NONSMOOTHING IN A SINGLE CONSERVATION LAW WITH MEMORY

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1 Elecronic Journal of Differenial Equaions, Vol. 2(2, No. 8, pp. 8. ISSN: URL: hp://ejde.mah.sw.edu or hp://ejde.mah.un.edu fp ejde.mah.sw.edu ejde.mah.un.edu (login: fp NONSMOOTHING IN A SINGLE CONSERVATION LAW WITH MEMORY G. GRIPENBERG Absrac. I is shown ha, provided he nonlineariy σ is sricly convex, a disconinuiy in he iniial value u (x of he soluion of he equaion ( u(, x+ k( s(u(s, x u (x ds + σ(u x(, x =, where >andx R, is no immediaely smoohed ou even if he memory kernel k is such ha he soluion of he problem where σ is a linear funcion is coninuous for >.. Inroducion and saemen of resuls The purpose of his paper is o show ha, provided he nonlineariy σ is sricly convex, a disconinuiy in he iniial value u(,x=u (x of he soluion of he equaion ( u(, x+ k( s ( u(s, x u (x ds + σ(u x (, x =, (. where > andx R, is no immediaely smoohed ou. Recall ha if he equaion is linear, i.e., σ(r = r, and if k is nonnegaive, nonincreasing, and locally inegrable wih lim k( =+, hen he soluion u is coninuous when >; see [8, Thm. 3], (where he noaion differs somewha from he one used here or see he argumen in he proofs below. If one has k =, hen i is well known and easy o check ha disconinuiies in he iniial value for he conservaion law u (, x+σ(u x (, x = are no smoohed ou and ha in fac disconinuiies can appear in he soluion even if he iniial value is smooh provided σ is no linear. (Bu here is of course no smoohing effec in he linear case. These resuls can also be esablished in he case where k is assumed o be of bounded variaion (and hus bounded. On he oher hand, i follows from [5, Thm., 2] ha he soluion of he equaion k( s ( u(s, x u (x ds + σ(u x (, x =, (.2 Mahemaics Subjec Classificaion. 35L65, 35L67, 45K5. Key words. conservaion law, disconinuous soluion, memory. c 2 Souhwes Texas Sae Universiy. Submied Sepember, 2. Published January, 2.

2 2 G. GRIPENBERG EJDE 2/8 is coninuous when > provided he iniial value has bounded variaion (bu is no necessarily coninuous, k is locally inegrable, nonnegaive, non-increasing, logconvex, and lim k( =+ and provided σ is coninuous and sricly increasing. Thus one sees ha for equaion (.2, in conras o equaion (., here is no much difference beween he coninuiy properies of he soluion in he linear and he nonlinear case. This paper does no answer he quesion wheher equaion (. has disconinuous soluions wih coninuous iniial values in he case lim k( =+ oo. Noe also ha he problem considered here is o a large exen only a model problem; he more ineresing cases are he ones where one considers sysems of conservaion laws and wans o invesigae he effec of memory erms. In paricular, in cerain cases hese sysems can be wrien as higher order equaions and hen one is led o consider he quesion of how he qualiaive properies of he soluions change when one goes from he wave equaion u = σ(u x x o he diffusion equaion u = σ(u x x by considering equaions of he form D α (u =σ(u x x (where D α is a fracional derivaive of order α (, or of he form γu + D α (u =σ(u x x which can be wrien in he form (γu (, x + k( s( u (s, x u (x ds = σ(u x x (, x. For some furher resuls on equaion (., see for example [3] and [4]. Here we prove he following resul. Theorem. Assume ha I. k L loc (R+ ; R is nonnegaive, non-increasing, and absoluely coninuous on (, wih sup α+ k ( <, << for some α (, ; II. σ C 2 (R; R is sricly increasing and sricly convex on [, ]. Then here is an enropy soluion of equaion (. wih iniial condiion u (x = χ (,] (x which has a disconinuiy along a curve saring a he origin. See [, Def. 6] for he definiion of an enropy soluion of (.; his definiion is, of course, a direc generalizaion of he concep of enropy soluions of a conservaion law, (i.e., he case k =. The idea of he proof is roughly he following. Firs we consider he case where k( < and hen i is quie sraighforward o show ha he soluion has a disconinuiy of he desired kind, because in his case he equaion is a sufficien nice perurbaion of he conservaion law u + σ(u x =. Nexwehaveoshow ha his disconinuiy does no disappear when k( grows o infiniy and for his purpose we consider a linearizaion of equaion (. ha can be solved explicily. The coefficien in he linear equaion mus be chosen appropriaely and here he Rankine-Hugonio relaion ha gives he speed of propagaion of a disconinuiy is crucial. We need he following (somewha srange looking resul ha can be used o obain an upper bound on he ime derivaive of he fundamenal soluion of he linear problem. Theorem 2. Assume ha I. ψ and Ψ are measurable funcions such ha ψ( Ψ( for all >;

3 EJDE 2/8 NONSMOOTHING IN A SINGLE CONSERVATION LAW 3 II. + Ψ(d< ; III. sup > sup s /2 Ψ(s Ψ( < ; IV. φ is a nonnegaive measurable funcion on R + such ha φ( =ψ(η+ φ(d and ψ( sφ(sds, >, (.3 where η [, ]. Then here are posiive consans C and T, which depend only on Ψ, such ha φ( C Ψ(, T. (.4 Noe ha i would be sufficien o assume ha he equaion holds almos everywhere, if one may change he values of u on a se of measure zero. As one easily sees he claim of his heorem is inuiively quie obvious, bu unforunaely I have no been able o find a proof ha does no involve oo many echnicaliies. 2. Proofs Proof of Theorem. We may wihou loss of generaliy assume ha σ( =, and since we will show ha u(, x for andx Rwe may assume ha σ is sricly increasing on R, and no jus on [, ]. In order o show ha here is an enropy soluion u of (. such ha u(, x = for >andx anduis nondecreasing in is firs variable and non-increasing in is second one, we argue as follows, see [] and [6]: If we le D(A ={v L (R + ;R σ(v AC(R + ; R,v( =, σ(v L (R + ;R}and define A(v = σ(v for v D(A, hen A is a closed, m-accreive operaor in L (R + ; R, ha in addiion saisfies cerain naural monooniciy properies, see [6, Lemma 3]. By [, Thm. ] here is a generalized soluion u C(R + ;L (R + ;R of he equaion u (+ d d k( su(sds+a(u(,, u( =. By he argumen used in he proof of [6, Thm. 5] one sees ha his funcion u(, x is nondecreasing in is firs and non-increasing in is second variable. We exend his soluion as for x and hen use he same argumen as in he proof of [, Thm. 7] o show ha we have an enropy soluion. If we replace u by he funcion (, x u(, λx we ge a soluion of equaion (. where σ has been replaced by λσ. Thus we may wihou loss of generaliy assume ha σ( < <σ (. Since σ is coninuously differeniable i follows ha here are consans µ and η (, such ha σ(y µ σ (y, y [ η, ]. (2. Suppose ha k C 2 (R + ; R. (2.2 Nex we shall show ha in his case he soluion has a disconinuiy (jus as in he case where k =. We wrie (. as u (, x+σ(u x (, x+g(u(, x,,x=, (2.3 where g(v,, x =k(v + k ( su(s, xds, (2.4

4 4 G. GRIPENBERG EJDE 2/8 and where we hus assume ha he funcion u appearing in he inegral is a known funcion. I is clear ha g(v,, x k(, g(u(, x,, x, v [, ],, x R. (2.5 If one uses he fac ha u is a weak soluion of (2.3, hen one sees ha he funcion x σ ( u(, x is Lipschiz coninuous in L ([,T]; R foreacht >. Since we are going o show ha here is a poin such ha u(, x {} [ η, ] when and x R, we may wihou loss of generaliy assume ha σ ( > and i follows ha he funcion x u(, x is Lipschiz coninuous in L ([,T]; R and herefore g is Lipschiz coninuous. Since he funcion x u(, x is non-increasing, we can apply he resuls from [2] and we recall ha a generalized characerisic saisfies he equaion σ ( u(, ξ( ξ (, ( if u(, ξ(+ = u(, ξ(, ( = σ u(,ξ(+ σ u(,ξ( u(,ξ(+ u(,ξ(, oherwise. (Noe ha in [2] i is assumed ha he funcion g is coninuously differeniable, bu i is sufficien for he conclusions needed here ha i is Lipschiz coninuous. Observe ha if u(, x =,heng ( u(, x,,x = as well and herefore i follows from (2.5 and from [2, Thm. 3.2, 3.3] ha if one follows a maximal backward characerisic from a poin (, x where< k( and u(, x >, hen one can conclude ha u(, x > k(. Thus we see ha if we define he funcion ϕ by ϕ( def = inf{ x> u(, x =}, hen we have u(, ϕ( k( when < k(. There is, by [2, Thm. 4.], a unique characerisic saring a (, and i mus be ϕ. We may and will assume ha u is coninuous from he lef in is second variable and hus we conclude ha for almos every >wehave ϕ (= σ( u(, ϕ(. (2.6 u(, ϕ( We le h be he inverse of ϕ and hence we have for almos every x: h (x = u(h(x,x σ (. (2.7 u(h(x,x Now we mus show ha here is a posiive number, independen of he assumpion (2.2 such ha u(, x η, x ϕ(,. (2.8 (Recall ha η was defined in (2.. Once we have done his, he desired conclusion follows from [, Thm., 7.(f]. Le v be he soluion of he linear problem ( v(, x+ k( s ( v(s, x v (x ds + v x (, x =, for >andx Rwhere v = χ R. Equivalenly one can consider he problem for x> wih boundary value v(, =. Thus v is nondecreasing in is firs variable and non-increasing in is second variable by he resul given above for he nonlinear case.

5 EJDE 2/8 NONSMOOTHING IN A SINGLE CONSERVATION LAW 5 The Laplace ransform of v (wih respec o is firs variable is given by ˆv(z,x = z e zx zˆk(zx, x,rz>. The derivaive of v wih respec o is hus a measure ha has Laplace-Sieljes ransform e zx zˆk(zx and we see ha i consiss of a ranslaion of a nonnegaive measure wih Laplace-Sieljes ransform e zˆk(zx, see [9, Prop. 4.2, 4.3]. This measure in urn has a poin-mass of size e k(x a and herefore we consider he measure w whose Laplace ransform is given by ŵ(z,x =e zˆk(zx e k(x, x. I is clear ha w does no have a nonzero poin-mass a. Nex we differeniae boh sides of his equaion wih respec o z and use he fac ha d dz (zˆk(z is he Laplace ransform of he funcion P ( def = k (. Since he convoluion of a locally inegrable funcion and a measure is a locally inegrable funcion we conclude ha w(d, x is given by a locally inegrable funcion and we ge (wih a sligh abuse of noaion ha w(, x =xp (e k(x + x The conclusion we can draw is ha P ( sw(s, xds. (2.9 v (d, x =w( x, xd+e k(x δ x (d. (2. When we resric ourselves o x> we can rewrie equaion (. as ( u(, x+ k( su(s, xds +u x (, x =F(, dx, where F = x( u σ(u. Take Laplace ransforms (wih respec o and solve he differenial equaion wih respec o x ha one ges. Then one sees ha u can be wrien in he form u(, x =v(, x+ v (ds, x yf ( s, dy. [,x] [,] We ake o be so small ha u(, x > η when x ϕ(. We noe ha F consiss of wo pars, one nonnegaive coming from he poins where h(x < so ha σ (u(, x and anoher non-posiive poin-mass when h(x =. The absolue value of his poin-mass is clearly less han. Since v is nondecreasing in is firs variable, we ge u(, x v(, x v (ds, x ϕ( s, (2. [( h(x +,] because if s< h(x, hen h(y h(x < swhen y [,x]. By (2.7, he fac ha u(h(x,x, and by he assumpion ha σ is convex we have h (x σ( µ > sohaif>h(xhen h(x s>h(x h(x s>s when s [,x] and i follows ha ϕ( s >x s. Tha is, we have s>x ϕ( s, s, > h(x.

6 6 G. GRIPENBERG EJDE 2/8 By (2. and (2. we herefore ge ( u(, x v(, x w s ( x ϕ( s,x ϕ( s ds, h(x > h(x. Since we assume ha u is coninuous from he lef and nonincreasing in is second variable and since he funcion u(, x is coninuous in L (R + ; R, we conclude ha lim h(x u(, x = u(h(x,x. Moreover, because h(x > x we have lim h(x v(, x = v(h(x,x. Thus we conclude afer changing variables s = h(x h(y ha x ( u(h(x,x v(h(x,x h (yw h(x h(y (x y,x y dy. (2.2 Le us now reurn o equaion (2.9. Since we only consider small values of we may wihou loss of generaliy assume ha here is a consan c (independen of he assumpion (2.2, such ha k ( c α, >. (2.3 Le x> and replace by x α we ge he equaion in (2.9. Afer a change of variables in he inegral φ( =xp (x α e k(x + xp ( ( sx α φ(sds, wherewehavedefinedφ( = x α w(x α,x. Now xp (x α c α and φ(d = ŵ(,x= e k(x and herefore i follows from Theorem 2 ha here is a consan c 2 depending only on c and α such ha φ( c 2 α when c 2. By he definiion of φ his implies ha By (2. we have v(, x = By (2.7 and (2.8 we have w(, x c 2 α x, c 2 x α. (2.4 w(s x, xds, > x >. (2.5 µ h (x σ( η, x ϕ(, (2.6 so ha h(x x µ when x ϕ(. If, in addiion, x<( µ µ c 2 α α and s h(x, hen s x>c 2 x α. Thus we ge from (2.5 v(, x c 2 (s x α x ds = c 2 α ( x α x c ( 2 µ α µ αx α, h(x, x min{ ϕ(, ( µ µ c 2 α α }. (2.7

7 EJDE 2/8 NONSMOOTHING IN A SINGLE CONSERVATION LAW 7 Wih he aid of (2.6 we conclude ha x ( h (yw h(x h(y (x y,x y dy σ( η x ( µ α y c α 2 dy = µ x min ( ασ( η c 2 { ( µ α x α, µ ϕ(, ( µ µ c 2 α α }. (2.8 When we combine (2.7 and (2.8 we conclude from (2.2 ha u(h(x,x > η for sufficienly small values of x, ha is, we have esablished (2.8 wih independen of k( and he smoohness assumpions on k. This complees he proof. Proof of Theorem 2. Le f ( = Ψ( + Ψ( sφ(sds,. I follows ha f L ([, ; R, wih an upper bound on he norm depending on Ψ only. Le ϕ be he soluion o he equaion ϕ( a.e. = f (+ Ψ( sϕ(sds,. (2.9 Such a soluion ϕ L ([, ; R exiss by [7, Thm , Cor ] when one observes ha sup s τ s Ψ( s d < for sufficienly large values of τ. Moreover, i follows from [7, Cor ] and he fac ha Ψ is nonnegaive ha we have φ( a.e. ϕ(,. (2.2 Now here is a poin [2, 3] (in fac a se wih posiive measure of such poins so ha ϕ( ϕ L ([,, and i follows from (2.9 ha Ψ( sϕ(sds ϕ L ([,. Using (III we see ha here is a consan c 3 such ha f 2 ( def = f (+ Ψ( sϕ(sds c 3,. Thus we have ϕ( a.e. = f 2 (+ Ψ( sϕ(sds,, (2.2 and i follows from [7, Thm , Cor ] (where we now use he fac ha Ψ( s ds < for sufficienly large values of τ ha here is a consan sup τ τ c 4 such ha ϕ( a.e. c 4,. (2.22 Thus we see ha he convoluion erm is coninuous, a leas when and herefore we may assume ha equaion (2.9 and inequaliies (2.2 and (2.22 hold for all values >.

8 8 G. GRIPENBERG EJDE 2/8 Since lim Ψ(sds= by (II we can choose a number T> such ha Ψ(sds<, T. ( Ψ(s sup > sup s /2 Ψ( Now i is clear ha here is a consan c 5 such ha Ψ( f 2 ( c 5, T, and herefore we have by (2.2, ϕ( c 5 Ψ( + max{t,/2} Ψ( sϕ(sds + Ψ( sϕ(sds, T. (2.24 max{t,/2} Using he fac ha ϕ is inegrable one sees by (III ha here is a consan c 6 such ha max{t,/2} Define v( = ϕ(/ψ(. Ψ( Ψ( sϕ(sds c 6, T. (2.25 Then one has by our choice of T in (2.23 and by inequaliies (2.24 and (2.25 ha v( c 5 + c max v(s, T, s [T,] and his inequaliy implies ha ϕ( 2(c 5 + c 6 Ψ(, T. The proof is complee. References. B. Cockburn, G. Gripenberg, and S-O. Londen, On convergence o enropy soluions of a single conservaion law, J. DifferenialEquaions28 (996, C.M. Dafermos, Generalized characerisics and he srucure of soluions of hyperbolic conservaion laws, Indiana Univ. Mah. J. 26 (977, E. Feireisl and H. Pezelová, On he long-ime behaviour of soluions o a conservaion law wih memory, Mah. Mehods Appl. Sci. 2 (997, , On compacness of bounded soluions o mulidimensional conservaion laws wih memory, NoDEA Nonlinear Differenial Equaions Appl. 5 (998, G. Gripenberg, Ph. Clémen, and S-O. Londen, Smoohness in fracional evoluion equaions and conservaion laws, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4 29 (2, G. Gripenberg and S-O. Londen, Fracional derivaives and smoohing in nonlinear conservaion laws, Differenial Inegral Equaions (995, G. Gripenberg, S-O. Londen, and O. Saffans, Volerra inegral and funcional equaions, Cambridge Universiy Press, Cambridge, Prüss J, Posiiviy and regulariy of hyperbolic volerra equaions in Banach spaces, Mah. Ann. 279 (987, , Evoluionary inegral equaions and applicaions, Birkhäuser, Basel, 995. Insiue of Mahemaics, Helsinki Universiy of Technology, P.O. Box, FIN- 25 HUT, Finland address: gusaf.gripenberg@hu.fi, ggripenb