Single shock solution for non convex scalar conservation laws
aa r X i v : . [ m a t h . A P ] J u l Single shock solution for non convex scalar conservation laws
Adimurthi ∗ and Shyam Sundar Ghoshal † Centre for Applicable Mathematics,Tata Institute of Fundamental Research,Post Bag No 6503, Sharadanagar,Bangalore - 560065, India.
Abstract
In this paper we study the finite time emergence of one shock for the solution ofscalar conservation laws in one space dimension with general flux f . We give a necessaryand sufficient condition to the initial data connecting to flux. The proof relies on thestructure theorem for the linear degenerate flux and the finer analysis of characteristiccurves. MSC (2010): 35B40, 35L65, 35L67.Keywords: conservation laws; characteristic lines; non convex flux; sin-gle shock solution, structure theorem.
Let f : IR → IR be a locally Lipschitz function and u ∈ L ∞ ( IR ). Consider the initial valueproblem u t + f ( u ) x = 0 , x ∈ IR, t > , (1.1) u ( x,
0) = u ( x ) , x ∈ IR. (1.2)Here f is the flux function.The above equation has a special importance, particularly in, mathematical physicsand fluid dynamics, for example in river flow, flow of gas, oil recovery, heterogeneous media,petroleum industry, modeling gravity, continuous sedimentation, modeling car in traffic flowon a highway, semiconductor industry, etc. Some of these applications finds from convexconservation laws, for example: very famous Burgers equation where f ( u ) = u , is animportant convex scalar conservation laws, which has several applications. On the otherhand, Buckley-Leverett equation is one of the most vital example of a non-convex scalar ∗ [email protected] † [email protected] f is given by u u + r (1 − u ) , for some constant r >
0, where the solution u could represent as a water saturation, and the fraction flow function could be the flux f and the constant r denotes the viscosity ratio of water and oil.(1.1), (1.2) admits a unique weak solution satisfying Kruzkov entropy condition [17,19, 35, 9, 21, 23, 25, 36, 38]. Through out this paper we mean the solution to (1.1), (1.2) inthe sense of Kruzkov. In general, even when the data u is smooth, the solution can havediscontinuities (shock), a shock curve is a locally Lipschitz curve. One of the fundamentalquestions in this area is to find the number of shock curves. When the flux f ∈ C , uniformlyconvex, there are many prior results to (1.1). Lax [26] obtained an explicit formula (one canalso see [33] and for the boundary value problem see [22, 27]) for the solution of (1.1) in hisseminal paper and also proved that if the support of u is compact, the solution behaves likean N -wave as t → ∞ . Also large time behaviour of the solution of (1.1) has been studiedin [14, 31]. Schaffer [35] showed later that there exists a set D ⊂ C ∞ ( IR ) of first categorysuch that if u ∈ D c , then the solution u admits finitely many shock curves. Tadmor andTassa [37] constructed explicitly a dense set D c for which the number of shocks are finite.Dafermos [13, 15, 16] introduced a landmark notion of generalized backward charac-teristics to study the structure of the solution for scalar conservation laws, whereas we givea notion of forward characteristics [1, 2, 3], namely R curves, has been introduced and adetailed study has been done without using the Filipov’s theory [20]. Previously, by usingthe forward characteristics as building blocks a detailed study of structure Theorem for thestrictly convex conservation laws has been obtained in [1]. In the present paper, we provedfirst a structure Theorem for a degenerate convex flux (see definition 4.1). This turns outto be an important step for proving finite time emergence of single shock solution for nonconvex scalar conservation laws.Another striking result which has fundamental importance in this paper was obtainedby Liu [30] and Dafermos-Shearer [17], in connection with the traffic flow problem. Todescribe it succinctly, let A < B, u ∈ L ∞ ( IR ) and ¯ u ∈ L ∞ ( A, B ) such that u ( x ) = u − if x < A, ¯ u if A < x < B,u + , if x > B, (1.3)where u ± are constants. Under the assumption u − > u + , (1.4)they showed that there exists a T > , x ∈ IR such that for t > T , u ( x, t ) = ( u − if x < x + f ( u + ) − f ( u − ) u + − u − ( t − T ) ,u + if x > x + f ( u + ) − f ( u − ) u + − u − ( t − T ) . (1.5)That is, they proved that after finite time there will be exactly one shock and the solutionwill become u − and u + , this is called a single shock solution. So if the left most charac-teristics speed of the data is bigger than the right most characteristics speed then these2 ( t ) : x − x = f ( u − ) − f ( u +) u − − u + ( t − T )( x , T )¯ u u − u + A Bu ( x, t ) = u − u ( x, t ) = u + Figure 1: Illustration for one shock solution from the point ( x , T ), T ≤ γ | A − B | two characteristics will dominate the other waves in finite time. The main ingredient ofthis proof is the comparison principle of the conservation laws. The case u − ≤ u + wasleft open in Dafermos-Shearer [17]. In [1], using the finer analysis of forward characteristiccurves, a unified approach has been developed to tackle both the cases, namely u − > u + and u − ≤ u + , for any C strictly convex flux.Very few results are known when one consider genuinely non linear and linear degen-eracy together. When the flux function is non convex not much literature is known in thissubject. Even solving a Riemann problem for flux having finitely many inflection points,one has to consider either a convex hull or a concave hull and it purely depends on thedata. Therefore even for a piecewise constant data it is difficult to keep into account theinteraction between waves. Thus it is one of the reasons, the structure of the solution fornon scalar convex conservation laws are less known.In this paper we have resolved the following important questions for nonconvex flux:1. Suppose f is not convex, what is the condition such that u becomes single shocksolution (see definition 1.1 and figures 2, 1) in finite time?2. Under which condition(s) u admits a single shock solution when one consider u ± tobe a function in L ∞ ( IR \ ( A, B )) instead of constant?To answer the above questions, we first proved the structure Theorem for the flux f ∈ C but need not be strictly convex. Also we give a proof of a generalized version of theone shock solution, namely we not only consider constant data away from a compact set,also we consider general data u − ( x ) and u + ( x ) and give a necessary and sufficient criteriathat in a finite time T , this will be separated by a Lipschitz curve. We have used heavilyfront tracking analysis, convex analysis for the degenerate convex flux and L contraction.One related topic in this direction is to understand the stability estimates. For the3 ( t )( x , T )¯ u u − ( x ) u + ( x ) A Bu ( x, t ) = u − u ( x, t ) ∈ Range ( u + )Figure 2: Illustration for one shock solution from the point ( x , T ), T ≤ γ | A − B | uniformly convex flux, L stability result has been obtained by [28] for the shock situation.He showed L norm of a perturbed solution can be bounded by the L norm of initialperturbation. Later in [4], L p stability for the shock and non-shock situation has beenstudied using the structure Theorem [1] for more general convex flux. Interested reader canalso see [29] for the system case.On a slightly different direction, structure, regularity of the entropy solution has beenstudied in [5, 6, 8, 10, 11, 18, 24, 32] and the references therein for the related work. Forthe flux with one inflection point, a class of attainable sets has been obtained in [7]. Before stating the main result, we need the following definitions and hypothesis.
DEFINITION 1.1 (Single shock solution) . Let u ± ∈ L ∞ ( IR ) and A ≤ B, ¯ u ∈ L ∞ ( A, B ) and f be a locally Lipschitz function. u ( x ) = u − ( x ) if x < A, ¯ u ( x ) if A < x < B,u + ( x ) , if x > B, (1.6) let u be the solution of (1.1) with the initial data (1.6). Then the pair ( u − ( x ) , u + ( x )) givesrise to a single shock solution if for every ¯ u ∈ L ∞ ( A, B ) , there exist ( x , t ) ∈ IR × (0 , ∞ ) , γ > , a Lipschitz curve r : [ T , ∞ ) → IR depending on u ± ( x ) , || u || ∞ such that for t > T , T ≤ γ | A − B | , (1.7) r ( T ) = x , (1.8) u ( x, r ( t )) ∈ Range of u − for x < r ( t ) , (1.9) u ( x, r ( t )) ∈ Range of u + for x > r ( t ) . (1.10)4bserve that if f is uniformly convex, then by Liu’s and Dafermos-Shearer’s result,( u + , u − ) gives rise to single shock solution provided it satisfies (1.4). α α C D β β L α ,β f Figure 3: Illustration of the condition to be one shock for the convex-convex case
DEFINITION 1.2 (Convex-convex type) . (see figure 3) Let f ∈ C ( IR ) and C ≤ D .Then ( f, C, D ) is said to be a convex-convex type triplet if f | ( −∞ ,C ] and f | [ D, ∞ ) (1.11) are convex functions. DEFINITION 1.3 (Convex-concave type) . (see figure 4) Let f ∈ C ( IR ) and C ≤ D .Then ( f, C, D ) is said to be a convex-concave type triplet if f | ( −∞ ,C ] is a convex function f | [ D, ∞ ) is a concave function . (1.12) Notation:
For a, b ∈ IR , let L a,b be the line joining ( a, f ( a )) and ( b, f ( b )) given by L a,b ( θ ) = f ( a ) + f ( b ) − f ( a ) b − a ( θ − a ) . (1.13)Let f ′ ( a − ) be the left derivative of f at a if it exists. Then define the tangent line L a at( a, f ( a )) by L a ( θ ) = f ( a ) + f ′ ( a − )( θ − a ) . (1.14) Hypothesis ( H ): Let f be a C function and α ≤ α < C ≤ D < β ≤ β . Assume thatthey satisfy [ α , α ] = { x < C : f ′ ( x ) ∈ [ f ′ ( α ) , f ′ ( α )] } (1.15)[ β , β ] = { x > D : f ′ ( x ) ∈ [ f ′ ( β ) , f ′ ( β )] } (1.16) α + ( β − β ) < C ≤ D < β − ( α − α ) . (1.17)5 α α C D β β L α L α L α ( β ) L α ,β f Figure 4: Illustration of the condition to be one shock for the convex-concave case
REMARK 1.1.
From the maximum principle, without loss of generality, we can assumethat lim | p |→∞ | f ( p ) || p | = ∞ . (1.18)Therefore, through out this paper we assume that f satisfies (1.18). We have the following THEOREM 1.1 (Main Theorem) . Let f, α , α , β , β , C , D satisfies ( H ) and u ± ( · ) ∈ L ∞ ( IR ) . Assume that they satisfy any one of the following conditions:I. Let ( f, C, D ) be of convex-convex type triplet and u ± satisfies u + ( x ) ∈ [ α , α ] , (1.19) u − ( x ) ∈ [ β , β ] . (1.20) For θ ∈ [ C, D ] f ( θ ) < min { L α ,β ( θ ) , L α ,β ( θ ) , L α + β − β ,β − ( α − α ) ( θ ) } (1.21) II. Let ( f, C, D ) be of convex-concave type triplet and u ± satisfies one of the followingconditions: ondition 1: u + ( x ) ∈ [ β , β ] , (1.22) u − ( x ) ∈ [ α , α ] . (1.23) For θ ∈ [ C, D ] f ( θ ) > max { L α ,β ( θ ) , L α ,β ( θ ) , L α + β − β ,β − ( α − α ) ( θ ) } , (1.24) L α ( β ) > f ( β ) . (1.25) Condition 2: u + ( x ) ∈ [ α , α ] . (1.26) u − ( x ) ∈ [ β , β ] . (1.27) For θ ∈ [ C, D ] f ( θ ) < min { L α ,β ( θ ) , L α ,β ( θ ) , L α + β − β ,β − ( α − α ) ( θ ) } , (1.28) L β ( α ) < f ( α ) . (1.29) Then ( u − ( · ) , u + ( · )) gives rise to a single shock solution (see figure 2, 3). REMARK 1.2.
In the main Theorem, for the case when the ( f, C, D ) is a convex-convextriplet, one cannot interchange the role of u + ( x ) and u − ( x ) , see second part of the Theorem2.2. Whereas for the convex-concave situation, we can interchange the role of u + ( x ) and u − ( x ) , due to the two different polarity. REMARK 1.3.
In the proof of the main Theorem we have not used the C regularity ofthe flux in ( C, D ) as Lipschitz regularity is good enough in ( C, D ) . As an immediate corollary to the main Theorem, we have the following generalizationof Liu [30] and Dafermos-Shearer [17].
COROLLARY 1.1.
Let α = α < C ≤ D < β = β . Assume one of the followingconditions holdI ′ . Let ( f, C, D ) be of convex-convex type such that f is strictly convex on ( −∞ , C ) ∪ ( D, ∞ ) . Assume that ( u − , u + ) = ( β , α ) such that for θ ∈ [ C, D ] f ( θ ) < L u − ,u + ( θ ) . (1.30) II ′ . Let ( f, C, D ) be of convex-concave type triplet and u ± satisfies one of the followingconditions:Condition 1: u + ( x ) = β , u − ( x ) = α ,f ( θ ) > L u − ,u + ( θ ) , f or θ ∈ [ C, D ] ,L u − ( u + ) > f ( u + ) . (1.31)7 ondition 2: u + ( x ) = α , u − ( x ) = β ,f ( θ ) < L u − ,u + ( θ ) , f or θ ∈ [ C, D ] ,L u − ( u + ) < f ( u + ) . (1.32) Then ( u − , u + ) gives rise to a single shock solution.Proof. From the strict convexity and concavity, it follows that (1.15) and (1.16) hold andhence ( H ). (1.21), (1.24), (1.25) follows from (1.30) and (1.31). Hence from the mainTheorem, ( u − , u + ) gives rise to a single shock solution. This proves the corollary. The paper is organized as follows:Section 2 deals with Lax-Oleinik formula for degenerate convex scalar conservation laws viaHamilton-Jacobi equations. There we introduce the notion of characteristics curves (forwardcharacteristic curve), shock packets from [1] and using them we prove the structure Theoremfor convex flux with bound. Getting this bounds is very crucial to obtain the final resultfor non convex flux.Section 3 concerns the proof of the main Theorems with the following main steps:Step 1. Since the Riemann problem solution involves the contact discontinuities, the convexhull or the concave hull of f admits degenerate parts. That is the hulls need not bestrictly convex or concave. Therefore, first we prove the Lax-Oleinik explicit formulaand structure Theorem for C convex flux with bounds using a blow up analysis. Thisversion of Structure Theorem is new to the literature. Using this, we first prove themain Theorem for the convex flux.Step 2. Using the front tracking and L -contraction, we complete the proof of the main The-orem, which contents two parts with the different polarity and therefore the proofsare of different nature. The main ingredient for the first part of the main Theorem isstructure Theorem with bound. First we prove the convex-convex situation and thenwe prove the case convex-concave situation by using the front tracking.Step 3. We need to use some elementary properties of convex and concave functions and forthe sake of completeness we are presenting their proof in the Appendix.Finally we give counter examples to show that the conditions (1.21), (1.24) and (1.25) areoptimal.In order to make the paper self contained, we give the proof of some important Lemmas toget the stability results using techniques from convex analysis in Section 4 (appendix).8 Structure Theorem for C -convex flux Let f be a locally Lipschitz function on IR . Let K ⊂ IR be a compact set and define Lip ( f, K ) = sup x,y ∈ Kx = y | f ( x ) − f ( y ) || x − y | . (2.1)Recall the facts from Kruzkov Theorem [21, 25]. Let u , v ∈ L ∞ ( IR ) and u, v be therespective solutions of (1.1) with initial data u and v . Let M = Max {|| u || ∞ , || v || ∞ } and K = [ − M, M ]. Then(i). Comparison principle: Assume that u ( x ) ≤ v ( x ) a.e. x ∈ IR , then for a.e. ( x, t ) ∈ IR × (0 , ∞ ), u ( x, t ) ≤ v ( x, t ) . (ii). L loc contraction: Let a ≤ b, then for t > Z ba | u ( x, t ) − v ( x, t ) | dx ≤ Z b + Mta − Mt | u ( x ) − v ( x ) | dx. (iii). Let u ∈ BV ( IR ) , then for 0 ≤ s < t, Z IR | u ( x, t ) − u ( x, s ) | dx ≤ Lip ( f, K ) | s − t | T V ( u ) , where T V ( u ) denotes the total variation semi norm of u .As a consequence of this we are stating the following well known approximation Lemma[17, 21, 9]. For the sake of completeness, we are presenting the proof in the appendix. LEMMA 2.1.
Let f k be a sequence of Lipschitz functions such that for any compact set K ⊂ IR sup n Lip ( f n , K ) < ∞ . (2.2) Assume that f k → f in C loc ( IR ) . Let u k , u be the solutions of (1.1) with the correspondingfluxes f k , f and the initial data u . Then u k → u in L loc ( IR × (0 , ∞ )) . Next we state the following front tracking Lemma of Dafermos [12] without proof [Seechapter 6 in [9], Lemma (2.6) in [23]].
LEMMA 2.2.
Let f be a piecewise affine continuous function. Let u be a piecewiseconstant function with finite number of jumps. Then for all t > , x → u ( x, t ) is a piecewiseconstant function with uniformly bounded number of jumps. .2 Structure Theorem for C -convex flux with bounds The structure Theorem is the main ingredient in the proof of the main Theorem. Sincethe flux f need not be strictly convex, one must prove the structure Theorem with bounds.The main ingredients are Hopf and Lax-Oleinik formula for the solution of Hamilton-Jacobiequation and the corresponding conservation laws. For the strictly convex C functions,structure Theorem has been proved in [1] when u ± are constants and without bounds.Here we prove the structure Theorem with bounds for C convex flux which exhibits finitenumber of degeneracies.Hmilton-Jacobi equation: In order to prove the structure Theorem, we need to prove theLax-Oleinik type explicit formula for the C convex flux. As in Lax-Oleinik, we establishthis via the Hamilton-Jacobi equations.Let f be a convex function and u ∈ L ∞ ( IR ) with M = || u || ∞ . For 0 ≤ s ≤ t, x ∈ IR, p ∈ IR , define f ∗ ( p ) = sup q { pq − f ( q ) } , (2.3) v ( x ) = Z x u ( θ ) dθ, (2.4) v ( x, t, f ) = inf y ∈ IR (cid:26) v ( y ) + tf ∗ (cid:18) x − yt (cid:19)(cid:27) , (2.5) w ( x, s, t, f ) = inf y ∈ IR (cid:26) v ( x, s, f ) + ( t − s ) f ∗ (cid:18) x − yt − s (cid:19)(cid:27) , (2.6) ch ( x, t, f ) = { minimizers in (2.5) } , (2.7) ch ( x, s, t, f ) = { minimizers in (2.6) } , (2.8) y + ( x, t, f ) = max { y : y ∈ ch ( x, t, f ) } , (2.9) y − ( x, t, f ) = min { y : y ∈ ch ( x, t, f ) } , (2.10) y + ( x, s, t, f ) = max { y : y ∈ ch ( x, s, t, f ) } , (2.11) y − ( x, s, t, f ) = min { y : y ∈ ch ( x, s, t, f ) } . (2.12)Points in ch ( x, t, f ) , ch ( x, s, t, f ) are called the characteristic points and the correspondingsets are called the characteristic sets. y ± are called the extreme characteristic points.Then we have the following stability Lemma. Most of the results here are known andfor the sake of completeness, we are sketching the proofs in the appendix. LEMMA 2.3 (Stability Lemma) . Let f be a convex function. Then1. lim | p |→∞ f ∗ ( p ) | p | = ∞ . (2.13)
2. Let { f n } be a sequence of convex functions such that f n → f in C loc ( IR ) and lim | p |→∞ inf n f n ( p ) | p | = ∞ . Then f ∗ n → f ∗ in C loc ( IR ) . . For ≤ s < t, x ∈ IR, v is a Lipschitz function with Lipschitz constant bounded by M = || u || ∞ . Let p > such that for | p | > p , f ∗ ( p ) − M | p | > f ∗ (0) , (2.14) then ch ( x, s, t, f ) = φ and v ( x, t, f ) = inf | x − yt − s | ≤ p (cid:26) v ( x, s, t ) + ( t − s ) f ∗ (cid:18) x − yt − s (cid:19)(cid:27) , (2.15) ch ( x, t, f ) = ch ( x, , t, f ) , (2.16) y ± ( x, t, f ) = y ± ( x, , t, f ) . (2.17)
4. For ≤ s < t, y ∈ IR , γ denotes the line joining ( x, t ) and ( y, s ) and is given by γ ( θ, x, s, t, y ) = x + (cid:18) x − yt − s (cid:19) ( θ − t ) . (2.18) If y ∈ ch ( x, s, t, f ) , then γ is called a characteristic line segment. Let x = x , for i = 1 , , ξ i ∈ ch ( x i , s, t, f ) , y i ∈ ch ( ξ i , s, f ) , then y i ∈ ch ( x i , t, f ) (2.19) x y ± ( x, s, t, f ) are non decreasing functions. (2.20) Furthermore if f ∗ is a strictly convex function, then no two different characteristicline segments intersect in the interior. That is for θ ∈ ( s, t ) γ ( θ, x , s, t, y ) = γ ( θ, x , s, t, y ) , (2.21)lim ξ ↑ x y + ( ξ, s, t, f ) = y − ( x , s, t, f ) , (2.22)lim ξ ↓ x y − ( ξ, s, t, f ) = y + ( x , s, t, f ) . (2.23)
5. For a sequence of sets E n ⊂ IR , let us denote the set of all cluster points of sequences { ρ n ∈ E n } by lim E n . Let { f n } be a sequence of convex functions such that f n → f in C loc ( IR ) and lim | p |→∞ inf n f n ( p ) | p | = ∞ . Let C > , then there exists p ≥ depending onlyon C such that for all n, | p | > p f ∗ n ( p ) | p | ≥ C + 1 , (2.24) v ( · , · , f n ) → v ( · , · , f ) in C loc ( IR × (0 , ∞ )) , (2.25)lim ch ( x, s, t, f n ) ⊂ ch ( x, s, t, f ) . (2.26)
6. As k → ∞ , let ( x k , t k ) → ( x, t ) . Let f, f k be as in (5). Let y k ∈ ch ( x k , s, t k , f k ) suchthat y k → y. Then y ∈ ch ( x, s, t, f ) .7. Let u ,k ⇀ u in L ∞ weak ∗ topology and let f, f k be as in (5). Let v ,k , v , v k , v , ch ( x, t, f k ) , ch ( x, t, f ) as in (2.4), (2.5), (2.7) associated to u ,k and u respectively.Let y k ∈ ch ( x, t, f k ) such that y k → y , then y ∈ ch ( x, t, f ) . THEOREM 2.1.
Let f be a C convex function and u ∈ L ∞ ( IR ) . Let v ( x, t ) = v ( x, t, f ) be the associated value function as in (2.5). Let u = ∂v∂x , then for t > and a.e. x ∈ IR , f ′ ( u ( x, t )) = x − y ( x, t ) t . (2.27) If u ∈ C ( IR ) ∩ L ∞ ( IR ) , then for a.e. x ∈ IR,u ( x, t ) = u ( y + ( x, t )) . (2.28) Proof.
From (4.1), choose a sequence { f n } ⊂ C ( IR ) of uniformly convex function such that f n → f in C loc( IR ) and lim | p |→∞ inf n f n ( p ) | p | = ∞ . Let v n ( x, t ) = v ( x, t, f n ), v ( x, t ) = v ( x, t, f ), y + ,n ( x, t ) = y + ( x, t, f n ), y + ( x, t ) = y + ( x, t, f ) and u n = ∂v n ∂x , u = ∂v∂x . Let D ( t ) be thepoints of discontinuities of y + . Then ch ( x, t ) = { y + ( x, t ) } if x / ∈ D ( t ) and thus from (2.26)lim n →∞ y + ,n ( x, t ) = y + ( x, t ) for x / ∈ D ( t ) . From Lemma 2.1, let u n → w in L loc( IR × (0 , ∞ ))and hence a.e. ( x, t ). Therefore from Lax-Oleinik [19] explicit formula, for a.e. t , a.e. x / ∈ D ( t ) , f ′ ( w ( x, t )) = lim n →∞ f ′ n ( u n ( x, t ))= lim n →∞ x − y + ,n ( x, t ) t = x − y + ( x, t ) t (2.29)Claim: u = w .From (2.25), v n → v in C loc( IR × [0 , ∞ )) , hence for φ ∈ C ∞ c ( IR × [0 , ∞ )) ∞ Z −∞ ∞ Z v ∂φ∂x dxdt = lim n →∞ ∞ Z −∞ ∞ Z v n ∂φ∂x dxdt = − lim n →∞ ∞ Z −∞ ∞ Z ∂v n ∂x φdxdt = − lim n →∞ ∞ Z −∞ ∞ Z u n φdxdt = − ∞ Z −∞ ∞ Z wφdxdt. Hence w = ∂v∂x = u . This proves the claim.First assume that u ∈ BV ( IR ) . Then for any s, t ≥
0, we have Z IR | u ( x, s ) − u ( x, t ) | dx ≤ Lip ( f, K ) | s − t | T V ( u ) . (2.30)12rom (2.29), (2.30), choose a sequence s k → t , a null set N ⊃ D ( t ) such that for all x / ∈ N , u ( x, s k ) → u ( x, t ) and f ′ ( u ( x, s k )) = x − y + ( x,s k ) t . Since x / ∈ D ( t ), ch ( x, t ) = { y + ( x, t ) } ,therefore from (6) of Lemma 2.3 y + ( x, s k ) → y + ( x, t ) and f ′ ( u ( x, t )) = lim k →∞ f ′ ( u ( x, s k ))= lim k →∞ x − y + ( x, s k ) t = x − y + ( x, t ) t . (2.31)Let u ∈ L ∞ ( IR ) and u ,k ∈ BV ( IR ) such that u ,k → u in L loc( IR ) and almost everywhere.Let u k be the solution of (1.1) with initial data u ,k , then from L loc contraction u k ( x, t ) → u ( x, t ) for a.e. x ∈ IR, t >
0. Let y + ,k , y + be as in (2.9) for u ,k and u respectively. Thenfrom (2.31) and (7) of Lemma 2.3, there exists a null set N ⊃ D ( t ) such that for x / ∈ N , y + ,k ( x, t ) → y + ( x, t ) and f ′ ( u ( x, t )) = lim k →∞ f ′ ( u k ( x, t ))= lim k →∞ x − y + ,k ( x, t ) t = x − y + ( x, t ) t . This proves (2.27).Let u ∈ C ( IR ) , then from the monotonicity of y + ,n and uniform convexity, it followsthat u n ( x, t ) = u ( y + ,n ( x, t )) . Hence as in the previous case for a.e. s , a.e. x, u n ( x, s ) → u ( x, s ) and y + ,n ( x, s ) → y + ( x, s ).Consequently u ( x, s ) = lim n →∞ u n ( x, s ) = lim n →∞ u ( y + ,n ( x, s )) = u ( y + ( x, s )) . Next assume that u ∈ BV ( IR ), then u ( x, s k ) → u ( x, t ) as s k → t , a.e. x . Therefore letting s k → t , x / ∈ N ⊃ D ( t ) such that y + ( x, s k ) → y + ( x, t ), we have u ( x, t ) = lim k →∞ u ( x, s k ) = lim k →∞ u ( y + ( x, s k ))= u ( y + ( x, s )) . Let u ∈ C , then approximate u by u ,k ∈ C ( IR ) ∩ BV ( IR ) in L loc norm. Then from L loc contraction, for a.e. x, u k ( x, t ) → u ( x, t ) , y + ,k ( x, t ) → y + ( x, t ). Thus u ( x, t ) = lim k →∞ u k ( x, t ) = lim k →∞ u ( y + ,k ( x, t )) = u ( y + ( x, t )) . This proves (2.28) and hence the Theorem.Next we pass onto quantitative version of the structure Theorem [1]. For this, let usrecall the definition of R ± curves called the characteristic curves.13 EFINITION 2.1.
Let f ∈ C ( IR ) be a convex function and y ± ( x, t ) = y ± ( x, t, f ) , y ± ( x, s, t ) = y ± ( x, s, t, f ) , ch ( x, s, t ) = ch ( x, s, t, sf ) , ch ( x, t ) = ch ( x, t, f ) as in (2.9) to(2.12). For α ∈ IR , ≤ s < t , define R + ( t, s, α, u ) = sup { x : y + ( x, s, t ) ≤ α } , (2.32) R − ( t, s, α, u ) = inf { x : y − ( x, s, t ) ≥ α } , (2.33) R ± ( t, α, u ) = R ± ( t, , α, u ) . (2.34)From the comparison principle and (2.15), R ± satisfies the following LEMMA 2.4.
Let f be C convex function having finite number of degeneracies. Also let u ∈ L ∞ ( IR ) , M = || u || ∞ and p > as in (2.14). Then1. t R ± ( t, α, u ) are uniformly Lipschitz continuous functions with R ± (0 , α, u ) = α, (2.35) (cid:13)(cid:13)(cid:13)(cid:13) dR ± dt (cid:13)(cid:13)(cid:13)(cid:13) ≤ p , (2.36) R − ( t, α, u ) ≤ R + ( t, α, u ) , (2.37) y − ( R ± ( t, α, u ) , t ) ≤ α ≤ y + ( R ± ( t, α, u ) , t ) . (2.38)
2. Let u ≤ w and y , ± , y , ± be the respective optimal characteristic points of u and w . Then y , ± ( x, t ) ≤ y , ± ( x, t ) , (2.39) R ± ( t, α, u ) ≤ R ± ( t, α, w ) . (2.40)
3. Let u ,n → u in L loc ( IR ) , then lim n →∞ R ± ( t, α, u ,n ) exist in C loc ( IR ) and satisfiesi. If for all n , R − ( t, α, u ,n ) ≤ R − ( t, α, u ) , then lim n →∞ R − ( t, α, u ,n ) = R − ( t, α, u ) . ii. If for all n , R + ( t, α, u ,n ) ≥ R + ( t, α, u ) , then lim n →∞ R + ( t, α, u ,n ) = R + ( t, α, u ) .
4. Let < s < t , then R + ( t, s, α, u ) = R − ( t, s, α, u ) , (2.41) R ± ( t, α, u ) = R ± ( t, s, R ± ( s, α, u ) , u ) . (2.42) If for some α, β and
T > , R + ( T, α, u ) = R + ( T, β, u ) or R + ( T, α, u ) = R − ( T, β, u ) ,then for t > T, R + ( t, α, u ) = R + ( t, β, u ) or R + ( t, α, u ) = R − ( t, β, u ) respectively. . Suppose for some T > , R − ( T, α, u ) < R + ( T, α, u ) , (2.43) then for R − ( T, α, u ) < x < R + ( T, α, u ) , α ∈ ch ( x, T ) and f ′ ( u ( x, T )) = x − αT . (2.44)
6. Let { u ,n } be a bounded sequence in L ∞ ( IR ) , { f n } be a sequence of C convex functionsand p > such that for | p | > p , for all nf ∗ n ( p ) − M | p | > sup m f ∗ m (0) , (2.45) u ,n ⇀ u in L ∞ weak ∗ , (2.46) f n → f in C loc ( IR ) . (2.47) Then for t > , a.e. x ∈ IR , lim n →∞ f ′ n ( u n ( x, t )) = f ′ ( u ( x, t )) , (2.48) where u n is the solution of (1.1), (1.2) with flux f n and initial data u ,n .
7. Let
T > , { y } = ch ( x , T ) , f ′ ( p ) = x − y T . Let γ be the characteristic line segmentdefined by γ ( θ ) = x + x − y T ( θ − T ) . (2.49) Let ˜ u ∈ L ∞ ( IR ) defined by ˜ u ( x ) = (cid:26) u ( x ) if x < y ,p if x > y (2.50) and ˜ u be the corresponding solution of (1.1), (1.2). Then for < t < T , ˜ u is given by ˜ u ( x, t ) = (cid:26) u ( x, t ) if x < γ ( t ) ,p if x > γ ( t ) . (2.51)
8. Let α ∈ IR , a ≤ a , b ≤ b and u be such that u ( x ) ∈ ( [ a , a ] if x > α, [ b , b ] if x < α. (2.52) Then for a.e. x , u ( x, t ) ∈ ( [ a , a ] if x > R + ( t, α, u ) , [ b , b ] if x < R − ( t, α, u ) . (2.53) Furthermore if b > a , then for all t > , s > R − ( t, α, u ) = R + ( t, α, u ) . min p ∈ [ b ,b ] q ∈ [ a ,a ] (cid:18) f ( p ) − f ( q ) p − q (cid:19) ≤ dR dt ( t, α, u ) ≤ max p ∈ [ b ,b ] q ∈ [ a ,a ] f ( p ) − f ( q ) p − q . (2.54)15 roof. (1) to (5) follows as in Lemma 4.2 in [1]. Let y + ,n = y + ( x, t, f n ), then from (7)of Lemma 2.3, lim n →∞ y + ,n ( x, t ) = y + ( x, t ) for all x / ∈ D ( t ). Hence from (2.27), for a.e. x ∈ IR, x / ∈ D ( t ) , lim n →∞ f ′ n ( u n ( x, t )) = lim n →∞ x − y + ,n ( x, t ) t = x − y + ( x, t ) t = f ′ ( u ( x, t )) . This proves (6).From (4) of Lemma 4.1, f ∗ is strictly convex and thus from (2.23) and (2.24) for s = 0 , we have lim ξ ↑ γ ( t ) y + ( ξ, t ) = y − ( x , t ) = y , (2.55)lim ξ ↓ γ ( t ) y − ( ξ, t ) = y + ( x , t ) = y . (2.56)First assume that f ′ is a strictly increasing function. Let w denote the right hand side of(2.51). From (2.27) and (2.55), we have for 0 < t < T ,lim ξ ↑ γ ( t ) f ′ ( w ( ξ, t )) = lim ξ ↑ γ ( t ) f ′ ( u ( ξ, t ))= lim ξ ↑ γ ( t ) ξ − y + ( ξ, t ) t = γ ( t ) − y t = x − y t = f ′ ( p ) . lim ξ ↓ γ ( t ) f ′ ( w ( ξ, t )) = f ′ ( p ) . Since f ′ is strictly increasing, w ( γ ( t ) − , t ) and w ( γ ( t )+ , t ) exist and w ( γ ( t ) − , t ) = w ( γ ( t )+ , t ) = p . Hence w is continuous across γ ( · ) and is a solution for x = γ ( t ) . Therefore w is the so-lution of (1.1) in IR × (0 , T ). Whence w = ˜ u in IR × (0 , T ) . For general f , let { f n } ⊂ C ( IR ) be a sequence of convex function converging to f in C loc( IR ) with lim | p |→∞ inf n f n ( p ) | p | = ∞ . Sincelim ch ( x , T, f n ) ⊂ ch ( x , T, f ) = { y } (2.57)and therefore from (2.55), (2.57) choose { x n } , { y n } such that ch ( x n , T, f n ) = { y n } , lim n →∞ ( x n , y n ) = ( x , y ) . Define for 0 < t < Tγ n ( t ) = x n + (cid:18) x n − y n T (cid:19) ( t − T ) , f ′ n ( p n ) = x n − y n T . u ,n ( x ) = (cid:26) u ( x ) if x < y n ,p n if x > y n , ˜ u n ( x, t ) = (cid:26) u n ( x, t ) if x < γ n ( t ) ,p n if x > γ n ( t ) , where u n is the solution of (1.1), (1.2) with flux f n and data u . Hence by previous analysis,for 0 < t < T , ˜ u n is the solution of (1.1), (1.2) with flux f n and initial data ˜ u ,n . FromLemma 2.1, u n → u a.e. ( x, t ) ∈ IR × (0 , ∞ ). Let w denotes the right hand side of (2.51).Then ˜ u n → w a.e. ( x, t ) ∈ IR × (0 , T ). Since ˜ u ,n → ˜ u , by dominated convergence Theorem,letting n → ∞ in the entropy inequality we obtain w is the solution of (1.1) and (1.2) withflux f and initial data ˜ u . Hence w = ˜ u in IR × (0 , T ). This proves (7).In order to prove (8), first assume that u is continuous for x < α. Let x < R − ( t, α, u ),then from (2.38), y + ( x, t ) < α and from (2.28), for a.e. x < R − ( t, α, u ) u ( x, t ) = u ( y + ( x, t )) ∈ [ b , b ] . (2.58)Similarly, if u is continuous for x > α , then for x > R + ( t, α, u ) .u ( x, t ) = u ( y + ( x, t )) ∈ [ a , a ] . (2.59)Let u ,n ∈ L ∞ such that u ,n is continuous in ( −∞ , α ) ∪ ( α, ∞ ) and u ,n → u in L loc( IR ).Let u n be the solution of (1.1), (1.2) with the initial data u ,n . Then for x
0. Since b > a , hence f ′ ( b ) > f ′ ( a )and therefore f ′ ( p ) / ∈ [ f ′ ( a ) , f ′ ( a )] ∩ [ f ′ ( b ) , f ′ ( b )] , which is a contradiction. Similarly if r ( t ) = R + ( t, α, u ) for some 0 ≤ t ≤ T . This proves (8) and hence the Lemma.In order to prove the Structure Theorem, let us recall the characteristic line and ASSPfrom [1]. Let f be C convex, u ∈ L ∞ and u be the solution of (1.1) and (1.2). Let a < b , p ∈ IR , γ ( t, a, p ) = a + tf ′ ( p ) , (2.72) D ( a, b, p ) = { ( x, t ) ∈ IR × (0 , ∞ ) : γ ( t, a, p ) < x < γ ( t, b, p ) } . (2.73)Let y ± ( x, t ) = y ± ( x, t, f ) and define1. Characteristic line: γ ( · , a, p ) is called a characteristic line if for all t > a = γ (0 , a, p ) ∈ ch ( γ ( t, a, p ) , t ) . (2.74)2. Asymptotically single shock packet (ASSP): D ( a, b, p ) is called an ASSP ifi. γ ( · , a, p ) , γ ( · , b, p ) are characteristic lines.ii. D ( a, b, p ) does not contain a characteristic line.iii. For α ∈ ( a, b ), R ± ( t, α, u ) ∈ D ( a, b, p ). LEMMA 2.5.
Let f ∈ C ( IR ) with finite number of degeneracies.1. Let γ ( · , a, p ) be a characteristic line. Then y ± ( γ ( t, a, p ) , t ) = a. (2.75)20 . Let A, α ≤ α , β ≤ β , u ∈ L ∞ ( IR ) are given. Assume that ¯ u , , ¯ u , ∈ L ∞ ( IR ) such that ¯ u , ( x ) ∈ [ β , β ] , ¯ u , ( x ) ∈ [ α , α ] and define u ( x ) = (cid:26) ¯ u , ( x ) if x < A,u ( x ) if x > A, (2.76) u ( x ) = (cid:26) u ( x ) if x < A, ¯ u , ( x ) if x > A. (2.77) Let u , u be the corresponding solutions of (1.1), (1.2) with initial data u and u .Then ( a ) . Assume that sup t> y + ( R − ( t, A, u ) , t ) < ∞ , (2.78) then there exist an A ≥ A , p − ∈ IR such that lim t →∞ y + ( R − ( t, A, u ) , t ) = A . (2.79) f ′ ( p − ) = lim t →∞ R − ( t, A, u ) − y + ( R − ( t, A, u ) , t ) t . (2.80) f ′ ( p − ) ∈ [ f ′ ( β ) , f ′ ( β )] . (2.81) γ ( · , A , p − ) is a characteristic line. (2.82)( a ) . Assume that inf t> y − ( R + ( t, A, u ) , t ) > −∞ , (2.83) then there exist an A ≤ A , p + ∈ IR such that lim t →∞ y − ( R + ( t, A, u ) , t ) = A . (2.84) f ′ ( p + ) = lim t →∞ R + ( t, A, u ) − y − ( R + ( t, A, u ) , t ) t . (2.85) f ′ ( p + ) ∈ [ f ′ ( α ) , f ′ ( α )] . (2.86) γ ( · , A , p + ) is a characteristic line. (2.87) Proof.
Note that from (4) of Lemma 4.1, f ∗ is a strictly convex function.1. Let γ ( s ) = γ ( s, a, p ) and v be the value function as in (2.5). Suppose for some s > + ( γ ( s ) , s ) > a , then for t > s , we have v ( γ ( t ) , t ) = v ( γ ( s ) , s ) + ( t − s ) f ∗ (cid:18) γ ( t ) − γ ( s ) t − s (cid:19) = v ( y + ( γ ( s ) , s )) + sf ∗ (cid:18) γ ( s ) − y + ( γ ( s ) , s ) s (cid:19) + ( t − s ) f ∗ (cid:18) γ ( t ) − γ ( s ) t − s (cid:19) > v ( y + ( γ ( s ) , s )) + tf ∗ (cid:18) γ ( t ) − y + ( γ ( s ) , s ) t (cid:19) ≥ v ( γ ( t ) , t ) , which is a contradiction. Hence y + ( γ ( s ) , s ) = a . Similarly y − ( γ ( s ) , s ) = a and thisproves (1).2. Assume (2.75) holds. Let R ( t ) = R − ( t, A, u ) , y ( t ) = y + ( R ( t ) , t ) and γ ( θ, t ) = y ( t ) + (cid:18) R ( t ) − y ( t ) t (cid:19) θ. (2.88)Let t > t . Then from (2.21), for all θ ∈ (0 , t ), either γ ( θ, t ) = γ ( θ, t ) or γ ( θ, t ) = γ ( θ, t ). From (2.38) y ( t ) ≥ A , hence γ ( θ, t ) ≥ γ ( θ, t ) for θ ∈ [0 , t ]. Hence y ( t ) ≥ y ( t ) . Therefore from (2.78), A exists satisfying (2.79). Let θ > t > θ , t → γ ( θ , t ) is a non decreasing function. Also from(2.15), n R ( t ) − y ( t ) t o is bounded. Hence p − exist and satisfies (2.80), lim t →∞ γ ( θ , t ) = A + f ′ ( p − ) θ . Thus from (7) of Lemma 2.3, A ∈ ch ( A + f ′ ( p − ) θ , θ ) for all θ > , therefore A ∈ ch ( γ ( θ, A , p − ) , θ ) for θ >
0. Hence γ ( · , A , p − ) is a characteristic line.This proves (2.82).Next we prove (2.81). From the strict convexity of f ∗ as in the proof of (2.75), wehave for all 0 ≤ θ ≤ t , { y ( t ) } = ch ( γ ( θ, t ) , θ ) . Therefore R ( θ ) ≤ γ ( θ, t ) for all 0 < θ ≤ t . Letting t → ∞ to obtain R ( θ ) ≤ γ ( θ, A , p − ) for all θ >
0. Define˜ u ( x ) = (cid:26) u ( x ) if x < A ,p − if x > A , ˜ u ( x, t ) = (cid:26) u ( x, t ) if x < γ ( t, A , p − ) ,p − if x > γ ( t, A , p − ) , then from (2.51), ˜ u is the solution of (1.1), (1.2) with initial data ˜ u .Case (i): Assume that there exists t > R ( t ) = γ ( t , A , p − ) . t > t it follows that R ± ( γ ( t, A , p − ) , t ) = γ ( t, A , p − )and hence R ( t ) = γ ( t, A , p − ). From (2.53), ˜ u ( x, t ) = u ( x, t ) ∈ [ β , β ] for x < R ( t ) = γ ( t, A , p − ) and t > t . Consequently from (2.27) for a.e. x < γ ( t, A , p − ), t > t , x − y + ( x, t ) t = f ′ (˜ u ( x, t ))= f ′ ( u ( x, t )) ∈ [ f ′ ( β ) , f ′ ( β )] . Letting x ↑ γ ( t, A , p − ) to obtain f ′ ( p − ) = γ ( t, A , p − ) − A t ∈ [ f ′ ( β ) , f ′ ( β )] . Case (ii): Assume that for all t > R ( θ ) < γ ( θ, A , p − ) . Suppose f ′ ( p − ) / ∈ [ f ′ ( β ) , f ′ ( β )] . Then choose an ǫ > , t > t ≥ t , (cid:12)(cid:12)(cid:12)(cid:12) R ( t ) − y ( t ) t − f ′ ( p − ) (cid:12)(cid:12)(cid:12)(cid:12) < ǫ/ f ′ ( p − ) − ǫ, f ′ ( p − ) + ǫ ] ∩ [ f ′ ( β ) , f ′ ( β )] = φ. (2.90)Then for R ( t ) < x < γ ( t , A , p − ) , y + ( x, t ) ≤ A and R ( t ) − y ( t ) t ≤ x − y + ( x, t ) t + y + ( x, t ) − y ( t ) t ≤ x − y + ( x, t ) t + A − y ( t ) t . Hence choose t large such that for R ( t ) < x < γ ( t o , A , p − ) with y ( x, t ) = y + ( x, t )and (cid:12)(cid:12)(cid:12)(cid:12) R ( t ) − y ( t ) t − x − y ( x, t ) t (cid:12)(cid:12)(cid:12)(cid:12) < ǫ/ . Thus (cid:12)(cid:12)(cid:12)(cid:12) x − y ( x, t ) t − f ′ ( p − ) (cid:12)(cid:12)(cid:12)(cid:12) < ǫ (2.91)and from (2.27) for a.e. x ∈ ( R ( t ) , γ ( t , A , p − )) | f ′ ( u ( x, t )) − f ′ ( p − ) | < ǫ. Let [ a, b ] = ( f ′ ) − [ f ′ ( β ) , f ′ ( β )][ c ( ǫ ) , d ( ǫ )] = ( f ′ ) − [ f ′ ( p − ) − ǫ, f ′ ( p − ) + ǫ ] . Then from (2.90), [ a, b ] ∩ [ c ( ǫ ) , d ( ǫ )] = φ . We have to consider two sub cases.23i): Suppose f ′ ( p − ) + ǫ < f ′ ( β ) . Then d ( ǫ ) < a and by convexity ( d ( ǫ ) , a ) containsopen sets on which f ′ is strictly increasing. Hencemin p ∈ [ a,b ] q ∈ [ c (0) ,d (0)] f ( p ) − f ( q ) p − q > f ′ ( p − ) . Therefore by continuity, there exists an ǫ > < ǫ ≤ ǫ , m = min p ∈ [ a,b ] q ∈ [ c ( ǫ ) ,d ( ǫ )] f ( p ) − f ( q ) p − q > f ′ ( p − ) + ǫ. Let 0 < ǫ < ǫ and choose t > u ( x, t ) at t = t , then from (2.70), thereexists a Lipschitz curve r ( · ) for t > t , such that for t > t , r ( t ) = R ( t ) , ˜ u ( x, t ) ∈ [ a, b ] , if x < r ( t ) , ˜ u ( x, t ) ∈ [ c ( ǫ ) , d ( ǫ )] , if x > r ( t ) ,drdt ( t ) ≥ m ≥ f ′ ( p − ) + ǫ. Hence for t > t , r ( t ) ≥ R ( t ) + ( t − t )( f ′ ( p − ) + ǫ ) = s ( t )and s ( t ), γ ( t, A , p − ) intersect at T given by R ( t ) + ( T − t )( f ′ ( p − ) + ǫ ) = γ ( T, A , p − )= γ ( t , A , p − ) + ( T − t ) f ′ ( p − ) T = t + γ ( t , A , p − ) − R ( t ) ǫ > t . Thus the curve r ( · ) , γ ( · , A , p − ) intersect at ˜ T ≤ T. From (8) and (4) of Lemma 2.4, R ( t ) = r ( t ) for t > t . Hence R ( ˜ T ) = r ( ˜ T ) = γ ( ˜ T , A , p − ). Therefore from case (i), f ′ ( p − ) ∈ [ f ′ ( β ) , f ′ ( β )] which is a contradiction.(ii): Suppose f ′ ( p − ) − ǫ > f ′ ( β ) , then b < c ( ǫ ) and from convexity of f as in case (i),we can choose ǫ sufficiently small and t large such that M = max p ∈ [ a,b ] q ∈ [ c (0) ,d (0)] f ( p ) − f ( q ) p − q ≤ f ′ ( p − ) − ǫ, and for t > t , drdt ( t ) ≤ M ≤ f ′ ( p − ) − ǫ. Hence r ( t ) ≤ R ( t ) + ( t − t )( f ′ ( p − ) − ǫ ) . t > t , r ( t ) = R ( t ) , we have f ′ ( p − ) = lim t →∞ R ( t ) − y ( t ) t = lim t →∞ r ( t ) − y ( t ) t ≤ lim t →∞ (cid:20) R ( t ) − y ( t ) t + (cid:18) − t t (cid:19) ( f ′ ( p − ) − ǫ ) (cid:21) = f ′ ( p − ) − ǫ, which is a contradiction. This proves (2.81).Proof of ( a ) follows in a similar way and this proves the lemma.Next we prove a quantitative result regarding the single shock situation by blow up analysis. LEMMA 2.6.
Let
A < B, u ± , ¯ u , u as in (1 . . Let u be the solution of (1.1), (1.2). Alsolet α ≤ α , β ≤ β such that (cid:26) u − ( x ) ∈ [ β , β ] if x < A,u + ( x ) ∈ [ α , α ] if x > B. (2.92) Assume that f ′ ( β ) > f ′ ( α ) . Then there exist ( x , T ) ∈ IR × (0 , ∞ ) , γ > and Lipschitz curve r ( · ) depending only on || u || ∞ , f ′ ( β ) − f ′ ( α ) such that for t > T r ( T ) = x , (2.93) T ≤ γ | A − B | , (2.94) (cid:26) u ( x, t ) ∈ [ β , β ] if x < r ( t ) ,u ( x, t ) ∈ [ α , α ] if x > r ( t ) . (2.95) Proof.
We prove this by blow up argument. Let( L ( t ) , R ( t )) = ( R − ( t, A, u ) , R + ( t, B, u )) , (2.96)( y + ( t ) , y − ( t )) = ( y + ( L ( t ) , t ) , y − ( R ( t ) , t )) . (2.97)Clearly L ( t ) ≤ R ( t ). Suppose L ( t ) < R ( t ) for all t >
0. Then A ≤ y + ( t ) ≤ y − ( t ) ≤ B forall t. Hence from (2) of Lemma 2.5, there exist p − and p + such that f ′ ( p − ) = lim t →∞ L ( t ) − y + ( t ) t ∈ [ f ′ ( β ) , f ′ ( β )] f ′ ( p + ) = lim t →∞ R ( t ) − y − ( t ) t ∈ [ f ′ ( α ) , f ′ ( α )]lim t →∞ ( y + ( t ) , y − ( t )) = ( A , B ) , γ ( · , A , p − ) and γ ( · , B , p + ) are characteristic lines. Since A ≤ B and f ′ ( p − ) > f ′ ( p + ),therefore γ ( · , A , p − ) and γ ( · , B , p + ) meet at ˜ T > L and R meet at some T >
0. Then from (4) of Lemma 2.4, L ( t ) = R ( t ) for t > T . Let r ( t ) = L ( t ), then from (2.53) and (2.54), we have for t > T (cid:26) u ( x, t ) ∈ [ β , β ] if x < r ( t ) ,u ( x, t ) ∈ [ α , α ] if x > r ( t ) . (2.98)We prove the uniform bounds on T by assuming [ A, B ] = [0 , A, B ]. Suppose not, then there exist ¯ u ,k ∈ L ∞ (0 , , α ,k ≤ α ,k , β ,k ≤ β ,k , δ > r k ( t ) , T k , convex C functions f k and f such thatsup k || u ,k || ∞ < ∞ , lim | p |→∞ inf k f k ( p ) | p | = ∞ , f k → f in C loc( IR ) , lim k →∞ ( α ,k , α ,k , β ,k , β ,k ) = ( α , α , β , β ) ,f ′ ( β ,k ) − f ′ ( α ,k ) ≥ δ and the solution u k of (1.1), (1.2) with flux f k and initial data u ,k satisfying (2.98) for t > T k .Let ( u − ,k , u + ,k , ¯ u ,k ) ⇀ ( u − , u + , ¯ u ) in L ∞ weak ∗ topology and u ( x ) = u − ( x ) if x < , ¯ u ( x ) if x ∈ (0 , ,u + ( x ) if x > . Then u − ( x ) ∈ [ β , β ] for x < u + ( x ) ∈ [ α , α ] if x >
1. Let u be the solution of (1.1)with initial data u . Let ( L k , R k , y ± ,k ) be as in (2.96), (2.97) for the solution u k and T k > L k and R k . That is L k ( t ) < R k ( t ) for 0 < t < T k ,L k ( T k ) = R k ( T k ) = x ,k ,L k ( t ) = R k ( t ) = r k ( t ) for t > T k , lim k →∞ T k = ∞ . Hence 0 ≤ y + ,k ( T k ) ≤ y − ,k ( T k ) ≤ . Let γ ,k ( θ ) = y + ,k ( T k ) + (cid:18) x ,k − y + ,k ( T k ) T k (cid:19) θ,γ ,k ( θ ) = y − ,k ( T k ) + (cid:18) x ,k − y − ,k ( T k ) T k (cid:19) θ, be the characteristic line segments. From (2.15), n x ,k − y ± ,k ( T k ) T k o , n dR k dt o , n dL k dt o are uni-formly bounded sequences, hence for subsequences as k → ∞ , let( R k , L k ) → ( R , L ) in C loc( IR ) , ( y + ,k ( T k ) , y − ,k ( T k )) → ( y + , y − ) ,x ,k − y ± ,k ( T k ) T k → f ′ ( p ± ) . γ ( θ ) , γ ( θ )) = lim k →∞ ( γ ,k ( θ ) , γ ,k ( θ ))= ( y + + f ′ ( p − ) θ, y − + f ′ ( p + ) θ ) . Therefore from (7) of Lemma 2.3, y + ∈ ch ( γ ( θ ) , θ ) , y − ∈ ch ( γ ( θ ) , θ ) for all θ >
0. Hence γ and γ are characteristic lines with respect to the solution u .Imitating the proof of (2.81) and (2.86), it follows that f ′ ( p − ) ∈ [ f ′ ( β ) , f ′ ( β )] and f ′ ( p + ) ∈ [ f ′ ( α ) , f ′ ( α )] . Since f ′ ( β ) > f ′ ( α ), therefore γ and γ must necessarily inter-sect, contradicting that γ and γ are characteristic lines. This proves (2.93) and (2.94) for[0 , A, B ], define w ( x, t ) = u ( A + x ( B − A ) , ( B − A ) t ) , then from the previous analysis, there exist a (˜ x , ˜ T ) ∈ IR × (0 , ∞ ) , γ >
0, ˜ r ( · ) a Lipschitzcurve such that (2.93) to (2.95) hold for w . Let x = A + ˜ x ( B − A ) , T = ( B − A ) ˜ T , r ( t ) = ˜ r (cid:16) tB − a (cid:17) , then (2.93) to (2.95) holds for u . This proves the Lemma. THEOREM 2.2 (Structure Theorem) . Let f ∈ C ( IR ) and convex with finite number ofdegeneracies. Assume that α ≤ α , β ≤ β , u ± ∈ L ∞ ( IR ) , ¯ u ∈ L ∞ ( IR ) , A < B such that (cid:26) u − ( x ) ∈ [ β , β ] if x < A,u + ( x ) ∈ [ α , α ] if x > B, (2.99) u ( x ) = u − ( x ) if x < A, ¯ u ( x ) if x ∈ ( A, B ) ,u + ( x ) if x > B (2.100) and let u be the solution of (1.1), (1.2). Then1. Shock case: { u − , u + } gives rise to a single shock solution if f ′ ( β ) > f ′ ( α ) , (2.101) and ( x , T ) , γ , r ( · ) exist as in (1.7) to (1.10) which depends on f ′ ( β ) − f ′ ( α ) , || u || ∞ , Lip ( f, [ −|| u || ∞ , || u || ∞ ]) and uniform growth of f ∗ ( p ) | p | as | p | → ∞ .
2. Rarefaction case: Assume that f ′ ( β ) ≤ f ′ ( α ) , (2.102) then there exist A ≤ A ≤ B ≤ B, p ± ∈ IR with f ′ ( p − ) ∈ [ f ′ ( β ) , f ′ ( β )] , f ′ ( p + ) ∈ [ f ′ ( α ) , f ′ ( α )] and a countable number of ASSP D i = D ( a i , b i , p i ) such thati. γ − ( t ) = A + tf ′ ( p − ) , γ + ( t ) = B + tf ′ ( p + ) are characteristic lines. i. D ( a i , b i , p i ) ⊂ { ( x, t ) ∈ IR × (0 , ∞ ) : γ − ( t ) < x < γ + ( t ) } .iii. Define F ± , D ± , R by F − = { ( x, t ) : x < R − ( t, A, u ) } ,F + = { ( x, t ) : x > R + ( t, B, u ) } ,D − = { ( x, t ) : R − ( t, A, u ) < x < γ − ( t ) } ,D + = { ( x, t ) : γ + ( t ) < x < R + ( t, B, u ) } ,R = { ( x, t ) / ∈ D − ∪ D − ∪ i D ( a i , b i , p i ) : γ − ( t ) < x < γ + ( t ) } , then f ′ ( u ( x, t )) ∈ [ f ′ ( β ) , f ′ ( β )] if ( x, t ) ∈ F − f ′ ( u ( x, t )) ∈ [ f ′ ( α ) , f ′ ( α )] if ( x, t ) ∈ F + iv. ( x, t ) ∈ R , then ( x, t ) lies on a characteristic line.v. If for some η > , u is continuous in [ a, a + η ) and ( b − η, b ] , then u ( a i ) = u ( b i ) = p i . vi. If u is monotone in [ a, a + η ) and ( b − η, b ] , then u is increasing in [ a, a + η ) and ( b − η, b ] .vii. b i − a i b i Z a i u ( x ) dx = p i . Proof.
Let L ( t ) = R − ( t, A, u ) , R ( t ) = R + ( t, B, u ) , y + ( t ) = y + ( R − ( t, A, u ) , t ), y − ( t ) = y − ( R + ( t, B, u ) , t ) .
1. Shock case follows from Lemma 2.6.2. Assume (2.102).If for all t > L ( t ) < R ( t ), then A ≤ y + ( t ) ≤ y − ( t ) ≤ B , hence existence of A , B , p − , p + follows from Lemma 2.5. Assume that there exists T > L ( T ) = R ( T ). Then from (4) of 2.4, L ( t ) = R ( t ) for all t ≥ T . From (2.53), for t > T , u ( x, t ) ∈ ( [ β , β ] if x < L ( t ) , [ α , α ] if x > L ( t ) . Let x ↑ L ( t ) , ξ ↓ R ( t ), then y + ( x, T, t ) → y − ( L ( t ) , T, t ) and y − ( ξ, T, t ) → y + ( R ( t ) , T, t ).Claim: y ± ( L ( t ) , T, t ) = L ( T ) and L ( t ) is a line segment for t > T .Suppose not, say y − ( L ( t ) , T, t ) < L ( T ) ≤ y + ( L ( t ) , T, t ), then L ( t ) − y − ( L ( t ) , T, t ) t − T = lim x ↑ L ( t ) x − y + ( x, T, t ) t − T = lim x ↑ L ( t ) f ′ ( u ( x, t )) ∈ [ f ′ ( β ) , f ′ ( β )] , (2.103)28 ( t ) − y + ( L ( t ) , T, t ) t − T = lim ξ ↓ L ( t ) ξ − y − ( ξ, T, t ) t − T = lim ξ ↓ L ( t ) f ′ ( u ( ξ, t )) ∈ [ f ′ ( α ) , f ′ ( α )] . (2.104)But L ( t ) − y − ( L ( t ) , T, t ) t − T > L ( t ) − y + ( L ( t ) , T, t ) t − T , which contradicts (2.102). Hence γ ( θ ) = L ( T ) + (cid:16) L ( t ) − L ( T ) t − T (cid:17) ( θ − T ) is the charac-teristic line segment joining ( L ( t ) , t ) and ( L ( T ) , T ). Thus L ( θ ) ≤ γ ( θ ) ≤ R ( θ ) for θ ∈ [ T, t ]. Since L ( θ ) = R ( θ ), for θ > T , which implies that L ( θ ) = γ ( θ ). Since t isarbitrary, this implies that L ( t ) is a straight line from ( L ( T ) , T ).Let x = L ( T ) = R ( T ) and γ ± ( θ ) = y ± ( x , T ) + (cid:16) x − y ± ( x ) T (cid:17) θ the characteristic linesegments at ( x , T ). Then by the strict convexity of f ∗ , the curves γ ( t ) = (cid:26) γ + ( t ) if 0 < t < T,L ( t ) if t > T,γ ( t ) = (cid:26) γ − ( t ) if 0 < t < T,L ( t ) if t > T, are characteristic lines and hence γ = γ = L = R. Therefore A = B = y ± ( x , T )and f ′ ( p − ) = f ′ ( p + )= f ′ ( β ) = f ′ ( α ). Hence either L ( t ) < R ( t ) for all t > L ( t ) = R ( t ) for all t > u in an ASSP. Since f is notstrictly convex, the proof does not follow from [1] and it is quite delicate, therefore weadopt a different procedure. In order to do this, we concentrated only on D ( a i , b i , p i )and by approximation procedure we prove (v), (vi) and (vii). For the sake of simplicitydenote an ASSP D ( a i , b i , p i ) by D ( a, b, p ). Define˜ u ( x ) = (cid:26) u ( x ) if x ∈ ( a, b ) ,p if x / ∈ ( a, b ) . ˜ u ( x, t ) = (cid:26) u ( x, t ) if ( x, t ) ∈ D ( a, b, p ) ,p if ( x, t ) / ∈ D ( a, b, p ) . Then from (2.50) and (2.52) ˜ u is the solution of (1.1) with initial data ˜ u . Moreover γ ( · , a, p ) and γ ( · , b, p ) are the characteristic lines and D ( a, b, p ) does not contain anyother characteristic line.Let f ǫ ∈ C ( IR ) be a uniformly convex flux converging to f in C loc( IR ) (see (4.1),(4.3) and (4.9)) with lim | p |→∞ inf ǫ f ǫ ( p ) | p | = ∞ . Then f ∗ ǫ → f ∗ in C loc( IR ). Let u ǫ bethe solution of (1.1) with flux f ǫ and initial data ˜ u . Let L ǫ ( t ) = R − ( t, a, f ǫ ) and R ǫ ( t ) = R + ( t, b, f ǫ ) be the extreme characteristic curves. Let y ǫ ( t ) = y + ( L ǫ ( t ) , t ),29 ǫ ( t ) = y − ( R ǫ ( t ) , t ). Then from the structure Theorem [1], there exist a ≤ a ǫ ≤ b ǫ ≤ b such that for i = 1 , γ ,ǫ , γ ,ǫ are characteristic lines wherelim t →∞ ( y ǫ ( t ) , Y ǫ ( t )) = ( a ǫ , b ǫ ) γ ,ǫ ( t ) = γ ( t, a ǫ , p ) = a ǫ + tf ′ ǫ ( p ) γ ,ǫ ( t ) = γ ( t, b ǫ , p ) = b ǫ + tf ′ ǫ ( p ) . Let for a subsequence { a ǫ , b ǫ } → { ˜ a, ˜ b } as ǫ →
0. Since the limits of characteristicsare characteristics, we obtain γ and γ are characteristics lines corresponding to ˜ u where ( γ ( t ) , γ ( t )) = lim ǫ → ( γ ,ǫ ( t ) , γ ,ǫ ( t )) = (˜ a + tf ′ ( p ) , ˜ b + tf ′ ( p )) . Claim 1: For a subsequencelim ǫ → ( L ǫ ( t ) , R ǫ ( t )) = ( γ ( t, a, p ) , γ ( t, b, p )) . Let for a subsequence lim ǫ → ( L ǫ ( t ) , R ǫ ( t )) = ( L ( t ) , R ( t )) in C loc( IR ). Since y − ( L ǫ ( t ) , t ) ≤ a ≤ y + ( L ǫ ( t ) , t ) , therefore by letting ǫ → ∞ to obtain y − ( L ( t ) , t ) ≤ a ≤ y + ( L ( t ) , t ) , and then γ ( t, a, p ) = R − ( t, a, f ) ≤ L ( t ) ≤ R + ( t, a, f ) = γ ( t, a, p ) , because γ ( t, a, p ) is a characteristic line. Hence L ( t ) = γ ( t, a, p ) and similarly R ( t ) = γ ( t, b, p ) and this proves the claim 1.Claim 2: Let for a subsequence lim ǫ → ( a ǫ , b ǫ ) = (˜ a, ˜ b ) , then ˜ a = a, ˜ b = b. If not, then let 0 < ˜ a ≤ b . If ˜ a < b , then γ ( t ) = lim ǫ → γ ,ǫ ( t ) = ˜ a + tf ′ ( t ) is acharacteristic line with respect to ˜ u , which is a contradiction since γ ( t ) ∈ D ( a, b, p )is an ASSP.Hence ˜ a = b . Let ǫ > < ǫ < ǫ , a ǫ > a + ( b − a ) . Then s ǫ ( t ) = R − ( t, a + b , f ǫ ) meet L ǫ at t = t ǫ and y − ( s ǫ ( t ǫ ) , t ǫ ) ≤ a + b ≤ y + ( s ǫ ( t ǫ ) , t ǫ ) . (2.105)Suppose lim ǫ → t ǫ = ∞ , then choose ˜ t ǫ < t ǫ such that lim ǫ → y + ( L (˜ t ǫ ) , ˜ t ǫ ) = b . Hence thecharacteristic line segments L (˜ t ǫ )+ y + ( L (˜ t ǫ ) , ˜ t ǫ ) − L (˜ t ǫ )˜ t ǫ ( t − ˜ t ǫ ) and L ( t ǫ )+ y − ( L ( t ǫ ) ,t ǫ ) − L ( t ǫ ) t ǫ ( t − t ǫ ) meet for some 0 < t < min(˜ t ǫ , t ǫ ) which contradicts the non intersection of char-acteristics lines. Hence { t ǫ } is bounded. Let t = lim ǫ → t ǫ and s ( t ) = lim ǫ r s ǫ ( t ) for0 ≤ t ≤ t . Then from (2.105), we have for 0 ≤ t ≤ t and from claim 1 R − (cid:18) t, a + b , f (cid:19) ≤ s ( t ) ≤ R + (cid:18) t, a + b , f (cid:19) ,s ( t ) = γ ( t , a, p ) . R − ( t, a + b , f ) meet γ ( t, a, p ) which contradicts that D ( a, b, p ) is anASSP, which proves claim 2. Let E ǫ ( t ) = { ( x, t ) : a ǫ + tf ′ ǫ ( p ) < x < b ǫ + tf ′ ǫ ( p ) , < t < t } , then integrating the equation (1.1) with flux f ǫ in E ǫ ( t ) to obtain b ǫ Z a ǫ u ( x ) dx = b ǫ + t f ′ ǫ ( p ) Z a ǫ + t f ′ ǫ ( p ) u ǫ ( x, t ) dx = b ǫ + t f ′ ǫ ( p ) Z a ǫ + t f ′ ǫ ( p ) ( f ′ ǫ ) − (cid:18) x − y + ( x, t , f ǫ ) t (cid:19) = b ǫ Z a ǫ ( f ′ ǫ ) − (cid:18) f ′ ǫ ( p ) + ξ − y + ( ξ + t f ′ ( p ) , t , f ǫ ) t (cid:19) , letting t → ∞ to obtain b ǫ Z a ǫ u ( x ) dx = p ( b ǫ − a ǫ ) . Now letting ǫ → b − a b Z a u ( x ) dx = p. This proves (vii). If u is continuous in [ a, a + η ) and ( b − η, b ] for some η > ǫ small and η small, u is continuous in [ a ǫ , a ǫ + η ) , ( b ǫ − η, b ǫ ] and hencefrom structure Theorem [1], we have u ( a ǫ ) = u ( b ǫ ) = p . Now letting ǫ → u ( a ) = u ( b ) = p and this proves (v). Furthermore if u is monotone in[ a, a + η ) , ( b − η, b ] then for ǫ small, u is monotone in [ a ǫ , a ǫ + η ) , ( b ǫ − η, b ǫ ]. Henceletting ǫ → u is increasing in [ a, a + η ) , ( b − η, b ] for some η > REMARK 2.1.
Decay estimates and N -wave follow exactly as in [1, 2] and therefore weomit here. First we give the proof of part I of the main Theorem using the convex modification (4.44)and the structure Theorem 2.2. 31
EMMA 3.1.
Let ( f, C, D ) be a convex-convex type triplet and α ≤ α ≤ ˜ α < C ≤ D < ˜ β ≤ β ≤ β . Assume that f ′ ( ˜ α ) < f ′ ( ˜ β ) , f ( θ ) < L ˜ α, ˜ β ( θ ) , for θ ∈ [ C, D ] . (3.1) Let
A < B, ¯ u ∈ L ∞ ( A, B ) such thateither ¯ u ( x ) ∈ ( −∞ , ˜ α ] for x ∈ [ A, B ] , (3.2) or ¯ u ( x ) ∈ [ ˜ β, ∞ ) for x ∈ [ A, B ] . (3.3) Let u ± ∈ L ∞ ( IR ) satisfies (1.19) and (1.20) and u be defined as in (1.6). Let u be thesolution of (1.1), (1.2). Then there exist a γ > , ( x , T ) ∈ IR × (0 , ∞ ) and a Lipschitzcurve r ( · ) in [ T , ∞ ) such that for t > T , r ( T ) = x , T ≤ γ | A − B | , (3.4) u ( x, t ) ∈ ( [ α , α ] if x > r ( t )[ β , β ] if x < r ( t ) . (3.5) Furthermore if ( g, ˜ C, ˜ D ) be another convex-convex type triplet such that g = f in a neigh-bourhood of ( −∞ , ˜ α ) ∪ ( ˜ β, ∞ ) , then u is also the solution of (1.1), (1.2) with flux g .Proof. Let ˜ α < α < C ≤ D < β < ˜ β such that for θ ∈ [ C, D ] ,f ′ ( α ) < f ′ ( β ) , f ( θ ) < L α,β ( θ ) (3.6)and let ˜ f be a convex modification of f as in (4.44). First assume that¯ u ( x ) ∈ ( −∞ , ˜ α ] , for x ∈ [ A, B ] . (3.7)Now consider a Riemann problem w ( x ) = (cid:26) a if x < z ,b if x > z , (3.8)where a ≤ ˜ α and b ≥ ˜ β . Then from Oleinik entropy condition and from (3.6), (4.40), thesolution to (1.1) with flux f and initial data (3.8) is same as the solution to (1.1) with flux˜ f and data (3.8). Hence by front tracking Lemma 2.2, the solution to (1.1) with flux f andthe data given by the hypothesis is same as that of (1.1) with flux ˜ f . Furthermore range of u is contained in ( −∞ , ˜ α ] ∪ [ ˜ β, ∞ ) and hence this is also the solution of (1.1) with flux g .From (4.40), f ′ ( ˜ α ) ≤ f ′ ( α ) < f ′ ( β ) ≤ f ′ ( ˜ β ) . Hence ˜ f ′ ( ˜ β ) > ˜ f ′ ( ˜ α ). Therefore (3.4),(3.5) follows from (1) of Theorem 2.2 with flux ˜ f . Similarly the result follows if ¯ u ( x ) ∈ [ ˜ β, ∞ ) for x ∈ ( A, B ). This proves the lemma.
Proof of the Theorem for convex-convex type flux.
Let ( f, C, D ) be a convex-convex typetriplet and α ≤ α < C < D < β ≤ β satisfying (1.15) to (1.17) and (1.21). Let u , u ± , ¯ u be as in (1.6), (1.19) and (1.20). Let m = min { α , inf ¯ u } , m = max { β , sup ¯ u } , ,m i ( x ) = (cid:26) u ( x ) if x / ∈ ( A, B ) ,m i if x ∈ ( A, B ) , and u i for i = 1 , u ,m i . Since u ,m ≤ u ≤ u ,m , therefore u ≤ u ≤ u . From Lemma 3.1, for i = 1 ,
2, there exist γ i , ( x i , T i ) ∈ IR × (0 , ∞ ), Lipschitz curves r i ( · ) such that for t > T i r i ( T i ) = x i , T i ≤ γ i | A − B | ,u i ( x, t ) ∈ ( [ α , α ] if x > r i ( t )[ β , β ] if x < r i ( t ) . Let T = max { T , T } , γ = max { γ , γ } , A = r ( T ) , B = r ( T ) , then from u ≤ u ≤ u ,to obtain u ( x, t ) ∈ [ α , α ] if x > B , [ α , β ] if A < x < B , [ β , β ] if x < A .T ≤ γ | A − B | , (3.9)where γ, T depending only on || u || ∞ and f ′ ( β ) − f ′ ( α ) . Hence without loss of generality we can assume that¯ u ( x ) ∈ [ α , β ] for x ∈ ( A, B ) . (3.10)From (1.17) and (1.21), choose an ǫ > , ( η , ξ ) , ( η , ξ ) in IR such that α + ǫ < C < D < β − ǫ, (3.11)( β − β ) + ( α − α ) < ǫ, (3.12) f ( θ ) < L α + ǫ,β − ǫ ( θ ) for θ ∈ [ C, D ] , (3.13) η < α < D < ξ < β − ǫ, (3.14) α + ǫ < η < C < β < ξ , (3.15)For θ ∈ [ C, D ] , f ( θ ) < min { L η ,ξ ( θ ) , L η ,ξ ( θ ) } . (3.16)Let u ,A = (cid:26) η if x > B,ξ if x < B.u ,B = (cid:26) η if x > A,ξ if x < A.l ,B ( θ ) = B + (cid:18) f ( η ) − f ( ξ ) η − ξ (cid:19) θ,l ,A ( θ ) = A + (cid:18) f ( η ) − f ( ξ ) η − ξ (cid:19) θ. u A and u B of (1.1) with respective initial data u ,A and u ,B are givenby u A ( x, t ) = (cid:26) η if x > l ,B ( t ) ,ξ if x < l ,B ( t ) .u B ( x, t ) = (cid:26) η if x > l ,A ( t ) ,ξ if x < l ,A ( t ) . Claim: There exist γ > , T > , A ≤ B , depending only on || u || ∞ and f ′ ( β ) − f ′ ( α )such that l ,B ( T ) ≤ A ≤ B ≤ l ,A ( T ) , (3.17) T ≤ γ | A − B | , (3.18) | A − B | ≤ β − α − ǫβ − α | A − B | , (3.19) u ( x, t ) ∈ [ α , α ] if x > B , [ α , β ] if A < x < B , [ β , β ] if x < A . (3.20)In order to prove the claim, define u , ( x ) = (cid:26) u ( x ) if x / ∈ ( A, B ) or u ( x ) < α + ǫ ,α + ǫ if x ∈ ( A, B ) , u ( x ) ≥ α + ǫ , (3.21) u , ( x ) = (cid:26) u ( x ) if x / ∈ ( A, B ) or u ( x ) ≥ β − ǫ ,β − ǫ if x ∈ ( A, B ) , u ( x ) ≤ β − ǫ (3.22)and u , u be the corresponding solutions of (1.1),(1.2) with respect to the initial data u , and u , . From (3.11), (3.16) and Lemma 3.1, there exist γ > , T > , A , B such that T ≤ γ | A − B | ,u ( x, T ) ∈ ( [ α , α ] if x > A , [ β , β ] if x < A .u ( x, T ) ∈ ( [ α , α ] if x > B , [ β , β ] if x < B . Since u , ≤ u ≤ u , , therefore u ≤ u ≤ u . This implies that A ≤ B and u ( x, T ) ∈ [ α , α ] if x > B , [ α , β ] if A < x < B , [ β , β ] if x < A . (3.23)34lso u ,A ≤ u , ≤ u , ≤ u ,B and hence u A ≤ u ≤ u ≤ u B . Therefore l ,B ( T ) ≤ A ≤ B ≤ l ,A ( T ) (3.24)and from L loc contraction, we have from (3.21), (3.22)( β − α ) | A − B | ≤ B Z A | u ( x, T ) − u ( x, T ) | dx ≤ B Z A | u , ( x ) − u , ( x ) | dx ≤ ( β − α − ǫ ) | A − B | . This gives | A − B | ≤ (cid:18) β − α − ǫβ − α (cid:19) | A − B | . (3.25)This proves the claim.Repeating the above procedure for t > T , by induction we can find γ >
0, sequence T n > T n +1 , A n ≤ B n , with A = A, B = B such that for n ≥ l ,B n − ( T n ) ≤ A n ≤ B n ≤ l ,A n − ( T n ) , (3.26) T n ≤ T n − + γ | A n − − B n − | , (3.27) | A n − B n | ≤ (cid:18) β − α − ǫβ − α (cid:19) | A n − − B n − | , (3.28) u ( x, T n ) ∈ [ α , α ] if x > B n , [ α , β ] if A n < x < B n , [ β , β ] if x < A n . (3.29)From (3.12), (3.27) and (3.28) we have δ = (cid:18) β − α − ǫβ − α (cid:19) < ,T n ≤ γ (1 + δ + δ + · · · + δ n − ) | A − B | ≤ γ − δ | A − B | , | A n − B n | ≤ δ n | A − B | . Hence { T n } is bounded, from (3.26), A n , B n are bounded and | A n − B n | → n → ∞ . Let ( x , T ) = lim n →∞ ( A n , T n ) . Then from (3.29), we have T ≤ γ − δ | A − B | , ( x, T ) ∈ ( [ α , α ] if x < x , [ β , β ] if x > x . From Lemma 3.1, there exists a Lipschitz curve r ( · ) such that r ( T ) = x and for t > T , u ( x, t ) ∈ ( [ α , α ] if x > r ( t ) , [ β , β ] if x < r ( t ) . This concludes the proof of the Theorem for the convex-convex type flux.Next we consider the convex-concave type flux and the method of the proof is differentfrom that of convex-convex type flux as they have different polarity. In order to prove thesecond part of the Theorem with condition 1 ((1.22) to (1.25)) we need to prove the followingLemmas whose proof depends on the front tracking and first part of the structure Theorem.Basically this Lemma reduces the problem to having ¯ u ∈ [ α , β ] . Second part of the Theorem with condition 2 follows in a similar way where one hasto use structure Theorem for concave flux instead of convex flux.From Lemma 2.3 we will prove the following Lemma which is essential to prove thesecond part of the main Theorem.Let ( f, C, D ) be a convex-concave type triplet, α , α , β , β and f satisfies α ≤ α < C < D < β ≤ β , (3.30) f ( θ ) > L α ,β ( θ ) , θ ∈ [ C, D ] , (3.31) f ( β ) < L α ( β ) . (3.32)From (3.32), choose α < α < D (see figure 4) such that f ( β ) = L α ( β ) . (3.33)Let u ± , ¯ u ∈ L ∞ ( IR ) such that u + ( x ) ∈ [ β , β ] , u − ( x ) ∈ [ α , α ] , (3.34) u ( x, t ) = u + ( x ) if x > B, ¯ u ( x ) if x ∈ ( A, B ) ,u − ( x ) if x < A, (3.35)and u be the solution of (1.1) and (1.2). Then LEMMA 3.2.
Let β ≥ β and assume that ¯ u satisfies one of the following conditions1. ¯ u ( x ) ∈ [ α , α ] if x ∈ ( A, B ) .2. ¯ u ( x ) ∈ [ β , β ] if x ∈ ( A, B ) . hen there exist ( x , T ) ∈ IR × (0 , ∞ ) , γ > and a Lipschitz curve r ( · ) depending only on || u || ∞ , L α ( β ) − f ( β ) such that T ≤ γ | A − B | , r ( T ) = x , (3.36) u ( x, t ) ∈ ( [ α , α ] if x < r ( t ) , [ β , β ] if x > r ( t ) . (3.37) Proof.
Without loss of generality, let ¯ u ( x ) ∈ [ α , α ] for x ∈ ( A, B ). Other case followssimilarly. First assume that u is piecewise constant with finite number of discontinuities.Define ξ = min (cid:26) f ( p ) − f ( q ) p − q : p ∈ [ α , α ] , q ∈ [ β , β ] (cid:27) ,ξ = max (cid:26) f ( p ) − f ( q ) p − q : p ∈ [ α , α ] , q ∈ [ β , β ] (cid:27) ,m = min (cid:26) f ( p ) − f ( q ) p − q : p, q ∈ [ α , α ] (cid:27) ,m = max (cid:26) f ( p ) − f ( q ) p − q : p, q ∈ [ α , α ] (cid:27) ,I = [ −|| u || ∞ , || u || ∞ ] ,E = { α , α , β , β , α , β , C, D } ∪ { jumps of u } . Let { f n } be a sequence of piecewise affine functions such that (see Lemma 4.1)i. f n → f in C loc( IR ) . ii. For some C >
0, for all n , Lip ( f n , I ) ≤ CLip ( f, I ).iii. E ⊂ corner points of f n (see definition 4.1), for all n .iv. ( f n , C, D ) is a convex-concave type triplet with f n ( θ ) > L α ,β ( θ ) for θ ∈ [ C, D ] f n ( β ) < L α ( β ) f n ( β ) = L α ( β ) = f ( α ) + f ′ n − ( α )( β − α ) , for all n. Let u n be the solution of (1.1), (1.2) with flux f n and initial data u . Define η n = f n ( u ( A − )) − f n ( u ( A +)) u ( A − ) − u ( A +) , (3.38) θ n = f n ( u ( B − )) − f n ( u ( B +)) u ( B − ) − u ( B +) , (3.39) l n ( t ) = A + η n t, L n ( t ) = B + θ n t. (3.40)37ow from (3.33), m > ξ and hence the lines A + m t and B + ξ t meet at ˜ T > T = B − Am − ξ . (3.41)Furthermore m ≤ η n = dl n dt ≤ m , ξ ≤ θ n = dL n dt ≤ ξ . (3.42)Let T n be the first point of interaction of waves of u n . Then for 0 < t < T n and fromOleinik entropy condition and (3.33), u n ( x, t ) ∈ [ α , α ] if x < l n ( t ) , [ α , α ] if l n ( t ) < x < L n ( t )[ β , β ] if x > L n ( t ) . Suppose l n ( T n ) = A n < B n = L n ( T n ) , then let T n > T n be the second time ofinteraction of waves of u n . Define η n = f n ( u n ( A n − , T n )) − f n ( u n ( A n + , T n )) u n ( A n − , T n ) − u n ( A n + , T n ) ,θ n = f n ( u n ( B n − , T n )) − f n ( u n ( B n + , T n )) u n ( B n − , T n ) − u n ( B n + , T n ) ,l n ( t ) = A n + η n ( t − T n ) , L n ( t ) = B n + θ n ( t − T n ) ,l n ( t ) = (cid:26) l n ( t ) if t < T n ,l n ( t ) if T n ≤ t ≤ T n , L n ( t ) = (cid:26) L n ( t ) if t < T n ,L n ( t ) if T n ≤ t ≤ T n . Then for 0 < t < T n , u n ( x, t ) ∈ [ α , α ] if x < l n ( t ) , [ α , α ] if l n ( t ) < x < L n ( t ) , [ β , β ] if x > L n ( t ) . (3.43) m ≤ dl n dt ≤ m , ξ ≤ dL n dt ≤ ξ , l n (0) = A, L n (0) = B. (3.44)If A n = l n ( T n ) < B n = L n ( T n ) , then repeat the above to obtain the curves l n , L n satisfying (3.43) and (3.44) for t ≤ T n k , where T n k is the k th time interaction of waves of u n . From (3.44) and (3.41), l n and L n meet at T n k ≤ ˜ T with x n = l n ( T n k ) = L n ( T n k ) . Then u n ( x, T n k ) ∈ ( [ α , α ] if x < x n , [ β , β ] if x > x n . and { x n } is bounded from (3.44). Furthermore for t > T n k ,u n ( x, t ) ∈ ( [ α , α ] if x < L n ( t ) , [ β , β ] if x > L n ( t ) . (3.45)38ence from Lemma 2.1 and Arzela Ascoli’s Theorem, for a subsequence x n → x , T n k → T ,u n → u in L loc( IR × (0 , ∞ )) . L n → r ( · ) as n → ∞ . Since T ≤ ˜ T , hence with γ = m − ξ ,and from (3.46), (3.36) and (3.37) follows. If u is arbitrary, approximate u by piecewiseconstant functions and from L contraction, the Lemma follows, since γ, x , T depends only || u || ∞ and L α ( β ) − f ( β ). This proves the Lemma. LEMMA 3.3.
Let α ≤ ˜ α < C ≤ D < ˜ β ≤ β such that f ( θ ) > L ˜ α, ˜ β ( θ ) , for θ ∈ [ C, D ] ,f ( ˜ β ) < L α ( ˜ β ) . (3.46) Assume ¯ u satisfies one of the following hypothesisi. range of ¯ u ⊂ [ α , ˜ α ] ii. range of ¯ u ⊂ [ ˜ β, β ] then there exist ( x , T ) ∈ IR × (0 , ∞ ) , γ > and a Lipschitz curve r ( · ) in [ T , ∞ ) dependingonly on || u || ∞ , f ′ ( α ) − f ′ ( α ) such that for t > T T ≤ γ | A − B | , r ( T ) = x , (3.47) u ( x, t ) ∈ ( [ α , α ] if x < r ( t ) , [ β , β ] if x > r ( t ) . (3.48) Proof.
We can assume that ¯ u ∈ [ α , ˜ α ] for all x ∈ ( A, B ). Other case follows in a similarway. Again we use front tracking. Let u is piecewise constant with finite number of jumps.Define E = { α , α , β , β , ˜ α, C, D } ∪ { Jumps of u } ,I = [ −|| u || ∞ , || u || ∞ ] ,ξ = min (cid:26) f ( p ) − f ( q ) p − q , p ∈ [ α , ˜ α ] , q ∈ [ β , β ] (cid:27) ,ξ = max (cid:26) f ( p ) − f ( q ) p − q , p ∈ [ α , ˜ α ] , q ∈ [ β , β ] (cid:27) ,m = f ′ ( α ) , m = f ′ ( ˜ α ) . Let { f n } be a sequence of piecewise affine functions such that (see Lemma 4.1)i. f n → f in C loc with f ′ n, − ( α ) = f ′ ( α ), Lip ( f n , I ) ≤ CLip ( f, I ), for some constant C > n , E ⊂ corner points of f n .iii. ( f n , C, D ) is a convex-concave type triplet with f n ( θ ) > L ˜ α, ˜ β ( θ ) , ∀ θ ∈ [ C, D ] . u n be the solution to (1.1), (1.2) with flux f n and initial data u . Define η n , θ n , l n , L n as in (3.37), (3.38) and (3.39). Since m > ξ , let ˜ T be the intersection point of A + m t and B + ξ t . Furthermore (3.41) holds.Let T n be the first time of interaction of waves of u n , then for 0 < t < T n , we have u ( x, t ) ∈ [ α , α ] if x < l n ( t ) , [ α , ˜ α ] if l n ( t ) < x < L n ( t ) , [ β , β ] if x > L n ( t ) . As in 3.2, suppose l n ( T n ) = A n < B n = L n ( T n ) , then by induction there exists a T n k and a piecewise affine curves l n ( t ) , L n ( t ) for 0 < t < T n k such that l n and L n satisfies (3.44)and u ( x, t ) ∈ [ α , α ] if x < l n ( t ) , [ α , ˜ α ] if l n ( t ) < x < L n ( t ) , [ β , β ] if x > L n ( t ) . From (3.44) and (3.40), there exists k such that x n = l n ( T n k ) = L n ( T n k ) and T n k ≤ ˜ T = B − Am − ξ . Hence from L -contraction and Arzela-Ascoli Theorem, for a subsequence,still denoted by n such that u n → u in L loc, x n → x , T n k → T , l n ( t ) → r ( t ), a Lipschitzcurve with r ( T ) = x , T ≤ γ | B − A | ,u ( x, t ) ∈ ( [ α , α ] if x < r ( t ) , [ β , β ] if x > r ( t ) . This proves the Lemma.
Convex modification of f : Define˜ f ( p ) = (cid:26) f ( p ) if p ≤ C, ( p − C ) + f ′ ( C )( p − C ) + f ( C ) if p > C. (3.49)Then ˜ f is a C convex function with ˜ f = f in ( −∞ , C ] andlim | p |→∞ ˜ f ( p ) | p | = ∞ . (3.50) LEMMA 3.4.
Let α be as in (3.33), m ≤ α and assume that ¯ u satisfies ¯ u ( x ) = m, (3.51) then there exist ( x , T ) ∈ IR × (0 , ∞ ) , γ > and a Lipschitz r ( · ) depending only on || u || ∞ and hypothesis (1.22), (1.25) such that r ( T ) = x , T ≤ γ | A − B | , (3.52) u ( x, t ) ∈ ( [ α , α ] if x < r ( t ) , [ β , β ] if x > r ( t ) . (3.53)40 roof. Proof is lengthy and we use the structure Theorem. Let g ( p ) = ˜ f ( p ) be the convexmodification of f as in (3.49).First assume that u is piecewise constant with finite number of discontinuities. Let E = { α , α , β , β , C, D } ∪ { points of discontinuities of u } . Let { f n } and { g n } be sequences of continuous piecewise affine functions such that g n isconvex satisfying (see Lemma 4.1)i. ( f n , g n ) → ( f, g ) in C loc( IR ) as n → ∞ . ii. f n = g n in ( −∞ , C ].iii. lim | p |→∞ inf n g n ( p ) | p | = ∞ . iv. For all x , E ⊂ { corner points of f n } .v. α , m lies in the interior of degenerate points of f n for all n .vi. f n ( θ ) < L α ,β ( θ ) for θ ∈ [ C, D ].vii. f n ( β ) = L α ( β ) . viii. I = [ −|| u || ∞ , || u || ∞ ] and for all n ,max( Lip ( f n , I ) , Lip ( g n , I )) ≤ Lip ( f, I ) , Lip ( g, I )) = J. Let w = (cid:26) u ( x ) if x < B,α if x > B, and u n , w n , w be the solutions of (1.1), (1.2) with respective fluxes f n , g n , g and initial data u , w , w . Define ξ n = min (cid:26) f n ( p ) − f n ( q ) p − q : p ∈ [ m, α ] , q ∈ [ α , α ] (cid:27) .ξ n = max (cid:26) f n ( p ) − f n ( q ) p − q : p ∈ [ m, α ] , q ∈ [ α , α ] (cid:27) .η n = g n ( u ( A − )) − g n ( m ) u ( A − ) − m .l n ( t ) = B + η n t, r ( t ) = B + f ′ ( m ) t, R ( t ) = B + f ′ ( α ) t. Then from (viii) we have for i = 0 , | ξ ni | , | η n | are bounded by J and from (v) to (vii), f ′ ( m ) < ξ = inf n ξ n ≤ η n . Hence the line r ( · ) and A + ξ t meet at ˜ T > T = B − Aξ − f ′ ( m ) . (3.54)41et T n be the first time of interaction of waves of w n . Let y ± be the extreme characteristicpoints with respect to w and g n . Then for 0 < t < T n , w n satisfies w n ( x, t ) ∈ [ α , α ] if x < l n ( t ) ,w n ( x, t ) = u n ( x, t ) if x < r ( t ) ,w n ( x, t ) = m if l n ( t ) < x < r ( t ) , rarefaction from m to α if r ( t ) < x < R ( t ) ,α if x > R ( t ) , (3.55)From (4.25) to (4.38) y − ( R ( t ) , t ) = A = y + ( R ( t ) , t ) ,y − ( l n ( t ) , t ) ≤ A ≤ y + ( l n ( t ) , t ) ,y − ( r ( t ) , t ) ≤ B ≤ y + ( r ( t ) , t ) , (cid:12)(cid:12)(cid:12)(cid:12) dl n dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ J, dl n dt ≥ ξ > f ′ ( m ) . (3.56)Let ( A n , B n ) = ( l n ( T n ) , r ( T n )) . Clearly A n ≤ B n . If A n < B n , again starting thefront tracking from T n , let T n > T n be the first time of interaction of waves for w n . Let η n = g ( u ( A n − , T n )) − g n ( m ) u ( A n − , T n ) − m , (3.57) l n ( t ) = A n + η n ( t − T n ) for T n ≤ t ≤ T n . (3.58)Then w n satisfies (3.55) and for T n ≤ t ≤ T n , y − ( l n ( t ) , T n , t ) ≤ A n ≤ y + ( l n ( t ) , T n , t ) ,y − ( r ( t ) , T n , t ) ≤ B n ≤ y + ( l n ( t ) , T n , t ) ,f ′ ( m ) < ξ ≤ dl n dt . (3.59)Hence from (2.19) for T n ≤ T n , y − ( l n ( t ) , t ) ≤ A ≤ y + ( l n ( t ) , t ) , (3.60) y − ( r ( t ) , t ) ≤ B ≤ y + ( r ( t ) , t ) , (3.61) y − ( R ( t ) , t ) = B = y − ( R ( t ) , t ) , (3.62)and from (3.59), l n ( t ) ≥ A + tξ for t ∈ [0 , T n ] , continuing the front tracking one canget T n k > T n k − > · · · > T n and A n k < B n k , l n ( · ) such that w n satisfies (3.55), (3.56) for0 < t < T n k . Since l n ( t ) ≥ A + tξ , from (3.54), there exists a k such that x n = A n k = B n k and l n ( T n k ) = r ( T n k ) , T n k ≤ ˜ T = B − Aξ − f ′ ( m ) . Let { m < v n < v n < · · · < v np < α } be thecorner points of f n in [ m, α ] . Then w n satisfies w n ( x, T n k ) ∈ [ α , α ] , if x < x n ,w n ( x, T n k ) = (cid:26) α if x > R ( T n k ) , rarefaction from v n to v nk if x n < x < R ( T n k ) . Hence from the front tracking Lemma and (2.19), we can extend the function l n ( t ) to amaximal time T n k ≤ ˜ T n ≤ ∞ such that (cid:26) l n ( t ) ≤ R ( t ) for t < ˜ T n ,l n ( ˜ T n ) = R ( ˜ T n ) if ˜ T n < ∞ . (3.63)42 w n ( x, t ) ∈ [ α , α ] , if x < l n ( t ) ,w n ( x, t ) = (cid:26) α if x > R ( t ) , rarefaction from v ns to v nk if l n ( t ) < x < R ( t ) , (3.64)where s depends on t and is a non decreasing function of t , with y − ( l n ( t ) , t ) ≤ A ≤ y + ( l n ( t ) , t ) , (3.65) y − ( R ( t ) , t ) ≤ B ≤ y + ( R ( t ) , t ) , (3.66) (cid:12)(cid:12)(cid:12)(cid:12) dl n dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ J. (3.67)Hence for a subsequence, let ˜ T n → T , l n → l in C loc( IR ), w n → w in L loc( IR n × (0 , ∞ )),where T ∈ [0 , ∞ ]. Then from (3.63), for lim n →∞ T n k ≤ t < T , w satisfies w ( x, t ) ∈ [ α , α ] if x < l ( t ) ,w ( x, t ) = α if x > R ( t ) . From (5), (6), (7) of Lemma 2.3 and from (3.64), y − ( l ( t ) , t ) ≤ A ≤ y + ( l ( t ) , t ) , (3.68) y − ( R ( t ) , t ) ≤ B ≤ y + ( R ( t ) , t ) , (3.69)where y ± are the extreme characteristic points corresponds to g and w . Hence by definition (cid:26) R − ( t, A ) ≤ l ( t ) ≤ R + ( t, A ) ,R − ( t, B ) ≤ R ( t ) ≤ R + ( t, B ) . (3.70)Since f ′ ( α ) < f ′ ( α ) for any α ∈ [ α , α ], from (1) of structure Theorem, R − ( · , A ) and R + ( · , B ) meet at (˜ x , ˜ T ) with ˜ x = R − ( ˜ T , A ) = R + ( ˜ T , B ).Hence l ( · ) and R ( · ) meets at T ≤ ˜ T ≤ γ | A − B | and for x = l ( T ) = R ( T ) ,w ( x, ˜ T ) ∈ (cid:26) [ α , α ] if x < x ,α if x > x . (3.71)Therefore from (3.63), choose n > n ≥ n ,˜ T n ≤ (˜ γ + 1) | A − B | . (3.72)Now coming back to u n , let θ n = f n ( u ( B +)) − f n ( m ) u ( B +) − m ,L n ( t ) = B + θ n t. Then from (v) to (viii), θ n ≤ f ′ ( α ) , | θ n | ≤ J. L n ( t ) ≤ R ( t ) . Let T n be the first time of interaction of the waves of u n . Let l n be asin the previous case. Then for 0 < t < T n , u n ( x, t ) ∈ [ α , α ] , if x < l n ( t ) ,u n ( x, t ) ∈ [ β , β ] , if x > L n ( t ) ,u n ( x, t ) = m if l n ( t ) < x < r ( t ) , rarefaction from m to v ns for some s depending on t if r ( t ) < x < L n ( t ) , (cid:12)(cid:12)(cid:12)(cid:12) dL n dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ J, L n ( t ) ≤ R ( t ) . From the front tracking Lemma and (3.72), this process can be continued till a time T n ≤ ˜ T n ≤ (˜ γ + 1) | A − B | x n = L n ( T n ) = l n ( T n ) ,R ( t ) ≥ L n ( t ) > l n ( t ) for t < T n . Then u n satisfies u n ( x, T n ) ∈ ( [ α , α ] if x < x n , [ β , β ] if x > x n . Now letting a subsequence n → ∞ and from Lemma 2.1 with γ = ˜ γ + 1, T n → T , x n → x , T satisfies (3.52) and u ( x, T ) ∈ ( [ α , α ] if x < x , [ β , β ] if x > x . Now from Lemma (3.2), r ( · ) exist satisfying (3.52) and (3.53).For a general u , approximate u by piecewise constant function in L loc norm andby L loc contraction, (3.52) and (3.53) follows. This proves the Lemma. Proof of the Theorem for convex-concave type.
Let m = min { α , inf ¯ u } , M = max { β , sup ¯ u } .u ,m ( x ) = (cid:26) u ( x ) if x / ∈ ( A, B ) ,m if x ∈ ( A, B ) ,u ,M ( x ) = (cid:26) u ( x ) if x / ∈ ( A, B ) ,M if x ∈ ( A, B ) , and u m , u M be the solutions of (1.1), (1.2) with respective initial data u ,m and u ,M . Thenfrom Lemma 3.4, Lemma 3.2, Lemma 3.3 there exist γ > T ≤ γ | A − B | , A , B suchthat u m ( x, T ) ∈ ( [ α , α ] if x < A , [ β , β ] if x > A . M ( x, T ) ∈ ( [ α , α ] if x < B , [ β , β ] if x > B . Since u ,m ≤ u ≤ u ,M , thus u m ≤ u ≤ u M . Therefore A ≤ B and u ( x, T ) ∈ [ α , α ] if x < A , [ α , β ] if A < x < B , [ β , β ] if x > B . (3.73)Therefore, without loss of generality we can assume that¯ u ( x ) ∈ [ α , β ] for x ∈ ( A, B ) . (3.74)From (1.17), (1.24), (1.25), choose ǫ > ( β − β ) + ( α − α ) < ǫ,f ( θ ) > L α + ǫ,β − ǫ ( θ ) for θ ∈ [ C, D ] ,L α ( β − ǫ ) > f ( β − ǫ ) . (3.75)Define u , = (cid:26) u ( x ) if x / ∈ ( A, B ) , or u ( x ) ≤ α + ǫ ,α + ǫ if x ∈ ( A, B ) , and u ( x ) ≥ α + ǫ ,u , = (cid:26) u ( x ) if x / ∈ ( A, B ) , or u ( x ) > β − ǫ ,β − ǫ if x ∈ ( A, B ) , and u ( x ) ≤ β − ǫ . Let u and u be the solutions of (1.1), (1.2) with respective initial data u , and u , . Thenfrom (3.75), Lemma 3.3, there exist γ > , T > , A , B such that T ≤ γ | A − B | ,u ( x, T ) ∈ ( [ α , α ] if x < A , [ β , β ] if x > A . (3.76) u ( x, T ) ∈ ( [ α , α ] if x < B , [ β , β ] if x > B . (3.77)Since u , ≤ u ≤ u , , consequently u ≤ u ≤ u and therefore from (3.76), (3.77), A ≤ B and u ( x, T ) ∈ [ α , α ] if x < A , [ α , β ] if A < x < B , [ β , β ] if x > B . L contraction,( β − α ) | A − B | ≤ B Z A | u ( x, T ) − u ( x, T ) | dx ≤ B Z A | u , ( x ) − u , ( x ) | dx ≤ ( β − α − ǫ ) | A − B | . This gives | A − B | ≤ ( β − α − ǫ )( β − α ) | A − B | = δ | A − B | , where δ = ( β − α − ǫ )( β − α ) < α ≤ α < ξ < C < D < ξ < β ≤ β , such that (cid:26) f ( θ ) > max { L α ,ξ ( θ ) , L ξ ,β ( θ ) } for θ ∈ [ C, D ] ,f ( ξ ) < L α ( ξ ) , and define w , = (cid:26) α if x < B,ξ if x > B.w , = (cid:26) ξ if x < A,β if x > A. Let w , w be the solutions of (1.1), (1.2) with respective initial data are given by w ( x, t ) = α if x < B + f ( α ) − f ( ξ ) α − ξ t = ρ ( t ) ,ξ if x > ρ ( t ) ,w ( x, t ) = ξ if x < A + f ( ξ ) − f ( β ) ξ − β t = ρ ( t ) ,β if x > ρ ( t ) . Since w , ≤ u , ≤ u ≤ u , ≤ w , , thus w ≤ u ≤ u ≤ u ≤ w . Hence ρ ( T ) ≤ A ≤ B ≤ ρ ( T ).By induction we can find T n > T n − > · · · > T > ρ ( T n ) ≤ A n ≤ B n ≤ ρ ( T n )such that | A n − B n | ≤ δ | A n − − B n − | ≤ δ n | A − B | T n ≤ T n − + γ | A n − − B n − |≤ γ (1 + δ + δ + · · · + δ n − ) | A − B | , ( x, T n ) ∈ [ α , α ] if x < A n , [ α , β ] if A n < x < B n , [ β , β ] if x > B n . Hence { T n } converges to T , x = lim A n = limB n , T ≤ γ − δ | A − B | and u ( x, T ) ∈ ( [ α , α ] if x < x , [ β , β ] if x > x . Hence from Lemma 3.2, there exist a Lipschitz curve r ( · ) with r ( T ) = x such that for t > T u ( x, T ) ∈ ( [ α , α ] if x < r ( t ) , [ β , β ] if x > r ( t ) . This proves the Theorem.
Counter examples:
1. Let f be a super linear function with two inflection points C < D . Then ( f, C, D ) is aconvex-convex type triplet. Let x ∈ ( C, D ) such that f ( x ) = max { f ( θ ) : θ ∈ [ C, D ] } . Let α < C < x < D < β such that f ( α ) = f ( x ) = f ( β ) and u ( x ) = β if x < A,x if x ∈ ( A, B ) ,α if x > B. Then the solution u to (1.1), (1.2) is given by u ( x, t ) = u ( x ) for all ( x, t ) ∈ IR × (0 , ∞ ),which is not a single shock solution.2. Let f be a superlinear function with one inflection point x . Let C < x < D , then( f, C, D ) is a convex-concave flux triplet. Let α < α < C < D < β is such that f ( β ) = L α ( β ) , f ′ ( α ) < f ′ ( α ) . Let u ( x ) = α if x < A,α if x ∈ ( A, B ) ,β if x > B. Let l ( t ) = A + f ′ ( α ) t, l ( t ) = A + f ′ ( α ) t, l ( t ) = B + f ′ ( α ) t. u of (1.1), (1.2) is given by u ( x, t ) = α if x < l ( t ) , ( f ′ ) − (cid:0) x − At (cid:1) if l ( t ) < x < l ( t ) ,α if l ( t ) < x < l ( t ) ,β if x > l ( t ) , which is not a shock solution. For the sake of completeness, we will prove some of the Lemmas stated earlier.
Proof of Lemma 2.1.
First assume that u ∈ BV ( IR ) . Then for any 0 ≤ s < t, T V ( u k ( · , t )) ≤ T V ( u ( · )) and Z IR | u k ( x, s ) − u k ( x, t ) | dx ≤ C | s − t | T V ( u ) . Hence from Helly’s Theorem, there exists a subsequence still denoted by { u k } such that u k → u in L ( IR × [0 , T ]) for any T > u ∈ L ∞ ( IR ) and u ,n ∈ BV ( IR ) such that u ,n → u in L loc( IR ). Let u nk be thesolution of (1.1) with flux f k and initial data u ,n . Let m = sup n || u ,n || ∞ and K = [ − m, m ], M = sup K Lip ( f k , K ). Then for t > L loc contraction, we have for T > L > L Z − L | u k ( x, t ) − u m ( x, t ) | dx ≤ L Z − L | u k ( x, t ) − u nk ( x, t ) | dx + L Z − L | u nk ( x, t ) − u nm ( x, t ) | dx + L Z − L | u nm ( x, t ) − u m ( x, t ) | dx ≤ L + Mt Z − L − Mt | u ( x ) − u ,n ( x ) | dx + L Z − L | u nk ( x, t ) − u nm ( x, t ) | dx, hence T Z L Z − L | u k ( x, t ) − u m ( x, t ) | dxdt ≤ T L + MT Z − L − MT | u ( x ) − u ,n ( x ) | dx + T Z L Z − L | u nk ( x, t ) − u nm ( x, t ) | dxdt. k, m → ∞ and n → ∞ to obtainlim k,m →∞ T Z L Z − L | u k ( x, t ) − u m ( x, t ) | dxdt = 0 . Let u k → w in L loc( IR × (0 , ∞ )) . Since u k are uniformly bounded, thus by Dominatedconvergence Theorem, w is the solution of (1.1), (1.2) and hence w = u . This proves theLemma. Properties of convex functions (see [34]):
Let 0 ≤ ρ ∈ C ∞ (( − , , ǫ > , ρ ǫ ( x ) = ǫ ρ ( x/ǫ ) be a mollifying sequence. For g ∈ L loc( IR ),let g ǫ ( p ) = ( ρ ǫ ∗ g )( p ) + ǫ | p | , (4.1)then g ǫ ∈ C ∞ ( IR ) and satisfies the followingi. g ǫ → g in C k loc( IR ) if g is in C k ( IR ) . ii. Let g be convex, then g ǫ is uniformly convex andsup <ǫ< Lip ( g ǫ , K ) < ∞ , for any compact set K ⊂ IR. (4.2)Furthermore if g is of superlinear growth, then for 0 < ǫ ≤ , | p | > ǫ , g ǫ ( p ) | p | = Z | q |≤ ρ ( q ) g ( p − ǫq ) | p − ǫq | | p − ǫq || p | dq ≥ (cid:18) inf | z − p |≤ ǫ g ( z ) | z | (cid:19) inf | q |≤ (cid:12)(cid:12)(cid:12)(cid:12) − ǫ q | p | (cid:12)(cid:12)(cid:12)(cid:12) ≥
12 inf | z − p |≤ ǫ g ( z ) | z | . Hence lim | p |→∞ inf <ǫ ≤ g ǫ ( p ) | p | = ∞ . (4.3) DEFINITION 4.1.
Let g be a convex function. Theni. g is said to be degenerate on an interval I = ( a, b ) if g is affine on I . That is thereexist α, m ∈ IR such that for all p ∈ Ig ( p ) = mp + α. (4.4) ii. g is said to have finite number of degeneracies if there is a finite number L of disjointintervals I i = ( a i , b i ) such that g is affine on each I i and g is a strictly increasingfunction on IR \ ∪ Li =1 I i . ii. g is said to have locally finite degeneracies if there exists a i < a i +1 , m i +1 > m i , α i ∈ IR such that lim i →∞ ( a i , m i ) = ( ∞ , ∞ ) , lim i →−∞ ( a i , m i ) = ( −∞ , −∞ ) , (4.5) and for p ∈ ( a i , a i +1 ) g ( p ) = m i p + α i . (4.6) iv. Corner points: Let g be a convex function. The collection of end points of maximalinterval on which g is affine is called set of corner points of g .For a convex function g , define the Fenchel’s dual g ∗ by g ∗ ( p ) = sup q { pq − g ( q ) } . (4.7)From now on we assume that functions under consideration are convex and of super-linear growth. Then we have the following LEMMA 4.1.
Let g, h, { g k } are convex functions having superlinear growth. Then1. lim | p |→∞ g ∗ ( p ) | p | = ∞ . (4.8)
2. Let g k → g in C loc ( IR ) and lim | p |→∞ inf k g k ( p ) | p | = ∞ , then g ∗ k → g ∗ in C loc ( IR ) . (4.9) Furthermore for any C ≥ , there exists a p ≥ such that for | p | > p and for all kg ∗ k ( p ) | p | ≥ C + 1 . (4.10)
3. Let g be C and strictly monotone in ( a, b ) , then g ∗ is differentiable in ( g ′ ( a ) , g ′ ( b )) and for p ∈ ( a, b ) , ( g ∗ ) ′ ( g ′ ( p )) = p. (4.11)
4. Let g be C and having finite number of degeneracies. J i = ( a i , b i ) for ≤ i ≤ L , with g ( p ) = m i p + α i for p ∈ J i . (4.12) Then g ∗ is strictly convex and C ( IR \ { m , · · · , m L } ) such that ( g ∗ ) ′ ( g ′ ( p )) = p, if p ∈ IR \ { m , · · · , m L } , (4.13) g ∗ ( g ′ ( p )) = pg ′ ( p ) − g ( p ) , for p ∈ IR. (4.14)50 . Let g be a locally finite degenerate convex function with a i < a i +1 , m i +1 > m i , α i ∈ IR such that (4.5) and (4.6) holds. Then g ∗ ( p ) = a i p − g ( a i ) for p ∈ [ m i − , m i ] . (4.15)
6. Assume that g is a convex function with finite number of degeneracies. Let E ⊂ IR be a finite set. Then there exists a sequence { g k } of convex functions having locallyfinite degeneracies such that g k → g in C loc ( IR ) , and for all k , E ∪ { corner points of g } ⊂ { corner points of g k } , (4.16)lim | p |→∞ inf k g k ( p ) | p | = ∞ . (4.17) Proof.
For q fixed, we have1. g ∗ ( p ) = sup r { pr − g ( r ) }≥ pq − g ( p ) . Hence lim | p |→∞ g ∗ ( p ) | p | ≥ ± q. Now letting ± q → ∞ to obtain (4.9).2. Let | p | ≤ p , then g ∗ k ( p ) = sup q { pq − g k ( q ) } ≥ − g k (0) . (4.18)From (4.9) choose q such that for all | q | ≥ q , | p | ≤ p , for all k , pq − g k ( q ) < − min k g k (0), pq − g ( q ) < − g (0). Then from (4.18), there exist | q k | ≤ q , | ˜ q | ≤ q suchthat g ∗ ( p ) = sup | q |≤ q { pq − g ( q ) } = p ˜ q − g (˜ q ) g ∗ k ( p ) = sup | q |≤ q { pq − g k ( q ) } = pq k − g ( q k ) . Hence g ∗ ( p ) − g ∗ k ( p ) ≥ pq k − g k ( q k ) − pq k + g ( q k )= g k ( q k ) − g ( q k ) ≥ − sup | q |≤ q | g ( q ) − g k ( q ) | and similarly g ∗ k ( p ) − g ∗ ( p ) ≥ − sup | q |≤ q | g k ( q ) − g ( q ) | . | p |≤ p | g ∗ k ( p ) − g ∗ ( p ) | ≤ sup | q |≤ q | g k ( q ) − g ( q ) | → , as k → ∞ . Thus g ∗ k → g ∗ in C loc( IR ) . Let q = ( C + 2) sign p | p | , then g ∗ k ( p ) | p | ≥ C + 2 − g k ( sign p | p | ( C + 2)) | p | . Let p > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g k ( sign p | p | ( C + 2)) | p | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < | p | > p , then for | p | > p ,g ∗ k ( p ) | p | ≥ C + 2 − C + 1 . This proves (4.10) and hence (2).3. Denote g ∗ ′ ± ( p ) the right and the left derivatives of g ∗ . Let g ǫ be as in (4.1), then g ǫ are uniformly convex, C function converging to g in C loc( IR ) . From (4.3) and (4.9), g ∗ ǫ → g ∗ in C loc( IR ). Let q ∈ ( a, b ) and a < q < r < q < r < q < b. Since g ∗ ′ ǫ ( g ′ ǫ ( θ )) = θ for all θ ∈ IR , we have by convexity q = g ∗ ′ ǫ ( g ′ ǫ ( q )) ≤ g ∗ ǫ ( g ′ ǫ ( r )) − g ∗ ǫ ( g ′ ǫ ( q )) g ′ ǫ ( r ) − g ′ ǫ ( q ) ≤ g ∗ ǫ ( g ′ ǫ ( r )) − g ∗ ǫ ( g ′ ǫ ( q )) g ′ ǫ ( r ) − g ′ ǫ ( q ) ≤ g ∗ ′ ǫ ( g ′ ǫ ( q )) = q . Now for q ∈ ( a, b ) and letting ǫ → , r ↑ q, r ↓ q to obtain q ≤ g ∗ ′ − ( g ′ ( q )) ≤ g ∗ ′ + ( g ′ ( q )) ≤ q . Letting q , q → q to obtain, g ∗ ′ ( g ′ ( q )) = q. This proves (3).4. From (4.11), g ∗ is in C ( IR \ { m , · · · , m L } ) and satisfies g ∗ ′ ( g ′ ( p )) = p, for p ∈ IR \ { m , · · · , m L } . Hence g ∗ is strictly convex. Let g ǫ be as in (4.1), then g ǫ satisfies g ∗ ǫ ( g ′ ǫ ( p )) = pg ′ ǫ ( p ) − g ǫ ( p ) . Then from (2), letting ǫ → g ∗ ( g ′ ( p )) = pg ′ ( p ) − g ( p ) . This proves (4). 52. From direct calculations, (4.15) follows.6. Let F = E ∪ { corner points of g } . For k >
1, define sequences { a ki } , { m ki } , { g k } suchthat a ki < a ki +1 , m ki ≥ m ki − lim i →±∞ a ki = ±∞ , F ⊂ { a ki } ∞ i = −∞ . If a ki and a ki +1 are not corner points of g , then choose | a ki − a ki +1 | < /k.m ki = g ( a ki ) − g ( a ki − ) a ki − a ki − .g k ( p ) = g ( a ki ) + m ki ( p − a ki ) if p ∈ ( a ki − , a ki ) . Then { g k } is the required sequence. This proves (6) and hence the Lemma. Proof of Lemma 2.3. (1) and (2) follows from (1) and (2) of Lemma 4.1.Let v ( x, t ) = v ( x, t, f ). Taking y = x to obtain v ( x, t ) ≤ v + tf ∗ (0) . (4.19) v ( y ) + tf ∗ (cid:18) x − yt (cid:19) = v ( x ) + v ( y ) − v ( x ) + tf ∗ (cid:18) x − yt (cid:19) ≥ v ( x ) − M | x − y | + tf ∗ (cid:18) x − yt (cid:19) = v ( x ) + t (cid:20) − M (cid:12)(cid:12)(cid:12)(cid:12) x − yt (cid:12)(cid:12)(cid:12)(cid:12) + f ∗ (cid:18) x − yt (cid:19)(cid:21) . Hence for (cid:12)(cid:12) x − yt (cid:12)(cid:12) > p , v ( y ) + tf ∗ (cid:18) x − yt (cid:19) > v ( x ) + tf ∗ (0) . (4.20)Hence from (4.19) and (4.20) v ( x, t ) = inf | x − yt | ≤ p (cid:26) v ( y ) + tf ∗ (cid:18) x − yt (cid:19)(cid:27) = min | x − yt | ≤ p (cid:26) v ( y ) + tf ∗ (cid:18) x − yt (cid:19)(cid:27) . Therefore ch ( x, t, f ) = φ . For 0 < s < t , (2.15), (2.16) follows from the Dynamic program-ming principle. This proves (3). 53et y ∈ ch ( x, t, f ) and v ( x, t ) = v ( x, t, f ) . Let 0 < s < t , r ( θ ) = r ( θ, x, y, t ) , r ( s ) = ξ .Then ξ ∈ ch ( x, s, t, f ) and y ∈ ch ( ξ, s, f ). Since x − ξt − s = x − yt = ξ − ys , thus v ( x, t ) = v ( y ) + tf ∗ (cid:18) x − yt (cid:19) = v ( y ) + sf ∗ (cid:18) ξ − ys (cid:19) + ( t − s ) f ∗ (cid:18) x − ξt − s (cid:19) ≥ v ( ξ, s ) + ( t − s ) f ∗ (cid:18) x − ξt − s (cid:19) ≥ v ( x, t ) . Hence ξ ∈ ch ( x, s, t, f ) and y ∈ ch ( ξ, s, t ).Let ξ ∈ ch ( x, s, t, f ) and y ∈ ch ( ξ, s, t ). Then y ∈ ch ( x, t, f ). Moreover if f ∗ is strictlyconvex, then ( x, t ) , ( ξ, s ) and ( y,
0) lies on the same straight line. For v ( x, t, f ) = v ( ξ, s ) + ( t − s ) f ∗ (cid:18) x − ξt − s (cid:19) = v ( y ) + sf ∗ (cid:18) ξ − ys (cid:19) + ( t − s ) f ∗ (cid:18) x − ξt − s (cid:19) ≥ v ( y ) + tf ∗ (cid:18) x − yt (cid:19) ≥ v ( x, t, f ) . Hence y ∈ ch ( x, t, f ) and if f ∗ is strictly convex, then ( x, t ) , ( ξ, s ) , ( y,
0) lie on the samestraight line.Let x < x and for i = 1 , y i ∈ ch ( x i , t, f ) and γ i ( θ ) = γ ( θ, x i , , t, y i ). Suppose y < y , then there exist 0 < s < t such that ξ = γ ( s ) = γ ( s ) . Then from the aboveanalysis, ξ ∈ ch ( ξ, s, f ). Hence y ∈ ch ( x , t, f ). Furthermore if f ∗ is strictly convex, then y ≤ y and hence γ and γ never intersect in (0 , t ).As a consequence of this, y + ( x , t, f ) ≤ y + ( x , t, f ). Similarly y − ( x , t, f ) ≤ y − ( x , t, f ) . This proves (2.19) to (2.23). Similar proof follows for 0 < s < t and this proves (4).Let p = sup n { f n ( C + 2) , f n ( − ( C + 2)) } , then for | p | > p , q = p | p | ( C + 2) , we have f ∗ n ( p ) | p | ≥ ( C + 2) − f n (cid:16) p | p | ( C + 2) (cid:17) | p | ≥ C + 1 . This proves (2.24).Let s = 0, y n ∈ ch ( x, t, f n ), v n ( x, t ) = v ( x, t, f n ) , v ( x, t ) = v ( x, t, f ). Since (cid:12)(cid:12) x − y n t (cid:12)(cid:12) ≤ , consequently for a subsequence let y n → y . Then for y ∈ IR , we have v ( y ) + tf ∗ (cid:18) x − y t (cid:19) = lim n →∞ (cid:26) v ( y n ) + tf ∗ n (cid:18) x − y n t (cid:19)(cid:27) ≤ lim n →∞ (cid:26) v ( y ) + tf ∗ n (cid:18) x − yt (cid:19)(cid:27) ≤ (cid:26) v ( y ) + tf ∗ (cid:18) x − yt (cid:19)(cid:27) . Hence y ∈ ch ( x, t, f ) and v n ( x, t ) → v ( x, t ) as n → ∞ . Since sup n Lip ( v n , IR ) ≤ || u || ∞ ,thus { v n } is an equicontinuous family. Hence from Arzela-Ascoli, v n → v in C loc( IR ) . Thisproves (2.24) to (2.26) when s = 0. Similar proof follows from the Dynamic programmingprinciple. This proves (5).Let s = 0 and ( x n , t n ) → ( x, t ) and y n ∈ ch ( x, t n , f n ) and y n → y as n → ∞ . Thenfrom (2.25) v ( x, t ) = lim n →∞ v n ( x n , y n )= lim n →∞ (cid:26) v ( y n ) + t n f ∗ n (cid:18) x − y n t n (cid:19)(cid:27) = (cid:26) v ( y ) + tf ∗ (cid:18) x − yt (cid:19)(cid:27) . Therefore y ∈ ch ( x, t, f ) . Similarly for s > u ,n ⇀ u in L ∞ weak ∗ topology, therefore v ,n ( x ) → v ( x ) for all x ∈ IR .Since sup n Lip ( v ,n , IR ) ≤ sup n || u ,n || ∞ < ∞ , thus { v ,n } is an equicontinuous sequence andhence converges in C loc( IR ) . Let y n ∈ ch ( x, t, f n ) such that y n → y . Then for any ξ ∈ IR ,we have v n ( x, t ) = v ,n ( y n ) + tf ∗ n (cid:18) x − y n t (cid:19) ≤ v ,n ( ξ ) + tf ∗ n (cid:18) x − ξt (cid:19) . Letting n → ∞ to obtain v ( y ) + tf ∗ (cid:18) x − yt (cid:19) ≤ v ( ξ ) + tf ∗ (cid:18) x − ξt (cid:19) . Thus y ∈ ch ( x, t, f ). This proves (7) and hence the Lemma. Riemann problem for piecewise convex flux:
Let a i < a i +1 , m i < m i +1 , α i ∈ IR such thatlim i →±∞ ( a i , m i ) = ( ±∞ , ±∞ ) (4.21) m i − a i + α i − = m i a i + α i . (4.22)55efine g ( p ) = m i p + α i , for p ∈ [ a i , a i +1 ] . (4.23)Then from (4.21), (4.22), g defines a super linear piecewise affine convex function with g ∗ ( p ) = a i p − g ( a i ) for p ∈ [ m i − , m i ] . (4.24) Riemann problem:
Let x , a, b, ∈ IR and u ( x ) = (cid:26) a if x < x ,b if x > x . (4.25) v ( x ) = (cid:26) ax − ax if x < x ,bx − bx if x > x . (4.26)Let u be the solution of (1.1), (1.2) with flux g and initial data u and v be the correspondingvalue function defined in (2.5). Then u is given byCase (i): Let a, b ∈ [ a i , a i +1 ]. Define L ( t, g ) = R ( t, g ) = x + m i t, then u ( x, t ) = (cid:26) a if x < L ( t, g ) ,b if x > L ( t, g ) . (4.27) ch ( x, t, g ) = { x − m i t } (4.28) y − ( x + m i t, t ) = y + ( x + m i t, t ) = x . (4.29)Case (ii): Let a j ≤ b < a j +1 < a i ≤ a ≤ a i +1 . Define m = f ( a ) − f ( b ) a − b , L ( t, g ) = R ( t, g ) = x + mt (4.30)Then u ( x, t ) = (cid:26) a if x < L ( t, g ) ,b if x > R ( t, g ) . (4.31) ch ( x, t, g ) = (cid:26) x − m i t if x < L ( t, g ) ,x − m j t if x > R ( t, g ) . (4.32) y − ( x + mt, t ) = x + ( m − m i ) t < x . (4.33) y + ( x + mt, t ) = x + ( m − m j ) t > x . (4.34)Case (iii): Let a i ≤ a < a i +1 < a j ≤ b ≤ a j +1 . Define L ( t, g ) = x + m i t, R ( t, g ) = x + m j t (4.35) l k ( t ) = x + m k t for i ≤ k ≤ j. (4.36)56hen u ( x, t ) = a if x < L ( t, g ) ,a k if l k − ( t ) < x < l k +1 ( t ) ,b if x > R ( t, g ) . (4.37) y − ( x + m i t, t ) = y + ( x + m j t, t ) = x . (4.38)All the above properties follows easily from the direct calculation. LEMMA 4.2.
Let ( f, C, D ) be a convex-convex type triplet and α < C < D < β such that f ( C ) < L α,β ( C ) , f ( D ) < L α,β ( D ) . (4.39) Then f ′ ( α ) < f ( β ) − f ( α ) β − α < f ′ ( β ) . (4.40) Proof.
Suppose f ( β ) − f ( α ) β − α ≤ f ′ ( α ) , then by convexity of f in ( −∞ , C ], for x ∈ [ α, C ]. f ( β ) − f ( α ) β − α ≤ f ′ ( α ) ≤ f ′ ( x ) . Integrating from α to C to obtain L α,β ( C ) = f ( α ) + f ( β ) − f ( α ) β − α ( C − α ) ≤ f ( C ) , contradicting (4.39).Suppose f ′ ( β ) ≤ f ( β ) − f ( α ) β − α , then by convexity of f in [ D, ∞ ), for all x ∈ [ D, β ], f ′ ( x ) ≤ f ′ ( β ) ≤ f ( β ) − f ( α ) β − α . Integrating from D to β to obtain f ( D ) ≥ f ( β ) + f ( β ) − f ( α ) β − α ( D − β ) = L α,β ( D ) . Contradicting (4.39). This proves the Lemma.From Lemma 4.2 we make a convex modification of ( f, C, D ) as follows.For i = 1 ,
2, let ( a i , b i ) ∈ IR such that a = a whenever b = b . (4.41)57 Q ( x ) C D x d x β L α L β L α,β f Figure 5: Illustration for the convex modification of f For b + b = 0 , x ∈ IR , define x = x + 2( a − a ) b + b , (4.42) Q ( x ) = b − b x − x ) ( x − x ) + b ( x − x ) + a . (4.43)Then ( Q ( x ) , Q ′ ( x )) = ( a , b ) Q ( x ) = ( b − b )2 ( x − x ) + b ( x − x ) + a = 12 ( b + b )( x − x ) + a = a − a + a = a .Q ′ ( x ) = b − b x − x + b = b . Let α < C < D < β satisfying (4.39). Let f ′ ( α ) = b , f ′ ( β ) = b l ( x ) = f ( α ) + f ′ ( α )( x − α ) l ( x ) = f ( β ) + f ′ ( β )( x − β ) d = f ( α ) − αf ′ ( α ) − f ( β ) + βf ′ ( β ) f ′ ( β ) − f ′ ( α )58 α d ˜ f = Q x β L α L β L β L α,β ˜ ff = ˜ f Figure 6: Convex modification ˜ f is the point of interaction of the tangent lines l and l . Observe that from (4.40) we have α < d < β, min( f ( α ) , f ( β )) < l ( d ) < max( f ( α ) , f ( β )) . Let α < x < d < x < β and a = l ( x ) , a = l ( x ) and define Q as in (4.43).Case (i): Suppose f ′ ( α ) + f ′ ( β ) = b + b = 0. Then choose l ( x ) = l ( x ). Then( Q ( x ) , Q ′ ( x )) = ( a , b ) and Q ( x ) = 12 ( b − b )( x − x ) + b ( x − x ) + l ( x )= ( b + b )( x − x ) + l ( x )= l ( x )= l ( x ) .Q ′ ( x ) = ( b − b ) + b = b . Hence ( Q ( x ) , Q ′ ( x ) , Q ( x ) , Q ′ ( x )) = ( a , b , a , b ) and uniformly convex function.Case (ii): Let f ′ ( α ) + f ′ ( β ) = b + b = 0. Then a − a = l ( x ) − l ( x )= f ( β ) − f ( α ) − βf ′ ( β ) − αf ′ ( α ) + f ′ ( β ) x − f ′ ( α ) x = − d ( f ′ ( α ) + f ′ ( β )) + f ′ ( β ) x − f ′ ( α ) x = ( f ′ ( β ) + f ′ ( α ))( x − d ) + f ′ ( α )(2 d − x − x )= f ′ ( β )( x + x − d ) + ( f ′ ( α ) + f ′ ( β ))( d − x ) . Suppose f ′ ( α ) + f ′ ( β ) >
0, then from (4.40), f ′ ( β ) >
0. Choose x and x such that x + x > d . Then a − a > . Suppose f ′ ( α ) + f ′ ( β ) <
0, then from (4.40), f ′ ( α ) < x and x such that x + x ≤ d . Then a − a >
0. Hence Q ( x ) as in (4.43) is auniformly convex function satisfying ( Q ( x ) , Q ′ ( x ) , Q ( x ) , Q ′ ( x )) = ( a , b , a , b ).Case (iii): f ′ ( β ) ≤ . Then from (4.40), f ( α ) > f ( β ), f ′ ( α ) < l ( x ) < l ( x ) for x > d .From Lemma 4.2, choose α < x < d < x < β , a = l ( x ), a = l ( x ) such that Q is asmooth convex function satisfying ( Q ( x ) , Q ′ ( x ) , Q ( x ) , Q ′ ( x )) = ( a , b , a , b ). Define ˜ f by ˜ f ( x ) = f ( x ) if x / ∈ ( α, β ) ,l ( x ) if α ≤ x ≤ x ,Q ( x ) if x ≤ x ≤ x ,l ( x ) if x ≤ x ≤ β. (4.44)Then ˜ f is in C ( IR ), convex and satisfy ˜ f ( x ) = f ( x ), for x / ∈ ( α, β ). Convex modification (see figures 5 and 6):
Given ( f, C, D ) of convex-convex type triplet and α < C < D < β satisfying (4.39). Then ˜ f constructed above is called a convex modificationof f with respect to α, β . Acknowledgements:
Both the authors would like to thank Gran Sasso Science Institute,L’Aquila, Italy, as the work has been initiated from there. First author acknowledge thesupport from Rajaramanna fellowship. The second author would like to thank Inspireresearch grant for the support.
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