On finite time BV blow-up for the p-system

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1 On finite time BV blow-up for the p-system Alberto Bressan ( ), Geng Chen ( ), and Qingtian Zhang ( ) (*) Department of Mathematics, Penn State University, (**) Department of Mathematics, University of Kansas, Lawrence, (***) Department of Mathematics, University of California, Davis s: bressan@mathpsuedu, gengchen@kuedu, qzhang@mathucdavisedu September 7, 07 Abstract The paper studies the possible blowup of the total variation for entropy weak solutions of the p-system, modeling isentropic gas dynamics It is assumed that the density remains uniformly positive, while the initial data can have arbitrarily large total variation (measured in terms of Riemann invariants) Two main results are proved (I) If the total variation blows up in finite time, then the solution must contain an infinite number of large shocks in a neighborhood of some point in the t-x plane (II) Piecewise smooth approximate solutions can be constructed whose total variation blows up in finite time For these solutions the strength of waves emerging from each interaction is exact, while rarefaction waves satisfy the natural decay estimates stemming from the assumption of genuine nonlinearity Introduction For hyperbolic systems of conservation laws in one space dimension, a major remaining open problem is whether, for large BV initial data, the total variation of entropy-weak solutions remains uniformly bounded or can blow up in finite time In the literature, BV bounds have been established by two main approaches: (I) Estimating the strength of new waves generated at each interaction, regardless of the order in which different wave-fronts cross each other For small initial data, this technique was introduced by Glimm [9] Under additional hypothesis, it can be applied also to solutions with large data See for example [, 6, 7, 8, 30] (II) Relying on the decay of rarefaction waves, due to genuine nonlinearity, to provide additional cancellations This approach first appeared in [0] and was then extended in [3, 7] On the other hand, some particular 3 3 hyperbolic systems have been constructed in [, 3], admitting solutions whose total variation blows up in finite time One should remark, however,

2 that these systems do not come from physical models and do not admit a strictly convex entropy In this paper we study the possible blowup for solutions to the p-system { v t u x = 0, u t + p(v) x = 0, () modeling isentropic gas dynamics in Lagrangian variables Here u is the velocity, ρ is the density, v = ρ is specific volume, while p = p(v) is the pressure In [5, 6], for a general class of pressure functions p( ), the authors constructed piecewise constant approximate solutions whose total variation grows without bound For these front tracking approximations, the strength of wave fronts emerging at each interaction is the same as in the exact solution, while the only source of error is in the wave speeds These examples confirm the analysis in [6], and show that uniform BV bounds cannot be established relying only on an accurate estimate of wave strengths across interations The next question, which we investigate in the present paper, is whether BV bounds for the p-system can be established by taking into account also the decay of rarefaction waves, stemming from the assumption p (v) > 0 We recall that Oleinik-type estimates on the decay of positive waves for genuinely nonlinear n n hyperbolic systems were proved in [4, 8, ] The analysis in the last section of [5] shows that, if this decay of rarefaction waves were taken into account, then the interaction patterns considered in [5, 6] would no longer yield a large amplification of total variation It is thus natural to ask: (Q) Consider a piecewise smooth approximate solution of () with large BV initial data Assume that at each interaction, the strengths of outgoing waves are the same as in an exact solution, rarefaction waves satisfy decay estimates as in [4, 8, ], due to genuine nonlinearity, the density remains uniformly positive Can the total variation still blow up in finite time? An example will be constructed, showing that finite time BV blowup for such approximate solutions is indeed possible Although our solutions are not exact, because some errors occur in the wave speeds, they possess all the qualitative properties known for exact solutions The present analysis thus provides some indication that finite time blowup of the total variation might be possible, for the p-system Our second main result yields a necessary condition for blowup Namely, we prove that if the total variation blows up in finite time, then the solution must contain an infinite number of large shocks, in a neighborhood of some point in the t-x plane This result should be compared with earlier literature, proving BV stability for various classes of initial data u(0, x) = ū(x), v(0, x) = v(x) ()

3 If ū, v have sufficiently small total oscillation, in the sense that all the initial values (ū(x), v(x)) are contained in a disc with sufficiently small radius, in the v-u plane, then the solution of () exists globally in time Moreover, bounds on the total variation can be provided, uniformly in time [0] If ū, v are a sufficiently small BV perturbation of some (possibly large) Riemann data, then again the solution exists globally in time and its total variation remains uniformly bounded [7, 4] Building upon these ideas, our present analysis shows that, for a solution containing only finitely many large shocks, the total variation remains bounded Indeed, the blow-up of the BV norm in finite time requires the presence of infinitely many large shocks in a bounded region of the t-x plane The remainder of the paper is organized as follows To keep the exposition self-contained, in Section we review some well known results on the interaction of elementary waves for the p- system Section 3 develops some estimates related to the decay of rarefaction waves, valid also for solutions with large oscillation In Section 4 we construct a piecewise constant approximate solution with a periodic interaction pattern, and where all rarefaction waves decay at the rate /t By a suitable modification of this basic pattern, in Section 5, we construct a piecewise smooth approximate solution whose BV norm blows up in finite time Section 6 contains the statement of our main theorem, providing a necessary condition for finite time blowup Details of the proof are then worked out in Sections 7 and 8 Elementary wave interactions for the p-system Throughout this paper we consider the p-system () with γ-law pressure p(v) = Av γ = Aρ γ, () for some constants γ > and A > 0 For this system one can define the Riemann invariants w and w by setting w = u h w = u + h, () where h = B v ( γ)/ = B ρ (γ )/, B = Aγ (3) γ For future use, we record the identities v = ( ) h /( γ) = B ( ) (γ )(w w ) /( γ) (4) 4 Aγ For any smooth solution, these Riemann invariants remain constant along forward and backward characteristics, respectively Namely where the (Lagrangian) wave speed c is w,t c w,x = 0, w,t + c w,x = 0, (5) c = p (v) = A/γ v (γ+)/ (6) 3

4 In the following, it will be convenient to express the wave speed in terms of the Riemann coordinates w, w in () Introducing the function c(s) = ( ) (γ ) s (γ+)/(γ ) A/γ 4, (7) Aγ by (4) and (6) the wave speed can be written as p (v) = c(w w ) Example In the special case where p = ρ 3 /3, one has the simple relation h = ρ By () and (6), the Riemann invariants and the wave speed are given by w = u ρ, w = u + ρ, c = ρ Elementary waves A solution to the p-system contains three types of waves: rarefactions, compressions, and shock waves In terms of the variable h at (3), left and right states will be denoted by respectively (u, h ) and (u +, h + ), (8) Recalling (), the signed wave strength will always be measured in terms of Riemann invariants: w,+ w for a -wave, w,+ w, for a -wave (9) We now recall the construction of basic wave curves See [4, 9] for details The rarefaction and compression waves satisfy the following equations: For a -wave (backward moving front), u + u = h h + where h + > h for a -compression wave and h + < h for -rarefaction wave For a -wave (forward moving front), u + u = h + h where h + < h for a -compression wave and h + > h for a -rarefaction wave 4

5 A shock wave with left state (u, ρ ) and right state (u +, ρ + ), traveling with speed λ, satisfies the Rankine-Hugoniot equations ( λ ) = u u +, ρ + ρ λ(u + u ) = p(v + ) p(v ) = Aρ γ + Aργ The Lax admissibility condition here yields u + < u for both -waves and -waves Hence ( u + u = ) (Aρ γ ρ + ρ +) Aργ, (0) and λ = ± ρ γ A ργ + () ρ + ρ For a -shock one has ρ + > ρ, while for a -shock one has ρ + < ρ The following observation will be useful Setting s = u u +, θ = ρ + ρ = ( h+ h ) γ, () from (0) it follows that, for any shock wave, s = h A( θ)( θ γ ), (3) B θ where B is the constant at (3) Small wave interactions Next, we review some well known results on wave interactions, for future use Note that, when a wave-front crosses a shock or a compression of the opposite family, the density ρ (and hence h as well) increases On the other hand, the density along a wave-front decreases when it crosses a rarefaction of the opposite family For any pairwise interaction between two small (shock or rarefaction) waves, one has the following estimates (see [4, 0, 9]) Proposition Call σ, σ the strengths of two interacting wave-fronts, and let σ, σ be the strengths of the outgoing waves of the first and second family, in the solution of the Riemann problem Then there exists a constant C 0 (uniformly valid as the state of the system ranges over a bounded set in the ρ-u plane, with ρ bounded away from zero) such that If σ is a -wave and σ is a -wave, then σ σ + σ σ C 0 σ σ ( σ + σ ) (4) 5

6 If both σ and σ belong to the first family, then σ (σ + σ ) + σ C 0 σ σ ( σ + σ ) (5) If both σ and σ belong to the second family, then σ + σ (σ + σ ) C 0 σ σ ( σ + σ ) (6) 3 A rarefaction or compression wave crosses a large shock h Q ε Q η(ε) σ Q ε η(ε) ε Pε P P ε ε P Q Figure : A small -rarefaction crosses a -shock In this configuration, the strength of the shock does not change, while the strength of the rarefaction increases from ε to some value η(ε) > ε To fix the ideas, consider a large -shock which crosses a small -wave (compression or rarefaction) of size σ = ε We seek an estimate on the size of the outgoing waves, up to leading order As shown in Fig, let P = (u, h ), Q = (u +, h + ) be the left and right states across the large -shock before the interaction, and let ( ) P ε = (u ε, h ε), Q ε = u + η(ε), h + η(ε) be the left and right states across the -shock after the interaction Set u s(ε) = (u ε) ( u + η(ε) ), θ(ε) = ( ) h+ η(ε) γ (7) h ε By (3), replacing P with P ε we can write Hence s(ε) = h ε B ψ(θ(ε)), with ψ(θ) = A(θ )(θ γ ) θ (8) η(ε) ε = s(ε) (u u + ) = h ε ψ(θ(ε)) (u u + ) (9) B 6

7 Differentiating wrt ε we obtain η (ε) = B θ (ε) = ψ(θ(ε)) + By (7), at ε = 0 the above expression reduces to h ε B ψ(θ(ε)) ψ (θ(ε))θ (ε), (0) γ [θ(ε)](3 γ)/ η (ε)(h ε) + (h + η(ε)) (h ε) () θ = γ θ(3 γ)/ η + θ (γ )/ () h Using () to compute the right hand side of (0), when ε = 0 we find η = B Solving for η, we finally obtain ψ(θ) + B ψ(θ) ( ψ (θ) γ θ(3 γ)/ η + θ (γ )/), (3) η = B ψ ψ + γ θψ B ψ + γ θ(3 γ)/ ψ = a(θ) (4) We observe that η is the factor by which an infinitesimal -wave (either a compression or a rarefaction) is amplified when it crosses the -shock According to (4), this ratio depends only on θ In particular, as θ remains bounded, the above amplification coefficient is a bounded number To compute the amplification of an arbitrary rarefaction or compression wave which crosses a large shock of the opposite family, we can simply integrate (4) and obtain η( ε) = ε 0 a(θ(ε)) dε (5) By a direct calculation we now prove a(θ) > In other words, as a compression or rarefaction wave crosses a shock of the opposite family, its strength always increases In view of (4), this will be a consequence of the two inequalities ( B ψ ψ + ) γ θψ B ψ + γ θ 3 γ ψ We begin by observing that θ = h + /h >, and hence ( B ψ + ) γ θ 3 γ ψ > 0, (6) > 0 (7) ψ(θ) = A(θ )(θγ ) θ > 0, ψ (θ) = Aθ [ +( γ)θ γ +γθ γ+ ] > 0, (8) 7

8 proving (7) Moreover, one has ( B ψ ψ + ) ( γ θψ B ψ + ) γ θ 3 γ ψ = ψ + γ θψ γ θ 3 γ ψ = A ( θ + θ γ θ γ ) + ( Aθ + A( γ)θ γ + Aγθ γ) ( θ γ ) γ = A γ ( θ γ+ ) [ ] γθ γ (θ ) + (θ γ ) > 0 (9) Indeed, to see that the last factor on the right hand side of (9) is positive for θ >, we set f(θ) = γθ γ (θ ) + θ γ = γθ γ+ + γθ γ + θ γ Then f() = 0, f (θ) = γθ γ 3 [ γ + θ + γ ] + θ γ+ = γθ γ 3 g(θ), where g(θ) = γ + θ + γ + θ γ+, g() = 0, g (θ) = γ + + γ + θ γ > 0 For θ > we thus have g(θ) > 0, hence f (θ) > 0 and f(θ) > 0 This completes the proof that a(θ) > for θ > With reference to Figure, the inequality η(ε) > ε implies that the h-components of the states P, Q, P ε, Q ε satisfy h Q h Q > h P h P (30) 4 A small shock crosses a large shock h Q η(ε) Q ε P P P ε Q P Q η(ε) u Figure : A small -shock crosses a large -shock 8

9 Let ε < 0 be the signed strength of the small shock Since shock and rarefaction curves have a second-order tangency [4, 9], with the notation used in Fig we have P = (u, h ), Q = (u +, h + ), ( ) P = (u ε + o(ε ), h ε + o(ε )), Q = u + η(ε) + o(ε ), h + η(ε) + o(ε ) Here and in the sequel, the Landau notation o(ε ) denotes an infinitesimal of higher order wrt ε Computing the derivative η at ε = 0, we thus recover exactly the same expression as in (3) Because of the second order tangency condition, the change in the strength of the big shock will also be of order o(ε ) We conclude with an estimate which will be used later Lemma Fix 0 < a < b and consider the Riemann problem determined by the interaction of a large -shock with another wave front of strength σ Assume that the left, middle, and right states remain in the region where a h(v) b Call σ the strength of the outgoing -wave generated by the interaction Then there exists a constant C γ depending only on a, b such that: (i) If the second wave impinges on the -shock from the left, then strength of the outgoing -wave satisfies σ C γ σ (ii) If the second wave is a -shock, or a small -compression or -rarefaction, impinging on the -shock from the right, then strength of the outgoing -wave satisfies σ σ Proof Part (i) is an immediate consequence of Proposition Toward a proof of (ii), we first recall a basic property of shock curves for the p-system Fix a left state (u, h ), and consider the curve of all points (u +, h + ) which can be connected to (u, h ) by a -shock Writing h = = h + (u + ), the slope of this curve satisfies dh + du + < 0 (3) Indeed, this inequality is established within the proof of Lemma 3 in [8] It is also found in Section 3 of [6] Next, assume that the left and right states across the large -shock are Two cases will be considered P = (u P, h P ), Q = (u Q, h Q ), CASE : The impinging -wave is a shock, connecting the states Q, Q As shown in Fig 3, the outgoing -rarefaction connects the states W, Q, where W = (u W, h W ) is the unique state along the -shock curve through P such that u Q h Q = u W h W 9

10 h Q' W Q W B Q' W P Q' A Q u P Q Figure 3: A small -shock impinges a large -shock In this case, call A and B the points in the u-h plane such that u Q h Q = u A h A = u B h B, h A = h Q, u B + h B = u Q + h Q We then have [strength of the outgoing -rarefaction] = (u Q + h Q ) (u W + h W ) (u B + h B ) (u A + h A ) = (u Q h Q ) (u B h B ) = (u Q h Q ) (u Q h Q ) = [strength of the incoming -shock] CASE : The incoming -wave is a compression previous inequalities remain valid In this case we have Q = B, and the h Q Q'' V V Q'' Q B A V P u P Q Q'' Figure 4: A small -rarefaction impinges a large -shock 0

11 CASE 3: The impinging -wave is a small rarefaction, connecting Q with a right state Q, of strength ε = (u Q h Q ) (u Q h Q ) In this case the interaction produces an outgoing -shock, connecting the states V and Q Here V is the state at the intersection of the -shock curve through P and the -shock curve through Q As shown in Fig 4, call A = (u A, h A ) the point such hat u A h A = u Q h Q, h A = h Q, and let B be the point at the intersection of the -shock curve through P and the segment AQ Recalling that the -shock curve through Q has a second order tangency with the segment AQ, and using the inequality for some δ 0 > 0 sufficiently small, we obtain dh+ du + δ 0 < 0 [strength of the outgoing -shock] = (u V + h V ) (u Q + h Q ) = (u B + h B ) (u Q + h Q ) + O(ε 3 ) = ( δ 0 ( δ 0 ) [ (u Q h Q ) (u Q h Q )] + O(ε 3 ) ) ε + O(ε 3 ) < ε = [strength of the incoming -rarefaction] Together, the above three cases prove part (ii) of Lemma 5 Wave measures Let now x (v(x), u(x)) be any profile with bounded variation As in Chapter 0 of [4], we can define the signed measures µ, µ describing strength of waves Namely, for i =,, The atomic part of µ i is supported on the countable set of points where v or u have a jump If x is one such point, then µ i ({ x}) is the signed strength of the i-th wave in the solution of the Riemann problem with left and right data (v, u)( x ), (v, u)( x+) The continuous part of µ i is defined as the continuous part of the distributional derivative of the scalar function x w i (x) Since w i has bounded variation, this is a bounded measure These measures can be decomposed into a positive and a negative part, so that Notice that: µ i = µ + i µ i, µ i = µ + i + µ i

12 µ + i accounts for i-rarefaction waves The continuous part of µ i accounts for i-compression waves The atomic part of µ i accounts for i-shocks As shown in [4], Glimm s functionals (originally defined for piecewise constant functions) can be extended to arbitrary BV functions The total strength of waves is defined as V = µ i (IR), (3) i=, while the interaction potential is Q = d µ (x) d µ (y) + µ i (IR) µ i x<y i=, (IR) (33) Notice that (33) accounts for the product of strengths of all couples of approaching waves We recall that two waves of the same family are approaching if at least one of them is a compression or a shock Next, consider a solution (v, u) of () defined for t [t 0, t ] and let V (t), Q(t) be the total strength of waves and the wave interaction potential at time t As shown in Chapter 0 of [4], these functionals satisfy the same estimates valid for Glimm or front-tracking approximations In particular, from the interaction estimates in Proposition it follows Lemma For any given K 0 and a, ε 0 > 0, there exists δ 0 > 0 such that the following holds Assume that (i) the density remains bounded away from zero: h(v(t, x)) a for all t [t 0, t ], x IR, (ii) the total strength of waves in the initial data satisfies V (t 0 ) K 0, and (iii) the solution (v, u) does not contain any shock of strength > δ 0 Then the function is non-increasing t V (t) + ε 0 Q(t), t [t 0, t ] (34) Given a BV solution U = (v, u) of (), we denote by λ (t, x) = c(v(t, x)), λ (t, x) = c(v(t, x)), (35) the two wave speeds at the point (t, x), as in (6) Following [4], for i =,, by a generalized i-characteristic we mean an absolutely continuous curve x = x(t) such that for ae t ẋ(t) [ λ i (t, x+), λ i (t, x ) ] (36)

13 For a given terminal point x we shall consider the minimal i-characteristic through x, defined as ξ(t, x) = min { x(t) ; x is an i-characteristic, x(t ) = x } As proved in [4], the curve ξ(, x) is itself an i-characteristic Indeed, for ae t the functions w, w and hence also the wave speed λ i are continuous at (t, ξ(t)) Therefore we can simply write ξ(t) = λ i (t, ξ(t)) In addition to the wave measures µ i, one can also introduce a scalar, positive measure µ int on the t-x plane bounding the amount of interaction, and hence the production of new waves More precisely, let (U ν ) ν be a sequence of piecewise constant front-tracking solutions, converging to the exact BV solution U = (v, u) For each ν we can also construct a purely atomic measure µ int ν by setting µ int ( ) ν {P } = σ σ (37) for every point P = ( t, x) at which two incoming fronts interact, with strengths σ, σ respectively By taking a suitable subsequence, we can achieve the weak convergence of measures µ int ν µ int (38) for some positive measure µ int, which we call a measure of wave interaction for the solution U Taking the limit of front tracking approximations one obtains a useful important property of this measure, namely: Lemma 3 For i {, }, let t ξ(t) and t ξ(t) be two minimal i-characteristics, with ξ(t) ξ(t) for t [t 0, t ] Then one has the estimate µ ± i ( [ξ(t ), ξ(t ) [) µ ± i ( [ξ(t0 ), ξ(t 0 ) [) + C µ int (Ω), (39) where Ω = { (t, x) ; } t [t 0, t ], x [ξ(t), ξ(t)[ (40) In other words, the total amount of (positive or negative) i-waves at time t contained in the interval [ξ(t ), ξ(t )[ can be estimated in terms of the old i-waves (positive or negative, respectively) present at time t 0 inside the interval [ξ(t 0 ), ξ(t 0 )[, plus some new waves generated by wave interactions occurring inside the domain Ω enclosed between the two characteristics The total strength of these new waves can be bounded in terms of the interaction measure µ int A precise value for the constant C in (39) can be determined using the interaction estimates (4) (6) In particular, by taking the limit of front tracking approximations, one obtains Lemma 4 For any given constants K 0, a and ε 0 > 0, one can find δ 0 > 0 such that, under the assumptions (i) (iii) of Lemma, one has 3

14 (i) the total amount of interaction satisfies (ii) the estimate (39) holds with C = ε 0 µ int( [t 0, t ] IR ) K 0, (4) 3 Decay of positive waves Differentiating (5) and writing the wave speed as c = c(w w ) with c as in (7), one obtains { w,xt cw,xx = c w,x + c w,x w,x, w,xt + cw,xx = c w,x w,x c w,x (3) The above system would be easy to integrate if we did not have the mixed terms w,x w,x To get rid of these terms, we first multiply each equation in (5) by a function φ = φ(w w ) and then differentiate For example, the second equation yields [φ w,x ] t + c[φ w,x ] x [ ] = φ c w,x w,x c w,x + φ (w,t w,t )w,x + c φ (w,x w,x )w,x [ ] = φ c w,x w,x + c w w,x c φ w,x w,x (3) provided that = φ c w,x, φ φ = c c = γ + γ Computing an explicit solution of (33) we find w w (33) φ(w w ) = (w w ) γ+ γ = (h) γ+ γ (34) In the end, this yields a decay estimate along any -characteristic t x(t) d dt (φw,x) ( t, x(t) ) = φ c w w,x C w,x, (35) for some constant C > 0 depending only on the upper and lower bounds for the density Of course, an entirely similar estimate holds for -rarefactions Next, let ξ (t) < ξ (t) be two -characteristics Calling c = c(w w ) the characteristics speed as a function of the Riemann coordinates, we have d dt (ξ (t) ξ (t)) = ξ (t) ξ (t) = ξ (t) ξ (t) c (w,x w,x ) dx (36) Notice that the above identity involves also the -waves inside the interval [ξ (t), ξ (t)] We seek an equivalent way to express the distance between two characteristics, which does not involve the contribution of intermediate -waves Toward this goal, consider the integral Z(t) = ξ (t) ξ (t) 4 ϕ dx, (37)

15 where ϕ = ϕ(w w ) = (w w ) γ+ γ d dt Z(t) = ξ ϕ(ξ ) ξ ξ ϕ(ξ ) + ϕ (w,t w,t ) dx ξ = c(ξ )ϕ(ξ ) c(ξ )ϕ(ξ ) = = ξ ξ ξ ξ ξ which satisfies c ϕ = cϕ We compute ξ ξ ϕ c (w,x + w,x ) dx [ c (w,x w,x )ϕ + c ϕ (w,x w,x ) ] ξ dx [ ] cϕ (w,x w,x ) + c ϕ (w,x w,x ) dx ξ ϕ c (w,x + w,x ) dx ξ ξ ϕ c (w,x + w,x ) dx = c ϕ w,x dx ξ (38) Notice that the last two equalities were obtained using the identity c ϕ = cϕ, which produces a cancellation of all terms involving -waves 4 A periodic interaction pattern As a preliminary to the blow-up example, in this section we construct a piecewise constant approximate solution with a periodic interaction pattern In the following (see Fig 5), we consider points P i = (u i, h i ) along the two lines γ 0 = {(u, h); h > 0, u h = 0}, γ = {(u, h); h > 0, u h = } Lemma There ( exists a point P) 0 γ 0 such that the following holds (Fig 5, right) Consider the point P = u 0 +, h 0 γ Let P 4 γ 0 be the point along the -shock curve through ( P and let P ) γ be the point along the -shock curve through P 4 Finally, call P 3 = u, h + γ 0 and let P 5 γ be the point along the -shock curve through P 3 Then h 5 < h 0 As a consequence, there is a left state L which can be connected to both P 0 and P 5 by -shocks ( Proof First, consider the -shock through P = ), This intersects the line γ at some point Q, say with h(q) = κ > 0 Next, for ε > 0 small consider the points P 0 = ( + ε, + ε ), P = ( + ε, ε) Starting from P, construct the corresponding point P 4 and then P = ( + η(ε), η(ε)) By (5) and the boundedness of the amplification factor a(θ), as ε 0+ we also have η(ε) 0 Finally, call P 3 = ( + η(ε), + η(ε)) and let P 5 γ be the point along the -shock curve through P 3 5

16 h γ 0 / h P 0 P 3 / P S Q γ P 4 0 / +κ u 0 P P P 5 L u +ε +η(ε) Figure 5: Left: the -shock curve through (, ) Right: the various points P i considered in Lemma By (30), it follows Therefore h 3 h 5 > h(p ) h(q) = κ h 0 h 5 = (h 0 h 3 ) + (h 3 h 5 ) for ε > 0 small enough We thus have This proves the first statement in Lemma ( ) = (h h ) + (h 3 h 5 ) > ε η(ε) + κ > 0 h 5 < h 0 < h 3, u 0 < u 3 < u 5 (4) It remains to prove that there exists a left state L = (u L, ρ L ) which is connected to both P 5 and P 0 by a -shock Referring to Fig 6 consider the -shock curve through P 5 Let Q = (u Q, h Q ) be any point on this curve Notice that, as u Q +, we have h Q 0 Next, let P (Q) = (ū, h) be the point where the -shock curve through Q intersects the line γ 0 = {h u = 0} Since this shock curve is concave down, one has lim inf u Q + h 5 h u 5 ū lim u Q + h Q h 5 u Q u 5 = 0 Therefore, as u Q + the point P (Q) approaches the point (u 5, h 5 ) On the other hand, as u Q u 5 one has P (Q) P 3 Since h 0 > h 5, by continuity, there is some choice of Q such that P (Q) = P 3 This completes the proof of Lemma Using Lemma we now construct a front tracking solution to the system () with a periodic interaction pattern Referring to Fig 7, at time t = τ the piecewise constant solution (u, h)(τ, ) takes the values L, P 5, P, P 3, P 4 As time increases, the following interactions take place, one after the other (i) The -rarefaction P P crosses the -compression P P 3 Afterwards, this -compression breaks into a -shock and a -rarefaction In the end, the states P and P 3 are connected by the -shock P P 4 followed by the -rarefaction P 4 P 3 6

17 h P 3 P 0 _ P(Q) P 5 0 Q u Figure 6: Constructing the left state L τ t τ S S L Figure 7: A periodic interaction pattern in the (x, t)-plane Here 0,,, 5 and L refer to the states P 0, P,, P 5 and to the left state L considered in Lemma The thick solid lines are shocks, the thin solid lines represent compressions, while the dashed lines are rarefaction fronts 7

18 (ii) The -compression P 5 P crosses the -shock P P 4 The Riemann problem is solved by the -shock P 5 P 3 and the -compression P 3 P 4 (iii) The -shock P 5 P 3 hits the -shock L P 5, generating the -shock L P 0 and the -rarefaction P 0 P 3 (iv) The -compression P 3 P 4 breaks into the -rarefaction P 3 P and the -shock P P 4 (v) The -rarefaction P 0 P 3 crosses the -rarefaction P 3 P, producing the -rarefaction P 0 P and the -rarefaction P P (vi) The -rarefaction P 0 P hits the -shock L P 0, producing the -shock L P 5 and the the -compression P 5 P (vii) The -shock P P 4 is canceled by the -rarefaction P 4 P 3, producing the -compression P P 3 At time t = τ we have reached the same configuration as at time τ, and the periodic pattern can be continued Remark If the initial data had small total variation, then the standard wave interaction estimates [4, 9, 9] would imply that the interaction potential approaches zero As proved in [5], the solution would converge to the solution of the Riemann Problem with left and right data (L, P 3 ) In the present interaction pattern, however, this does not happen because wave strengths are large In particular, notice that the -compression P 5 P is greatly amplified when it crosses the large -shock P P 4 5 An example with finite time blow-up of the total variation In this section we provide an affirmative answer to the question (Q) considered in the Introduction Namely, we construct a piecewise smooth approximate solution of () such that: (C) At each interaction, the strengths of outgoing waves is the same as in an exact solution (C) For some constant C 0 > 0, all rarefaction waves satisfy a decay estimate of the form d dt (φw i,x)(t, x i (t)) C 0 w i,x, i =, (5) Here ϕ is the function at (34) and t x i (t) is any i-characteristic (C3) The density ρ remains uniformly positive (C4) The total variation blows up in finite time 8

19 S S t L h P 4 P 0 P 3 P P 6 P P 5 u L Figure 8: Left: a modified periodic pattern Compared with the pattern in Fig 7, the compression wave joining P 5 to P is now split in two parts This creates an additional state, which we call P 6 Right: the location of the new states P 0,, P 6 and L, in the u-h plane 5 Outline of the construction Then to construct an approximate solution of () whose total variation blows up in finite time, the periodic pattern constructed in the previous section will be modified in two ways: (i) By slightly changing the wave speeds, the interaction pattern can be repeated on a sequence of shorter and shorter time intervals [τ n, τ n ], with lim n τ n = T < (ii) In the original pattern the -shock connecting P with P 4 is entirely cancelled by the -rarefaction connecting P 4 with P 3 We slightly change the speeds of these two waves so that they do not entirely cancel each other More precisely, for every n large enough, at the terminal time T the solution will still contain a -shock and a -rarefaction, both of strength α n, connecting the states P (n+) 3 and P (n) 3 Here α is a fixed positive constant These are the remaining portions of the -shock and -rarefaction which are not completely cancelled by the n-th iteration of the basic pattern The total strength of all these waves is n N α n = +, providing the blow-up of the total variation as t T Because of (ii), it is clear that the intermediate states P 0, P 5 generated by this interaction pattern can no longer repeat cyclically, but will slightly change after each round of interactions Still, as t T, a periodic interaction pattern will be approached The construction of the approximate solution will be achieved in the next three steps 9

20 h P 3 P 0 P 0 P 4 P 5 P P P 6 L u Figure 9: Constructing a perturbed periodic pattern Here the states P 0, P,, P 5 are the same as in Fig 7 The states P 0, P,, P 6 and L are those in the new pattern shown in Fig 8 5 A perturbed periodic interaction pattern To construct our approximate solution, we begin by defining a slightly different periodic interaction pattern As shown in Fig 8, the compression wave between the states P and P 5 is now split in two parts A small portion breaks at the same point where the -compression waves merge into a large -shock This portion is thus completely cancelled by the interaction The remaining portion eventually forms a large -shock, as in the previous periodic pattern As a result, the new periodic approximate solution will contain an additional constant state P 6 between the two portions of this -compression wave Notice that, as P 6 P, the new pattern becomes identical to the old one Being able to partition the -compression into two separate waves adds one more degree of freedom to the construction of a periodic pattern This will be used to achieve more easily a convergence estimate A periodic pattern as in Fig 8, can be obtained by a perturbation argument, starting with the pattern constructed in the previous section, and using the implicit function theorem Our construction is better explained with the aid of Fig 9 We start from our earlier periodic example in Fig 5 including states P 0,, P 5 and L Furthermore, by the proof of Lemma, we have h(p 0 ) > h(p 5 ) We can thus find a state P 0 along the line segment P 4P 0, such that Note that P 0 can be chosen arbitrarily close to P 0, h(p 0) > h(p 5 ) (5) We then call P 6 the intersection between the -wave curve through P 0 and the -wave curve through P Notice that the -wave with left state P and right state P 6 is a compression wave 0

21 S P 3 k+ P k 3 t L P k 3 k 3 P k P 3 k k+ 3 P 3 P 3 P P 0 P 4 P P 6 P P 5 u L Figure 0: Left: by slightly perturbing the periodic pattern in Fig 8, we obtain a new pattern where, at the k-th iteration, an additional pair of shock and rarefaction waves is produced, each with strength α/k Right: the sequence of left, middle, and right states P k+ 3, P k 3, and P k 3 Finally, by (5) and using the same argument as in the proof of Lemma, we can find a new left state L which is connected to both P 0 and P 5 by a -shock One now checks that the states P 0, P,, P 6 and L produce the periodic pattern in Fig 9 In particular, notice that the Riemann problem with left and right states P, P 3 is still solved by the -shock P P 4 and the -rarefaction P 4 P 3 53 A sequence of nearly periodic patterns Next, we slightly modify the previous periodic interaction pattern by assuming that, at the k-th iteration, the -shock P P 4 is not entirely cancelled by the -rarefaction P 4 P 3 Instead, a pair of -waves survive, namely (see Fig0): a -shock of strength α/k, joining the left state P k+ 3 with an intermediate state P k 3, a -rarefaction, also of strength α/k, joining the intermediate state P 3 k state P3 k to the right Since we require that these two shock and rarefaction waves have exactly the same strength (measured in Riemann invariants), all states P3 k, k =,, must lie along the same -wave curve through P 3 Recalling that shock and rarefactions curves coincide up to second order [4, 9], for some constant C we have P k+ 3 P k 3 C k 3 (53)

22 Therefore, by choosing α > 0 small enough, we can uniquely determine the states P k 3 so that P k 3 P 3 as k (54) In turn, we claim that all other intermediate states Pi k, with i {0,,, 4, 5, 6} and k, can be uniquely determined as well Indeed, these can be constructed in the following order: P k 3 P k 5 P k P k 0 P k 4 P k P k 6 P5 k (k) is the state at the intersection of the -shock curve with right state P 3 and the -shock curve through L P k is the state at intersection of the -wave curve through P 3 and the -wave curve through P5 k P0 k is the state at the intersection of the -wave curve through P 3 k and the -shock curve through L P4 k k+ is the state at the intersection of the -shock curve with right state P and the -wave curve through P3 k P k is the state at the intersection between the -wave curve through P 5 k curve with right state P4 k and the -shock Finally, P k 6 is the state at the intersection between the -wave curve through P k 0 and the -wave curve through P k+ We observe that, by choosing α > 0 small, all points P3 k will lie in a suitably small neighborhood of P 3 By the implicit function theorem, all the states P5 k, k, are well defined and lie in a suitably small neighborhood of P 5 In turn, again by the implicit function theorem, it follows that all the states P k defined and lie in a small neighborhood of P are well After six steps, all sequences of points P3 k, P 5 k, P k, P 0 k, P 4 k, P k, P 6 k, k =,, are thus uniquely determined, provided that α > 0 was chosen sufficiently small Moreover, we have the convergence lim P i k = P i i = 0,,, 6 (55) k We remark that, by the convergence P k P and P6 k P 6, it follows that (by possibly shrinking the value of α) the states P6 k k+ and P are always connected by a -compression (not a -rarefaction) The previous analysis achieves the construction of the modified interaction pattern shown in Fig 0 54 An approximate solution with finite time BV blow-up The approximate solution constructed in the previous step (Fig 0) contains a sequence of -shocks followed by a -rarefaction, both of strength α/k, k =,, Clearly, the total strength of all these waves is infinite

23 t τ T 0 h x λt λt+h y Figure : After the transformation of the t-x coordinates defined at (56), from the interaction pattern shown on the left (same as the one in Fig 0) we obtain a new approximate solution where all wave interactions take place within the time interval [0, T ] At the terminal time τ = T, the total variation becomes infinite To provide an example where blow up of the total variation occurs in finite time, it suffices to slightly modify the wave speeds, so that the interaction cycles repeat over shorter and shorter time intervals [τ k, τ k+ ], with τ k T as k To fix the ideas, assume that in the previous construction the basic interaction cycle takes place on the parallelograms { } Γ k = (t, x) ; t [k, k + ], x [ λt, λt + h], k =,, for some h > 0 (see Fig, left) Fix T > 0 sufficiently large and define ε = ( ln ) ( = ln + ) T T Observe that, as T +, we have Consider the transformation (see Fig ) defined for t 0, x IR εt = + O() T τ = ( e εt ) T, y = (x + λt + λt )e εt, (56) If now x = ξ(t) is the equation of a wave front in the t-x coordinates, let y = ζ(τ) be the corresponding equation in the τ-y coordinates Differentiating wrt t the identity y(t, ξ(t)) = ζ ( τ(t, ξ(t)) ), 3

24 we compute ζ (τ) = y t + y x ξ τ t + τ x ξ (t) = λe εt (x + λt + λt )εe εt + e εt φ T εe εt = λ( εt ) + ε(x + λt) + ξ (t) εt = ξ (t) + O() T, (57) as long as x + λt [0, h] In other words, by choosing the blow up time T large enough, the speeds of all waves contained in the strip {x + λt [0, h]} are almost unchanged by the coordinate transformation Furthermore, we impose that each pair of -shocks and - rarefactions (created at each interaction cycle) travels with the same speed in the τ-y as in the old t-x coordinates (see Fig ), right) In view of (57), this approximate solution in the τ-y variables satisfies all conditions (C) (C4) stated at the beginning of this section 6 A necessary condition for blowup In the second part of this paper, we prove that, if the total variation blows up in finite time, then the solution must contain an infinite number of large shocks in a neighborhood of some point P = (T, x) Since the p-system with p(v) = v γ admits a group of rescalings, a precise meaning of large shock must be given in terms of the upper and lower bounds on the gas density ρ We recall that, as proved in [], for any b > 0 the domain D b = {(h, u) ; h 0, u b h} (6) is positively invariant for the system (), with p and h as in (), (3) In the following we shall assume that the density ρ remains uniformly positive, and hence the same holds for h We thus consider a solution taking values in the domain D ab = {(h, u) ; u b h, h a}, (6) for some 0 < a < b In the following, the total variation of the vector-valued function x ( h(t, x), u(t, x) ) on an (possibly unbounded) interval I IR is defined as TotVar { (h, u)(t, ) ; I } ( = sup h(t, x i ) h(t, x i ) + u(t, x i ) u(t, x i ) ), i where the supremum is taken over all finite increasing sequences of points x 0 < x < < x N contained in I Observe that, as long as the solution takes values inside the compact domain D ab, a bound on the total variation of (v, u) is equivalent to a bound on the total variation of (h(v), u) In turn, this is also equivalent to a bound on the total strength of waves, measured in Riemann invariants, as in (9) Theorem 6 For any two constants b > a > 0, there exists δ 0 > 0 such that the following holds Consider an entropy weak solution (v, u) of () such that (h(v), u) D ab for all t, x, 4

25 and assume that the total variation is initially bounded but blows up at a finite time T Then there exists a point x such that every neighborhood of (T, x) in the t-x plane contains infinitely many shocks with strength δ 0 A proof of this theorem will be completed in the next two sections We observe that, since the initial data ( v, ū) have bounded variation, for every ε 0 > 0 there exists R 0 > 0 sufficiently large such that TotVar { ( v, ū) ; ], R 0 [ } < ε 0 TotVar { ( v, ū) ; ]R 0, + [ } < ε 0, For a solution taking values in the domain D ab, the characteristic speeds ±c in remain uniformly bounded above and below Indeed, since a h b, by (6) and (3) it follows c = p (v) = A/γ v (γ+)/ A/γ (b/b) (γ+)/(γ ) = ˆλ (63) By choosing ε 0 small enough, by the Glimm interaction estimates it follows that for any t > 0 the total variation of the solution on the two domains ], R 0 ˆλt[, ]R 0 + ˆλt, + [, (64) remains uniformly small Here ˆλ is the upper bound on all characteristic speed, computed at (63) Hence, if the total variation blows up at time T, this must happen within the compact interval [ R 0 ˆλT, R 0 + ˆλT ] We conclude this section with a preliminary lemma Lemma 6 For any BV function x (v(x), u(x)) with ( h(v(x)), u(x) ) Dab for all x IR, (65) the following holds (i) For any compact interval I, one has ( ) [total strength of all waves in I] b + [total strength of rarefaction waves in I] (66) (ii) There exists δ > 0 such that, for every subinterval J IR of length δ one has [total strength of all waves in J] b (67) Proof To prove (i), call µ, µ the corresponding wave measures, defined as in Section 5 Moreover, call µ + i the positive part of µ i Then (66) means that ) µ (I) + µ (I) (b + µ + (I) + µ+ (I) (68) To prove (68) we observe that the u component has a downward jump at every point of shock More precisely, for an i-shock located at a point x, one has u( x+) u( x ) w i ( x+) w i ( x ) < 0 5

26 If I = [α, β], since u takes values inside D ab we have the inequalities b u(β+) u(α ) µ (I) + µ (I), This yields (68) µ (I) µ + (I) + µ (I) µ + (I) b Next, if (ii) fails, then we can find a sequence of intervals J n = [α n, β n ] with lengths β n α n = n, such that µ (J n ) + µ (J n ) > b for every n By taking a subsequence we can assume the convergence α n x I This implies µ ({ x}) + µ ({ x}) b (69) As defined in Section 5, the left hand side of (69) is the total strength of the two waves in the solution of the Riemann problem with left and right states (v, u)( x ), (v, u)( x+) Since this solution takes values in the domain D ab, the sum of these two strengths must be (b a) We thus reach a contradiction with (69), proving the second part of the lemma 7 Wave decay estimates In this section we prove two estimates on the decay of rarefaction waves, extending the analysis in Section 3 to general BV solutions t } } Γ t 0 x y y x x Figure : The two types of rarefaction waves which can cross the upper boundary of the trapezoidal domain Γ at (7) 7 Solutions without large shocks We first study the simpler case where no large shock is present Consider a BV solution of () Fix a time step t and a space step x, such that x = ˆλ t, (7) and consider a domain of the form (see Fig ) { Γ = (t, x) ; t [t 0, t ], x [x + ˆλ(t t 0 ), x ˆλ(t } t 0 )], (7) 6

27 with t t 0 = t, x x = x (73) We seek an estimate on the total amount of rarefaction waves at time t, along the upper boundary [y, y ] = [ x + ˆλ(t t 0 ), x ˆλ(t t 0 ) ] Notice that, by (7) and (73), this interval has length y y = x As shown in Fig, these rarefactions can be of two types: ) Old rarefactions which were already present along the basis [x, x ] at time t 0 The total amount of these waves can be controlled because, as discussed in Section 3, their density has decayed during the entire time interval [t 0, t ] Roughly speaking, we have [total amount of old rarefactions] (y y ) [maximum density] (y y ) O() t t 0 = O() ˆλ ) New rarefactions produced by wave interactions inside the domain Γ Assuming that the total strength of all waves at the initial time t 0 is K 0 and all shocks in Γ have size δ 0, the total strength of these new waves will be of order O() δ 0 K 0 Lemma 7 Let D ab be the domain in (6) Then one can find a constant K ab such that, for any given K 0, there exists δ 0 > 0 for which the following holds Let (v, u) be a BV solution of () taking values inside D ab and let Γ be the trapezoid defined at (7) Assume that (i) at time t 0 the total strength of all waves contained inside the lower boundary [x, x ] is K 0, and (ii) all shocks inside Γ have strength δ 0 Call ˆµ i the measures of i-waves in the solutions at time t Then the total strength of all rarefactions waves contained inside the upper boundary [y, y ] satisfies (ˆµ + + ˆµ+ )( [y, y ] ) K ab (74) Notice that here the constant K ab can be large, but is independent of K 0 This implies that, if the initial data contain a large amount of waves but the solution does not develop large shocks, then most of the rarefaction waves present at time t 0 will disappear during the time interval [t 0, t ], being canceled by waves of the same family but opposite sign Compared with the decay estimate proved in Chapter 0 of [4], the main difference is that here the total strength of waves can be large However, thanks to Proposition, the total strength of new waves produced by interactions can be made arbitrarily small by choosing δ 0 > 0 small enough 7

28 Proof Consider any interval I = [a, b] [y, y ] Call t ξ (t), t ξ (t) respectively the minimal backward -characteristics passing through a, b at time t Motivated by (38), we define ξ (t) Z(t) = ϕ(t, x) dx, (75) where ξ (t) ϕ(t, x) = ϕ ( w (t, x) w (t, x) ) = ϕ(h(t, x)) is the function defined at (34) As long as the solution (v, u) takes values in D ab we have h(v) a, hence ϕ remains bounded and uniformly positive The integral (75) thus provides an equivalent way to measure the distance ξ (t) ξ (t) between the two characteristics Since the total variation is bounded and all characteristic speeds are bounded by ˆλ, the function t Z(t) is Lipschitz continuous Its time derivative Ż = ξ ϕ(ξ ) ξ ξ ϕ(ξ ) + D t ϕ (76) ξ is well defined for ae time t Notice that here D t ϕ is a bounded measure Its atomic part is supported on the set of shocks In the following, for any given time t we denote by S the set of all shocks contained inside the interval [a(t), b(t)] and call k α {, } the family of the shock located at x α (t) Moreover, by D c x be denote the continuous (ie, non atomic) part of a distributional derivative wrt x, Motivated by (38), denoting by c(x) = c(v(t, x)) the wave speed and using (5), (76), we compute Ż ξ = c(ξ )ϕ(ξ ) c(ξ )ϕ(ξ ) ϕ c (D xw c + Dxw c ) ξ ẋ α [ϕ(x α +) ϕ(x α ) ] α S ξ = (c ϕ + cϕ ) (D xw c Dxw c ) + [ c(xα +)ϕ(x α +) c(x α )ϕ(x α ) ] ξ α S ξ ξ ϕ c (D c xw + D c xw ) α S ẋ α [ϕ(x α +) ϕ(x α ) ] (77) For each shock α S, two cases must be considered CASE : The shock at x α belongs to the first family By definition, its strength is σ α = w (x α ) w (x α +) [0, δ 0 ] In this case, we have ẋ α = c(x α+) + c(x α ) + O() σα, w (x α +) w (x α ) = O() σ 3 α, ϕ(x α +) ϕ(x α ) = ϕ (x α )σ α + O() σ α, c(x α +) c(x α ) = c (x α )σ α + O() σ α, 8

29 Using the fundamental relation c ϕ + cϕ = 0 we thus obtain [ c(xα +)ϕ(x α +) c(x α )ϕ(x α ) ] ẋ α [ϕ(x α +) ϕ(x α ) ] = [ c (x α )ϕ(x α ) + c(x α )ϕ (x α ) ] σ α + c(x α )ϕ (x α )σ α + O() σ α (78) = O() σ α CASE : The shock at x α belongs to the second family By definition, its strength is σ α = w (x α ) w (x α +) [0, δ 0 ] In this case, we have ẋ α = c(x α+) + c(x α ) + O() σα, w (x α +) w (x α ) = O() σα 3, ϕ(x α +) ϕ(x α ) = ϕ (x α )σ α + O() σ α, c(x α +) c(x α ) = c (x α )σ α + O() σ α, In this case we obtain [ c(xα +)ϕ(x α +) c(x α )ϕ(x α ) ] ẋ α [ϕ(x α +) ϕ(x α ) ] = [ c (x α )ϕ(x α ) + c(x α )ϕ (x α ) ] σ α c(x α )ϕ (x α )σ α + O() σ α (79) = c (x α )ϕ(x α )σ α + O() σ α From (77), using (78)-(79) and the relation c ϕ + cϕ = 0 one obtains Ż = ξ ξ c ϕ Dxw c + c (x α )ϕ(x α ) σ α + O() σα (70) α S Here the first summation ranges over the set S of all shocks of the second family, while the second summation ranges over the set of all shocks (of both families) α S Call ˆµ the measure of -waves in the solution at time t For any ε > 0 we can find finitely many intervals [a l, b l ], l =,, m, whose union contains nearly all positive -waves, and very few negative -waves More precisely: ( ˆµ + [y, y ] \ ) ( ) [a l, b l ] ε, ˆµ [a l, b l ] ε (7) l l For each such interval, let ξ l (t), ξ l (t) be the minimal backward -characteristics through a l, b l, respectively Setting Z l (t) = ξl (t) ξ l (t) ϕ(t, x) dx and applying (70) to each subinterval [ξ l, ξ l ] we obtain Ż l (t) = l ξl ξ l c ϕ Dxw c + c (x α )ϕ(x α ) σ α + O() σα, (7) α S α S 9

30 where now S and S refer to the shocks contained in the union of the intervals [ξ l (t), ξ l (t)] For convenience, we introduce the constants 0 < κ min = min c ϕ, κ max = max c ϕ, ϕ max = max ϕ, defined by taking the minimum and the maximum values of the functions c ϕ and ϕ over the domain D ab Using Lemmas 3 and 4, the amounts of positive and negative -waves contained in the union of the intervals [ξ l (t), ξ l (t)] at any time t [t 0, t ] can be estimated as ( ) ( ) µ + [ξ l (t), ξ l (t)] µ + [ξ l (t ), ξ l (t )] O() δ 0 µ int (Γ), (73) µ l ( ) [ξ l (t), ξ l (t)] l µ l ( ) [ξ l (t ), ξ l (t )] + O() δ 0 µ int (Γ) (74) l Combining (7) with (73)-(74) we obtain ( ) ( ) Ż l κ min µ + [ξ l (t), ξ l (t)] κ max µ [ξ l (t), ξ l (t)] + O() σα l ( l l α S ) ( ) κ min ˆµ + [a l, b l ] κ max ˆµ [a l, b l ] O() δ 0 µ int (Γ) O() δ 0 V (t) ( l ) l κ min ˆµ + ([y, y ]) ε κ max ε O() δ 0 µ int (Γ) O() δ 0 K 0 κ min ˆµ + ([y, y ]) O() δ 0 K0 O() ε Observing that Z l (t 0 ) 0, Z l (t ) (y y ) ϕ max, l l (75) and integrating (75) over the time interval [t 0, t ], we obtain (t t 0 )κ min ˆµ + ([y, y ]) (y y ) ϕ max + O() (t t 0 )δ 0 K 0 + O() (t t 0 )ε (76) 3 Since y y = ˆλ(t t 0 ) and ε > 0 can be taken arbitrarily small, (76) yields an a priori bound on the total amount of positive -waves at the terminal time t, namely ˆµ + ([y, y ]) ˆλ ϕ max κ min + C δ 0 K 0, (77) for a suitable constant C Of course, an entirely similar estimate is valid for rarefaction waves of the first family For any given K 0, we can now choose δ 0 > 0 so that ˆµ + ([y, y ]) + ˆµ + ([y, y ]) ˆλ ϕ max κ min + C δ 0 K 0 3ˆλ ϕ max κ min = Kab (78) With the above definition of the constant K ab, the conclusion of the Lemma is achieved 30

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