 2,M+1 TV{G ; w} G ; w L 1 gw G ; w L 1 G ; w L gw L 1 G ; w L g L 2,M+1 w 1 L 1 e κ g L M, 228 2,M+1 G, w L G, w L gw L G, w L M g L 2,M+1 eκ

11 We cn represent the open set { ; w w h} {I k ; k 1} s disjoint nion of open intervls I k k, b k Using 226 nd we ths obtin G; w G h; w σ gw h d h g w b k h G b k g w k h G k k gw L TV{G, w} + TV{gw} G, w L g 2 L 2,M+1 M eκ TV{w} + g 2 L 2,M+1 M eκ TV{w} g 2 L 2,M+1 2M eκ TV{w} 230 Together with 224, this yields TV{w} TV{z} εc 1 TV{w} Therefore, ssming εc 1 < 1, we conclde TV{w} TV{z} 1 ε2m + 2e κ g 2 L 2,M+1 TV{z} 1 εc vi - L 1 continity in time If w E ε z, reclling 223 nd we obtin w z L 1 ε g wg; w + λ gwg w d + ε ; w d ε 2λ w d + ε gw L TV{G ; w} ε 2λ + Me κ g L TV{w} 2,M+1 2ε 2λ + κe κ TV{z} 232 Here the lst ineqlity follows from 231, provided tht εc 1 1/2 vii L 1 stbility Assme z D 0 nd let w Eε z, so tht w z + ε g w G ; w + ελw 233 By possibly incresing the vles of M, U 0 we cn ssme tht both z nd z stisfy the ineqlities in 23, with these constnts We then hve w w L 1 z z L 1 + ε sign w w g w G ; w gwg; w d +ε λ sign w w w w d, 11

12 where the lst term vnishes Therefore, we hve w w L 1 z z L 1 ε sign w w g w G ; w gwg; w d ε sign w w {g w gw}g ; w d +ε sign w w gw {G ; w G; w} d ε sign w w gw {G ; w G; w} d g ε ww {G ; w gw G; w} d + ε {G ; w G; w } d ε g L 2,M+1 TV{w} G ; w G ; w L +ε gw L TV {G ; w G ; w} 234 The definition of G t 25 nd the bond G e κ imply G ; w G ; w L e κ g L 2,M+1 w w L Moreover, { } TV G; w G ; w G; wgw G ; w g w d G; w gw g w G; d + w G ; w g w d G ; w L g L 2, M+1 w w L 1 + gw L 1 G ; w G ; w L e κ g L 2, M+1 w w L 1 + g L 2, M+1 w 1 L 1 e κ g L 2,M+1 w w L 1 e κ g L 2,M κ w w L Using 235 nd 236 in 234 we obtin n estimte of the form 1 εc w w L 1 z z L 1 for sitble constnt C, depending on g nd on the constnt M, bt not on ε viii - Entropy ineqlity Let c > 0 be n rbitrry constnt, nd let ψ Cc be positive test fnction From 211 it follows { gw } w c z c + ε gc G; w + λw c εgcgwg; w 237 Note tht the condition 29 implies signw c sign gw gcg; w + λw c

13 Mltiplying 237 by signw c ψ nd integrting in, we obtin w c z c ψ d signw c gw gcg; w + λw c ε ψ d signw cgcgwg; wψ d By sing integrtion by prts on the first integrl on the right-hnd side, nd then pplying 238, we obtin 215 This completes the proof 3 A projection opertor The bckwrd Eler step mps positive fnction z D 0 + to fnction w E ε z D 0 which my lso tke negtive vles We now introdce projection opertor π, mpping D 0 bck into D 0 +, nd determine some of its properties For nottionl convenience, in this section by f L 1 loc we denote generic fnction, not to be confsed with the erosion fnction Consider the sets X { } f L 1 loc IR ; lim fx 1, f 1 L x 1 M 31 nd For given f X, define Notice tht this implies X + {f X ; fx 0} 32 F x F x hence F is bsoltely continos nd x y 0 0 x Let F be the lower convex envelope of F, nmely F x min 0 fs ds dy 33 fs ds, 34 F fx for e x 35 { } θf + 1 θfb ; θ 0, 1, x θ + 1 θb 36 For f X, we denote by } K f {x IR ; F x F x 37 the closed set where F coincides with its lower convex envelope We observe tht { } K f x IR ; F y F x y xf x 0 for ll y IR { y s } x IR ; fr dr ds 0 for ll y IR 38 x x 13

14 F γ F * x Figre 2: A fnction F nd its lower convex envelope F One hs F x F x if nd only there exists line γ spporting the grph of F t the point x This is the cse if nd only if x K f Moreover, the ssmption lim x ± fx 1 implies F x fx > 1/2 whenever x is sfficiently lrge Hence the complement of K f is bonded open set, possibly empty The projection opertor π : X X + is now defined by setting πfx F x { fx if x K f, 0, if x / K f Since F is convex, its second derivtive is non-negtive Hence πf X + The next lemm collects the min properties of this opertor 39 Lemm 31 Let π : X X + be the opertor defined t 39 Then the following holds i πf f for every f X + ii For ny, b K f one hs Moreover b b x πfx dx πfy dy dx b b x fx dx, 310 fy dy dx 311 ξ x πfy dy dx ξ x fy dy dx for ll ξ IR 312 iii monotonicity If f, g X nd fx gx for e x, then πfx πgx for e x, 313 nd πg πf L 1 g f L

15 iv L 1 -contrctivity For ny f, g X, we hve πf πg L 1 f g L In prticlr, πf 1 L 1 f 1 L v BV stbility For ny f X hving bonded totl vrition, one hs TV{πf} TV{f} 317 Proof i If f X +, then F is convex Hence F F nd πf f ii The ssmption, b K f implies Therefore F F, F b F b, F F, F b F b b πfx dx b F x dx F b F F b F b fx dx, proving 310 Next, still for, b K f we hve b x πfy dy dx b x F y dy dx b F x F dx F b F b F F b F b F b x fy dy dx This proves 311 Finlly, for ny ξ IR the ineqlity 312 follows from ξ x ξ πfy fy dy dx ξ x F y F y dy dx F x F x F F dx ξ F ξ F ξ F F F ξ F ξ 0 iii If f g e, then F x F x dx y s x x fr dr ds y s x x gr dr ds, for ll x, y IR Hence, by the chrcteriztion 38, the corresponding sets stisfy K f K g To prove 313 we consider two cses If x K f, then x K g, hence πfx fx gx πgx Otherwise, if x / K f, then πfx 0 πgx 15

16 To prove 314, we choose n intervl, b so lrge tht, b K f, b K g IR Since πfx πgx for ll x, sing 310 we obtin b πg πf L 1 πgx πfx dx + πgx πfx dx IR\,b IR\,b iv Let f, g X, nd denote gx fx dx + b gx dx b f g mx{f, g}, f g min{f, g} Since the opertor π preserves the ordering, for every x IR we hve πfx πgx Then, 315 follows becse πf πg L 1 IR IR πf gx πf gx πf gx πf gx dx fx dx g f L 1 f gx f gx dx f g L 1 Finlly, by tking g 1 in 315, we obtin 316 v Since the projection π commtes with trnsltions, sing the contrctivity property 315 one obtins the estimte completing the proof TV{πf} lim sp h 0+ lim sp h 0+ 1 h 1 h πfx + h πfx dx fx + h fx dx TV{f}, We now stdy how the projection opertor behves in connection with fmily of convex entropies For f X, define the fnction Θ f x x πfy fy dy, 318 so tht Θ f is n bsoltely continos fnction which vnishes for x lrge nd stisfies Θ f xx πfx fx 319 The following properties of Θ f follow immeditely from Lemm 31 Lemm 32 Let f X, nd ssme, b K f Then nd Θ f Θ f b 0, x b Θ f y dy 0, 320 Θ f y dy 0 for ll x IR

17 The next lemm shows tht the projection opertor is dissiptive wrt fmily of convex entropies Lemm 33 Let f X For ny constnt c > 0 nd ny non-negtive test fnction ψ Cc IR one hs πfx c ψx dx fx c ψx dx IR IR signπfx cθ f x ψ x x dx 322 IR Proof By 319 we hve πfx c fx c + Θ f xx Mltiplying both sides by signπfx c ψ nd integrting over IR, we obtin πfx c ψx dx fx c ψx dx + signπfx c Θ f xψx dx, IR IR To hndle the lst term, we observe tht Θ f is spported on the region where πf 0, hence signπfx c 1 An integrtion by prts yields signπfx cθ f xψx dx Θ f xx ψx dx IR IR signπfx cθ f x ψ x x dx, completing the proof 4 Approximte soltions by flx-splitting lgorithm Combining the bckwrd Eler opertor nd the projection opertor introdced in the previos sections, we now constrct fmily of pproximte soltions Let n initil dt z0, z D 0 be given Fix time step ε > 0 nd let z ε : IR D 0 be the niqe fnction sch tht { zε t z if t 0, z ε t π Eε 41 z ε t ε if t > 0 Here Eε is the bckwrd Eler step introdced in Definition 1, nd π is the projection opertor defined t 39 It is nderstood tht, in the constrction of Eε, we choose λ > 0 sfficiently lrge so tht 29 holds Thnks to Lemm 22, the constnt λ cn be chosen s in 213, depending on the initil dt z bt not on ε Notice tht z ε cn be constrcted throgh discrete time itertions Consider the times t k kε, k 0, 1, 2, IR IR 17

18 We begin by setting z 0 z 42 Next, given z k z ε t k,, the fnction z k+1 z ε t k+1, is compted by setting The soltion of 41 is then w k+1 E ε z k, z k+1 πw k+1, 43 z ε t, z k, if t t k, t k+1 44 To stdy the projection opertor t every time step t k, it is convenient to introdce the fnctions Θ k 1 z k w k d, Θ k ε 1 z k w k, 45 ε nd Θ ε t, Θ k, Θ ε t, Θ k if t t k, t k+1 46 Combining the properties of the bckwrd Eler opertor proved in Lemm 22 nd the properties of the projection opertor in Lemm 31 nd Lemm 33, we obtin similr estimtes for z ε Lemm 41 Consider initil dt z, z D 0 where D 0 is defined in 23, nd let M { } mx z 1 L 1, z 1 L 1, TV{ z}, TV { z } Let z ε t,, z εt, be the corresponding soltions of 41, with λ chosen s in 213 Then, for every ε > 0 sfficiently smll nd every t 0, the following estimtes hold i sp {z ε t, } sp { z} ii z ε t, 1 L 1 z 1 L 1 M iii 0 < C 1 G ; z ε t C iv TV {z ε t, } e Ct TV { z} v z ε t, z εt, L 1 e Ct z z L 1 vi z ε t j, z ε t i, L 1 C e Ct j t j t i for ny integers 0 i < j vii Krzhkov entropy ineqlity For ny c > 0 nd ny positive test fnction ψ Cc 0, T IR, we hve z ε c ψ t d dt gz ε gcg; z ε + λz ε c ψ d dt + signz ε c gcgz ε G; z ε ψ d dt + signz ε c Θ ε ψ ddt Cε 47 Here C is sitble constnt independent of ε 18

19 viii Θ ε hs niformly bonded spport For ny given T, R 0 > 0 there exist R, δ > 0 sch tht the following holds If z 1 δ for > R 0, then for every ε > 0 smll enogh one hs z ε t, > 1 2 for ll t 0, T, > R 48 In prticlr, the spport of Θ ε is contined in 0, T R, R Here the constnt C depends only on M nd on the fnction g, while K depends on M, g, c, T, nd on ψ C 2 The constnts R, δ depend on M, g, c, T, nd on R 0 Proof 1 The properties i iv re strightforwrd conseqences of the corresponding properties in Lemm 22 for the bckwrd Eler step nd in Lemm 31 for the projection opertor 2 To prove v, we observe tht the 214 nd the contrction property of the projection π imply z ε t k+1 zεt 1 k+1 L 1 1 Cε z εt k zεt k L 1, for some constnt C depending only on M nd for ny ε < C 1 k 0, 1, 2 we obtin v, with possibly different constnt C By indction on 3 To prove vi we observe tht, by 232, { Eε z k z k L εc 1 0 TV z k} 49 Here C 0 is constnt depending only on sp z k nd on z k 1 L 1 In ddition, since z k πz k nd π is contrction, we hve πeε z k Eε z k L πe 1 ε z k πz k L πz + k E 1 ε z k L 1 2 Eε z k z k L Ptting together nd sing the estimte iv on the totl vrition of z k zt k, we obtin { z k+1 z k L πe 1 ε z k z k L 3εC 0 TV z k} 3εC 1 0 e Ct k TV{ z} 411 We now write z ε t j, z ε t i, L 1 j 1 z k+1 z k L 1 nd se 411 to estimte ech term This yields vi, for sitble constnt C nd ll ε > 0 sfficiently smll 4 To prove the Krzhkov entropy ineqlity, we se property viii in Lemm 22 with z z k z ε t k,, w w k+1, nd then Lemm 33 with f w k+1, πf z k+1 z ε t k+1, Choose N so lrge tht T < Nε By ssmption, the test fnction ψ ki 19

20 vnishes for t 0 nd for t Nε Smming over k 0,, N 1, by stndrd smmtion-by-prts techniqe we obtin N z ε k+1 c ψ t k+1, ψ t k, d ε k0 N 1 k0 N k0 N k0 z k c z k c z k+1 c ψ t k, d w k+1 c w + k+1 c z k+1 c ψ t k, d ε g w k+1 gc G ; w k+1 + λ w k+1 ψ c t k, d + + N k0 N k0 ε ε sign w k+1 c gcg w k+1 G ; w k+1 ψ t k, d sign z k+1 c Θ ε t k+1, ψ t k, d 412 As ε 0, the difference between the left hnd side of 47 nd the left hnd side of 412 is bonded by constnt mltiple of ε Similrly, compring the right hnd side of 47 with C 0 with the right hnd side of 412, we see tht the difference is gin bonded by constnt mltiple of ε Therefore the ineqlity in 47 follows from 412, for sitble constnt C 5 The bond 48 will be estblished by comprison rgment For every t t k kε we will prove by indction tht zt, stisfies bonds of the form ξt if > Rt, zt, 0 if Rt, Rt, 413 ηt, if < Rt See Figre 3 Here the fnctions t ξt, t Rt nd ηt, re defined s ξt 1 δ 2δt, Rt R 0 + Mt, ηt, 1 δe 2Ct e +Rt 2ɛ 414 The vle of δ will be chosen sfficiently smll, s specified lter At t 0, the bonds 413 hold by ssmption Now ssme tht, t time t t k, zt, stisfies the bonds 413 We will show tht 413 holds t t t k+1 t + ε On the intervl Rt k+1, Rt k+1, 413 holds trivilly, becse z k+1 0 Next, consider the hlf line { > Rt k+1 } We clim tht w k+1 ξt k 2εδ ξ t k+1 for ll > R t k

21 1 zt, 1 δ Rt 0 Rt Figre 3: The lower estimtes on flx splitting pproximtion zt, Indeed, if w k+1 chieves locl min t, one then hs w k+1 0 if > Rt k, while w k+1 0 if Rt k Then, from 216 we obtin which is eqivlent to w k+1 z k εg mx L 2 1 w k+1 2, 2 1 w k+1 1 z k + εg mx L 2 1 w k+1, 416 where G mx provides n n pper bond on G nd L is constnt strictly lrger thn the Lipschitz constnt of g We compre 416 to the problem b + εmb 2, 0 < 4εM 05, b 1 1 4εM 2εM < + 2εM 2, where in the lst ineqlity we sed the reltion 1 1 x < 1 2 x x2 for 0 < x < 05 By stndrd comprison rgment, choosing ε nd δ sfficiently smll sch tht 4εG mx L 1 2, δ G mxl T we hve 2 1 w k+1 1 z k + 2εG mx L 2 1 z k Applying the ssmption 1 z k δ1 + 2t nd the condition 417, we get 1 w k+1 δ1 + 2t + 2εG mx L 2 δ t 2 δ1 + 2t + 2εδ 1 ξt k + ε, proving 415 The projection opertor cold move the spport of Θ ε frther to the right Thnks to the properties ii nd vi in Lemm 22, we hve the estimte R t k+1 R t k MC 0 ε 1 ξ t k+1 2MC 0ε Finlly, we consider the hlf line < Rt k+1 Sppose z k ηt k,, we first show tht, for some constnt M it holds w k+1 η 1 δe 2Ct k+ε e +Rt k+ Mε/2ε

22 From 216 nd property vi in Lemm 22, we hve w k+1 wk+1 z k ε + C 3 1 w k+1, w k+1 Rt k C 0 ε 419 We proceed by contrdiction Assme tht 418 fils nd let < Rt k be the rightmost point where the eqlity holds: { } mx Rt k : w k+1 η This yields contrdiction provided tht Using 419, t we hve w k+1 < η 1 +Rt k +Mε/2ε 2ε e 420 w k+1 η ηt k, + C 1 η ε 1 δe 2Ct k e 2Cε 1 + e +Rt k + Mε/2ε 1 e M/2 ε +C δe 2Ct k+1 + e +Rt k + Mε/2ε δe 2Ct k e 2Cε 1 Ce 2Cε 1 e M 2 Cε e +Rt k + Mε/2ε ε ε Here the first term is negtive, nd the constnt in the second term is bonded by 1 e M/2 Cε ε < 1 2ε for ε sfficiently smll nd M sfficiently lrge, ths 420 holds, providing the contrdiction We choose δ smll sch tht δe 2CT < 1/3 By the property of the exponentil fnction, there exist constnts C nd C, sch tht 0 C ε 2/3 e /2ε d Cε, η > 1 3, Rt k Mε Cε The projection step will psh Rt k Mε frther to the left Thnks the properties ii nd vi in Lemm 22, nd the properties of the exponentil fnction, we hve, z k+1 η for Rt k Mε C ε Finlly, letting M M + C, we conclde tht z k+1 ηt k+1, for Rt k+1, completing the indctive step 5 A semigrop of wek soltions Tking seqence of flx-splitting pproximtions, s the time step ε 0, in the limit we expect to recover semigrop of wek soltions Before stting the min reslt in this direction, we give precise definition of entropy wek soltion 22

23 Definition 51 Given time intervl 0, T, n entropy wek soltion to the Cchy problem 111 is bonded, mesrble fnction z zt, 0 with the following properties P1 The mp t zt, is continos from 0, T into L 1 loc IR Moreover, z0, z 0 P2 There exist mesrble fnction Θ Θt, with compct spport in 0, T IR, sch tht zt, > 0 Θt, 0 zt, > 0, zt, b > 0 b Θt, d 0 51 Moreover, for ny constnt c 0 nd every non-negtive test fnction ψ C c 0, T IR, the following entropy ineqlity holds: T 0 IR T T z c ψ t d dt + 0 IR T 0 0 IR signz c gz gc G; zt ψ d dt signz c gc gz G; zt ψ d dt IR signz c Θt, ψ d dt 52 Remrk 52 Since the fnction Θ is spported on the set where z 0, for c > 0 in 52 we lwys hve signz c Θt, Θt, Remrk 53 According to 52, for every c 0 the fnction z zt, stisfies the ineqlity z c t signz c gz gc G; zt Θ 0 53 in distribtionl sense, for some mesrble fnction Θt, stisfying 51 We now stte the min reslt on the globl existence of BV soltions to the Cchy problem Consider the domin D + {z : IR 0, ; z is bsoltely continos nd } z 1 L 1 <, TV{z } <, 54 nd, for ny M > 0, the sbdomin D M {z D + ; z 1 L 1 < M } 55 Theorem 54 Let the fnction g stisfy the ssmptions A2 Then for ny M > 0 there exists mp S : D + M 0, D+ with the following properties 23

24 i For every z D M, the trjectory t S t z is n entropy wek soltion to the Cchy problem 111 in the sense of Definition 51 ii For ny M, one cn find constnt C sch tht, if z, z D M, TV{ z} M, TV{ z } M, then for ll t > s 0 one hs S t z S s z L 1 Ce Ct t s, 56 S t z S t z L 1 Ce Ct z z L 1 57 Proof 1 The domin D M is seprble metric spce, with the L 1 distnce In prticlr, we cn select contble sbset D D M D 0 + sch tht the following holds P For every z D M, there exists seqence of elements z n D sch tht z n z L 1 0, lim sp TV{ z n } TV{ z} 58 n 2 Let the constnts κ, λ be s in 213, depending on g nd on the constnt M Let n initil condition z D be given Consider seqence ε n 0 Let t z εn t be the corresponding soltions to 41 Observe tht, s t rnges over ny compct intervl 0, T, the totl vrition of z εn t, remins niformly bonded Next, let Z εn : 0, D 0 + be the piecewise ffine fnction which coincides with z εn t the discrete times t k k εn Then the mps t Z εn t re niformly Lipschitz continos wrt the L 1 distnce, on bonded intervls of time By Helly s compctness theorem see for exmple 5, we cn extrct sbseqence ε ν ν 1 sch tht the fnctions Z εν converge in L 1 loc, nd hence the sme holds for the fnctions z ε ν By stndrd digonlistion rgment, we cn ssme tht the sme seqence ε ν 0 chieves convergence for every z D If now z εν z in L 1 loc, we define the fnction S t z by setting S t z zt, λt 59 3 For initil dt z, z D, the estimtes follow from the corresponding estimtes v-vi in Lemm 41 We cn now extend the definition of S from D 0, to the whole domin D M 0,, by continity Indeed, given z D M, there exists seqence of initil dt z n D sch tht 58 holds We then define S t z lim S t z n 510 n The ineqlities 56-57, vlid for z, z D, grntee tht the limit exists nd is independent of the pproximting seqence By continity, the estimtes remin vlid lso for z, z in the lrger domin D M This lredy proves prt ii of the theorem 4 It remins to prove tht ech trjectory t S t z is n entropy wek soltion We fix z D M By choosing frther sbseqence, we cn ssme the wek convergence 24

25 Θ εn Θ By constrction, the mp t zt is Lipschitz continos from 0, T into L 1 IR nd stisfies the initil condition z0 z 0 Moreover, by vii in Lemm 41, s ε 0 we hve ψ z c ψ t d dt g z gcg; z + λ z c d dt sign z c gcg zg; z ψ d dt + sign z c Θ 511 ψ d dt, for ny c > 0 nd ny positive test fnction ψ Cc 0, T IR Defining zt, zt, + λt, Θt, Θt, + λt 512 we obtin n entropy wek soltion of the Cchy problem 111 Remrk 55 We cnnot constrct the flow generted by the eqtion 111 simltneosly for ll initil dt z D + This is becse the bckwrd Eler pproximtions re defined for the conservtion lw 28, where the shift λ mst be chosen lrge enogh so tht the chrcteristic speed is strictly negtive This choice of λ depends on zt 1 L 1 nd on sp zt, As shown in Lemm 22, both of these qntities do not increse in time, hence their pper bonds re determined by the initil conditions Remrk 56 If we ssme tht g0 0, then for ny z 0 the soltion w Eε z to the bckwrd Cchy problem cn never become negtive In this cse, in 43 one hs z k+1 πw k+1 Eε z k Hence the projections π cn be omitted, nd trjectories of the semigrop S cn be constrcted simply s limits of bckwrd Eler pproximtions Under the ssmption g0 0, niqe soltion of the Cchy problem 11 with Lipschitz continos initil dt ū ws constrcted in 2, 9 In prticlr, it ws proved tht the -profile never develops shocks In terms of the trnsformed vribles, this mens tht z > 0 for ll IR zt, > 0 for ll IR, t 0 6 Uniqeness of entropy wek soltions We re now redy to stte the niqeness Theorem Theorem 61 For ny initil dtm z D +, the entropy wek soltion of the Cchy problem 111 is niqe Proof We implement dobling of vribles rgment to show tht the entropy ineqlity 52 implies niqeness Let ẑ nd z be two entropy wek soltions of 111 ccording to Definition 51, nd let Θ, Θ be the corresponding fnctions in Let φ Cc 0, T IR 0, T IR be ny positive test fnction, nd let c, c 0 be two rbitrry constnts Then ẑ ẑt, stisfies ẑ c φ t d dt + signẑ c gẑ gcg; ẑ + Θt, φ d dt signẑ cgcgẑg; ẑ φ d dt 61 25

26 Similrly, sing s, v s independent vribles, the soltion z zs, v stisfies z c φ s dv ds + signz c gz gc Gv; z + Θs, v φ v dv ds signz c gc gzgv; z φ dv ds 62 Choosing c zs, v in 61 nd c ẑt, in 62, integrting wrt ll vribles, nd smming the reslting ineqlities, we obtin L 1 + L 2 + L 3 + L 4 d dt dv ds 0 63 where L 1 ẑ z φ t + φ s, 64 L 2 signẑ zgẑ gz G; ẑ φ + Gv; z φ v, 65 L 3 signẑ z gzgẑ G; ẑ Gv; z φ, 66 L 4 Θt, φ + Θs, vφ v 67 Here for L 4 we sed the fct tht signẑ c 1 where ever Θt, is non-zero, nd signz c 1 where ever Θs, v is non-zero See Remrk 52 Let δ ρ nd η ρ be two stndrd one-dimensionl mollifiers, nd let t + s φ φt,, s, v ψ 2, + v 2 To shorten the nottion, in the following we write t + s ψ ψ 2, + v t s, δ ρ δ ρ 2 2 etc One hs δ ρ t s 2 η ρ v 2, η ρ η ρ v 2 68 η ρ v η ρ, ψ v ψ φ ψ δ ρ η ρ + ψδ ρ η ρ, v φ v ψ δ ρ η ρ ψδ ρ η ρ, t+s φ t ψ + s ψ δ ρ η ρ, v+ φ ψ + v ψ δ ρ η ρ With the choice 68 of the test fnction φ, the bove terms L 1, L 2 L 21 + L 22, L 3 nd L 4 L 41 + L 42 in tke the form L 1 ẑ z ψ t + ψ s δ ρ η ρ, L 21 signẑ zgẑ gz G; ẑ + Gv; z ψ δ ρ η ρ, L 22 signẑ zgẑ gz G; ẑ Gv; z ψ δ ρ η ρ, L 3 signẑ zgzgẑ G; ẑ Gv; z ψ δ ρ η ρ, L 41 Θt, + Θs, v ψ δ ρ η ρ, L 42 Θt, Θs, v ψ δ ρ η ρ, 26

27 Tking the limit s ρ 0 nd writing ψ ψt,, one obtins L 1 ẑ z ψ t d dt, L 21 signẑ zgẑ gz 1 G; ẑ + G; z ψ d dt, 2 L 3 signẑ zgzgẑ G; ẑ G; z ψ d dt, 1 L 41 Θ + Θ ψ d dt 2 Concerning the term L 22, n integrtion by prts yields L 22 signẑ zgẑ gzψ gẑg; ẑ δ ρ η ρ d dt + signẑ zgẑ gzψ G; ẑ Gv; z δ ρ η ρ d dt Tking the limit ρ 0 nd the integrting by prts, we get L 22 signẑ zgẑ gzψ G; ẑ G; z signẑ zgẑ gzψ gẑg; ẑ d dt d dt signẑ zgẑ gzψ G; ẑ G; z d dt signẑ zgẑ gzψ gẑg; ẑ d dt signẑ zgẑ gzψ gzg; z d dt Therefore, we hve L 22 + L 3 signẑ zgz gẑg; ẑ gzg; z d dt The term L 42 is treted in similr wy s for L 22 An integrtion by prts yields L 42 Θ Θψ δ ρ η ρ + Θ Θ ψ δ ρ η ρ d dt, Tking the limit ρ 0 nd se integrtion-by-prts for the second term, we get L 42 Θ Θψ d dt Θ Θ ψ d dt Θ Θψ d dt + Θ Θψ d dt 0 Combining the bove expressions one obtins ẑ z ψ t + signẑ zgẑ gz 1 2 G; ẑ + G; z ψ d dt signẑ z gẑg; ẑ gzg; z gzψ d dt 1 Θ + Θ ψ d dt

28 Since both ẑ nd z stisfy the conditions in Definition 51, we cn find constnt M sch tht ẑt, 0, M, zt, 0, M, G; ẑt M, G; zt M 610 for ll t, Since g is niformly Lipschitz continos on the intervl 0, M, setting λ M g L 0,M, for ll t, we obtin gẑ gz 1 2 G; ẑ + G; z λ ẑ z 611 Concerning the first term on the right hnd side of 69, sing , we hve the estimte gẑg ; ẑ gzg; z gẑ gz G; ẑ + gz G; ẑ G; z e κ g L 2,M+1 ẑ z + gz ek g L 2,M+1 ẑ z L 1 Let 0 < t 1 < t 2 < T, nd consider the domin Γ {t, ; t t 1, t 2, R λt}, where R is chosen lrge enogh so tht Θt, Θt, 0 whenever R λt This is possible becse, ccording to Definition 51, Θ nd Θ hve compct spport in 0, T IR Following well estblished techniqe, we now consider test fnctions ψ which pproximte the chrcteristic fnction of the domin Γ Thnks to the choices of R nd λ in , in the limit one obtins R λt2 R+λt 2 ẑt 2, zt 2, d t2 t 1 M 2 ẑt, zt, L 1 dt R λt1 Letting R, for ny t 2 t 1 > 0 we obtin R+λt 1 ẑt 1, zt 1, d ẑt 2, zt 2, L 1 ẑt 1, zt 1, L 1 M 2 t2 t 1 ẑt, zt, L 1 dt By Gronwll s lemm, for ny 0 < t T this implies ẑt, zt, L 1 e M 2t ẑ0, z0, L 1 This shows the continos dependence on initil dt, nd ths the niqeness of entropy wek soltions 28

29 7 Eqivlence with the originl problem By Theorem 61, for every nonnegtive initil dt z D + defined t 54, the entropy soltion of 111 is niqe In prticlr, it does not depend on the constnt λ > 0 chosen to constrct the bckwrd Eler pproximtions Ptting together the estimtes proved in the previos sections, we ths obtin Theorem 71 Let the fnction g stisfy the ssmptions A2 Then there exists continos semigrop S : D + 0, D + sch tht, for every z D +, the trjectory t S t z is the niqe entropy wek soltion to the Cchy problem 111, in the sense of Definition 51 In this finl section we stdy the eqivlence between soltions z zt, of the eqtion 111 nd soltions t, x of the originl problem 11 Before stting precise reslt, some definitions re needed Recll in 15 we define the prmeter η lim f p g0 > 0 p Given n incresing fnction : IR IR, cll µ µ + µ s the decomposition of the mesre µ D x into n bsoltely continos nd singlr prt wrt Lebesge mesre Motivted by 11, we introdce the flx fnction { } Φ x exp f x y dy + µ s x, η 71 x Definition 72 Consider mesrble fnction t, x, sch tht t, is strictly incresing for every fixed time t We sy tht is wek soltion of 11 if the mp t t, is continos with vles in L 1 loc nd ϕ t Φ t ϕ x dxdt + ūxϕ0, x dx 0 72 for every test fnction ϕ C 1 c IR 2 0 To chieve niqeness of soltion, one clerly needs to impose dditionl entropy conditions We shll do this by ssming some dditionl BV reglrity nd by imposing the Lx dmissibility conditions t ech point of pproximte jmp RC There exists fnction w wt, x 0 sch tht, for every t 0, the mp x wt, x is lower semicontinos bonded vrition niformly wrt time, nd stisfies { 0 if x Sppµ s t, wt, x x t, x 1 73 for e x IR Here µ s t is the singlr prt of the mesre D x t, Remrk 73 The vrible w introdced here is essentilly the sme s z in Theorem 54 However, we prefer to keep different nottions to stress the fct tht w wt, x while z zt, 29

30 Following 4, 5, 11, we sy tht t, x is point of pproximte jmp for the fnction if there exist, +, λ sch tht, setting Us, y { if y x < λs t, + if y x > λs t, 74 one hs lim ε 0 1 ε 2 s, y Us, y dyds 0 75 s t 2 +y x 2 <ε 2 We denote by J, J w the jmp points of nd w, respectively Since, w BV, clssicl strctre theorem 4, 11 implies tht the sets J, J w re rectifible, ie they re contined in the nion of contbly mny Lipschitz continos crves, together with set whose one-dimensionl Hssdorff mesre is zero We cn now impose dditionl dmissibility conditions on the soltion of 11 AC There exists set of times N with mesre zero sch tht, t ech point of jmp of or w with t / N, the following holds i Let t, x J, so tht hold Then the speed of the jmp is greter thn or eql to the chrcteristic speed of the right stte Nmely λ f x t, x+ Φ t x+ 76 ii Let t, x J w \ J Then the Lx dmissibility condition holds: wt, x > wt, x+ 77 The next theorem provides bsic correspondence between soltions of 11 nd entropy wek soltions to the xiliry eqtion 111 Theorem 74 i Let z zt, be n entropy wek soltion of 111, with z D + Fix ny constnt C nd define Xt, zt, ξ 1 dξ + C 78 Let x t, x be the inverse fnction of Xt, nd cll ū the inverse fnction of X0, Then t, x provides wek soltion to the Cchy problem 11 stisfying the reglrity condition RC nd the dmissibility conditions AC ii Vicevers, let t, x be soltion to wek soltion to 11 stisfying the reglrity nd dmissibility conditions RC-AC For ech t 0, let Xt, be the inverse fnction of x t, x Then the fnction zt, X t, provides the niqe wek entropy soltion of 111, with z X 0, By the niqeness of entropy soltions to 111, the bove theorem implies 30

31 Corollry 75 Let ū : IR IR be n incresing fnction sch tht the inverse fnction X : IR IR is Lipschitz continos nd its derivtive stisfies X BV, X 1 L 1 < + 79 Then the Cchy problem 11 hs niqe wek soltion stisfying the reglrity nd dmissibility conditions RC-AC b τ,x ε b x Figre 4: Proving the continity of the inverse fnction in L 1 loc Proof of Theorem 74 1 We strt by proving i Let z zt, be wek entropy soltion to 111, ccording to Definition 51 Since zt, 1 L 1 IR, the fnction Xt, in 78 is well defined For simplicity we ssme C 0, which is not restrictive Since 0 zt, X t, < M for some constnt M nd ll t,, for ech t 0 the inverse fnction t, is well defined nd strictly incresing Since t X t, is Lipschitz continos with vles in L 1 IR, by integrting wrt we see tht t Xt, is Lipschitz continos with vles in L IR Fix τ 0 For ny intervl, b, let sp x< τ, x, b inf x>b τ, x Given ε > 0, choose δ > 0 so tht Xt, Xτ, L ε whenever t τ δ An elementry rgment see Fig 4 shows tht the inverse fnction stisfies t, τ, L 1,b b + 2Mε ε Since ε > 0 is rbitrry, this shows the continity of the mp t t, with vles in L 1 loc 2 Let ϕ C 1 c IR 2 Chnging the vribles of integrtion from t, x to τ, nd writing φτ, ϕτ, Xτ,, we compte { t τ, x Xτ,, { φτ ϕ t + ϕ x X τ, φ ϕ x X, dx dt X d dτ 31

32 Next, we observe tht for e x the flx in 71 is eqivlently compted by { } + Φ t x exp gzt, d G; zt, 710 t,x where z X Observing tht X Xτ, is loclly Lipschitz continos wrt both vribles nd the sme is tre for t, G; zt, we compte t, xϕ t t, x Φ t xϕ x t, x dxdt 0 φ τ X φ X τ G; X τφ ddτ 0 { } X φ τ X τ φ + X τ φ G; X τφ ddτ 0 { } Xφ τ + G; X τφ ddτ 0 { } zτ, ξ 1 dξ φ τ + G; zτ 1 φ ddτ Introdce the test fnction ψ by setting φτ, v dv if N, ψτ, 0 if N + 1, nd sch tht ψτ, is ffine for N, N + 1 Observe tht ψ is Lipschitz continos with compct spport, nd ψ φ for < N Let N be lrge enogh so tht the spport of Θ is contined on the set where < N Using 52 we then obtin zψ τ ddτ gzg; zψ ddτ Θψ ddτ Indeed, ψ τ, φτ, is constnt on every intervl, b where zτ, 0 By 51, the integrl of Θ over this intervl is zero Integrting by prts, for N sfficiently lrge, the right hnd side of 711 cn now be estimted by N 0 N 0 { zτ, ξ 1 dξ φ τ + N zψ τ ddτ gzg; zψ ddτ 0 zτ, ξ 1 dξ ψ τ + } G; zτ 1 φ ddτ G; zτ dτ 1 ψ 0 N A N + B N

33 Assming tht φ, ψ vnish for t / 0, T, s N one hs T N+1 T N+1 A N zψ τ ddτ + gzg; zψ ddτ 0 N 0 N N+1 { } ψ L TV z, ; 0, T d B N N T N+1 + G L φ L g, z ddτ, 0 N T zτ, ξ 1 dξ ψ τ + G; zτ 1 ψ dτ 0 N ψ τ L sp Xτ, N N + φ L sp GN; zτ 1 τ 0,T τ 0,T From the bove, it is cler tht A N, B N 0 s N This shows tht the left hnd side of 711 vnishes Hence t, x provides wek soltion to the Cchy problem 11 3 Since z hs bonded vrition, nd cn be rendered lower semicontinos by chnge on set of mesre zero, the reglrity condition RC is clerly stisfied It remins to prove tht the dmissibility conditions AC re stisfied s well Consider first point t, x J w where is continos This mens tht x hs jmp, bt the limits t, x t, x+ 0 coincide To fix the ides, ssme tht t, 0 is point of jmp for z, with left nd right sttes z, z + nd speed λ 0 The continity ssmption on implies tht Gt, is continos t t, 0 Hence, from 53 we dedce see for exmple 5, p84 λ 0 z + c z c G 0, zt signz + cgz + gc signz cgz gc 714 Choosing c 1 2 z+ + z we obtin z 0 G 0, zt signz + z gz + + gz + + z 2g Since g is concve down, this implies signz + z < 0 Reclling tht z 1/ x w, we conclde tht 77 holds 4 Next, ssme t, x J nd let, + nd ẋ λ be s in see Fig 5 Then zt, 0 for, + By removing set of times of mesre zero, it is not restrictive to ssme tht t, nd t, + re jmp points for z zt,, sy with speeds, +, respectively For nottionl convenience, define G + G + ; zt, G G ; zt e + g0 G By the projection property, the fnction Θ is non-zero on the intervl, +, with Θ t, g 2 0G; zt g 2 0e + g0 G + > 0,, +, 33

34 while Θt, hs jmps t nd + By sing the conditions in 51, we cn obtin n explicit forml for Θ In prticlr the sizes of jmps t nd + re given s B Θt, + Θt, G G + + g0g, 717 B + Θt, + + Θt, + G G g0g+ 718 Note tht B < 0 nd B + < 0 Writing z zt, nd z + zt, + +, where z > 0 nd z + > 0, sing 717 nd the Rnkine-Hgoniot condition, we obtin the two jmp speeds: gz g0 z G + B z, gz+ g0 z + G + B+ z Consider the jmp in z t t,, where z zt, > 0 while zt, + 0 A direct compttion shows tht the Lx condition < gz g0 z G < g z G 721 is lwys stisfied, becse g is strictly concve zt, t,x z + + z _ + x Figre 5: A point where t, hs jmp right corresponds to n intervl, + where zt, 0 left Next, consider the jmp t +, where z + zt, + + > 0 while zt, + 0 Choosing 0 < c < z +, from 52 one obtins z + c c 0 + gz + gcg + + g0 gcg + + B + Adding z + + to both sides nd sing 720 we obtin, for ll c with 0 < c < z +, 2z + c + z gz + + g0 2gc G + + B + 2gz + 2gc G + 34

35 Since g is strictly concve, the bove condition is eqivlent to the constrint { gz + + } gc min 0<c<z + z + G + g z + G c Combining 722 with 720 nd 718, then sing 716, we obtin G z + + gz + g0g + G g0g+ z + g z + G +, 723 nd e g gz + z + g z We now retrn to the originl t, x coordintes Reclling 71, let Φ + Φ t x+ G + nd Φ Φ t x G be the flxes to the right nd to the left of the point of jmp It will be sefl to recll the identities 1 f gz 1 z z, f gz zg z, lim z f p g0 p By the Rnkine-Hgoniot condition, sing 724 we find tht the speed of the jmp stisfies ẋt Φ+ Φ + ef Φ + eg G + gz + z + g z + f x t, x+ Φ t x+ This proves the dmissibility condition 76, completing the proof of prt i of the Theorem 5 From now on, we work on prt ii Let t, x be soltion to 11 which stisfies the reglrity nd dmissibility conditions RC-AC For ech fixed time t 0, let Xt, be the inverse fnction By RC, the derivtive zt, X t, is well defined e nd hs bonded vrition Up to modifiction on set of mesre zero, we cn ssme tht z is lower semicontinos We clim tht the fnction z provides n entropy soltion to 111, ccording to Definition 51 As first step, we show tht the mp t zt, is continos with vles in L 1 loc IR By the continity of the mp t t, with vles in L 1 loc, it is cler tht the inverse fnction t Xt, is lso continos with vles in L 1 loc Consider ny convergent seqence of times t j τ By the reglrity condition RC, the fnctions zt j, re niformly bonded nd hve niformly bonded totl vrition Hence, by extrcting sbseqence, we cn ssme the convergence zt j, z in L 1 loc, for some BV fnction z From the convergence Xt j, Xτ, it now follows the identity z X τ, zτ, The bove rgment shows tht from every seqence t j τ one cn extrct sbseqence t jk k 1 sch tht zt jk, zτ, in L 1 loc This proves continity property P1 in Definition 51 6 In the remining steps we will prove tht P2 in Definition 51 lso holds For ech t 0, the mesre µ t on the right hnd side of 111 is determined s follows If 35

36 t, is continos, then µ t 0 In generl, let t, hve jmps t contbly mny points x i nd cll + i t, x i +, i t, x i, { G + i Φ + i Φ t x i +, G i Φ i Φ t x i exp{ + i i f + }Φ + i, G e + i g0 G + i, i, + i Restricted to ech intervl i, + i, the mesre µt is the sm of n bsoltely continos mesre with density G, pls two point msses t + i, i The sizes of these msses re given by B i +, B i, where B i + G i G + i + i i + g0g + i, Bi G i G + i + i i g0g i 725 Notice tht these msses re chosen so tht µ t i, + i 0, while the brycenter of the positive prt of µ t coincides with the brycenter of the negtive prt Eqivlently, one cn define µ t D Θt,, where Θt, Bi + g 2 0 Gξ dξ if Θ i, Θ i i < < + i, i i 0 otherwise With the bove definitions, one esily checks tht the properties in 51 hold 7 For ech fixed time t 0, let Xt, be the inverse of the mp x t, x By the reglrity ssmption RC, the derivtive x is positive nd niformly bonded wy from zero Since the flx fnction in 11 remins niformly bonded, we conclde tht the mp t, Xt, is niformly continos nd therefore hs prtil derivtives X t, X defined pointwise for e t, x Since provides soltion to 11, it follows tht X stisfies the PDE X t t, g X t, exp g X t, v dv Θt, pointwise lmost everywhere In trn, the BV fnction z X provides distribtionl soltion to the eqtion z t gz exp gzt, v dv Θ t, In prticlr, 52 is stisfied s n eqlity in the specil cse where c 0 To prove tht the ineqlity 52 holds for every non-negtive test fnction ψ C 1 c 0, T IR nd every constnt c, we recll tht z is BV fnction of the two vribles t, By well known strctre theorem 4, 11, for lmost every c IR the sets Ω + c {t, ; zt, > c}, Ω c {t, ; zt, < c}, 36