1. Introduction. Consider a strictly hyperbolic system of n conservation laws. 1 if i = j, 0 if i j.

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1 SIAM J MATH ANAL Vol 36, No, pp c 4 Society for Industrial and Applied Mathematics A SHARP DECAY ESTIMATE FOR POSITIVE NONLINEAR WAVES ALBERTO BRESSAN AND TONG YANG Abstract We consider a strictly hyperbolic, genuinely nonlinear system of conservation laws in one space dimension A sharp decay estimate is proved for the positive waves in an entropy weak solution The result is stated in terms of a partial ordering among positive measures, using symmetric rearrangements and a comparison with a solution of Burgers s equation with impulsive sources Key words hyperbolic conservation laws, positive nonlinear waves, Burgers s equation AMS subject classifications 35L65 DOI 11137/S Introduction Consider a strictly hyperbolic system of n conservation laws 11) u t + fu) x = and assume that all characteristic fields are genuinely nonlinear Call λ 1 u) < < λ n u) the eigenvalues of the Jacobian matrix Au) = Dfu) We shall use bases of left and right eigenvectors l i u), r i u) normalized so that 1) λ i u) r i u) 1, l i u) r j u) = { 1 if i = j, if i j Given a function u : R R n with small total variation following [BC], [B], one can define the measures µ i of i-waves in u as follows Since u BV, its distributional derivative D x u is a Radon measure We define µ i as the measure such that 13) µ i = li u) D x u restricted to the set where u is continuous, while, at each point x where u has a jump, we define 14) µ i {x} ) = σi, where σ i is the strength of the i-wave in the solution of the Riemann problem with data u = ux ), u + = ux+) In accordance with 1), if the solution of the Riemann problem contains the intermediate states u = ω,ω 1,,ω n = u +, the strength of the i-wave is defined as 15) σ i = λi ω i ) λ i ω i 1 ) Observing that σ i = l i u + ) u + u )+O1) u + u, Received by the editors May 13, 3; accepted for publication in revised form) October 4, 3; published electronically August 7, 4 SISSA, Via Beirut 4, Trieste 3414, Italy bressan@sissait) The research of this author was supported by Italian MIUR research project 1719, Equazioni iperboliche e paraboliche non lineari Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong Matyang@ mathcityueduhk) The research of this author was supported by CityU Direct Allocation Grant

2 66 ALBERTO BRESSAN AND TONG YANG we can find a vector l i x) such that ) 16) l i x) l i ux+) = O1) ux+) ux ), 17) σ i = l i x) ux+) ux ) ) We can thus define the measure µ i equivalently as 18) µ i = li D x u, ) where l i x) =l i ux) at points where u is continuous, while li x) is some vector which satisfies 16) 17) at points of jump For all x R there holds ) 19) l i x) l i ux) = O1) ux+) ux ) We call, µ i, respectively, the positive and negative parts of µ i, so that 11) µ i = µ i, µ i = + µ i It is our purpose to prove a sharp estimate on the decay of the density of the measures This will be achieved by introducing a partial ordering within the family of positive Radon measures In the following, measa) denotes the Lebesgue measure of a set A Definition 1 Let µ, µ be two positive Radon measures We say that µ µ if and only if 111) sup µa) measa) s sup µ B) for every s> measb) s In some sense, the above relation means that µ is more singular than µ Namely, it has a greater total mass, concentrated on regions with higher density Notice that the usual order relation 11) µ µ if and only if µa) µ A) for every A R is much stronger Of course µ µ implies µ µ, but the converse does not hold Following [BC], [B], together with the measures µ i, we define the Glimm functionals 113) V u) = i µ i R), 114) Qu) = i<j µ j µ i ){ x, y); x<y } + i µ i µ i ){ x, y); x y } Now let u = ut, x) be an entropy weak solution of 11) If the total variation of u is small and the constant C is large enough, it is well known that the quantities 115) Qt) = Q ut) ), Υt) = V ut) ) + C Q ut) ) are nonincreasing in time The decrease in Q controls the amount of interaction, while the decrease in Υ controls both the interaction and the cancellation in the solution An accurate estimate on the measure t of positive i-waves in ut, ) will be obtained by a comparison with a solution of Burgers s equation with source terms

3 SHARP DECAY ESTIMATE FOR POSITIVE NONLINEAR WAVES 661 Theorem 1 For some constant κ and for every small BV solution u = ut, x) of the system 11) the following holds Let w = wt, x) be the solution of the scalar Cauchy problem with impulsive source term 116) w t +w /) x = κ sgnx) d dt Q ut) ), 117) Then, for every t, 118) w,x) = sgnx) sup measa)< x t D x wt) A) As shown in the next section, the initial data in 117) represents the odd rearrangement of the function v i x) = ) ],x] The above theorem improves the earlier estimate derived in [BC] For a scalar conservation law with strictly convex flux, a classical decay estimate was proved by Oleinik [O] In the case of genuinely nonlinear systems, results related to the decay of nonlinear waves were also obtained in [GL], [L1], [L], [L3], [BG] An application of the present analysis can be found in [BY], where Theorem 1 plays a key role in the estimate of the rate of convergence of vanishing viscosity approximations Lower semicontinuity Let µ be a positive Radon measure on R, so that µ = D x v is the distributional derivative of some bounded, nondecreasing function v : R R We can decompose µ = µ sing + µ ac as the sum of a singular and an absolutely continuous part, wrt Lebesgue measure The absolutely continuous part corresponds to the usual derivative z = v x, which is a nonnegative L 1 function defined at ae point We shall denote by ẑ the symmetric rearrangement of z, ie, the unique even function such that 1) ) ẑx) =ẑ x), ẑx) ẑx ) if <x<x, {x } ) {x } ) meas ; ẑx) >c = meas ; zx) >c for every c> Moreover, we define the odd rearrangement of v as the unique function ˆv such that Figure 1) 3) 4) ˆv x) = ˆvx), ˆv+) = 1 µsing R), ˆvx) =ˆv+) + x zy) dy for x> By construction, the function ˆv is convex for x< and concave for x> The relation between the odd rearrangement ˆv and the partial ordering 11) is clarified by the following result, which is an easy consequence of the definitions Proposition 1 Let µ = D x v and µ = D x v be positive Radon measures Call ˆv, ˆv the odd rearrangements of v, v, respectively Then µ D xˆv µ and moreover 5) 6) µa) ˆvx) = sgnx) sup, measa) x µ µ if and only if ˆvx) ˆv x) for all x>

4 66 ALBERTO BRESSAN AND TONG YANG v v^ x x Fig 1 Two more results will be used in what follows By the restriction of a measure µ to a set J, we mean the measure µ J)A) = µa J) Proposition Let µ, µ be positive measures Consider any finite partition R = J 1 J N If the restrictions of µ, µ to each set J l satisfy µ J l µ J l, then µ µ Proposition 3 Assume that µ D s w for some nondecreasing odd function w If µ µ R) ε, then [ µ D s w + sgns) ε ] The next result is concerned with the lower semicontinuity of the partial ordering wrt weak convergence of measures Proposition 4 Consider a sequence of measures µ converging weakly to a measure µ Assume that the positive parts satisfy µ + Dw for some odd, nondecreasing functions s w s), concave for s> Letw be the odd function such that ws) = lim inf w s) for s> Then the positive part of µ satisfies 7) µ + D s w Proof By possibly taking a subsequence, we can assume that w s) ws) for all s Moreover, we can assume the weak convergence µ + µ +, µ µ for some positive measures µ +, µ We thus have 8) µ = µ + µ, µ + µ +, µ µ By 8) it suffices to prove that µ + D s w, ie, 9) meas A) s = µ + A) ws) for every s> and every Borel measurable set A R If 9) fails, there exists s> and a set A such that meas A) =s, µ + A) > ws) = lim w s)

5 SHARP DECAY ESTIMATE FOR POSITIVE NONLINEAR WAVES 663 Since w is continuous for s>, we can choose an open set A A such that, setting s = meas A )/, one has ws ) < µ + A) By the weak convergence µ + µ + one obtains µ + A ) lim inf µ+ A ) ws ) < µ + A), reaching a contradiction Hence 9) must hold Toward the proof of Theorem 1 we shall need a lower semicontinuity property for wave measures, similar to what was proved in [BaB] In the following, C is the same constant as in 115) Lemma 1 Consider a sequence of functions u with uniformly small total variation and call the corresponding measures of positive i-waves Let s w s), 1, be a sequence of odd, nondecreasing functions, concave for s>, such that [ 1) D s w + C sgns) Q Qu ) )] for some Q Assume that u u and w w in L 1 loc Then the measure of positive i-waves in u satisfies 11) D s [w + C sgns) Q Qu) )] Proof The main steps follow the proof of Theorem 11 in [B] 1 By possibly taking a subsequence we can assume that u x) ux) for every x and that the measures of total variation converge weakly, say, 1) µ = Dx u µ for some positive Radon measure µ In this case one has µ µ, in the sense of 11) Let any ε> be given Since the total mass of µ is finite, one can select finitely many points y 1,,y N such that 13) µ {x} ) <ε for all x/ {y 1,,y N } We now choose disjoint open intervals I k =]yk ρ, y k + ρ[ such that 14) µ I k \{y k } ) < ε N, k =1,,N Moreover, we choose R> such that 15) N I k [ R, R], k=1 µ ], R] [R, [ ) <ε Because of 13), we can now choose points p < R <p 1 < <R<p r which are continuity points for u and for every u, such that 16) µ {p h } ) =, u p h ) up h ) for all h =,,r and such that either 17) p h p h 1 < ε N, p h 1 <y k <p h, [p h 1,p h ] I k

6 664 ALBERTO BRESSAN AND TONG YANG for some k {1,,N}, or else 18) µ [p h 1,p h ] ) µ [p h 1,p h ] ) <ε Call J h =[ph 1,p h ] If 18) holds, by weak convergence for some sufficiently large one has 19) µ J h ) <ε for all On the other hand, if 17) holds, from 14) it follows that ) µ J h \{y k } ) µ J h \{y k } ) < ε N In the remainder of the proof, the main strategy is as follows On the intervals J hk) containing a point y k of large oscillation, we first replace each u by a piecewise constant function ū having a single jump at y k The relations between the corresponding measures µ i and µ i are given by Lemma 1 in [B] Then we take the limit as On the remaining intervals J h with small oscillation, we replace the left eigenvectors l i u ) by a constant vector l i u h ) Then we use Proposition 4 to estimate the limit as 3 We first take care of the intervals J h containing a point y k of large oscillation, so that 17) holds For each k =1,,N, let h = hk) {1,,r} be the index such that y k J h =[ph 1,p h ] For every 1 consider the function ū x) = u x) if x/ k J hk), u p hk) 1 ) if x ]p hk) 1,y k [, u p h ) if x [y k,p hk) ] Observe that all functions u, ū are continuous at every point p,,p r and have jumps at y 1,,y N Call µ i, i =1,,n, the corresponding measures, defined as in 18) with u replaced by ū Clearly µ i = µ i outside the intervals J hk) of large oscillation By Lemma 1 at page 3 in [B], there holds Qū ) Qu ), Vū )+C Qū ) V u )+C Qu ), As a consequence, from 1) we deduce R) R) C [ Qu ) Qū ) ] 1) D s [T ε w + C sgns) Q Qū ) )], where T ε ws) = Indeed, all the mass which in Ω = { ws + ε/) if s>, ws ε/) if s < lies on the set N J hk), J h =[ph 1,p h ] k=1

7 SHARP DECAY ESTIMATE FOR POSITIVE NONLINEAR WAVES 665 is replaced in by point masses at y 1,,y N We obtain 1) by observing [ that, by 17), measω) <ε Moreover, the increase in the total mass is C Qu ) Qū ) ] Since u p h ) up h ) for every h, there holds µ i {y k } ) µ i {yk } ) { = O1) uy k ) up hk) 1 ) + uy k +) up hk) ) + up hk) 1 ) u p hk) 1 ) + up hk) ) u p hk) ) } ) = O1) ε N for each k =1,,N and all sufficiently large By construction we also have 3) µ i J hk) \{y k } ) =, µ i J hk) \{y k } ) ε = O1) N 4 Next, call S = { h ; µ J h ) <ε } the family of intervals where the oscillation of every u is small, so that 18) holds If h S, for every x, y J h and sufficiently large, one has u x) u y) µ J h ) <ε, ux) uy) µ Jh ) µ J h ) <ε Set u h = up h ) By the pointwise convergence u p h ) up h ) and the two above estimates it follows that u x) u h <ε, ux) u 4) h <ε for all x Jh 5 We now introduce the measures ˆµ i such that ˆµ i = l i u h) D x u restricted to each interval J h, h S, where the oscillation is small, while ˆµ i = µ i on each interval J h = J hk) where the oscillation is large Observe that the restriction of ˆµ i to J hk) consists of a single mass at the point y k Namely, ˆµ i {yk } ) is precisely the size of the ith wave in the solution of the Riemann problem with data u = u p hk) 1 ), u + = u p hk) ) We define ŵ as the nondecreasing odd function such that 5) ŵ s) = sup measa) s ˆ A), s > By possibly taking a further subsequence we can assume the convergence Qū ) Q, ˆµ i ˆµ i, ŵ s) ŵs) Using 16), we can apply Proposition 4 on each interval J h and obtain 6) ˆ D s ŵ

8 666 ALBERTO BRESSAN AND TONG YANG 6 Observe that, by 4) and 19), 7) ˆµ i µ i J h )=O1) εµ J h ), h S From 1) and the definition of ŵ at 5) it thus follows that 8) ŵ s) T ε w s)+c [ Q Qū ) ] + O1) ε, s > Letting we obtain 9) 3) ŵs) T ε ws)+c [Q Q]+O1) ε, s >, Q = lim Qū ) lim Qu ) O1) ε Qu) O1) ε, because of the lower semicontinuity of the functional u Qu) From 6), 9), and 3) we deduce ˆ D s [T ε w + sgns) C [Q Qu)] + O1) ε )] By ) 4), our construction of the measure ˆµ i achieves the property ˆ R) =O1) ε Hence, by Proposition 3, D s [T ε w + sgns) C [Q Qu)] + O1) ε )] Since ε> was arbitrary, this proves 11) 3 A decay estimate The second basic ingredient in the proof is the following lemma, which refines the estimate in [BC] Lemma For some constant κ>the following holds Let u = ut, x) be any entropy weak solution of 11), with initial data u,x) =ūx) having small total variation Then the measure t of positive i-waves in ut, ) can be estimated as follows Let w :[,τ[ R R be the solution of Burgers s equation 31) w t +w /) x = with initial data 3) Set w,x) = sgnx) sup measa) x A) 33) wτ,x)=wτ, x)+κ sgnx) [Qū) Q uτ) )] Then 34) τ D x wτ) Proof The main steps follow the proof of Theorem 13 in [B] We first prove the estimate 33) under the following additional hypothesis

9 SHARP DECAY ESTIMATE FOR POSITIVE NONLINEAR WAVES 667 H) There exist points y 1 < < y m such that the initial data ū is smooth outside such points, constant for x<y 1 and x>y m, and the derivative component l i u) u x is constant on each interval ]y l,y l+1 [ Moreover, the Glimm functional t Q ut) ) is continuous at t = τ 1 The solution u = ut, x) can be obtained as the limit of front tracking approximations In particular, we can consider a particular converging sequence u ) 1 of ε -approximate solutions with the following additional properties: i) Each i-rarefaction front x α travels with the characteristic speed of the state on the right: ẋ α = λ i uxα +) ) ii) Each i-shock front x α travels with a speed strictly contained between the right and the left characteristic speeds: 35) λ i uxα +) ) < ẋ α <λ i uxα ) ) iii) As, the interaction potentials satisfy 36) Q u, ) ) Qū) Let u be an approximate solution constructed by the front tracking algorithm By a generalized) i-characteristic we mean an absolutely continuous curve x = xt) such that ẋt) [ λ i u t, x )), λ i u t, x+)) ] for ae t If u satisfies the above properties i) ii), then the i-characteristics are precisely the polygonal lines x : [,τ] R for which the following holds For a suitable partition = t <t 1 < <t m = τ, on each subinterval [t j 1,t j ] either ẋt) =λ i u t, x) ), or else x coincides with a wave front of the ith family For a given terminal point x we shall consider the minimal backward i-characteristic through x, defined as yt) = min { xt); x is an i-characteristic, xτ) = x } Observe that y ) is itself an i-characteristic By 35), it cannot coincide with an i-shock front of u on any nontrivial time interval In connection with the exact solution u, we define an i-characteristic as a curve t xt) = lim x t) which is the limit of i-characteristics in a sequence of front tracking solutions u u 3 Let ε> be given If the assumption H) holds, the measure τ of i-waves in uτ) is supported on a bounded interval and is absolutely continuous wrt Lebesgue measure We can thus find a piecewise constant function ψ τ with jumps at points x 1 τ) < x τ) < < x N τ) such that d τ dx xj+1τ) ) ψτ dµ i+ dx<ε, τ x jτ) dx ψτ dx =, j =1,,N 1 37) To prove the lemma in this special case, relying on Proposition, it thus suffices to find i-characteristics t x j t) such that the following hold Figure ):

10 668 ALBERTO BRESSAN AND TONG YANG τ x I t) j x 1 xj x j+1 x Fig ] i) For each j =1,,N, the function ψ τ is constant on the interval xj τ), x j+1 τ) [ and 37) holds Moreover, either x j ) = x j+1 ), or else the derivative component ψ = l i u)u x, ) is constant on the interval ] x j ), x j+1 ) [ ii) [ An estimate corresponding to 33) 34) holds restricted to each subinterval xj τ), x j+1 τ) [ We need to explain in more detail this last statement Define I j t) = [ x j t), x j+1 t) [, j = { t, x); t [,τ], x Ij t) } For each j, we denote by Γ j the total amount of wave interaction within the domain j This is defined as in [B], first for a sequence of front tracking approximations u, then taking a limit as Furthermore, we define the constant values = ψ τ x), x I j τ), Call ψ τ j ψ j = ψ x), x I j ) σ j = lim t + µi+ I j t) ) the initial amount of positive i-waves inside the interval I j For each interval I j, we consider on one hand the function wj τ 3) 33), namely, { } wj τ s) = min σj s, τ +ψj + κγ j sgns) ) 1 corresponding to Here ψj ) 1 = in the case where xj ) = x j+1 ) This may happen when the initial data has a jump at x j ), and the corresponding measure has a Dirac mass with infinite density) at that point On the other hand, we look at the nondecreasing, odd function η j such that η j s) = { min ψj τ s, ψj τ [ xj+1 τ) x j τ) ]}, s > Our basic goal is to prove that Figure 3) 38) η j s) w τ j s) for all s>

11 SHARP DECAY ESTIMATE FOR POSITIVE NONLINEAR WAVES 669 w s) j σ j +C Γ j η js) κγ j s* j s Fig 3 Indeed, by 37), for s> one has with sup measa) s τ Proving 38) for each j will thus imply A Ij τ) ) ε j <ε j η j s)+ε j τ wτ,x)=wτ,x)+κsgnx) [Qū) Q uτ) ) + O1) ε ] Since ε>was arbitrary, this establishes the lemma under the additional assumptions H) 4 We now work toward a proof of 38), in three cases Case 1: σj = Case : x j ) = x j+1 ) and σj > Case 3: x j ) <x j+1 ) and σj = x j+1 ) x j ) ) ψj > In Case 1 the proof is easy Indeed, the total amount of positive i-waves in I j τ) is here bounded by a constant times the total amount of interaction taking place inside the domain j, ie, for some constant C On the other hand τ Ij τ) ) C Γ j w τ j s) =κγ j sgns) Choosing κ>c we achieve 38) 5 Since Case can be obtained from Case 3 in the limit as x j+1 x j, we shall only give a proof for Case 3 We can again distinguish two cases If the amount of interaction Γ j is large compared with the initial amount of i-waves, say Γ j 1 6C σ j,

12 67 ALBERTO BRESSAN AND TONG YANG then the bound 38) is readily achieved choosing κ>8c Indeed, for s>wehave η j s) 1 µi+ τ Ij τ) ) C Γ j + σj 7C Γ j The more difficult case to analyze is when Γ j is small, say 39) Γ j <σ j /6C Looking at Figure 3, it clearly suffices to prove 38) for the single value Equivalently, calling s = s j = x j+1τ) x j τ) z j t) = x j+1 t) x j t) the length of the interval I j t) and = Ij τ) ) = z j τ) ψj τ σ τ j the total amount of positive i-waves inside I j τ), we need to show that { } 31) σj τ κγ j + min σj, τ s j τ +ψ j ) 1 By the approximate conservation of i-waves over the region j, we can write 311) σ τ j σ j + C Γ j Using 311) in 31), our task is reduced to showing that 31) s j σj τ κγ j + τ +ψj ) 1 for a suitably large constant κ Because of 311), it suffices to show that z j τ) σj C Γ j ) τ +ψj ) 1) = [ z j ) + τσj ] C 313) τ +ψj ) 1) Γ j for a suitable constant C 6 We now prove 313) Notice that, by genuine nonlinearity and the normalization 1), if no other waves were present in the region j we would have Γ j = and d dt z jt) σj In this case, the equality would hold in 313) To handle the general case, we represent the solution u as a limit of front tracking approximations u, where for each 1 the function u, ) contains exactly rarefaction fronts equally spaced along the interval I j ) Figure 4) Each of these fronts has initial strength σ α ) = σ j / For α =1,,, let y αt) I j t) bethe

13 SHARP DECAY ESTIMATE FOR POSITIVE NONLINEAR WAVES 671 τ I τ) j x α σ β I ) j x Fig 4 location of one of these fronts at time t [,τ], and let σ α t) > be its strength Moreover, call J α t) = [ y α t),y α+1 t) ], α = { t, x); t [,τ], x Jα t) }, and let Γ α be the total amount of interaction in u taking place inside the domain α We define a subset of indices I {1,,} by setting α I if 314) 5C Γ α >σ α ) = σ j / Observe that, if α/ I, then σ α t) σ α ) 1 < 1 for all t [,τ] In particular, if α, α +1 / I, then the interval J α t) is well defined for all t [,τ] Its length satisfies the differential inequality 315) for some constant C 1 Here 316) z α t) = y α+1 t) y α t) d dt z αt) W α t) C 1 β C αt) σ β W α t) = [ amount of i-waves inside the interval J α t) ] σ α ) C Γ α,

14 67 ALBERTO BRESSAN AND TONG YANG while C α t) refers to the set of all wave fronts of different families which are crossing the interval J α at time t Calling W α the total amount of waves of families i which lie inside J α ), we can now write 317) τ β C αt) ) σ β dt max z αt) t [,τ] σ j zj ) + 1 Γ α + O1) τγ α + O1) ) W α Indeed, by strict hyperbolicity, every front σ β of a different family can spend at most a time O1) z α inside J α Either it is located inside J α already at time t =, or else, when it enters, it crosses y α or y α+1 In this case, since α, α +1 / I, by 314) it will produce an interaction of magnitude σ β σ α σ β σj / The second term on the right-hand side of 317) takes care of the new wave fronts which are generated through interactions inside J α The last term takes into account wave fronts of different families that initially lie already inside J α at time t = Integrating 315) over the time interval [,τ] and using 316) 317) one obtains z α τ) z α )+τ σ ) j O1) τγ α O1) max z αt) t [,τ] 318) 7 To proceed in our analysis, we now show that σ j zj )+1 Γ α O1) ) W α 319) max t [,τ] z αt) z α τ) Indeed, let τ [,τ] be the time where the maximum is attained If our claim 319) does not hold, there would exist a first time τ [τ,τ] such that z α τ )=z α τ )/ From 315) and the assumption W α t) it follows that 3) τ τ C 1 β C αt) σ β dt z ατ ) Using the smallness of the total variation, a contradiction is now obtained as follows Call Φt) = C Qt)+ k β i φ kβ t, xβ t) ) σ β, where the sum ranges over all fronts of strength σ β located at x β, of a family k β i The weight functions φ j are defined as φ j t, x) = if x>y α+1 t), y α+1t) x y α+1t) y αt) if x [ y α t), y α+1 t) ], 1 if x<y α t) in the case j>i, while φ j t, x) = 1 if x>y α+1 t), x y αt) y α+1t) y αt) if x [ y α t), y α+1 t) ], if x<y α t)

15 SHARP DECAY ESTIMATE FOR POSITIVE NONLINEAR WAVES 673 in the case j < i Because of the term C Qt), the functional Φ is nonincreasing at times of interactions Moreover, outside interaction times a computation entirely similar to the one on page 13 of [B] now yields 31) d dt Φt) β C αt) σ β c zt) for some small constant c > related to the gap between different characteristic speeds From 3) and 31), respectively, we now deduce τ τ β C αt) σ β dt z ατ ) C 1, τ τ β C αt) σ β dt τ τ dφt) dt zατ ) dt Φτ ) z α τ ) c c Since Φt) =O1) TotVar { ut) }, by the smallness of the total variation we can assume Φτ ) < C 1 /c In this case, the two above inequalities yield a contradiction 8 Using 319), from 318) we obtain 3) z j τ) = z α τ) z α τ) 1 α α/ I { zα ) + τσj / zj )+1 1+C /σ α/ I j )Γ O1) τγ j O1) α ) z α ) + τ σ j α/ I ) z α ) + τ σ j α/ I By 314) the cardinality of the set I satisfies ) } W α ) 1 C σj Γ α O1) τγ j O1) zj)+1 z j ) C σj Γ j O1) τγ j O1) zj)+1 hence σ j #I 5C Γ α Γ j ; α I #I 5C Γ j In turn, this implies ) z α ) + τ σ j z j ) + τσj α/ I 33) z j ) + τσj σ j ) 1 #I ) ) z j ) 5C Γ j σj Γ j 5C τγ j

16 674 ALBERTO BRESSAN AND TONG YANG Using 33) in 3), observing that z j ) σ j and letting, we conclude = x j+1) x j ) σ j =ψ j ) 1, z j τ) z j ) + τσ j ) O1) ψ j ) 1 Γ j O1) τγ j This establishes 313) for a suitable constant C 9 In the general case, without the assumptions H), the lemma is proved by an approximation argument We construct a convergent sequence of initial data ū ū which satisfy H) and such that ū ū, Qū ) Qū), Calling w the solution of 31) with initial data, µi+ w,x) = sgnx) sup measa) x, A), by the previous analysis we have,τ D x [w τ ) + sgnx) [Qū ) Qu τ )) ]] Observe that w τ ) wτ ) inl 1 loc Choosing κ C, by the lower semicontinuity result stated in Lemma 1 we now conclude τ D x [wτ )+κ sgnx) [Qū) Quτ)) ]] 4 Proof of the main theorem Using the previous lemmas, we now give a proof of Theorem 1 For a given interval [,τ], the solution of the impulsive Cauchy problem 117) 118) can be obtained as follows Consider a partition = t <t 1 < <t N = τ Construct an approximate solution by requiring that w,x)=ˆv i x), 41) w t +w /) x = on each subinterval [t k 1,t k [, while 4) wt k,x)=wt k,x)+κ sgnx) [Qt k 1 ) Qt k ) ] We then consider a sequence of partitions = t <t 1 < <t N = τ, and the corresponding solutions w If the mesh of the partitions approaches zero, ie, lim sup k t k t k 1 =, then the approximate solutions w converge to a unique limit, which yields the solution of 117) 118) Call F the set of nondecreasing odd functions, concave for x> This set is positively invariant for the flow of Burgers s equation 41) Moreover, this flow is order preserving Namely, if w, w Fare solutions of 41) with initial data such that w,x) w,x) for all x>, then also wt, x) w t, x) for all t, x >

17 Equivalently, SHARP DECAY ESTIMATE FOR POSITIVE NONLINEAR WAVES 675 D x w) D x w ) = D x wt) D x w t) for every t> For each fixed, we can apply Lemma on each subinterval [t k 1,t k ] and obtain t k By induction on k, this yields D x w t k) = t k+1 D x w t k+1) 43) τ D x w τ), where w is the approximate solution constructed according to 41) 4) Letting and using Lemma 1, we achieve a proof of Theorem 1 5 Examples Example 1 Consider first the scalar case Let u = ut, x) be a solution of Burgers s equation with smooth initial data u t +u /) x =, u,x)=ūx) Define the function w as the solution to w t +w /) x =, w,x) = sgnx) sup measa)< x A ū x y) This corresponds to 116) 117) with Q According to Oleinik s estimate we now have u x t, x) 1/t for all t>, and ae x R Of course, this reflects the fact that, along each characteristic with ẋ = u t, xt) ), the gradient satisfies dy 51) d dt u ) 1 x t, xt) = ) t, xt) u x A better estimate on u x t, xt) ) when it is positive, based on 51), is 5) u x t, xt) ) 1 t + [ ū x x)) ] 1 According to 118), for every t> we have the relation 53) µ + t D x wt), which includes the additional information 5) This relation is sharp in the sense that the converse inequality D x wt) µ + t also holds, as long as no positive waves are cancelled by interacting with shocks Analogous results are valid for scalar equations with more general flux: u t + fu) x =, u,x)=ūx),

18 676 ALBERTO BRESSAN AND TONG YANG t 1 σ η σ σ" x Fig 5 under the assumption of genuine nonlinearity f > Notice that in this case the normalization 1) yields ru) =1/f u), lu) =f u) As long as the solution ut, ) remains smooth, the corresponding wave measure is thus defined as µ t A) = f ut, x) ) u x t, x) dx A Example Consider the p-system in Lagrangean coordinates see [Sm]) v t u x =, u t +K/v γ ) x = Here K> and γ 1 are constants Consider a solution which initially contains two approaching 1-shocks and a -rarefaction Figure 5) Assume that at time t = 1 the two shocks interact As shown in [Sm], the Riemann problem is then solved in terms of a 1-shock and a -rarefaction Let us look at the measure of positive -waves in the solution Let η be the strength of the centered rarefaction During the interval t [, 1[ the density of rarefaction waves decays, as in the scalar case At time t =1 a new centered rarefaction is created by the interaction Calling σ,σ the strengths of the incoming shocks, the strength σ of this new rarefaction will satisfy 54) σ κ σ σ = σ for a suitable constant κ> Notice that the decrease in the interaction potential at time t = 1 is Q = σ σ The values of the corresponding function w = wt, x) in 116) 118) at various times are illustrated in Figure 6 For t [, 1[ the estimate 118) is sharp, in the sense that 55) D x wt) µ + t D x wt) The first relation in 55) will fail for t 1if σ<σ The accuracy of our estimate in this case depends essentially on the careful choice of the constant κ in 54) In particular, if we could choose κ so that σ = κ σ σ, then both relations in 55) would remain valid for all times t Remark Concerning compression waves, an estimate of the form 118) could be derived also for the negative part of the measures µ i t In this case, µ i t can be compared with the gradient D x w of an odd, nonincreasing solution of a perturbed Burgers s equation However, the result does not appear to be very interesting Indeed, as time progresses negative waves become ever more singular and a bound such as 118)

19 SHARP DECAY ESTIMATE FOR POSITIVE NONLINEAR WAVES 677 η/ w) η+σ)/ ~ w1) w) w1 ) σ/ ~ x x Fig 6 retains little content For large times, a much better way to estimate negative waves is to analyze their cancellation with positive waves of the same family, as in [L1], [L] REFERENCES [BaB] P Baiti and A Bressan, Lower semicontinuity of weighted path length in BV, in Geometrical Optics and Related Topics, F Colombini and N Lerner, eds, Birkhäuser, Boston, 1997, pp [B] A Bressan, Hyperbolic Systems of Conservation Laws The One Dimensional Cauchy Problem, Oxford University Press, New York, [BC] A Bressan and R M Colombo, Decay of positive waves in nonlinear systems of conservation laws, Ann Scuola Norm Sup Pisa Cl Sci 4), ), pp [BG] A Bressan and P Goatin, Oleinik type estimates and uniqueness for n n conservation laws, J Differential Equations, ), pp 6 49 [BY] A Bressan, and T Yang, On the convergence rate of vanishing viscosity approximations, Comm Pure Appl Math, submitted [GL] J Glimm and P Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Mem Amer Math Soc, ) [L1] T P Liu, Decay to N-waves of solutions of general systems of nonlinear hyperbolic conservation laws, Comm Pure Appl Math, ), pp [L] T P Liu, Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws, Comm Pure Appl Math, ), pp [L3] T P Liu, Admissible solutions of hyperbolic conservation laws, Mem Amer Math Soc, ) [O] O Oleinik, Discontinuous solutions of nonlinear differential equations, Amer Math Soc Transl ), ), pp [Sm] J Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983