A quadratic interaction estimate for conservation laws: motivations, techniques and open problems*

Transcription

1 Bull Braz Mat Soc, New Series 47(2), , Sociedade Brasileira de Matemática ISSN: (Print) / (Online) A quadratic interaction estimate for conservation laws: motivations, tecniques and open problems* Stefano Modena Abstract. In a series of joint works wit S. Biancini [3, 4, 5], we proved a quadratic interaction estimate for general systems of conservation laws. Aim of tis paper is to present te results obtained in te tree cited articles [3, 4, 5], discussing ow tey are related wit te general teory of yperbolic conservation laws. To tis purpose, first we explain wy tis quadratic estimate is interesting, ten we give a brief overview of te tecniques we used to prove it and finally we present some related open problems. Keywords: conservation laws, Glimm potentials, interaction estimates, Glimm sceme. Matematical subject classification: 35L65. 1 Introduction A system of conservation laws in one space dimension (see [8]) is a system of PDEs of te form u t + f (u) x = 0, (1.1) were u :[0, ) R R n is te unknown and f : R n R n is a given smoot (C 3 ) map, called flux, defined on a neigborood of te origin and satisfying te strict yperbolicity condition, i.e. te Jacobian Df(u) of f as n distinct eigenvalues λ 1 (u) < <λ n (u) in eac point u of its domain. Trougout tis paper, we will assume witout loss of generality (w.l.o.g.) tat λ k (u) [0, 1] for any k and for any u. (1.2) Received 26 Marc *Tis paper was written wile te autor was a P.D. student at Sissa, Trieste, Italy.

2 590 STEFANO MODENA Tis can always be acieved by a cange of variable in te (t, x)-plane. As it is customary, denote by r 1 (u),...,r n (u) te rigt eigenvalues (normalized to 1) associated to λ 1 (u),...,λ n (u) respectively: Df(u)r k (u) = λ k (u)r k (u), for any k = 1,...,n and for any u. Equation (1.1) is usually coupled wit an initial datum u(t = 0) =ū, (1.3) were ū : R R n is a given map, wit sufficiently small total variation. W.l.o.g. we assume also tat ū as compact support. It is well known tat classical (smoot) solutions to te Caucy problem (1.1), (1.3) are in general not defined on te wole time interval [0, ), evenifte initial datum is smoot, because tey develop discontinuities in finite time. On te oter side, te notion of distributional solution is too weak to guarantee te uniqueness. For tis reasons te notion of solution wic is typically used is te following one. Definition 1. Amapu :[0, ) R R n belonging to L 1 loc weak solution of te Caucy problem (1.1), (1.3) if: is said to be a (1) u satisfies equation (1.1) in te sense of distributions; (2) u is continuous as a map [0, ) L 1 loc (R; Rn ); (3) at time t = 0, u(0, x) =ū(x); (4) u satisfies someadditionaladmissibilitycriteria, wic comefrom pysical or stability considerations and guarantee te uniqueness of te solution. Many admissibility criteria ave been proposed in te literature: just to name a few, te Lax-Liu condition on socks (see [14, 16, 17]), te entropy condition (see [15]), te vanising viscosity criterion (see [2]). We do not want to enter into details: te interested reader can refer to te cited literature. Te existence of solutions to te Caucy problem (1.1), (1.3) can be proved troug several approximation algoritms, for example te wavefront tracking algoritm and te Glimm sceme. One of te main problems in te construction of te approximate solutions and in te proof of te convergence of te approximate solutions to te exact one is to find suitable bounds on some quantities, usually called amounts of interaction, wic, rougly speaking, measure ow strong te interaction between two colliding wavefronts is. In particular, new results were recently obtained by S. Biancini and te autor [3, 4, 5], about a

3 A QUADRATIC INTERACTION ESTIMATE 591 quadratic interaction estimate (see Teorem 1 below) on te cange in speed of te waves in a Glimm approximate solution. Tis quadratic estimate can be used to prove sarp results on te convergence rate of te Glimm sceme and could ave applications in regularity and stability analysis. Aim of tis paper is to present te results obtained in te tree cited articles [3, 4, 5], discussing ow tey are related wit te general teory of yperbolic conservation laws. To tis purpose, first we explain wy tis quadratic estimate is interesting, ten we give a brief overview of te tecniques we used to prove it and finally we present some related open problems. In particular, after introducing some notations about te Riemann problem in Section 2, in Section 3 we describe te main results on te convergence of te Glimm sceme in te genuinely non linear/linearly degenerate case. In Section 4 we see ow tese results can be extended to te case in wic no assumption on f is made except its strict yperbolicity. In Section 5 we summarize wat is done in te tree papers [3, 4, 5], stating our main result, namely Teorem 1, and presenting te main tecniques used in te proof. Finally Section 6 concludes te paper wit some open problems for wic te new tools introduced in [3, 4, 5] could be of elp. 2 Te Riemann problem Te basic ingredient to solve te Caucy problem (1.1), (1.3) is te solution of te Riemann problem, i.e. te Caucy problem wen te initial datum as te simple form { u L if x < 0, u(0, x) =ū(x) = (2.1) u R if x 0. Te solutionof te Riemann problem (1.1)-(2.1) was obtained first by P. Lax in 1957 [14], under te assumption tat eac caracteristic field is eiter genuinely non linear (GNL), i.e. λ k (u) r k (u) = 0foranyu or linearly degenerate (LD), i.e. λ k (u) r k (u) = 0foranyu. In tis case, if u R u L 1, using Implicit Function Teorem, one can find intermediate states u L = ω 0,ω 1,...,ω n = u R suc tat eac pair of adjacent states (ω k 1,ω k ) can be connected by eiter a sock or a rarefaction wave or a contact discontinuity of te k-t family. Te complete solution is now obtained by piecing togeter te solutions of te n Riemann problems (ω k 1,ω k ) on different sectors of te (t, x)-plane. In te general case (ere and in te rest of te paper, by general case we mean tat no assumption on f is made besides strict yperbolicity) te solution to te Riemann problem (u L, u R ) was obtained by S. Biancini and A. Bressan in [2]. Tey first construct, for any left state u L and for any family k = 1,...,n, a curve s Ts ku L of admissible rigt states, defined for s R small enoug,

4 592 STEFANO MODENA suc tat te Riemann problem (u L, Ts ku L ) can be solved by (countable many) admissible socks (in te sense of limit of viscosity approximations), contact discontinuities and rarefactions waves. Ten, as in te GNL/LD case, te global solution of (u L, u R ) is obtained by piecing togeter te solutions of n Riemann problems, one for eac family: u R = Ts n n Ts 1 1 u L. 3 Glimm approximate solutions in te GNL/LD case Te first result about existence of solutions to te Caucy problem (1.1), (1.3) can be found in te celebrated paper by J. Glimm [13] in 1965, in wic te existence of solutions is proved under te assumption tat eac caracteristic field is eiter GNL or LD. In [13], for any ε > 0 an approximate solution u ε (t, x) is constructed by recursion as follows. First of all, take any sampling sequence {ϑ i } i N [0, 1]. Te algoritm starts coosing, at time t = 0, an approximation ū ε of te initial datum ū, suc tat ū ε is compactly supported, rigt continuous, piecewise constant wit jumps located at point x = mε, m Z. We can tus separately solve te Riemann problems located at (t, x) = (0, mε), m Z. Tanks to (1.2), te solution u ε (t, x) can now be prolonged up to time t = ε. At t = ε a restarting procedure is used. Te value of u ε at time ε is redefined as u ε (ε+, x) := u ε (ε, mε + ϑ 1 ε), if x [mε, (m + 1)ε). (3.1) Te solution u(ε, ) is now again piecewise constant, wit discontinuities on points of te form x = mε, m Z. If te sizes of te jumps are sufficiently small, we can again solve te Riemann problem at eac point (t, x) = (ε,mε), m Z and tus prolong te solution up to time 2ε, were again te restarting procedure (3.1) is used, wit ϑ 2 instead of ϑ 1. Te above procedure can be repeated on any time interval [iε, (i + 1)ε], i N, as far as te size of te jump at eac point (iε, mε), i N, m Z, remains small enoug, or, in oter words, as far as Tot.Var.(u ε (t); R) 1. (3.2) In order to prove (3.2), Glimm introduces a uniformly bounded decreasing functional t Q Glimm (t) O(1)Tot.Var.(ū) 2, suc tat at any time iε, i N, Tot.Var.(u ε (iε+); R) Tot.Var.(u(iε ); R) O(1) ( Q Glimm (iε ) Q Glimm (iε+) ) (3.3). HereandintefollowingO(1) denotes a constant wic depends only on te flux f. As an immediate consequence, we get Tot.Var.(u ε (t); R) O(1)Tot.Var.(u ε (0); R) 1

5 A QUADRATIC INTERACTION ESTIMATE 593 and tus te solution u ε (t, x) can be defined on te wole (t, x)-plane [0, ) R. Te uniform bound on te Tot.Var.(u ε (t); R) yields a compactness on te family {u ε } ε : we can tus extract a converging subsequence, wic turns out to be, for almost every sampling sequence {ϑ i } i, a weak admissible solution of te Caucy problem (1.1), (1.3). In 1977 T.-P. Liu [18] improved Glimm s result, sowing tat if te sampling sequence is equidistributed, tat means tat for any λ [0, 1], {i N 1 i j and ϑ i [0,λ]} lim = λ, j j ten te subsequence extracted from {u ε } ε converges to a weak admissible solution of (1.1), (1.3). A different approac wic relies on results about te stabilty of te solution of (1.1), (1.3) wit respect to (w.r.t.) te initial datum ū led to te introduction of te notion of standard Riemann semigroup. Definition 2. A standard Riemann semigroup for te system of conservation laws (1.1) is a map S : D [0, ) D, defined on a domain D L 1 (R; R n ) containing all functions wit sufficiently small total variation, wit te following properties: 1. for some Lipscitz constants L, L, S t ū S s v 1 L ū v 1 + L t s, for any ū, v D, t, s 0; (3.4) 2. if ū D is piecewise constant, ten for t > 0 sufficiently small S t ū coincides wit te solution of (1.1), (1.3), wic is obtained by piecing togeter te standard self-similar solutions of te corresponding Riemann problems. In te GNL/LD case it is proved (see, among oters, [9], [20], [11]) tat any system of conservation laws admits a standard Riemann semigroup and tat at any time t 0 te solution u(t) obtained as limit of Glimm approximations u ε (t), for te initial datum ū, coincides wit te semigroup S t ū. We will discuss in te next section te general case. Relying on te existence of te standard Riemann semigroup for GNL/LD systems, in 1998 A. Bressan and A. Marson [12] furter improved te Glimm sampling metod, constructing an equidistributed sequence {ϑ i }, satisfying te

6 594 STEFANO MODENA additional assumption: sup λ {i N j 1 i < j 2 and ϑ i [0,λ]} λ [0,1] j 2 j 1 C 1 + log( j 2 j 1 ) j 2 j 1. (3.5) Using tis sequence, tey are able to prove tat te rate of convergence of te Glimm approximate solutions u ε (t) to te exact weak admissible solution u(t) = S t ū at any time t is given by lim ε 0 u ε (t, ) S t ū L 1 log ε ε = 0. (3.6) Te tecnique used in [12] to prove (3.6) is as follows. Tanks to te Lipscitz property of te semigroup (3.4), in order to estimate te distance u ε (t, ) u(t, ) L 1 = u ε (t, ) S t ū L 1, we can partition te time interval [0, t] in subintervals J r := [t r, t r+1 ] and estimate te error u ε (t r+1 ) S tr+1 t r u ε (t r ) L 1 (3.7) on eac interval J r. Te error (3.7) on J r comes from two different sources: 1. first of all tere is an error due to te fact tat in a Glimm approximate solution, rougly speaking, we give eac wave eiter speed 0 or speed 1 (according to te sampling sequence {ϑ i } i ), wile in te exact solution it would ave a speed in [0, 1], but not necessarily equal to 0 or 1; 2. secondly, tere is an error due to te fact tat some waves can be created at times t > t r, some waves can be canceled at times t < t r+1 and, above all, some waves, wic are present bot at time t r and at time t r+1, can cange teir speeds, wen tey interact wit oter waves. Te first error source is estimated by coosing te intervals J r sufficiently large in order to use estimate (3.5) wit j 2 j 1 1. Te second error source can be estimated (coosing te intervals J r not too large) if we are able to bound te cange in speed of te waves present in te approximate solution. In te GNL/LD case, tis was acieved by Liu in [18], were e provides a wave tracing algoritm wic splits eac wavefront in te approximate solution into a finite number of discrete waves, wose trajectories

7 A QUADRATIC INTERACTION ESTIMATE 595 can be traced and wose canges in speed at any interaction time are bounded by te corresponding decrease of te functional Q Glimm. As ε 0, it is convenient to coose te asymptotic size of te intervals J r in suc a way tat te errors in (1) and (2) ave approximately te same order of magnitude. In particular, te estimate (3.6) is obtained by coosing J r ε log log ε. 4 Glimm approximate solutions in te general case All te results in te previous section were obtained under te assumption tat eac caracteristic field is eiter GNL or LD. In tis section we consider now te general case, wen tis assumption is removed and te only property of f is its strict yperbolicity. Te problem of finding a suitable decreasing potential to bound te increase of t Tot.Var.(u ε (t); R) for a Glimm approximate solution u ε (see (3.3)) was solved first by T.-P. Liu in [19] for fluxes wit a finite number of inflection points. Later, in [1], Biancini solved te problem for general yperbolic fluxes, introducing te cubic functional t Q cubic (t) := σ(t, s) σ(t, s ) dsds O(1)Tot.Var.(u ε (t)) 3, (4.1) were s, s are two waves in te approximate solution at time t and σ(t, s), σ(t, s ) denote teir speed. In [2] Biancini and Bressan also proved tat any strictly yperbolic f admits a standard Riemann semigroup (see Definition 2 above) of vanising viscosity solutions wit small total variation obtained as te (unique) limits of solutions to te viscous parabolic approximations u t + f (u) x = μu xx, (4.2) wen te viscosity μ 0. About te rate of convergence of te Glimm sceme wen an equidistibuted sequence satisfying (3.5) is used, te main problem in extending te proof by Bressan and Marson to te general case is to partitioneac wavefront in te approximate solution in waves wose trajectories can be traced and wose canges in speed are bounded by some suitable decreasing functional. Indeed, if suc a functional is provided, considerations similar to te ones in previous section can be made (see Points (1)-(2) above and te consequent analysis). Wile Glimm potential Q Glimm fits appropriately in te GNL/LD case to control te cange in speed of te waves, it is not apt anymore in te general case. We try to explain

8 596 STEFANO MODENA te problem as follows. At any interaction time t = iε, te cange in speed of te waves, integrated over all te possible waves, σ ( iε, s ) σ ( (i 1)ε, s ) ds (4.3) satisfies waves waves σ ( iε, s ) σ ( (i 1)ε, s ) ds Lip( f )Tot.Var.(u ε (t)) 2 i.e. it is quadratic w.r.t. te total variation of te solution at time t, tanks to te Lipscitzianity of f. Glimm potential Q Glimm is also quadratic w.r.t. Tot.Var.(u ε (t)) and tus it is reasonable to tink tat it can be used to bound not only te increase of te Total Variation (3.3), but also te cange in speed of te waves (4.3). On te oter and, te potential Q cubic in (4.1) is cubic w.r.t. Tot.Var.(u ε (t)) and tus tere is no ope to use it in order to bound a quadratic quantity suc (4.3). 5 A new quadratic interaction functional In a series of joint works wit S. Biancini ([3] for te scalar case, [4] for a simple system case, [5] for te general system case, see also [21] for a summary of te work done in te tree cited papers), we construct a new quadratic functional t Q quadr (t), defined for any Glimm approximate solution u ε (t, x), wose decrease at eac interaction controls te cange in speed (4.3) of te waves. More precisely, we first construct a wave tracing algoritm (wic we call Lagrangian representation) wic splits eac wavefront in te approximate solution u ε into infinite many infinitesimal waves as follows: for any family k = 1,...,n, we define a set of real numbers W k R, called te set of (infinitesimal) waves; foranywavew W k, we define its sign S(w) { 1, 1}, its time of creation t cr (w) [0, ) and its time of cancellation t canc (w) (0, ]; foranywavew W k and any time t [t cr (w), t canc (w)) we define te position x(t,w)of te wave w at time t and te speed σ(t,w)of te wave w at time t. Te speed σ(t,w) and te position x(t,w) are defined by recursion on time intervals [iε, (i + 1)ε], i N and are related to eac oter as described below:

9 A QUADRATIC INTERACTION ESTIMATE 597 te speed σ(t,w)is te speed given to te wave w by te Riemann problem located at point (t, x(t,w)); te position x(t,w)of te wave w at time t is related to te speed σ(t,w) by te formula for any t [iε, (i + 1)ε), x(t,w)= { x(iε, w) if σ(iε, w) ϑ i+1 x(iε, w) + t iε if σ(iε, w) > ϑ i+1, wic takes into account te sampling metod used to construct te Glimm approximate solution u ε. Ten, for any grid point (iε, mε) Nε Zε, wedefinetecange in speed of te waves wic at time iε ave position mε and wic exist also at time (i 1)ε: σ(iε, mε) := n k=1 were for any time t and point x W k (iε,mε) W k ((i 1)ε) σ(iε, w) σ((i 1)ε, w) dw W k (t) := { w W k t [t cr (w), t canc (w)) }, W k (t, x) := { w W k (t) x(t,w)= x }. Te teorem we prove in [5] is te following. Teorem 1. It olds + σ(iε, mε) O(1)Tot.Var.(ū; R) 2. (5.1) i=1 m Z Recall tat O(1) isa constant wicdepends onlyon te flux f. Te proof of Teorem 1 follows a classical approac used in yperbolic systems of conservation laws in one space dimension. We first prove a local estimate. For te couple of Riemann problems (u L, u M ), (u M, u R ), we define te quantity + n ( =1 A quadr A(u L, u M, u R ) := A trans (u L, u M, u R ) (u L, u M, u R ) + A canc (u L, u M, u R ) + A cubic (u L, u M, u R ) ), (5.2)

10 598 STEFANO MODENA wic we will call te global amount of interaction of te two merging RPs (u L, u M ), (u M, u R ). Tree of te terms in te r..s. of (5.2) ave already been introduced in te literature, namely A trans (u L, u M, u R ) is te transversal amount of interaction (see [13] and Definition 2.5 in [5]) and measures te strengt of te interactions between waves of different families; A canc (u L, u M, u R ) is te amount of cancellation of te -t family (see Definition 2.8 in [5]) and measures ow many waves of te -t family are canceled, if two waves of te -t family wit different sign interact; A cubic (u L, u M, u R ) is te cubic amount of interaction of te -t family (see [1] and Definition 2.6 in [5]) and measures ow many waves of te -t family are canceled and created in te interaction because of te full nonlinearity of te flux f. Te term A quadr (u L, u M, u R ) is te quadratic amount of interaction of te -t family (see Definition 3.1 in [5]) and it is introduced for te first time in [5]; it measures te cange in speed of te waves wen two wavefronts of te same family and aving te same sign interact. Te local estimate we prove is te following: at any grid point (iε, mε), te cange in speed is bounded by σ k (u L, u M, u R ) O(1)A(u i,m 1, u i 1,m 1, u i,m ) =: A(iε, mε), (5.3) were, for any j N and r Z, u j,r := u ε ( jε, rε).seesection3in[5]. Next we sow a global estimate, based on a new interaction potential. For any grid point (iε, mε) define A trans (iε, mε), A canc (iε, mε), A cubic (iε, mε), A quadr (iε, mε) as te transversal amount of interaction, te amount of cancellation, te cubic amount of interaction and te quadratic amount of interaction respectively of te two Riemann problems (u i,m 1, u i 1,m 1 ), (u i 1,m 1, u i,m ) interacting at te grid point (iε, mε). We introduce a new interaction potential ϒ wit te following properties: (a) it is uniformly bounded at time t = 0: in fact, ϒ(0) O(1)Tot.Var.(ū; R) 2 ; (b) it is constant on time intervals [(i 1)ε, iε);

11 A QUADRATIC INTERACTION ESTIMATE 599 (c) at any time iε, it decreases at least of 1 2 m Z A(iε, mε). It is fairly easy to see tat Points a, b, c above, togeter wit inequality (5.3), imply Teorem 1. Te potential ϒ is constructed as follows. We define a quadratic functional t Q quadr (t), constant in te time intervals [(i 1)ε, iε) and bounded by O(1)Tot.Var.(ū; R) 2 at t = 0, wic satisfies te following inequality: Q quadr (iε) Q quadr ((i 1)ε) n A quadr (iε, mε) m Z =1 + O(1)Tot.Var.(ū; R) A(iε, mε) m Z = ( 1 O(1)Tot.Var.(ū; R) ) n m Z =1 + O(1)Tot.Var.(ū; R) m Z A trans (iε, mε) A quadr (iε, mε) (5.4) + O(1)Tot.Var.(ū; R) m Z n ( =1 A canc (iε, mε) + A cubic It is well known (see [13], [1]) tat te decreasing potential ) (iε, mε). Q known (t) := c 1 Tot.Var.(u ε (t)) + c 2 Q trans (t) + c 3 Q cubic (t) at eac time iε satisfies [ n ( A trans (iε, mε) + m Z =1 A canc ) ] (iε, mε) + A cubic (iε, mε) Q known ((i 1)ε) Q known( iε ). (5.5) Here Q trans (t) is te transversal part of te Glimm potential Q Glimm (t) (see Section 2.4 in [5] for a precise definition). It is straigtforward to see from (5.4), (5.5) tat we can find a constant C big enoug, suc tat te potential ϒ(t) := Q quadr (t) + CQ known (t) satisfies Properties (a)-(c) above, provided tat Tot.Var.(ū; R) 1. We would like to stress te main differences between our constructions (te Lagrangian representation and te quadratic functional Q quadr ) and all te wave tracing algoritms and functionals already present in te literature.

12 600 STEFANO MODENA First of all, our Lagrangian representation is continuous: eac wavefront in te approximate solution is partitioned into an infinite number of waves wose strengt is infinitesimal, wile all previous wave tracing algoritm are based on te partition provided by Liu in [18], were eac wavefront in te approximate solution is partitioned into a finite number of discrete waves. Secondly, our functional is non-local in time. All previous functionals t Tot.Var.(u(t)), Q Glimm (t), Q cubic (t) are local in time: in order to compute teir value at a fixed time t it is enoug to know te approximate solution u ε ( t) at time t. On te contrary, to compute te value of our functional Q quadr ( t) at time t, one must know te beavior of te solution u ε at any time t [0, ). A non-local in time functional is needed for te following reason. In te GNL/LD case, te functional Q Glimm fits appropriately to estimate te cange in speed (4.3), because two waves w, w wic interact at time t (i.e. x( t,w)= x( t,w )), will ave te same position at any time t t: x(t,w) = x(t,w ) for any t t. No splitting can occur. Te main problem in te general case is tat tis property is not true anymore. Indeed two waves wic interact at time t can split after time t, due to an interaction wit a wave of te same family but different sign, or due to an interaction wit a wave of a different family. Later tey can interact again, ten tey can split again and so on. Tis penomenon can appen an unbounded number of times. Te non-locality in time of Q quadr takes into account tis kind of beavior. Let us sketc now briefly te definition of te functional t Q quadr (t). We set n Q quadr (t) := q(t,w,w )dwdw, k=1 W k (t) W k (t) were q(t,w,w ) is called te weigt of te pair of waves (w, w ) at time t and its value is determined by te past and future common istory of (w, w ), from te last time t split (t,w,w ) t before time t wen w, w interact, to te first time t int (t,w,w )>t after time t wen tey interact again. More precisely, we define te interval of waves I(t,w,w ) as te smallest interval wic contains all te waves wic ave te same position as w, w at time t split (t,w,w ) and wic ave not been canceled up to time t. Associated to tis interval, we define a partition P(t,w,w ) of I(t,w,w ): te elements of tis partition are waves wit te same past istory, from te time t split (t,w,w ). Te weigt q(t,w,w ) can tus be defined (rougly speaking) as q k (t,w,w ) difference in speed of w,w, assigned troug te partition P(t,w,w ) of te interval I(t,w,w ) wit an appropriate flux function f eff k lengt of te interval I(t,w,w. )

13 A QUADRATIC INTERACTION ESTIMATE 601 Te precise form is sligtly more complicated, in order to minimize te oscillations of q k in time. Te flux function f eff k at time t split (t,w,w ) is defined as te reduced flux function wic splits w, w ; at any later time t > t split (t,w,w ) it canges taking into account all te transversal interactions wic occur in te time interval (t split (t, s, s ), t]. See Section 6 in [5] for a precise definition. 6 Open problems Even if Teorem 1 as been proved to extend formula (3.6) to a general system wit a strictly yperbolic flux f, it is also interesting by itself, since we tink tat it could be used to prove furter results on conservation laws. In particular, in a work under preparation [6], S. Biancini and te autor are constructing a Lagrangian representation for te exact solution S t ū, passing to te limit te Lagrangian representation for te Glimm approximate solutions u ε and using Teorem 1. Tis limit Lagrangian representation can be viewed as a sort of generalized caracteristics metod, or, in oter words, as a continuous wavefront tracking, wic leads to a deeper understanding of te fine structure of te solutions. Bot Teorem 1 and te limit Lagrangian representation could be used to investigate te following problems. 6.1 Regularity of solutions to systems of conservation laws It is well known tat te solution u(t, x) is a BV function of te two variables (t, x). Hence is sares te regularity properties of general BV functions. In particular eiter u is approximately continuous or it as an approximate jump at eac point (t, x), wit teexception ofaset N woseone-dimensional Hausdorff measure is zero. However it seems tat muc stronger regularity properties old. In te GNL/ LD case it is proved (see, for instance, [10] and [7]) tat te set N is countable and u is continuous (not just approximately continuous) outside N and outside a countable family Ɣ of Lipscitz sock curves; moreover at any point Ɣ \ N, te solution as left and rigt limits (not just approximate limits). We tink tat te Lagrangian representation we introduce in [6] can be of great elp to investigate similar regularity properties for a solution to a general Caucy problem (1.1), (1.3), witout any assumption on f except te strict yperbolicity.

14 602 STEFANO MODENA 6.2 Structural stability of solutions to systems of conservation laws Anoter interesting problem,reletated to te previous one concerns te structural stability of te solutions to system of conservation laws: one expects tat te strengt and te location of a large sock is not affected too muc by a small perturbation of te initial datum. Various results in tis direction, but still in te GNL/LD case, can be found in [10, 7]. It would be interesting to extend suc results to te general case. 6.3 L 1 stability of solutions to system of conservation laws via yperbolic metods As observed in Section 4, te paper [2] by Biancini and Bressan provides also te L 1 stability of te solutions to systems of conservation laws. However, in teir work te autors use stability estimates on te parabolic equation (4.2) to get te stability result (3.4) for te yperbolic system. In several previous works by different people (see, among many, [9], [20]), te stability of genuinely non linear/linearly degenerate systems is proved using purely yperbolic tecniques. One of tese tecniques relies on a functional (t) = (u(t), v(t)) defined on pairs of solutions, equivalent to te L 1 norm, wit te fundamental property tat it is almost decreasing in time. Tis functional works only in te GNL/LD setting, wile it is not clear ow to define a similar functional in te general case. Te tools we developed in [3, 4, 5] could be of elp in answering tis question. 6.4 Quadratic interaction estimate for viscous conservation laws Finally, a very interesting question concerns te proof of a quadratic estimate similar to (5.1) for te conservation law (4.2) wit a viscosity term μu xx,in wic we can assume w.l.o.g. μ = 1 (see [2]). Wile te building block of te solution of te yperbolic equation (1.1) is te Riemann problem, in te viscous case te building block is te viscous traveling profile, i.e. a solution of te form wic satisfies te second order ODE u(t, x) = U(x λt) (6.1) U = (Df(U) λ)u. In tis case te velocity of te viscous profile is λ and it olds λ = u t u x

15 A QUADRATIC INTERACTION ESTIMATE 603 In te inequality (5.1) te l..s. is te sum over all grid points (iε, mε), i N, m Z on te (t, x) plane of te cange in speed of te wavefronts present at point (iε, mε) multiplied by teir strengt. Hence its equivalent in te viscous setting is te integral over te (t, x) plane of te cange in speed multiplied by te strengt of te viscous profile at point (t, x),i.e. u t t [0, ) R u ux dxdt = u tt u t u tx x [0, ) R u dxdt (6.2) x Wat would be nice to obtain is an uniform estimate of (6.2) in terms of te total variation of te initial datum. However, toug te term to estimate can be written down in a very clean form (6.2), it is not clear at all wic tecnique can be used in order to estimate it. Indeed, te metod we use to prove (5.1) relies eavily on te wave tracing algoritm we developed, in particular on te notion of position of a given wave w at a given time t: in te viscous case it is not clear any more wat tis concept means, since te traveling profiles (6.1) are not localized in space, and tus it is not at all obvious ow to define a suitable functional wic can play te role tat Q quadr as in te yperbolic case. It would be tus nice to develop an analog teory on viscous conservation laws wic leads to estimate te term (6.2). Acknowledgments. Te autor wises to tank prof. Stefano Biancini and prof. Fabio Ancona for many elpful discussions about te topics of tis paper. References [1] S. Biancini. Interaction Estimates and Glimm Functional for General Hyperbolic Systems. Discrete and Continuous Dynamical Systems, 9 (2003), [2] S. Biancini and A. Bressan. Vanising viscosity solutions of nonlinear yperbolic systems. Annals of Matematics, 161 (2005), [3] S. Biancini and S. Modena. On a quadratic functional for scalar conservation laws. Journal of Hyperbolic Differential Equations, 11(2) (2014), [4] S. Biancini and S. Modena. Quadratic interaction functional for systems of conservation laws: a case study. Bulletin of te Institute of Matematics, Academia Sinica (New Series), 9(3) (2014), [5] S. Biancini and S. Modena. Quadratic interaction functional for general systems of conservation laws, accepted for publicationon Comm. Mat. Pys., (2015). [6] S. Biancini and S. Modena. Lagrangian representation for solution to general systems of conservation laws, in preparation. [7] S. Biancini and L. Yu. Global Structure of Admissible BV Solutions to Piecewise Genuinely Nonlinear, Strictly Hyperbolic Conservation Laws in One Space Dimension. Comm. PDE, 39(2) (2014),

16 604 STEFANO MODENA [8] A. Bressan. Hyperbolic Systems of Conservation Laws. Te One Dimensional Caucy problem. Oxford University Press (2000). [9] A. Bressan, G. Crasta and B. Piccoli. Well posedness of te Caucy problem for n n conservation laws.amer.mat.soc.memoir,694 (2000). [10] A. Bressan and P. LeFloc. Structural stability and regularity of entropy solutions to yperbolic systems of conservation laws. Indiana Univ. Mat. J., 48 (1999), [11] A. Bressan, T.-P. Liu and T. Yang. L 1 stability estimates for n n conservation laws. Arc. Rat. Mec. Anal., 149 (1999), [12] A. Bressan and A. Marson. Error Bounds for a Deterministic Version of te Glimm Sceme. Arc. Rational Mec. Anal., 142 (1998), [13] J. Glimm. Solutions in te Large for Nonlinear Hyperbolic Systems of Equations. Comm. Pure Appl. Mat., 18 (1965), [14] P.D. Lax. Hyperbolic systems of conservation laws II. Comm. Pure Appl. Mat., 10 (1957), [15] P.D. Lax. Sock waves and entropy, in Contribution to Nonlinear Functional Analysis, E. Zarantonello Ed., Academic Press, New York, (1971), [16] T.P. Liu. Te Riemann problem for general 2 2 conservation laws. Trans. Amer. Mat. Soc., 199 (1974), [17] T.P. Liu. Te Riemann problem for general systems of conservation laws. J.Diff. Eqs., 18 (1975), [18] T.-P. Liu. Te deterministic version of te Glimm sceme. Comm. Mat. Pys., 57 (1977), [19] T.-P. Liu. Admissible solutions of yperbolic conservation laws.amer.mat.soc. Memoir, 240 (1981). [20] T.-P. Liu and T. Yang. A new entropy functional for scalar conservation laws. Comm. Pure Appl. Mat., 52 (1999), [21] S. Modena. Quadratic Interaction Estimate for yperbolic conservation laws, an overview, accepted for publication on te Proceedings of te DFDE2014 Conference in Contemporary Matematics. Fundamental Directions (2015). Stefano Modena Matematisces Institut Universität Leipzig Augustusplatz 10/ Leipzig GERMANY Stefano.Modena@mat.uni-leipzig.de