New interaction estimates for the Baiti-Jenssen system
NNEW INTERACTION ESTIMATES FOR THE BAITI-JENSSENSYSTEM
Laura Caravenna
Dipartimento di Matematica,Universit`a degli Studi di Padova,via Trieste 63, 35121 Padova, Italy
Laura V. Spinolo
IMATI-CNR,via Ferrata 1, 27100 Pavia, Italy
Abstract.
We establish new interaction estimates for a system introducedby Baiti and Jenssen. These estimates are pivotal to the analysis of the wavefront-tracking approximation. In a companion paper we use them to con-struct a counter-example which shows that Schaeffer’s Regularity Theorem forscalar conservation laws does not extend to systems. The counter-example weconstruct shows, furthermore, that a wave-pattern containing infinitely manyshocks can be robust with respect to perturbations of the initial data. Theproof of the interaction estimates is based on the explicit computation of thewave fan curves and on a perturbation argument. Introduction
We deal with the system of conservation laws(1) ∂ t U + ∂ x (cid:2) F η ( U ) (cid:3) = 0 . The unknown U = U ( t, x ) attains values in R : U : [0 , + ∞ [ × R → R ( t, x ) (cid:55)→ U = uvw and the flux function F η : R → R is defined as(2) F η ( U ) := (cid:2) ( v − u − w (cid:3) + ηp ( U ) v (cid:110) v ( v − u − ( v − w (cid:111) + ηp ( U ) . In the previous expression, the parameter η attains values in the interval [0 , / p and p by setting p ( U ) = 2 uw − u ( v − , (3) p ( U ) = w − u ( v − v. (4) Mathematics Subject Classification.
Primary: 35L65.
Key words and phrases. conservation laws, regularity, shock formation, Schaeffer Theorem,counter-example. a r X i v : . [ m a t h . A P ] O c t LAURA CARAVENNA AND LAURA V. SPINOLO
Note, however, that for some of the results discussed in the following the preciseexpression of the functions p and p is irrelevant.System (1),(2) was introduced by Baiti and Jenssen in [3, 19] and it was usedto construct an example of a Cauchy problem where the initial data have finite,but large, total variation and the L ∞ -norm of the admissible solution blows up infinite time. More recently, the authors of the present paper used the Baiti-Jenssensystem (1) to exhibit an explicit counter-example which shows that Schaeffer’s reg-ularity result for scalar conservation laws does not extend to systems, see [11]. Thecounter-example we construct shows, furthermore, that a wave-pattern containinginfinitely many shocks can be robust with respect to perturbations of the initialdata. We refer to § U (0 , · ) = U . We refer to [5, 14, 18] foran extended discussion on the wave front-tracking approximation. Here we onlymention that the wave front-tracking algorithm is based on the construction of apiecewise constant approximation of the Cauchy problem. Under suitable conditionson the initial datum U and on the flux function F η , one can show that the wavefront-tracking approximation converges to an admissible solution of the Cauchyproblem, see in particular the analysis in [5]. In [11] we construct wave front-tracking approximations of the Cauchy problems obtained by coupling (1) withsuitable initial data. We then rely on the wave front-tracking approximation toestablish qualitative properties of the limit solutions. In the following we do notconsider all the possible interactions one has to handle when constructing the wavefront-tracking approximation. We only discuss those that we encounter in [11] andthat cannot be handled by relying on straightforward considerations on the structureof the flux F η .Before going into the technical details, we make some further remarks. First, inthe present note we fix a very specific system in the wider class considered in [3].The motivation for this choice is twofold: i) it simplifies the notation and ii) theanalysis in the present note is sufficient for the applications in [11]. Note, moreover,that in the proof of Lemma 1.1 we use (although not in an essential way) the exactexpression of the function F η evaluated at η = 0. However, we are confident thatour results can be extended to wider classes of systems of the type considered in [3].Second, in this note the only occurrences where we explicitly use the preciseexpression of the functions p and p is in the results discussed in § η = 0and then we show that the same holds provided η is sufficiently small. The proof ofLemma 1.2 is completely independent of the specific expression of p and p . In theperturbation argument in the proof of Lemma 1.1 we use some results from § p and p .Third, the Baiti-Jenssen (1) system in not physical, in the sense that it does notadmit strictly convex entropies, see [3] for a proof. It is natural to wonder whether ornot the results established in the present note can be extended to physical systems.Very loosely speaking, by combining Lemmas 1.2 and 1.1 below with the analysisin [11, § NTERACTION ESTIMATES 3 waves generated at the interactions between two shocks are shock waves, or, moreprecisely, no rarefaction waves are generated at the interaction between two shocks.There are actually several physical systems that share this property: for instance,one can consider the 2 × §
5] and assumethat the data have sufficiently small total variation. We refer to [6, §
4] for theanalysis of shock interactions for this system. On the other hand, a much morechallenging question is whether or not there is any physical system that exhibitthe same behaviors as those discussed in [3, 11]. In other words, one can wonderwhether or not a physical system can i) exhibit finite time blow up or ii) violatethe regularity prescribed, for scalar conservation laws, by Schaeffer’s Theorem. Tothe best of the authors’ knowledge, the answers to the above questions is presentlyopen.We now give some technical details about the estimates we establish. First, wepoint out that the Baiti-Jenssen system (1) is strictly hyperbolic in the unit ball,which amounts to say that the Jacobian matrix DF η admits three real and distincteigenvalues(5) λ ( U ) < λ ( U ) < λ ( U )for every U such that | U | <
1. Also, if η > (cid:126)r , . . . , (cid:126)r denote the right smooth eigenvectors asso-ciated to the eigenvalues λ , λ , λ . Then(6) ∇ λ i ( U ) · (cid:126)r i ( U ) ≥ c > c > i = 1 , , | U | <
1. In thefollowing, we distinguish three families of shocks: we term a given shock 1-, 2- or3-shock depending on whether the speed of the shock is close to λ , λ or λ .We also point out that establishing interaction estimates for system (1) boilsdown to the following. Consider the so-called Riemann problem, namely the Cauchyproblem obtained by coupling (1) with an initial datum in the form(7) U (0 , x ) := (cid:26) U (cid:96) x < U r x > , where U (cid:96) , U r ∈ R are constant states. The above problem admits, in general, in-finitely many distributional solutions: we term admissible the solution constructedby Lax in the pioneering work [21], see § U (cid:96) and U r satisfy suitable structuralassumptions.The first case we consider is the case of the interaction of two 2-shocks, seeFigure 1, left part. In other words, we assume that there is a state U m ∈ R suchthat • U (cid:96) and U m are the left and the right states of a Lax admissible 2-shock, • U m and U r are the left and the right states of a Lax admissible 2-shock and • the shock between U (cid:96) and U m has higher speed than the shock between U m and U r .We now give an heuristic formulation of our interaction estimate and we refer to § strength of a shock is a quantity defined in § LAURA CARAVENNA AND LAURA V. SPINOLO
Figure 1.
Left: interaction between two 2-shocks. Right: inter-action between a 2-shock and a 1-shock. U ℓ U m U r U ℓ U m U r Lemma 1.1.
Fix a constant a such that < a < / and set U (cid:93) := ( a, , − a ) .Consider the interaction between two 2-shocks and assume that the states U (cid:96) and U r are sufficiently close to U (cid:93) . If the strengths of the interacting 2-shocks are sufficientlysmall, then the admissible solution of the Riemann problem (1) - (7) is obtained bypatching together a 1-shock, a 2-shock and a 3-shock. We remark that the relevant point in the above result is that the solution of theRiemann problem that we consider in the statement contains no rarefaction wave.The second case we consider is the case of the interaction between a 1-shock anda 2-shock, see Figure 1, right part. In other words, we assume that there is a state U m ∈ R such that • U (cid:96) and U m are the left and the right states of a Lax admissible 2-shock, • U m and U r are the left and the right states of a Lax admissible 1-shock.The case of the interaction of a 3-shock with a 2-shock is analogous. We now givean heuristic formulation of our result and we refer to § Lemma 1.2.
Consider the interaction between a -shock and a -shock and assumeboth shocks have sufficiently small strength. Then the admissible solution of theRiemann problem (1) - (7) is obtained by patching together a 1-shock, a 2-shock anda 3-shock. Also, we establish quantitative bounds from above and from below on thestrength of the outgoing shocks, see formulas (35) . Note that the fact that the three outgoing waves are shocks follows from theanalysis in [3]. Also, the bound from above on the strength of the outgoing 3-shocks follows from by now classical interaction estimates, see [5, Page 133, (7.31)]:the main novelty in Lemma 1.2 is that we have a new bound from below on thestrength of the outgoing 3-shock, see the left hand side of formula (35). Thisestimate is important for the analysis in [11].This note is organized as follows. In § § § § § § NTERACTION ESTIMATES 5 Overview of previous results
For the reader’s convenience, in this section we go over some previous results.More precisely: § § § Counter-examples based on the Baiti-Jenssen system.
This paragraphis organized as follows: § § η > i = 1 , ,
3. This is a remarkable property becauseloosely speaking systems where all the characteristic field are genuinely nonlinearare usually better behaved than general systems. For instance, the celebrated decayestimate by Ole˘ınik [23], which applies to scalar conservation laws with convexfluxes, has been extended to systems of conservation laws where all the characteristicfield are genuinely nonlinear, see for instance the works by Glimm and Lax [17], byLiu [22] and, more recently, by Bressan and Colombo [7], Bressan and Goatin [8]and Bressan and Yang [9], while for balance laws we refer to Christoforou andTrivisa [12].2.1.1.
Finite time blow up of admissible solutions with large total variation.
Con-sider the general system of conservation laws(8) ∂ t U + ∂ x (cid:2) F ( U ) (cid:3) = 0 , where the unknown U ( t, x ) attains values in R N , the variables ( t, x ) ∈ [0 , + ∞ [ × R and the flux function F : R N → R N is smooth and strictly hyperbolic (5). Considerfurthermore the Cauchy problem obtained by coupling (8) with the initial condition(9) U (0 , · ) = U . Under some further technical assumption on the structure of the flux, Glimm [16]established existence of a global in time solution of the Cauchy problem providedthat TotVar U , the total variation of the initial datum, is sufficiently small. Underthe same assumptions, Bressan and several collaborators established uniquenessresults, see [5] for a detailed exposition.The requirement that the total variation TotVar U is small is highly restrictive,but necessary to obtain well-posedness results unless additional assumptions areimposed on the flux function F . Indeed, explicit examples have been constructedof systems and data U where TotVar U is finite, but large, and the admissiblesolution blows up in finite time. In particular, in [3] Baiti and Jenssen constructedan initial datum for system (1) such that the L ∞ -norm of the admissible solutionblows up in finite time. The solution is admissible in the sense that it is piecewiseconstant and every shock is Lax admissible. For further examples of finite timeblow up, see the references in [3] and [14]. LAURA CARAVENNA AND LAURA V. SPINOLO
Schaeffer’s Regularity Theorem does not extend to systems.
In [24] Schaefferestablished a regularity result which can be loosely speaking formulated as follows.Consider a scalar conservation law with strictly convex flux, namely equation (8)in the case when U ( t, x ) attains real values and F : R → R is uniformly convex, i.e. F (cid:48)(cid:48) ≥ c > c >
0. The work by Kruˇzkov [20] establishes existenceand uniqueness of the so-called entropy admissible solution of the Cauchy problemposed by coupling (8) and (9). It is known that, even if U is smooth, the entropyadmissible solution can develop shocks, namely discontinuities that propagate in the( t, x )-plane. Schaeffer’s Theorem states that, for a generic smooth initial datum,the number of shocks of the entropy admissible solution is locally finite. The word“generic” is here to be interpreted in a suitable technical sense, which is related tothe Baire Category Theorem, see [24] for the precise statement.In [11] we discuss whether or not Schaeffer’s Theorem extends to systems ofconservation laws where every characteristic field is genuinely nonlinear, namely (6)holds. Note that the assumption that every characteristic field is genuinely nonlinearcan be loosely speaking regarded as the analogous for systems of the condition(which applies to scalar equations) that the flux is strictly convex. Indeed, regularityresults for scalar equations with strictly convex fluxes have been extended to systemswhere every characteristic field is genuinely nonlinear: as we mentioned before, thisis the case of Ole˘ınik’s [23] decay estimate, see for instance [7, 8, 9, 12, 17, 22] forpossible extensions to systems. Also, the SBV regularity result by Ambrosio andDe Lellis [1], which applies to scalar conservation laws with strictly convex fluxes,has been extended to systems where every characteristic field is genuinely nonlinear,see [2, 4, 13].Despite the above considerations, in [11] we exhibit an explicit example whichrules out the possibility of extending Schaeffer’s Theorem to systems of conservationlaws where every characteristic field is genuinely nonlinear. More precisely, weconstruct a “big” set of initial data such that the corresponding solutions of theCauchy problems for the Baiti-Jenssen system (1) develop infinitely many shockson a given compact set of the ( t, x )-plane. The term “big” is to be again interpretedin a suitable technical sense, which is related to the Baire Category Theorem, see [11]for the technical details.2.2.
The Lax solution of the Riemann problem.
We consider a system ofconservation laws (8) and we assume that F : R → R is strictly hyperbolic (5) andthat every characteristic field is genuinely nonlinear, namely (6) holds for i = 1 , , U (0 , x ) := (cid:26) U − x < U + x > , where U + and U − are given states in R . In [21], Lax constructed a solution of theRiemann problem (8)-(10) under the assumptions that the states U + and U − aresufficiently close: we now briefly recall the key steps of the analysis in [21].We fix i = 1 , , U ∈ R and we define the i -wave fan curve through ¯ U bysetting(11) D i [ s, ¯ U ] := (cid:26) R i [ s, ¯ U ] s ≥ S i [ s, ¯ U ] s < NTERACTION ESTIMATES 7
In the previous expression, R i is the i -rarefaction curve through ¯ U and S i is the i -Hugoniot locus through ¯ U . The i - rarefaction curve R i is the integral curve of thevector field (cid:126)r i , namely the solution of the Cauchy problem(12) dR i ds = (cid:126)r i (cid:0) R i (cid:1) R i [0 , ¯ U ] = ¯ U .
The i -th Hugoniot locus S i is the set of states that can be joined to ¯ U by a shockwith speed close to λ i ( ¯ U ). The i -Hugoniot locus S i is determined by imposingthe Rankine-Hugoniot conditions. We term the value | s i | strength of the i -waveconnecting the states ¯ U (on the left) and D i [ s, ¯ U ] (on the right). Note that, owingto (11), when s i > i -wave is a i -th rarefaction wave, when s i < i -wave isan i -shock satisfying the so-called Lax admissibility criterion . The solution of theRiemann problem (8)-(10) is computed by imposing U + = D (cid:104) s , D (cid:2) s , D [ s , U − ] (cid:3)(cid:105) and by using the Local Invertibility Theorem to solve for ( s , s , s ). From thevalue of ( s , s , s ) one can reconstruct a solution of the Riemann problem (8)-(10),see [21] for the precise construction. This solution is obtained by patching togetherrarefaction waves and shocks that satisfy the Lax admissibility criterion. In thefollowing, we refer to this solution as the Lax solution of the Riemann problem (8)-(10).2.3.
The wave fan curves of the Baiti-Jenssen system.
We collect in thisparagraph some features of the Baiti-Jenssen system. For the proof, we refer to [3,11].The first result states that in the unit ball the Baiti-Jenssen system is strictlyhyperbolic whenever 0 ≤ η < /
4. Also, when η > η = 0 this last condition is lost becausetwo characteristic fields became linearly degenerate. See [3] or [11] for the explicitcomputations. Lemma 2.1.
Assume that ≤ η < / and that U varies in the unit ball, | U | < .Then the Baiti-Jenssen system with flux (2) is strictly hyperbolic, namely (5) holdstrue. If we also have η > then every characteristic field is genuinely nonlinear,namely (6) is satisfied for i = 1 , , . We now discuss the structure of the wave fan curves. We start by giving theexplicit expression of the 1- and the 3-wave fan curve. In the statement of thefollowing result, we denote by (¯ u, ¯ v, ¯ w ) the components of the state ¯ U ∈ R . Lemma 2.2.
Consider the flux function (2) , assume that < η < / and fix ¯ U ∈ R such that | ¯ U | < . Then the following properties hold true.i) The 1-wave fan curve D [ σ, ¯ U ] is a straight line in the plane v = ¯ v , moreprecisely D [ σ, ¯ U ] = ¯ U + σ(cid:126)r ( ¯ U ) , (13) where (cid:126)r ( ¯ U ) = v . LAURA CARAVENNA AND LAURA V. SPINOLO
Note that (cid:126)r ( ¯ U ) is the first eigenvector of the Jacobian matrix DF ( ¯ U ) . Also,the states ¯ U (on the left) and D [ σ, ¯ U ] (on the right) are connected by a wavewhich is – a 1-rarefaction wave when σ > , – a Lax admissible 1-shock when σ < .ii) The 3-wave fan curve D [ τ, ¯ U ] is a straight line in the plane v = ¯ v , moreprecisely D [ τ, ¯ U ] = ¯ U + τ(cid:126)r ( ¯ U ) , (14) where (cid:126)r ( ¯ U ) = v − . The vector (cid:126)r ( ¯ U ) is the third eigenvector of the Jacobian matrix DF ( ¯ U ) .Also, the states ¯ U (on the left) and D [ τ, ¯ U ] (on the right) are connected bya wave which is – a 3-rarefaction wave when τ < , – a Lax admissible 3-shock when τ > . Note that, for the 3-wave fan curve, the positive values of τ correspond to shocks,the negative values to rarefaction waves. This is the contrary with respect to (11)and it is a consequence of the fact that we use the same notation as in [3, 11] and wechoose the orientation of (cid:126)r in such a way that when η > ∇ λ · (cid:126)r < . We now turn to the structure of the 2-wave fan curve. In the following statement,we use the notation U − = u − v − w − , U + = u + v + w + . Also, we consider entropy admissible solutions of scalar conservation laws, in theKruˇzkov [20] sense.
Lemma 2.3.
Assume that U is a Lax solution of the Riemann problem (8) - (10) .Then the second component v is an entropy admissible solution of the Cauchy prob-lem (15) ∂ t v + ∂ x [ v ] = 0 v (0 , x ) = (cid:26) v − x < v + x > . Also, we can choose the eigenvector (cid:126)r and the parametrization of the 2-wave fancurve D [ s, ¯ U ] in such a way that the second component of D [ s, ¯ U ] is exactly ¯ v + s . Interaction of two 2-shocks
We first rigorously state Lemma 1.1
Lemma 3.1.
There is a sufficiently small constant ε > such that the followingholds. Fix a constant a such that < a < / and set U (cid:93) := ( a, , − a ) . Assumethat | U (cid:96) − U (cid:93) | ≤ εa, ≤ η ≤ εa,s , s < , s , s ∈ [ − εa, . NTERACTION ESTIMATES 9
Assume furthermore that (16) U r = D (cid:104) s , D [ s , U (cid:96) ] (cid:105) . Then there are σ < and τ > such that (17) U r = D (cid:104) τ, D (cid:2) s + s , D [ σ, U (cid:96) ] (cid:3)(cid:105) . Note that by combining (17) with the inequalities σ < τ > s + s < § § Proof of Lemma 3.1: first step.
We start with some preliminary consid-erations. Assume that the states U (cid:96) and U r satisfy (16). Next, solve the Riemannproblem between U (cid:96) (on the left) and U r (on the right): owing to [21], this amountsto determine by relying on the Local Invertibility Theorem the real numbers σ , s and τ such that(18) U r = D (cid:104) τ, D (cid:2) s, D [ σ, U (cid:96) ] (cid:3)(cid:105) . Establishing the proof of Lemma 3.1 amounts to prove that s = s + s < σ < τ > s = s + s we recall Lemma 2.3 and the fact that the v componentis constant along the 1-st and the 3-rd wave fan curves D and D . We concludethat s = v r − v (cid:96) = s + s < v r and v (cid:96) are the second component of U r and U (cid:96) .We are left to prove that σ < τ >
0. We first introduce some notation: weregard σ and τ as functions of η , s and s and U (cid:96) and we write σ η ( s , s , U (cid:96) ) and τ η ( s , s , U (cid:96) ) to express this dependence. Note that σ and τ depend on η becausethe wave fan curve D depends on η .Owing to the Implicit Function Theorem, the regularity of σ η ( s , s , U (cid:96) ) and σ η ( s , s , U (cid:96) ) is at least the same as the regularity of the functions D , D and D . Also, note that the Lax Theorem [21] (see also [5, p.101]) states that the wavefan curves D , D and D are C . The reason why we can achieve C ∞ regularityis because we are actually considering the wave fan curves in regions where theyare C ∞ . To see this, we first point out that, owing to (13) and (14), the wave fancurves D , D are straight lines and hence they are C ∞ . Next, we point out that weare only interested in negative values of s + s . Hence, we can replace the 2-wavefan curve D defined as in (11) with the 2-Hugoniot locus S . We recall that the2-Hugoniot locus S [ s, ¯ U ] contains all the states that can be connected to ¯ U by ashock, namely all the states such that the couple ( ¯ U , S [ s, ¯ U ]) satisfies the Rankine-Hugoniot conditions. The 2-Hugoniot locus S [ s, ¯ U ] is C ∞ and by combining allthe previous observations we can conclude that σ η ( s , s , U (cid:96) ) and τ η ( s , s , U (cid:96) ) areboth C ∞ with respect to the variables ( η, s , s , U (cid:96) ).Next, we discuss the partial derivatives of σ η ( s , s , U (cid:96) ) and τ η ( s , s , U (cid:96) ) withrespect to ( s , s ) at the point ( η, , , U (cid:96) ). By arguing as in the proof of estimate(7.32) in [5, p.133] we conclude that • for every U (cid:96) , for every η > k ≥ ∂ k σ η ∂s k (cid:12)(cid:12)(cid:12)(cid:12) (0 , ,U (cid:96) ) = ∂ k σ η ∂s k (cid:12)(cid:12)(cid:12)(cid:12) (0 , ,U (cid:96) ) = ∂ k τ η ∂s k (cid:12)(cid:12)(cid:12)(cid:12) (0 , ,U (cid:96) ) = ∂ k τ η ∂s k (cid:12)(cid:12)(cid:12)(cid:12) (0 , ,U (cid:96) ) = 0 . (19) • For every U (cid:96) and for every η > ∂ σ η ∂s ∂s (cid:12)(cid:12)(cid:12)(cid:12) (0 , ,U (cid:96) ) = ∂ τ η ∂s ∂s (cid:12)(cid:12)(cid:12)(cid:12) (0 , ,U (cid:96) ) = 0 . This implies that σ η and τ η admit the following Taylor expansions σ η ( s , s , U (cid:96) ) = 12 ∂ σ η (0 , , U (cid:96) ) ∂s ∂s s s + 12 ∂ σ η (0 , , U (cid:96) ) ∂s ∂s s s + o ( | ( s , s ) | ) s s ( s + s ) τ η ( s , s , U (cid:96) ) = 12 ∂ τ η (0 , , U (cid:96) ) ∂s ∂s s s + 12 ∂ τ η (0 , , U (cid:96) ) ∂s ∂s s s ++ o ( | ( s , s ) | ) s s ( s + s )(20)In § η = 0 and U (cid:96) = U (cid:93) the functions σ and τ admit theTaylor expansions (cid:18) σ ( s , s , U (cid:93) ) τ ( s , s , U (cid:93) ) (cid:19) = a (cid:18) − (cid:19) s s ( s + s ) + o ( | ( s , s ) | ) . (21a)Next, we use the Lipschitz continuous dependence of the derivatives of third orderwith respect to η and U (cid:96) and we conclude that (cid:12)(cid:12)(cid:12)(cid:12) ∂ σ η (0 , , U (cid:96) ) ∂s ∂s − a (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∂ σ η (0 , , U (cid:96) ) ∂s ∂s − a (cid:12)(cid:12)(cid:12)(cid:12) < Cεa (cid:12)(cid:12)(cid:12)(cid:12) ∂ τ η (0 , , U (cid:96) ) ∂s ∂s + a (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∂ τ η (0 , , U (cid:96) ) ∂s ∂s + a (cid:12)(cid:12)(cid:12)(cid:12) < Cεa provided that 0 ≤ η ≤ εa and | U (cid:96) − U (cid:93) | ≤ εa . In the above expression, C denotes auniversal constant. By plugging the above expressions into (20) and recalling that s , s < ε is sufficiently small, then σ η ( s , s , U (cid:96) ) < a s s ( s + s ) < ,τ η ( s , s , U (cid:96) ) > − a s s ( s + s ) > . The proof of the lemma is complete.3.2.
Proof of formula (21) . The proof of the Taylor expansion (21) is dividedinto two parts: § S [ s, U ] § η = 0 because formula (21) dealswith this case. NTERACTION ESTIMATES 11
The 2-Hugoniot locus.
Before giving the technical results, we introduce somenotation. First, we recall that we term F the flux function F η in (2) in the casewhen η = 0. In the following, we will mostly focus on the behavior of the first andthe third component of U . Hence, it is convenient to term ˆ U and ˆ F the vectorsobtained by erasing the second components of U and F , respectively. We have therelation(22) (cid:98) F ( U ) = 4 (cid:18) v − − v ( v −
2) 1 − v (cid:19) (cid:18) uw (cid:19) = (cid:98) J ( v ) · (cid:98) U , where we have also introduced the 2 × (cid:98) J ( v ).Finally, we recall that we term S [ s, ¯ U ] the 2-Hugoniot locus passing through ¯ U ,namely the set of states that can be connected to ¯ U by a (possibly not admissible)shock of the second family. Also, as usual we denote by ¯ u , ¯ v and ¯ w the first, secondand third component of ¯ U , respectively. We use the notation (cid:98) ¯ U = (¯ u, ¯ w ). Lemma 3.2.
Fix η = 0 and assume that | v + s | < , then the 2-Hugoniot locusthrough ¯ U has the following expression: the second component of S [ s, ¯ U ] is ¯ v + s while the first and third components are (cid:99) S [ s, ¯ U ] = (cid:98) ¯ U + E (¯ v, s ) (cid:98) ¯ U (23) where the × matrix E (¯ v, s ) is E (¯ v, s ) = 4 s (2¯ v + s ) − (cid:18) s + 4 − v s + 4)( s −
2) + 4¯ v s − v (cid:19) . Proof.
By Lemma 2.3 the second component of S [ s, ¯ U ] is ¯ v + s . To construct S [ s, ¯ U ] we use the Rankine-Hugoniot conditions, which are a system of 3 equations.Owing to Lemma 2.3, the second equation reads γs = (¯ v + s ) − ¯ v and this implies that the speed γ of the 2-shock is(24) γ = 2¯ v + s. We define the vector A ( s, ¯ U ) by setting A ( s, ¯ U ) := (cid:99) S [ s, ¯ U ] − (cid:98) ¯ U and we point out that to establish Lemma 3.2 we are left to show that(25) A ( s, ¯ U ) = E (¯ v, s ) (cid:98) ¯ U .
The first and the third equations in the Rankine-Hugoniot conditions can be writtenas γ A ( s, ¯ U ) = (cid:98) J (¯ v + s ) (cid:104) (cid:98) ¯ U + A ( s, ¯ U ) (cid:105) − (cid:98) J (¯ v ) (cid:98) ¯ U , (26)where (cid:98) J is the same as in (22). Next, we introduce the 2 × A ( v, γ ) = γ I − (cid:98) J ( v )= (cid:18) γ γ (cid:19) − (cid:18) v − − v ( v −
2) 1 − v (cid:19) , and we rewrite (26) as A (¯ v + s, γ ) A ( s, ¯ U ) = (cid:104) (cid:98) J (¯ v + s ) − (cid:98) J (¯ v ) (cid:105) (cid:98) ¯ U , which implies (25) provided that E (¯ v, s ) = A − (¯ v + s, γ ) (cid:2) (cid:98) J (¯ v + s ) − (cid:98) J (¯ v ) (cid:3) By recalling that γ = 2¯ v + s we can compute the explicit expression of the abovematrices: A (¯ v + s, v + s ) = (cid:18) − s − v − ¯ v − s )(¯ v + s ) 6¯ v + 5 s − (cid:19) , (cid:98) J (¯ v + s ) − (cid:98) J (¯ v ) = 4 s (cid:18) s + 2¯ v − − (cid:19) . The determinant of the matrix A (¯ v + s, v + s ) isdet := (2¯ v + s ) − | v + s | <
4. We can now complete thelemma by computing the explicit expression of E , namely E (¯ v, s ) = 1det (cid:18) v + 5 s − − v + s − v + s ) 4 − s − v (cid:19) · s (cid:18) s + 2¯ v − − (cid:19) = 4 s det (cid:18) s + 4 − v s + 4)( s −
2) + 4¯ v s − v (cid:19) . (cid:3) Conclusion of the proof of formula (21) . We are now ready to establish (21).We first recall some notation: we consider the system of conservation laws withflux F , see (2). We consider the collision between two 2-shocks and we assumethat U (cid:93) = ( a, , − a ), U m and U r are the left, middle and right states before theinteraction. This means that for some s < s < U r = D [ s , U m ] = D (cid:2) s , D [ s , U (cid:93) ] (cid:3) = S (cid:2) s , S [ s , U (cid:93) ] (cid:3) . (27)In the above expression, S represents the 2-Hugoniot locus. To establish the lastequality we used the fact that s and s are both negative. We plug (23) into (27)and we use the equality v (cid:93) = 0: we arrive at (cid:99) U r = (cid:104)(cid:99) U (cid:93) + E (0 , s ) (cid:99) U (cid:93) (cid:105) + E ( s , s ) (cid:104)(cid:99) U (cid:93) + E (0 , s ) (cid:99) U (cid:93) (cid:105) = (cid:99) U (cid:93) + (cid:104) E (0 , s ) + E ( s , s ) + E ( s , s ) E (0 , s ) (cid:105)(cid:99) U (cid:93) . (28)Next, we focus on the states after the interaction. By arguing as at the begin-ning of § σ = σ ( s , s , U (cid:93) ) and τ = τ ( s , s , U (cid:93) ) such that U r = D (cid:104) τ, D (cid:2) s + s , D [ σ, U (cid:93) ] (cid:3)(cid:105) . By the explicit expression of D and D and by applying Lemma 3.2 we infer thatthe above equality implies (cid:98) U r = (cid:104)(cid:99) U (cid:93) + σ (cid:98) (cid:126)r (0) (cid:105) + E (0 , s + s ) (cid:104)(cid:99) U (cid:93) + σ (cid:98) (cid:126)r (0) (cid:105) + τ (cid:98) (cid:126)r ( s + s )= (cid:99) U (cid:93) + E (0 , s + s ) (cid:99) U (cid:93) + (cid:104) I + E (0 , s + s ) (cid:105) σ (cid:98) (cid:126)r (0) + τ (cid:98) (cid:126)r ( s + s )= (cid:99) U (cid:93) + E (0 , s + s ) (cid:99) U (cid:93) + H ( s + s ) (cid:18) στ (cid:19) . (29) NTERACTION ESTIMATES 13
In the previous expression we denote by (cid:98) (cid:126)r and (cid:98) (cid:126)r the vectors obtained from (cid:126)r and (cid:126)r by erasing the second component. Also, we introduced the matrix H : itsfirst column is (cid:2) I + E (0 , s + s ) (cid:3) (cid:98) (cid:126)r (0), the second column is (cid:98) (cid:126)r ( s + s ). In thefollowing, we will prove that H ( s + s ) is invertible provided that s and s areboth sufficiently close to 0. By comparing (28) and (29) we then obtain(30) (cid:18) στ (cid:19) = H − ( s + s ) (cid:104) E (0 , s ) + E ( s , s ) + E ( s , s ) E (0 , s ) − E (0 , s + s ) (cid:105)(cid:124) (cid:123)(cid:122) (cid:125) G ( s , s ) (cid:99) U (cid:93) . Assume that we have established the following asymptotic expansion for G :(31) G ( s , s ) = 132 (cid:18) (cid:19) s s ( s + s ) + o ( | ( s , s ) | ) . Then by plugging both (31) and (cid:99) U (cid:93) = ( a, − a ) into (30) we obtain the asymptoticexpansion (21). Hence, to conclude the proof of (21) we are left to establish (31).First, we point out that, owing to the expression of E in the statement ofLemma 3.2, E (0 , s ) = 4 ss − (cid:18) s + 4 4( s + 4)( s −
2) 3 s − (cid:19) . This implies that when s = s = 0, the matrix E (0 , s + s ) vanishes and hence H − (0) = (cid:16) ˆ (cid:126)r (0) | ˆ (cid:126)r (0) (cid:17) − , (14) = (cid:18) − (cid:19) − = (cid:18) / − / (cid:19) . We compute now the asymptotic expansion of E (0 , s ) + E (0 + s , s ) − E (0 , s + s ) + E (0 + s , s ) E (0 , s ) . By directly computing the sum of the above matrices, we obtain that we can factorthe term 4 s s ( s + s )( s − s + s ) − s + s ) − , which multiplies the matrix with coefficientsCoeff , : ( s + 4)( s + s + 4)(6 s + 5 s − , : 4(5 s + 13 s s + 9 s − , : 2( s + 4)( s + s + 4)(4 − s + 2 s − s + 4 s s + 2 s )Coeff , : 192 − s − s + 26 s − s − s s + 65 s s − s + 55 s s + 16 s . By combining the above computations we obtain the following asymptotic expan-sion: G ( s , s ) = − (cid:18) / − / (cid:19) (cid:18) − · − · · · (cid:19) · s s ( s + s ) + o ( (cid:107) ( s , s ) (cid:107) )= 132 (cid:18) (cid:19) s s ( s + s ) + o ( (cid:107) ( s , s ) (cid:107) ) . (32) This establishes (31) and hence concludes the proof of (21). (cid:3) Interaction of a 1-shock and a 2-shock
We first rigorously state Lemma 1.2.
Lemma 4.1.
There is a sufficiently small constant ε > such that if ≤ η ≤ ε ,then the following holds. Assume that the states U (cid:96) , U r ∈ R satisfy (33) U r = D (cid:104) σ, D [ s, U (cid:96) ] (cid:105) for real numbers s , σ such that σ, s < , | s | , | σ | < . Furthermore, assume that | U (cid:96) | < / . Then there are real numbers σ (cid:48) and τ (cid:48) suchthat (34) U r = D (cid:104) τ (cid:48) , D (cid:2) s, D [ σ (cid:48) , U (cid:96) ] (cid:3)(cid:105) and (35) 2 σ ≤ σ (cid:48) ≤ σ, σs ≤ τ (cid:48) ≤ σs. Note that (35) implies σ (cid:48) < τ (cid:48) >
0. If we combine these inequalitieswith (34) and s < § σ (cid:48) , s (cid:48) , τ (cid:48) such that U r = D (cid:104) τ (cid:48) , D (cid:2) s (cid:48) , D [ σ (cid:48) , U (cid:96) ] (cid:3)(cid:105) . (2) By combining (13), (14) and Lemma 2.3 we obtain that s (cid:48) = s .To establish (35) we proceed as follows: § η = 0. § F η in (2) smoothly depends on η weshow that (35) holds provided η is sufficiently small.Note that, as we have mentioned in the introduction, the precise expression of thefunction p and p plays no role in the proof of Lemma 4.1, what is actually relevantis that estimates (35) hold at η = 0 with strict inequalities and that η is sufficientlysmall.4.1. Proof of Lemma 4.1: the case η = 0 . We establish (35) in the case η = 0.This part of the proof is actually the same as in [3, p. 844-845], but for completenesswe go over the main steps.We term σ (cid:48) , τ (cid:48) the real numbers satisfying (34) when η = 0. Let v r and v (cid:96) denote the second components of U r and U (cid:96) , respectively. We term γ the speedof the incoming 2-shock (which is the same as the speed of the outgoing 2-shock),we recall Lemma 2.3 and the fact that the second component varies only across2-shocks. We conclude that γ = v r − v (cid:96) v r − v (cid:96) = v r + v (cid:96) = 2 v (cid:96) + s. Since by assumption | U (cid:96) | < / | s | < /
4, then(36) | γ | < . NTERACTION ESTIMATES 15
By imposing the Rankine-Hugoniot conditions on the incoming and outgoing 2-shocks and by arguing as in [3, pp. 844-845], with the choice c = 4, we arrive atthe following system: (cid:26) ( γ + 4) σ (cid:48) + ( γ − τ (cid:48) = ( γ + 4) σv (cid:96) ( γ + 4) σ (cid:48) + ( v (cid:96) + s − γ − τ (cid:48) = ( v (cid:96) + s )( γ + 4) σ If we set(37) A := (cid:18) γ + 4 γ − v (cid:96) ( γ + 4) ( v (cid:96) + s − γ − (cid:19) and(38) X = (cid:18) σ (cid:48) τ (cid:48) (cid:19) , Y = (cid:18) γ + 4( v (cid:96) + s )( γ + 4) (cid:19) σ, then the above linear system can be recast as AX = Y . The explicit expression ofthe matrix A − is(39) 1(4 − γ )( − s + 2) (cid:18) ( v (cid:96) + s − γ − − ( γ − − v (cid:96) ( γ + 4) ( γ + 4) (cid:19) We solve for σ (cid:48) and τ (cid:48) and we obtain(40) σ (cid:48) = 2 − s + 2 σ, τ (cid:48) = γ + 4(4 − γ )( − s + 2) sσ By using (36) and the inequality | s | < /
4, we obtain(41) 23 < − s + 2 < , < γ + 4(4 − γ )( − s + 2) < η = 0.4.2. Proof of Lemma 4.1: the case η > . We are now ready to complete theproof of Lemma 4.1. We proceed as follows: § § Preliminary considerations.
We first introduce some notation. We term U m the intermediate state before the interaction, namely(42) U m := D [ s, U (cid:96) ] . Also, we term U (cid:48) m and U (cid:48)(cid:48) m the intermediate states after the interaction, namely U (cid:48) m := D [ σ (cid:48) , U (cid:96) ] ,U (cid:48)(cid:48) m := D [ s, U (cid:48) m ] = D (cid:2) − τ (cid:48) , U (cid:3) = D (cid:2) − τ (cid:48) , D [ σ, U m ] (cid:3) = D (cid:2) − τ (cid:48) , D [ σ, D [ s, U (cid:96) ]] (cid:3) (43)Next, we use [3, eq. (5.3)-(5.4)] and we recast the Rankine-Hugoniot conditions for2-shocks as a nonlinear system in the form(44) AX + η F ( X, U (cid:96) , s, σ ) = Y, where A and Y are as in (37) and (38), respectively. Also, the vector X is definedby setting X := (cid:18) σ (cid:48) τ (cid:48) (cid:19) and the nonlinear term F ( X, U (cid:96) , s, σ ) is equal to(45) (cid:18) p ( U (cid:48)(cid:48) m ) − p ( U m ) − p ( U (cid:48) m ) + p ( U (cid:96) ) p ( U (cid:48)(cid:48) m ) − p ( U m ) − p ( U (cid:48) m ) + p ( U (cid:96) ) , (cid:19) . In the above expression, the functions p and p are the same as in (3). Note,however, that the precise expression of p and p plays no role in the proof, theonly relevant point is that p and p are both regular (say twice differentiable withLipschitz continuous second derivatives). Note furthermore that we can regard F as a function of X , U (cid:96) , s and σ because, owing to (42) and (43), U m , U (cid:48) m and U (cid:48)(cid:48) m are functions of X , U (cid:96) , s and σ . Next, we rewrite equation (44) as(46) X = X − ηA − F ( X, U (cid:96) , s, σ ) , where the vector X = A − Y is given by (38) and (40).We now fix s , σ , η and | U (cid:96) | satisfying the assumptions of Lemma 4.1 and wedefine the closed ball(47) K := (cid:8) X = ( σ (cid:48) , τ (cid:48) ) ∈ R : | X − X | ≤ kησs (cid:9) . In the above expression, k > σ (cid:48) and τ (cid:48) are defined by (40). We also define the function T : R → R by setting(48) T ( X ) := X − ηA − F ( X, U (cid:96) , s, σ ) . Assume that T is a strict contraction from K to K . Then the proof of Lemma 4.1 iscomplete: indeed, owing to (46) the fixed point X satisfies the Rankine-Hugoniotconditions (44). Also, owing to (41) and to (47) we infer that the inequalities (35)are satisfied provided that the parameter η is sufficiently small.4.2.2. Conclusion of the proof of Lemma 4.1.
In this paragraph we prove that themap T defined by (48) is a strict contraction on the closed set K defined by (47).First, we make some remarks about notation. To simplify the exposition, inthe following we denote by C a universal constant: its precise value can vary fromoccurrence to occurrence. Also, in the following we will determine the constant k in (47) and then choose the constant η in such a way that kη ≤
1. This choiceimplies in particular that, when X belongs to the set K defined as in (47) and thehypotheses of Lemma 4.1 are satisfied, then the map F attains values on a boundedset and so F and all its derivatives are bounded by some constant C . Finally, notethat, if X ∈ K and the hypotheses of Lemma 4.1 are satisfied, then | A − | ≤ C .We now proceed according to the following steps. Step 1: we point out that to show that the map T is a contraction it suffices toshow that(49) |F ( X, U (cid:96) , s, σ ) | ≤ Cσs provided that X ∈ K and the hypotheses of Lemma 4.1 hold. Indeed, assumethat (49) holds, then | T ( X ) − X | (48) ≤ η | A − F ( X, U (cid:96) , s, σ ) | (49) ≤ Cησs
NTERACTION ESTIMATES 17 and hence T attains values in the set K defined as in (47) provided that k is largeenough. Also, | T ( X ) − T ( X ) | (48) ≤ η | A − ||F ( X , U (cid:96) , s, σ ) − F ( X , U (cid:96) , s, σ ) |≤ ηC | X − X | ≤ | X − X | provided that the constant η is sufficiently small. This implies that T is a contractionand concludes the proof of Lemma 4.1. Step 2: we establish (49). First, we point out that, if X ∈ K and the hypothesesof Lemma 4.1 are satisfied, then | p ( U (cid:48)(cid:48) m ) − p ( U m ) − p ( U (cid:48) m ) + p ( U (cid:96) ) | ≤ C (cid:16) | U (cid:48)(cid:48) m − U m | + | U (cid:48) m − U (cid:96) | (cid:17) ≤ C (cid:16) | U (cid:48)(cid:48) m − U r | + | U r − U m | + | U (cid:48) m − U (cid:96) | (cid:17) ≤ C (cid:16) | τ (cid:48) | + | σ | + | σ (cid:48) | (cid:17) ≤ C (cid:16) | σ | + | X (cid:48) | + | X − X (cid:48) | (cid:17) (40) , (47) ≤ C (cid:16) | σ | + ηkσs (cid:17) ≤ C | σ | . By using an analogous argument, we control the second component of F and wearrive at(50) |F ( X, U (cid:96) , s, σ ) | ≤ C | σ | . Next, we point out that when s = 0 we have U (cid:48) m = U (cid:48)(cid:48) m and U (cid:96) = U m and by usingagain the Lipschitz continuity of the functions p and p we conclude that(51) |F ( X, U (cid:96) , s, σ ) | ≤ C | s | . Finally, we use the regularity of the function F and, by arguing as in the proof of [5,Lemma 2.5, p. 28], we combine (50) and (51) to obtain (49). This concludes theproof of Lemma 4.1. (cid:3) Acknowledgments
The authors wish to thank the anonymous referees for their useful remarks thathelped improve the exposition and allowed to shorten and improve the proof ofLemma 4.1. The second author wish to thank the organizers of the conference“Contemporary topics in Conservation Laws” for the invitation to give a talk andpresent results related to the topic of the present paper. Both authors are membersof the Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Appli-cazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) andare supported by the PRIN national project “Nonlinear Hyperbolic Partial Differ-ential Equations, Dispersive and Transport Equations: theoretical and applicativeaspects”. This work was also supported by the EPSRC Science and Innovationaward to the OxPDE (EP/E035027/1).
References [1] L. Ambrosio and C. De Lellis, A note on admissible solutions of 1D scalar conservationlaws and 2D Hamilton-Jacobi equations,
J. Hyperbolic Differ. Equ. , (2004), 813–826, URL http://dx.doi.org/10.1142/S0219891604000263 .[2] F. Ancona and K. T. Nguyen, in preparation . [3] P. Baiti and H. K. Jenssen, Blowup in L ∞ for a class of genuinely nonlinear hyperbolicsystems of conservation laws, Discrete Contin. Dynam. Systems , (2001), 837–853, URL http://dx.doi.org/10.3934/dcds.2001.7.837 .[4] S. Bianchini and L. Caravenna, SBV regularity for genuinely nonlinear, strictly hyperbolicsystems of conservation laws in one space dimension, Comm. Math. Phys. , (2012), 1–33,URL http://dx.doi.org/10.1007/s00220-012-1480-5 .[5] A. Bressan, Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem ,vol. 20 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press,Oxford, 2000.[6] A. Bressan and G. M. Coclite, On the boundary control of systems of conservation laws,
SIAM J. Control Optim. , (2002), 607–622 (electronic), URL http://dx.doi.org/10.1137/S0363012901392529 .[7] A. Bressan and R. M. Colombo, Decay of positive waves in nonlinear systems of conservationlaws, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) , (1998), 133–160, URL .[8] A. Bressan and P. Goatin, Oleinik type estimates and uniqueness for n × n conservation laws, J. Differential Equations , (1999), 26–49, URL http://dx.doi.org/10.1006/jdeq.1998.3606 .[9] A. Bressan and T. Yang, A sharp decay estimate for positive nonlinear waves, SIAMJ. Math. Anal. , (2004), 659–677 (electronic), URL http://dx.doi.org/10.1137/S0036141003427774 .[10] L. Caravenna, A note on regularity and failure of regularity for systems of conservation lawsvia Lagrangian formulation, Bull. Braz. Math. Soc. (N.S.), to appear. Also arXiv:1505.00531 .[11] L. Caravenna and L. V. Spinolo, Schaeffer’s Regularity Theorem for scalar conservation lawsdoes not extend to systems, preprint. Also arXiv:1505.00609 .[12] C. Christoforou and K. Trivisa, Sharp decay estimates for hyperbolic balance laws,
J. Differen-tial Equations , (2009), 401–423, URL http://dx.doi.org/10.1016/j.jde.2009.03.013 .[13] C. M. Dafermos, Wave fans are special, Acta Math. Appl. Sin. Engl. Ser. , (2008), 369–374,URL http://dx.doi.org/10.1007/s10255-008-8010-4 .[14] C. M. Dafermos, Hyperbolic conservation laws in continuum physics , vol. 325 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathemati-cal Sciences], 3rd edition, Springer-Verlag, Berlin, 2010, URL http://dx.doi.org/10.1007/978-3-642-04048-1 .[15] R. J. DiPerna, Global solutions to a class of nonlinear hyperbolic systems of equations,
Comm.Pure Appl. Math. , (1973), 1–28.[16] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. PureAppl. Math. , (1965), 697–715.[17] J. Glimm and P. D. Lax, Decay of solutions of systems of nonlinear hyperbolic conserva-tion laws , Memoirs of the American Mathematical Society, No. 101, American MathematicalSociety, Providence, R.I., 1970.[18] H. Holden and N. H. Risebro,
Front tracking for hyperbolic conservation laws , vol. 152 ofApplied Mathematical Sciences, Springer-Verlag, New York, 2002, URL http://dx.doi.org/10.1007/978-3-642-56139-9 .[19] H. K. Jenssen, Blowup for systems of conservation laws,
SIAM J. Math. Anal. , (2000),894–908.[20] S. N. Kruˇzkov, First order quasilinear equations with several independent variables., Mat. Sb.(N.S.) ,
81 (123) (1970), 228–255.[21] P. D. Lax, Hyperbolic systems of conservation laws. II,
Comm. Pure Appl. Math. , (1957),537–566.[22] T. P. Liu, Decay to N -waves of solutions of general systems of nonlinear hyperbolic conser-vation laws, Comm. Pure Appl. Math. , (1977), 586–611.[23] O. A. Ole˘ınik, Discontinuous solutions of non-linear differential equations, Uspehi Mat. Nauk(N.S.) , (1957), 3–73.[24] D. G. Schaeffer, A regularity theorem for conservation laws, Advances in Math. , (1973),368–386. E-mail address : [email protected] E-mail address ::