Hyperbolic Conservation Laws and Hydrodynamic Limit for Particle Systems
aa r X i v : . [ m a t h . A P ] O c t HYPERBOLIC CONSERVATION LAWS WITH DISCONTINUOUS FLUXESAND HYDRODYNAMIC LIMIT FOR PARTICLE SYSTEMS
GUI-QIANG CHEN, NADINE EVEN, AND CHRISTIAN KLINGENBERG
Abstract.
We study the following class of scalar hyperbolic conservation laws with discontin-uous fluxes: ∂ t ρ + ∂ x F ( x, ρ ) = 0 . (0.1)The main feature of such a conservation law is the discontinuity of the flux function in thespace variable x . Kruzkov’s approach for the L -contraction does not apply since it requires theLipschitz continuity of the flux function; and entropy solutions even for the Riemann problemare not unique under the classical entropy conditions. On the other hand, it is known that,in statistical mechanics, some microscopic interacting particle systems with discontinuous speedparameter λ ( x ), in the hydrodynamic limit, formally lead to scalar hyperbolic conservation lawswith discontinuous fluxes of the form: ∂ t ρ + ∂ x ( λ ( x ) h ( ρ )) = 0 . (0.2)The natural question arises which entropy solutions the hydrodynamic limit selects, thereby lead-ing to a suitable, physical relevant notion of entropy solutions of this class of conservation laws.This paper is a first step and provides an answer to this question for a family of discontinuousflux functions. In particular, we identify the entropy condition for (0.1) and proceed to show thewell-posedness by combining our existence result with a uniqueness result of Audusse-Perthame(2005) for the family of flux functions; we establish a compactness framework for the hydrody-namic limit of large particle systems and the convergence of other approximate solutions to (0.1),which is based on the notion and reduction of measure-valued entropy solutions; and we finallyestablish the hydrodynamic limit for a ZRP with discontinuous speed-parameter governed by anentropy solution to (0.2). Introduction
We are concerned with the following class of scalar hyperbolic conservation laws with discon-tinuous fluxes: ∂ t ρ + ∂ x F ( x, ρ ( t, x )) = 0 (1.1)and with initial data: ρ | t =0 = ρ ( x ) , (1.2)where F ( · , ρ ) is continuous except on a set of measure zero.The main feature of (1.1) is the discontinuity of the flux function in the space variable x .This feature causes new important difficulties in conservation laws. Kruzkov’s approach in [18]for the L -contraction does not apply; entropy solutions even for the Riemann problem of (1.1)are not unique under the classical entropy conditions; several admissibility criteria have beenproposed in [1, 3, 8, 15, 17] and the references cited therein. In particular, a uniqueness theoremwas established in Baiti-Jenssen [3] when F ( x, · ) is monotone and Audusse-Perthame [1] for moregeneral flux functions that especially include non-monotone functions F ( x, · ) in (1.1) under theirnotion. However, the existence of entropy solutions for the non-monotone case under the notion Mathematics Subject Classification.
Key words and phrases. hyperbolic conservation laws, discontinuous flux functions, measure-valued, entropysolutions, entropy conditions, uniqueness, hydrodynamic limits, microscopic, particle systems, zero range process,process of misanthropes, compactness framework. of Audusse-Perthame [1] has not been established, and the entropy conditions proposed in theliterature in general are not equivalent.On the other hand, in statistical mechanics, some microscopic interacting particle systems withdiscontinuous speed parameter λ ( x ), in the hydrodynamic limit, formally lead to scalar hyperbolicconservation laws with discontinuous flux of the form ∂ t ρ + ∂ x ( λ ( x ) h ( ρ )) = 0 (1.3)and with initial data (1.2), where λ ( x ) is continuous except on a set of measure zero and h ( ρ )is Lipschitz continuous. Equation (1.3) is equivalent to the following 2 × (cid:26) ∂ t ρ + ∂ x ( λh ( ρ )) = 0 ,∂ t λ = 0 . (1.4)In particular, when h ( ρ ) is not strictly monotone, system (1.4) is nonstrictly hyperbolic, one of themain difficulties in conservation laws (cf. [5, 7]). The natural question is which entropy solutionthe hydrodynamic limit selects, thereby leading to a suitable, physical relevant notion of entropysolutions of this class of conservation laws. This paper is a first step and provides an answer to thisquestion for a family of discontinuous flux functions via an interacting particle system, namely,the attractive zero range process (ZRP). This ZRP leads to a conservation law of the form (1.3)with λ ( x ) > h ( ρ ) being monotone in ρ , and its hydrodynamic limit naturally gives rise toan entropy condition of the type described in [1, 3] in the formal level.Motivated by the hydrodynamic limit of the ZRP, in this paper, we adopt the notion of entropysolutions for a class of conservation laws with discontinuous flux functions, including the non-monotone case in the sense of Audusse-Perthame [1], and establish the existence of such an entropysolution via the method of compensated compactness in Section 3. This completes the well-posedness by combining a uniqueness result established in [1] for this class of conservation lawsunder the notion of entropy solutions.In order to establish the hydrodynamic limit of large particle systems and the convergence ofother approximate solutions to (1.1) rigorously, we establish another compactness framework for(1.1)–(1.2) in Section 2. This mathematical framework is based on the notion and reduction ofmeasure-valued entropy solutions developed in Section 2, which is also applied for another proofof the existence of entropy solutions for the monotone case in Section 3.In Section 4, we establish the hydrodynamic limit for a ZRP with discontinuous speed-parameter λ ( x ) governed by the unique entropy solution of the Cauchy problem (1.2)–(1.3).2. Notion and Reduction of measure-valued entropy solutions
In this section, we first develop the notion of measure-valued entropy solutions and establishtheir reduction to entropy solutions in L ∞ (provided that they exist) of the Cauchy problem(1.1)–(1.2) satisfying(H1) F ( x, ρ ) is continuous at all points of ( R \N ) × R with N a closed set of measure zero;(H2) ∃ continuous functions f, g such that, for any x ∈ R and large ρ , f ( ρ ) ≤ | F ( x, ρ ) | ≤ g ( ρ )with f ( ρ ) ≥ f ( ±∞ ) = ∞ ;(H3) There exists a function ρ m ( x ) from R to R and a constant M such that, for x ∈ R \N , F ( x, ρ ) is a locally Lipschitz, one to one function from ( −∞ , ρ m ] and [ ρ m , ∞ ) to [ M , ∞ )(or ( −∞ , M ]) with F ( x, ρ m ( x )) = M ;or (H3’) For x ∈ R \N , F ( x, · ) is a locally Lipschitz, one to one function from R to R .One example of the flux function satisfying (H1)–(H2) and (H3) or (H3’) is F ( x, ρ ) = λ ( x ) h ( ρ ) , (2.1) YPERBOLIC CONSERVATION LAWS AND HYDRODYNAMIC LIMIT FOR PARTICLE SYSTEMS 3 where λ ( x ) is continuous in x ∈ R with 0 < λ ≤ λ ( x ) ≤ λ < ∞ for some constants λ and λ , except on a closed set N of measure zero, h ( ρ ) is locally Lipschitz and is either monotone orconvex (or concave) with h ( ρ m ) = 0 for some ρ m in which case M = 0.It is easy to check that, if the flux function F ( x, ρ ) satisfies (H1)–(H3), then, for any constant α ∈ [ M , ∞ ) (or α ∈ ( −∞ , M ]), there are two steady-state solutions m + α from R to [ ρ m ( x ) , ∞ )and m − α from R to ( −∞ , ρ m ( x )] of (1.1) such that F ( x, m ± α ( x )) = α. (2.2)In the case (H1)–(H2) and (H3’), m + α ( x ) = m − α ( x ) which is even simpler.2.1. Notion of measure-valued entropy solutions.
First, the notion of entropy solutions in L ∞ introduced in Audusse-Perthame [1] and Baiti-Jenssen [3] can be further formulated into thefollowing. Definition 2.1 (Notion of entropy solutions in L ∞ ) . We say that an L ∞ function ρ : R := R + × R R is an entropy solution of (1.1) – (1.2) provided that, for each α ∈ [ M , ∞ ) (or α ∈ ( −∞ , M ] ) and the corresponding two steady-state solutions m ± α ( x ) of (1.1) , Z (cid:16) | ρ ( t, x ) − m ± α ( x ) | ∂ t J + sgn ( ρ ( t, x ) − m ± α ( x )) (cid:0) F ( x, ρ ( t, x )) − α (cid:1) ∂ x J (cid:17) dtdx + Z | ρ ( x ) − m ± α ( x ) | J (0 , x ) dx ≥ for any test function J : R R + . It is easy to see that any entropy solution is a weak solution of (1.1)–(1.2) by choosing α suchthat m + α ( x ) ≥ k ρ k L ∞ and m − α ( x ) ≤ −k ρ k L ∞ , respectively, for a.e. x ∈ R .From the uniqueness argument in Audusse-Perthame [1] (also see [6]), one can deduce that, forany L >
0, lim t → Z | x |≤ L | ρ ( t, x ) − ρ ( x ) | dx = 0 . (2.4)Following the notion of entropy solutions, we introduce the corresponding notion of measure-valued entropy solutions. We denote by P ( R ) the set of probability measures on R . Definition 2.2 (Notion of measure-valued entropy solutions) . We say that a measurable map π : R → P ( R ) is a measure-valued entropy solution of (1.1) – (1.2) provided that h π ,x ; k i = ρ ( x ) for a.e. x ∈ R and, for each α ∈ [ M , ∞ ) (or α ∈ ( −∞ , M ] ) and the corresponding two steady-state solutions m ± α ( x ) of (1.1) , Z (cid:0) h π t,x ; | k − m ± α ( x ) |i ∂ t J + h π t,x ; sgn ( k − m ± α ( x )) ( F ( x, k ) − α ) i ∂ x J (cid:1) dxdt + Z | ρ ( x ) − m ± α ( x ) | J (0 , x ) dx ≥ for any test function J : R R + . If a measure-valued entropy solution π t,x ( k ) is a Dirac mass with the associated profile ρ ( t, x ),i.e. π t,x ( k ) = δ ρ ( t,x ) ( k ), then ρ ( t, x ) is an entropy solution of (1.1)–(1.2), which is unique as shownin [1].Note that, when the flux function F ( x, ρ ) is locally Lipschitz in both variables ( x, ρ ), one canuse the Kruzkov entropy inequality, instead of (2.5), to formulate the following notion of measure-valued solutions: ∂ t h π t,x ; | k − c |i + ∂ x h π t,x ; sgn( k − c ) ( F ( x, k ) − F ( x, c )) i + h π t,x ; sgn( k − c ) ∂ x F ( x, c ) i ≤ GUI-QIANG CHEN, NADINE EVEN, AND CHRISTIAN KLINGENBERG in the sense of distributions and to establish their reduction as in DiPerna [12]. One of the newfeatures in our formulation (2.5) in Definition 2.2 is that the constant c in (2.6) is replaced bythe steady-state solutions m ± α ( x ) so that the additional third term in (2.6) vanishes, as in [1, 3],and thereby allows the discontinuity of the flux functions on a closed set of measure zero formeasure-valued entropy solutions.2.2. Reduction of measure-valued entropy solutions.
In this section we first establish thereduction of measure-valued entropy solutions of (1.1)–(1.2) and prove that any measure-valuedentropy solution π t,x ( k ) in the sense of Definition 2.2 is the Dirac solution such that the associatedprofile ρ ( t, x ) is an entropy solution in the sense of Definition 2.1. That is, our goal is to establishthat, when π ,x ( k ) = δ ρ ( x ) ( k ), π t,x ( k ) = δ ρ ( t,x ) ( k ) , (2.7)where ρ : R → R is the unique entropy solution determined by (2.3). The reduction proof isachieved by two theorems. We start with the first theorem. Theorem 2.1.
Assume ρ : R → R is the unique entropy solution of (1.1) – (1.2) with initialdata ρ ∈ L ∞ ( R ) . Assume that there exists a measure-valued entropy solution π : R → P ( R ) of (1.1) in the sense of Definition with π t,x having a fixed compact support for a.e. ( t, x ) and π ,x ( k ) = δ ρ ( x ) ( k ) for a.e. x ∈ R . Then Z (cid:0) h π t,x ; | k − ρ ( t, x ) |i ∂ t J + h π t,x ; sgn ( k − ρ ( t, x ))( F ( x, k ) − F ( x, ρ ( t, x ))) i ∂ x J (cid:1) dxdt ≥ for any test function J : R R + .Proof. The proof is divided into six steps.
Step 1.
We first rewrite E := ∂ t (cid:10) π t,x ; | k − ρ ( t, x ) | (cid:11) + ∂ x (cid:10) π t,x ; sgn ( k − ρ ( t, x )) (cid:0) F ( x, k ) − F ( x, ρ ( t, x )) (cid:1)(cid:11) in the entropy inequality (2.8). We notice the following: • Under the assumption (H3’), F ( x, ρ ( t, x )) is continuous in x a.e. Then we can define afunction ˜ ρ ( s, y, x ) for a.e. ( s, y, x ) ∈ R + × R such that, for fixed ( s, y ), F ( x, ˜ ρ ( s, y, x )) := F ( x, m F ( y,ρ ( s,y )) ( x )) = F ( y, ρ ( s, y )) , (2.9)where the last equality follows from (2.2). Thus, we define β ( s, y ) := F ( y, ρ ( s, y )) so that ˜ ρ ( s, y, x ) = m β ( s,y ) ( x ) . • For the case (H3), we define the sign of the difference between the tilda function and ρ m ( y )to be the same as the sign of the corresponding solution. Since ρ m ( y ) is the minimum (ormaximum) point of the flux function with F ( y, ρ m ( y )) = M , then, for˜ ρ ( s, y, x ) := m + β ( s,y ) ( x )sgn + ( ρ ( s, y ) − ρ m ( y )) + m − β ( s,y ) ( x )sgn − ( ρ ( s, y ) − ρ m ( y )) , (2.10)we have as in (2.9) F ( x, ˜ ρ ( s, y, x )) := F ( x, m + β ( s,y ) ( x )sgn + ( ρ ( s, y ) − ρ m ( y )) + m − β ( s,y ) ( x )sgn − ( ρ ( s, y ) − ρ m ( y ))= F ( y, ρ ( s, y )) = β ( s, y ) . With these notations, we set˜ E := ∂ t (cid:10) π t,x ; | k − ˜ ρ ( s, y, x ) | (cid:11) + ∂ x (cid:10) π t,x ; sgn( k − ˜ ρ ( s, y, x )) (cid:0) F ( x, k ) − β ( s, y ) (cid:1)(cid:11) . (2.11)Then, to obtain the inequality E ≤
0, it suffices to show that lim x → y ˜ E = E . Step 2.
We now show that˜ ρ ( s, y, x ) x −→ y −→ ˜ ρ ( s, y, y ) = ρ ( s, y ) for a.e. ( s, y ) ∈ R . (2.12) YPERBOLIC CONSERVATION LAWS AND HYDRODYNAMIC LIMIT FOR PARTICLE SYSTEMS 5
For the case (H3’), since the flux function is continuous outside a negligible set N , then, for x ∈ R \N , F ( x, ˜ ρ ( s, y, y )) x → y −→ F ( y, ˜ ρ ( s, y, y )) . On the other hand, we have F ( y, ˜ ρ ( s, y, y )) = F ( x, ˜ ρ ( s, y, x )). Therefore, we have F ( x, ˜ ρ ( s, y, x )) − F ( x, ˜ ρ ( s, y, y )) x → y −→ , and (2.12) is a consequence of the fact that F ( x, · ) is a one to one function. The case (H3) is clearfrom the definition of ˜ ρ ( s, y, x ) in (2.10). Step 3.
With Steps 1–2, to achieve inequality (2.8), it suffices by choosing α = β ( s, y ) in (2.5)to show the following inequality:lim τ,ω → Z ∂ t J ( t, x ) ¯ H τ ( t − s ) H ω ( x − y ) h π t,x ; | k − ˜ ρ ( s, y, x ) |i dtdxdsdy + lim τ,ω → Z ∂ x J ( t, x ) ¯ H τ ( t − s ) H ω ( x − y ) h π t,x ; sgn ( k − ˜ ρ ( s, y, x )) (cid:0) F ( x, k ) − β ( s, y ) (cid:1) i dtdxdsdy + lim τ,ω → Z J (0 , x ) ¯ H τ ( − s ) H ω ( x − y ) | ρ ( x ) − ˜ ρ ( s, y, x ) | dxdsdy ≥ J ∈ C ∞ ( R ) and verify that ˜ ρ ( s, y, x ) can be replaced by ρ ( t, x ) in the limitas τ, ω →
0. Here the two families of functions ¯ H τ , H ω ∈ C ∞ ( R ) are defined as¯ H τ ( z ) = 1 τ ¯ H ( zτ ) and H ω ( z ) = 1 ω H ( zω ) for τ, ω > , for a positive, compactly supported function H ∈ C ∞ ( R ) and a positive function ¯ H ∈ C ∞ ( R )with compact support in ( − , −
1) such that R R H ( z ) dz = 1 and R ( − , − ¯ H ( z ) dz = 1.This can be easily seen by first choosing the test function in (2.5) as J ( t, x ) ¯ H τ ( t − s ) H ω ( x − y ) ≥ s, y ) and then integrating with respect to ( s, y ). We now estimate the three terms of(2.13) in Steps 4–6, respectively. Step 4.
We show that, as τ, ω →
0, the first term converges to Z ∂ t J ( t, x ) h π t,x ; | k − ρ ( t, x ) |i dtdx. Observe that (cid:12)(cid:12)(cid:12) Z ∂ t J ( t, x ) ¯ H τ ( t − s ) H ω ( x − y ) h π t,x ; (cid:12)(cid:12) k − ˜ ρ ( s, y, x ) (cid:12)(cid:12) i dtdxdsdy − Z ∂ t J ( t, x ) ¯ H τ ( t − s ) H ω ( x − y ) h π t,x ; (cid:12)(cid:12) k − ˜ ρ ( s, y, y ) (cid:12)(cid:12) i dtdxdsdy (cid:12)(cid:12)(cid:12) ≤ Z | ∂ t J ( t, x ) | ¯ H τ ( t − s ) (cid:0) Z H ω ( x − y ) (cid:12)(cid:12) ˜ ρ ( s, y, x ) − ˜ ρ ( s, y, y ) (cid:12)(cid:12) dx (cid:1) dtdsdy → ω → , (2.14)by the dominated convergence theorem and the fact that R H ω ( x − y ) (cid:12)(cid:12) ˜ ρ ( s, y, x ) − ˜ ρ ( s, y, y ) (cid:12)(cid:12) dx → ω → s, y ) ∈ R since ˜ ρ ( s, y, x ) x → y −→ ˜ ρ ( s, y, y ) = ρ ( s, y ) by (2.12). Then, to find thelimit of the first term of (2.13), it suffices to compute the limit of Z ∂ t J ( t, x ) ¯ H τ ( t − s ) H ω ( x − y ) h π t,x ; (cid:12)(cid:12) k − ρ ( s, y ) (cid:12)(cid:12) i dtdxdsdy. (2.15)Thus, it suffices to show that ρ ( s, y ) can be replaced by ρ ( t, x ) in (2.15), i.e., as τ, ω → Z ∂ t J ( t, x ) ¯ H τ ( t − s ) H ω ( x − y ) (cid:12)(cid:12) ρ ( t, x ) − ρ ( s, y ) (cid:12)(cid:12) dtdxdsdy = Z ∂ t J ( t, x ) ¯ H ( − r ) H ( − z ) (cid:12)(cid:12) ρ ( t, x ) − ρ ( t + τ r, x + ωz ) (cid:12)(cid:12) dtdxdrdz → . (2.16) GUI-QIANG CHEN, NADINE EVEN, AND CHRISTIAN KLINGENBERG
This is guaranteed by the fact thatlim τ,ω → Z (cid:12)(cid:12) ρ ( t, x ) − ρ ( t + τ r, x + ωz ) (cid:12)(cid:12) dtdx = 0 , and the dominated convergence theorem since all the functions involved are bounded. This impliesthat, in (2.15), we can indeed replace ρ ( s, y ) by ρ ( t, x ) to arrive at the result. Step 5.
We show that the second term of (2.13) converges to Z ∂ x J ( t, x ) h π t,x ; sgn ( k − ρ ( t, x )) (cid:0) F ( x, k ) − β ( t, x ) (cid:1) i dtdx as τ, ω → . The hypothesis (H2) on F ( x, ρ ) implies (cid:12)(cid:12)(cid:12) sgn ( k − ˜ ρ ( s, y, x )) (cid:0) F ( x, k ) − β ( s, y ) (cid:1) − sgn ( k − ˜ ρ ( s, y, y )) (cid:0) F ( x, k ) − F ( x, ˜ ρ ( s, y, y )) (cid:1)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) sgn ( k − ˜ ρ ( s, y, x )) (cid:0) F ( x, k ) − F ( x, ˜ ρ ( s, y, x )) (cid:1) − sgn ( k − ρ ( s, y )) (cid:0) F ( x, k ) − F ( x, ρ ( s, y )) (cid:1)(cid:12)(cid:12)(cid:12) ≤ C | ˜ ρ ( s, y, x ) − ρ ( s, y ) | . Integrating the last expression with respect to x against the function H ω ( x − y ) yields its conver-gence to zero by Step 2 as ω →
0. Therefore, the limit of the second term of (2.13) is the same asthe limit of Z ∂ x J ( t, x ) ¯ H τ ( t − s ) H ω ( x − y ) h π t,x ; sgn ( k − ρ ( s, y )) (cid:0) F ( x, k ) − F ( x, ρ ( s, y )) (cid:1) i dtdxdsdy, and it suffices to prove that, as τ, ω → R ∂ x J ( t, x ) ¯ H τ ( t − s ) H ω ( x − y ) h π t,x ; (cid:12)(cid:12) sgn ( k − ρ ( s, y )) (cid:0) F ( x, k ) − F ( x, ρ ( s, y )) (cid:1) − sgn ( k − ρ ( t, x )) (cid:0) F ( x, k ) − F ( x, ρ ( t, x )) (cid:1)(cid:12)(cid:12) i dtdxdsdy → . Using the Lipschitz property and fact (2.16), we achieve the result for the second term of (2.13).
Step 6.
We now show that the third term of (2.13) converges to zero if τ, ω →
0. Note that Z J (0 , x ) ¯ H τ ( − s ) H ω ( x − y ) (cid:12)(cid:12)(cid:12) | ρ ( x ) − ˜ ρ ( s, y, x ) | − | ρ ( x ) − ˜ ρ ( s, y, y ) | (cid:12)(cid:12)(cid:12) dxdsdy ≤ Z J (0 , x ) ¯ H τ ( − s ) H ω ( x − y ) | ˜ ρ ( s, y, x ) − ˜ ρ ( s, y, y ) | dxdsdy. For the same reason as in the first and the second term of (2.13), the right hand side converges tozero if τ, ω →
0. We therefore next compute the limit as τ, ω → Z J (0 , x ) ¯ H τ ( − s ) H ω ( x − y ) | ρ ( x ) − ρ ( s, y ) | dxdsdy. As before, lim τ,ω → R J (0 , x ) H τ ( − s ) H ω ( x − y ) | ρ ( s, x ) − ρ ( s, y ) | dxdsdy = 0. Therefore, the nextgoal is to compute the limit of Z J (0 , x ) ¯ H τ ( − s ) H ω ( x − y ) | ρ ( x ) − ρ ( s, x ) | dxdsdy = Z J (0 , x ) ¯ H ( − r ) | ρ ( x ) − ρ ( τ r, x ) | dxdr. (2.17)Since all the functions are bounded and supp H ⊂ ( − , − ρ ( t, x ), this converges to 0 as τ → (cid:3) Then Theorem 2.1 yields the L -contraction between the measure-valued entropy solution π t,x and the unique entropy solution ρ ( t, x ) of (1.1)–(1.2). YPERBOLIC CONSERVATION LAWS AND HYDRODYNAMIC LIMIT FOR PARTICLE SYSTEMS 7
Theorem 2.2 ( L -contraction) . The function R h π t,x ; | k − ρ ( t, x ) |i dx is non-increasing in t > ,which implies π t,x ( k ) = δ ρ ( t,x ) ( k ) when π ,x ( k ) = δ ρ ( x ) ( k ) for a.e. x ∈ R .Proof. In expression (2.8), we choose the test function as the product test function J j ( t ) H ( x ), with J j ( t ) converging to the indicator function [ t ,t ] ( t ) as j → ∞ for t > t ≥
0. Then (2.8) is equalto Z H ( x ) h π t ,x ( k ); | k − ρ ( t , x ) |i dx − Z H ( x ) h π t ,x ( k ); | k − ρ ( t , x ) |i dx + Z t t Z H ′ ( x ) h π t,x ( k ); sgn ( k − ρ ( t, x )) (cid:0) F ( x, k ) − β ( t, x ) (cid:1) i dxdt ≥ . (2.18)In (2.18), we choose H ( x ) = e − γ √ | x | χ ( xN ) , γ, N > , for χ ∈ C ∞ ( − ,
2) with χ ( x ) = 1 when x ∈ [ − ,
1] and χ ( x ) ≥
0. Letting N → ∞ first and γ → t > t ≥ Z h π t ,x ; | k − ρ ( t , x ) |i dx − Z h π t ,x ; | k − ρ ( t , x ) |i dx ≤ . In particular, when t = t > , t →
0, then π ,x ( k ) = δ ρ ( x ) ( k ) implies Z h π t,x ; | k − ρ ( t, x ) |i dx ≤ π t,x ( k ) = δ ρ ( t,x ) ( k ) for any t > (cid:3) Existence of entropy solutions
In this section, we establish the existence of entropy solutions (1.1)–(1.2) in the sense of Defi-nition 2.1, as required for the reduction of measure-valued entropy solutions. More precisely, foreach fixed ε > ρ ε denotes the unique Kruzkov solution of (1.1)–(1.2) in the sense (3.3), wherethe flux function depends smoothly on the space variable x ; then it is shown that the sequence ρ ε converges to an entropy solution of (1.1)–(1.2).3.1. Existence of entropy solutions when F is smooth. Define F ε ( x, ρ ) the standard molli-fication of F ( x, ρ ) in x ∈ R : F ε ( x, ρ ) := ( F ( · , ρ ) ∗ θ ε )( x ) → F ( x, ρ ) a.e. as ε → , (3.1)with θ ε ( x ) := θ ( xε ) , θ ( x ) ≥ , supp θ ( x ) ⊂ [ − , R − θ ( x ) dx = 1. For fixed ε >
0, considerthe following Cauchy problem: ( ∂ t ρ + ∂ x F ε ( x, ρ ) = 0 ,ρ | t =0 = ρ ( x ) ≥ . (3.2)Kruzkov’s result in [18] indicates that there exists a unique solution ρ ε of (3.2) satisfying theKruzkov entropy inequality: ∂ t | ρ ε ( t, x ) − c | + ∂ x (cid:0) sgn ( ρ ε ( t, x ) − c )( F ε ( x, ρ ε ( t, x )) − F ε ( x, c )) (cid:1) + ( sgn ( ρ ε ( t, x ) − c ) ∂ x F ε ( x, c ) ≤ ρ ε also satisfies (2.3). Proposition 3.1.
Let ρ ε ( t, x ) be a solution of the Cauchy problem (3.2) satisfying the Kruzkoventropy inequality (3.3) . Then ρ ε ( t, x ) also satisfies the entropy inequality (2.3) with steady-statesolutions m ± α = m ε, ± α . GUI-QIANG CHEN, NADINE EVEN, AND CHRISTIAN KLINGENBERG
Proof.
In (3.3), we choose the constant c = m ε, ± α ( y ) for any α ∈ [ M , ∞ ) (or α ∈ ( −∞ , M ]),integrate against a test function J ω ( t, x, y ) := J ( t, x + y ) H ω ( x − y ) with H ω as in the proof ofTheorem 2.1, integrate by parts in the term involving H ′ ω ( x − y ) with respect to dy , and observethat ( ∂ x + ∂ y ) J ω ( t, x, y ) = ∂ x J ( t, x + y ) H ω ( x − y ) to obtain from (3.3) that Z (cid:12)(cid:12) ρ ε ( t, x ) − m ε, ± α ( y ) (cid:12)(cid:12) H ω ( x − y ) ∂ t J dtdxdy + Z sgn (cid:0) ρ ε ( t, x ) − m ε, ± α ( y ) (cid:1) (cid:0) F ε ( x, ρ ε ( t, x )) − F ε ( x, m ε, ± α ( y )) (cid:1) H ω ( x − y ) ∂ y J dtdxdy − Z sgn (cid:0) ρ ε ( t, x ) − m ε, ± α ( y ) (cid:1) ∂ y F ε ( x, m ε, ± α ( y )) H ω ( x − y ) J dtdxdy − Z sgn (cid:0) ρ ε ( t, x ) − m ε, ± α ( y ) (cid:1) ∂ x F ε ( x, m εα ( y )) H ω ( x − y ) J dtdxdy + Z (cid:12)(cid:12) ρ ε (0 , x ) − m ε, ± α ( y ) (cid:12)(cid:12) H ω ( x − y ) J (0 , x + y dxdy ≥ , (3.4)where we have used that Z (cid:0) F ε ( x, ρ ε ( t, x )) − F ε ( x, m ε, ± α ( y )) (cid:1) H ω ( x − y ) J d y (cid:0) sgn( ρ ε ( t, x ) − m ε, ± α ( y )) (cid:1) dxdt = 0 . As in the proof of Theorem 2.1, we can replace ρ ε ( t, x ) by ρ ε ( t, y ) in the first term as ω → ρ ε (0 , x ) by ρ ε (0 , y ) in the last term as ω → Z sgn (cid:0) ρ ε ( t, x ) − m ε, ± α ( y ) (cid:1) (cid:0) F ε ( x, ρ ε ( t, x )) − F ε ( x, m ε, ± α ( y )) (cid:1) H ω ( x − y ) ∂ y J dtdxdy.
By the hypothesis (H3) on the flux function, F ε ( · , ρ ) is a Lipschitz function from ( −∞ , ρ m ] and[ ρ m , ∞ ) to [ M , ∞ ) (or ( −∞ , M ]), which implies (cid:12)(cid:12) sgn ( ρ ε ( t, x ) − ρ ε ( t, y )) (cid:0) F ( x, ρ ε ( t, x )) − F ( x, ρ ε ( t, y )) (cid:1)(cid:12)(cid:12) ≤ C | ρ ε ( t, x ) − ρ ε ( t, y ) | . One can show in a similar way as in the first term thatlim ω → Z | ρ ε ( t, x ) − ρ ε ( t, y ) | H ω ( x − y ) ∂ y J dtdxdy = 0 . This means that, in the second term of (3.4), one can replace F ( x, ρ ε ( t, x )) by F ( x, ρ ε ( t, y )). Since F ε is also a smooth function with respect to the first variable, the second term converges to Z sgn (cid:0) ρ ε ( t, y ) − m ε, ± α ( y ) (cid:1) (cid:0) F ε ( y, ρ ε ( t, y )) − α (cid:1) ∂ y J ( t, y ) dtdy. In the third and fourth term in (3.4), for z ∈ R , we havelim ω → Z ( ∂ x + ∂ y ) F ε ( x, m ε, ± α ( y )) ω H ω ( ωz ) J ( t, y + 12 ωz ) dz = lim ω → Z ( ∂ x + ∂ y ) F ε ( x, m ε, ± α ( y )) H ( z ) J ( t, y + 12 ωz ) dz = J ( t, y ) ∂ y F ε ( y, m ε, ± α ( y ) (cid:1) = J ( t, y ) ∂ y α = 0 . With these results, as ω →
0, inequality (3.4) becomes (2.3) for F ε ( x, ρ ) = ( F ( · , ρ ) ∗ θ ε )( x ) withsteady-state solutions m ± α = m ε, ± α . (cid:3) Thus we conclude the existence of an entropy solution ρ ε ( t, x ) in the sense of Definition 2.1 foreach F ε with fixed ε > YPERBOLIC CONSERVATION LAWS AND HYDRODYNAMIC LIMIT FOR PARTICLE SYSTEMS 9
Remark . Notice that the sequence of approximate entropy solutions converges to a measure-valued entropy solution as ε →
0: First, since ρ ∈ L ∞ , we find that, for α big enough, m ε, − α ( x ) ≤ ρ ( x ) ≤ m ε, + α ( x ) for all x ∈ R . From [1], it then follows that m ε, − α ( x ) ≤ ρ ε ( t, x ) ≤ m ε, + α ( x ) , which implies the uniform boundedness of ρ ε ( t, x ) in ε since m ε, ± α ( x ) are uniformly bounded in ε . Then there exists a compactly supported family of probability measures π t,x on R (i.e. Youngmeasures; see Tartar [22]) and a subsequence (still denoted by) ρ ε ( t, x ) such that, for any continuousfunction f ( ρ ), f ( ρ ε ( t, x )) ∗ ⇀ h π t,x , f ( k ) i as ε → . (3.5)On the other hand, by Section 3.1, the sequence ρ ε ( t, x ) satisfies the entropy inequality (2.3) for F ε ( x, ρ ) and the steady-state solutions m ± α = m ε, ± α . In particular, we use (3.5) and the definitionof the sequence F ε ( x, ρ ) in (3.1) to conclude that, as ε →
0, the compactly supported family ofprobability measures π t,x satisfies that, for any test function J : R R + , Z (cid:0) h π t,x ; (cid:12)(cid:12) k − m ± α ( x ) (cid:12)(cid:12) i ∂ t J + (cid:10) π t,x ; sgn (cid:0) k − m ± α (cid:1) (cid:0) F ( x, k ) − α (cid:1)(cid:11) ∂ x J (cid:1) dxdt + Z (cid:12)(cid:12) ρ ( x ) − m ± α ( x ) (cid:12)(cid:12) J (0 , x ) dx ≥ . (3.6)Thus, π t,x is a measure-valued entropy solution of (1.1)–(1.2) with compact support for a.e. ( t, x ) ∈ R in the sense of Definition 2.2.3.2. Existence of entropy solutions when F is discontinuous in x . We are now ready tostate the main theorem of this section.
Theorem . Let F ( x, ρ ) be strictly convex or concave in ρ for a.e. x ∈ R and satisfy (H1)–(H3) ,or let F ( x, ρ ) satisfy (H1)–(H2) and (H3’) . Let ρ ( x ) ∈ L ∞ . Then the sequence of entropy solutions ρ ε of the Cauchy problem (3.2) (in the sense of Definition ) converges to the unique entropysolution of the Cauchy problem (1.1) – (1.2) in the sense of Definition .Proof. We consider the two cases separately.For the case (H1)–(H2) and (H3’), that is, the flux function F is monotone in ρ , we apply thecompactness framework established in Section 2 to establish the convergence. For this case, theexistence of entropy solutions has been established in [3]. In Remark 3.1, we have shown that thelimit of the entropy solutions ρ ε is determined by a measure-valued entropy solution π t,x . Then,by Theorems 2.1–2.2, π t,x is the Dirac measure concentrated on the unique entropy solution ρ ( t, x )of (1.1)–(1.2) in the sense of Definition 2.1, which implies the whole sequence converges.For the case (H1)–(H3), since we have not established the existence of an entropy solution, weemploy the compensated compactness method to establish the convergence of the entropy solutionsof the Cauchy problem (3.2), which also yields the existence of a unique entropy solution of theCauchy problem (1.1)–(1.2).From Remark 3.1, we have known that ρ ε is uniformly bounded in L ∞ which implies that thereexists a subsequence ρ ε converging weakly to a compactly supported family of probability measures ν t,x on R + such that, for any function f ( ρ, t, x ) that is continuous in ρ for a.e. ( t, x ), f ( ρ ε ( t, x ) , t, x ) ∗ ⇀ h ν t,x , f ( k, t, x ) i as ε → . (3.7)In particular, ρ ε ( t, x ) ∗ ⇀ h ν t,x , k i =: ρ ( t, x ) ∈ L ∞ . (3.8) Our goal is to prove the strong convergence of ρ ε ( t, x ) to ρ ( t, x ) a.e., equivalently, ν t,x = δ ρ ( t,x ) ,which implies that ρ ( t, x ) is an entropy solution of (1.1)–(1.2), that is, ρ ( t, x ) satisfies the entropyinequality in Definition 2.1.By Section 3.1, we have known that the sequence ρ ε exists and satisfies E ε := ∂ t | ρ ε ( t, x ) − ˆ ρ ε ( s, y, x ) | + ∂ x (cid:0) sgn ( ρ ε ( t, x ) − ˆ ρ ε ( s, y, x )) ( F ε ( x, ρ ε ( t, x )) − γ ( s, y )) (cid:1) ≤ ρ ε ( s, y, x ) := m + ,εγ ( s,y ) ( x )sgn + ( ρ ( s, y ) − ρ m ( y )) + m − ,εγ ( s,y ) ( x )sgn − ( ρ ( s, y ) − ρ m ( y )) . Notice that γ ( s, y ) := F ( y, ρ ( s, y )) is independent of ε . Thus, for fixed ( s, y ), we have the strongconvergence of m ± ,εγ ( s,y ) ( x ) to a steady-state solution m ± γ ( s,y ) ( x ) of (1.1)–(1.2) as ε →
0. In particular, k ˆ ρ ε k L ∞ ≤ M, M independent of ε ;and, for a.e. ( s, y, x ) ∈ R × R ,ˆ ρ ε ( s, y, x ) → ˆ ρ ( s, y, x ) := m + γ ( s,y ) ( x )sgn + ( ρ ( s, y ) − ρ m ( y )) + m − γ ( s,y ) ( x )sgn − ( ρ ( s, y ) − ρ m ( y )) , as ε →
0. By Schwartz’s lemma, E ε is a sequence of measures; by Murat’s lemma [20], E ε isuniformly bounded measure sequence in the measure space. This implies that E ε is compact in W − ,ploc ( R ) for any p ∈ (1 , . (3.9)On the other hand, since the vector field sequence( (cid:12)(cid:12) ρ ε ( t, x ) − m ± ,εγ ( s,y ) ( x ) (cid:12)(cid:12) , sgn (cid:0) ρ ε ( t, x ) − m ± ,εγ ( s,y ) ( x ) (cid:1) ( F ε ( x, ρ ε ( t, x )) − γ ( s, y )))is uniformly bounded in ε for any fixed ( s, y ), it follows that E ε is bounded in W − , ∞ loc ( R ) . (3.10)With (3.9)–(3.10), we obtain by a compact interpolation theorem in [4, 11] that E ε is compact in H − loc ( R ) . (3.11)On the other hand, ∂ t ρ ε + ∂ x F ε ( x, ρ ε ) = 0 which is automatically compact in H − loc ( R ) . (3.12)Moreover, since ˆ ρ ε ( s, y, x ) strongly converges a.e., then we find that, as ε → η ε ( ρ ε , t, x, s, y ) := | ρ ε ( t, x ) − ˆ ρ ε ( s, y, x ) | ∗ ⇀ h ν t,x ( k ); | k − ˆ ρ ( s, y, x ) |i =: h ν t,x ; η ( k, t, x, s, y ) i ,q ε ( ρ ε , t, x, s, y ) := sgn ( ρ ε ( t, x ) − ˆ ρ ε ( s, y, x )) ( F ε ( x, ρ ε ) − γ ( s, y )) ∗ ⇀ h ν t,x ( k ); sgn ( k − ˆ ρ ( s, y, x )) ( F ( x, k ) − γ ( s, y )) i =: h ν t,x ; q ( k, t, x, s, y ) i ,η ε ( ρ ε ( t, x )) := ρ ε ( t, x ) ∗ ⇀ h ν t,x ( k ); k i = ρ ( t, x )=: h ν t,x ; η ( k ) i ,q ε ( ρ ε ( t, x ) , x ) := F ε ( x, ρ ε ) ∗ ⇀ h ν t,x ( k ); F ( x, k ) i := h ν t,x ; q ( k, x ) i , (3.13)and (cid:12)(cid:12)(cid:12)(cid:12) η ( ρ ε ( t, x ) , s, y, x ) q ( ρ ε ( t, x ) , s, y, x ) η ( ρ ε ( t, x )) q ( ρ ε ( t, x ) , x ) (cid:12)(cid:12)(cid:12)(cid:12) ∗ ⇀ (cid:28) ν t,x ; (cid:12)(cid:12)(cid:12)(cid:12) η ( k, s, y, x ) q ( k, s, y, x ) η ( k ) q ( k, x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:29) , (3.14) YPERBOLIC CONSERVATION LAWS AND HYDRODYNAMIC LIMIT FOR PARTICLE SYSTEMS 11 where ( η ( k, t, x, s, y ) , q ( k, t, x, s, y )) = ( | k − ˆ ρ ( s, y, x ) | , sgn ( k − ˆ ρ ( s, y, x ))) ( F ( x, k ) − γ ( s, y ))) , ( η ( k ) , q ( k, x )) = ( k, F ( x, k )) . Together (3.11)–(3.12) with (3.13)–(3.14), we apply the Div-Curl lemma (see Tartar [22] and Murat[19]) to obtain (cid:28) ν t,x ; (cid:12)(cid:12)(cid:12)(cid:12) η ( k, s, y, x ) q ( k, s, y, x ) η ( k ) q ( k, x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:29) = (cid:12)(cid:12)(cid:12)(cid:12) h ν t,x ; η ( k, s, y, x ) i h ν t,x ; q ( k, s, y, x ) ih ν t,x ; η ( k ) i h ν t,x ; q ( k, x ) i (cid:12)(cid:12)(cid:12)(cid:12) for all ( s, y ) , ( t, x ) ∈ R \M with M a set of measure zero in R . Thus, we have h ν t,x ; | k − ˆ ρ ( s, y, x ) | F ( x, k ) − k sgn ( k − ˆ ρ ( s, y, x )) ( F ( x, k ) − γ ( s, y )) i = h ν t,x ; | k − ˆ ρ ( s, y, x ) |i h ν t,x ; F ( x, k ) i − h ν t,x , k i h ν t,x ; sgn ( k − ˆ ρ ( s, y, x ))) ( F ( x, k ) − γ ( s, y )) i . Equivalently, we have h ν t,x ; | k − ˆ ρ ( s, y, x ) | ( F ( x, k ) − h ν t,x ; F ( x, k )) ii− h ν t,x ; ( k − ρ ( t, x ))sgn ( k − ˆ ρ ( s, y, x )) ( F ( x, k ) − F ( y, ρ ( s, y ))) i = 0 . Since this is true for all ( s, y ) and ( t, x ) except on a set M of measure zero, we then choose( s, y ) = ( t, x ) for ( t, x ) ∈ R \M to obtain h ν t,x ; | k − ρ ( t, x ) | ( F ( x, k ) − h ν t,x ; F ( x, k )) ii− h ν t,x ; ( k − ρ ( t, x ))sgn ( k − ρ ( t, x )) ( F ( x, k ) − F ( x, ρ ( t, x ))) i = 0 , that is, h ν t,x ; | k − ρ ( t, x ) |i ( F ( x, ρ ( t, x )) − h ν t,x ; F ( x, k ) i ) = 0 . (3.15)There are two possibilities:When h ν t,x ; | k − ρ ( t, x ) |i = 0, then we have ν t,x ( k ) = δ ρ ( t,x ) ( k ).When h ν t,x ; F ( x, k ) i − F ( x, ρ ( t, x )) = 0, we note that h ν t,x ; F ( x, k ) i − F ( x, ρ ( t, x )) = h ν t,x ; F ( x, k ) − F ( x, ρ ( t, x )) i = h ν t,x ; F ρ ( x, ρ )( k − ρ ) + 12 F ρρ ( x, ξ )( k − ρ ) i = F ρ ( x, ρ ) h ν t,x ; k − ρ i + 12 h ν t,x ; F ρρ ( x, ξ )( k − ρ ) i = 12 h ν t,x ; Z θF ρρ ( x, θρ + (1 − θ ) k ) dθ ( k − ρ ) i . Since F ( x, ρ ) is strictly convex or concave in ρ , we conclude ν t,x ( k ) = δ ρ ( t,x ) ( k ) for ( t, x ) a.e. (3.16)Therefore, we have ρ ε ( t, x ) → ρ ( t, x ) a.e. as ε → . Since the limit is unique via the uniqueness result in [1], the whole sequence ρ ε ( t, x ) stronglyconverges to ρ ( t, x ) a.e. It is easy to check that ρ ( t, x ) is the unique entropy solution of the Cauchyproblem (1.1)–(1.2) in the sense of Definition 2.1. (cid:3) Remark . The conditions on the flux function F ( x, ρ ) in Theorem 3.1 for the non-monotonecase can be relaxed as follows: F ( x, ρ ) satisfies (H1)–(H3) and is convex or concave with L { ρ : F ρρ ( x, ρ ) = 0 } = 0 for a.e. x ∈ R , where L is the one-dimensional Lebesgue measure. Hydrodynamic Limit of a Zero Range Process with DiscontinuousSpeed-Parameter
In Section 2, we have established a compactness framework for approximate solutions via thereduction of measure-valued entropy solutions of (1.1)–(1.2) in the sense of Definition 2.1. In thissection we focus on a microscopic particle system for a Zero Range Process (ZRP) with discontinu-ous speed-parameter λ ( x ). We apply the compactness framework to show the hydrodynamic limitfor the particle system, when the distance between particles tend to zero, to the unique entropysolution of the Cauchy problem ∂ t ρ + ∂ x ( λ ( x ) h ( ρ )) = 0 (4.1)and with initial data: ρ | t =0 = ρ ( x ) ≥ , (4.2)where h ( ρ ) is a monotone function of ρ , and λ ( x ) is continuous in x ∈ R with 0 < λ ≤ λ ( x ) ≤ λ < ∞ for some constants λ and λ , except on a closed set N of measure zero. Then m + α = m − α := m α for α ∈ [0 , ∞ ).Rezakhanlou in [21] first established the hydrodynamic limit of the processus des misanthropes(PdM) with constant speed-parameter. Covert-Rezakhanlou [10] provided a proof of the hydro-dynamic limit of a PdM with nonconstant continuous speed-parameter λ . In both proofs, themost important step is to show an entropy inequality at microscopic level, which then implies the(macroscopic) Kruzkov entropy inequality, when the distance between particles tends to zero, andthereby implies the uniqueness of limit points. In this section, we generalize this to the case whenthe speed-parameter λ has jumps for the attractive Zero Range Process (ZRP). In § § ε = ε ( N ) = N − σ , σ ∈ (0 , N → ∞ .In § Some properties of the microscopic interacting particle system.
We consider a systemof particles with conserved total mass and evolving on a one-dimensional lattice Z according to aMarkovian law. With the Euler scaling factor N , the microscopic particle density is expected toconverge to a deterministic limit as N → ∞ , which is characterized by a solution of a conservationlaw. Under the Euler scaling, N represents the distance between sites. Obviously we have twospace scales: The discrete lattice Z as embedded in R with “vertices” uN and u ∈ Z . In this way,the distances between particles tend to zero if N increases to infinity. Sites of the microscopic scale Z are denoted by the letters u, v and correspond to the points uN , vN in the macroscopic scale R .Points of the macroscopic space scale R are denoted by the letters x, y and correspond to the sites[ xN ], [ yN ] in the microscopic space scale, where [z] is the integer part of z. We denote by η t ( u )the number of particles at time t > u . Then the vector η t = ( η t ( u ) : u ∈ Z ) is called aconfiguration at time t with configuration space NZ .In general, the ZRP can be described as follows: Infinitely many indistinguishable particles aredistributed on a 1-dimensional lattice. Any site of the lattice may be occupied by a finite numberof particles. Associated to a given site u there is an exponential clock with rate λ ε ( uN ) g ( η ( u ))depending on the macroscopic spatial coordinates. Each time the clock rings on the site u , oneof the particles jumps to the site v chosen with probability p ( u, v ). The elementary transitionprobabilities p : Z [0 ,
1] are supposed to be(i) translation invariant: p ( x, y ) = p (0 , y − x ) =: p ( y − x );(ii) normalized: P y p ( x, y ) = 1, p ( x, x ) = 0;(iii) assumed to be of finite range: p ( x, y ) = 0 for | y − x | sufficiently large;(iv) irreducible: p (0 , > YPERBOLIC CONSERVATION LAWS AND HYDRODYNAMIC LIMIT FOR PARTICLE SYSTEMS 13
Without loss of generality, we assume that P z p ( z ) z = γ = 1; otherwise, for γ = 1, we replacethe function h ( ρ ) by h ( ρ ) /γ in the following argument. The rate g : N → R + is a positive,nondecreasing function with g (0) = 0, g (+ ∞ ) = + ∞ , and g ( k ) k → k → ∞ . (4.3)With this description, the Markov process η t is generated by N L Nε f ( η ) = N X u,v λ ε ( uN ) g ( η ( u )) p ( v − u )( f ( η u,v ) − f ( η )) . (4.4)Here N comes from the Euler scaling factor speeding the generator, thus η t denotes a configurationon which this speeded generator N L Nε has acted for time t , and η u,v represents the configuration η where one particle jumped from u to v : η u,v ( w ) = ( η ( w ) if w = u, v,η ( u ) − w = u,η ( v ) + 1 if w = v. For any ε = ε ( N ) > α ≥
0, we define a product measure given by˜ ν Nα ( η ) := Y u Z (cid:0) α/λ ε ( uN ) (cid:1) α η ( u ) ( λ ε ( uN )) η ( u ) g ( η ( u ))! := Y u ˜ ν Nα ( η ( u )) , (4.5)where Z is a partition function equal to Z (cid:0) αλ ε ( uN ) (cid:1) = ∞ X n =0 α n (cid:0) λ ε ( uN ) (cid:1) n g ( n )! . (4.6)Then the expected value of the occupation variable η ( u ) is equal to E ˜ ν Nα [ η ( u )] = αλ ε ( uN ) Z ′ (cid:0) αλ ε ( uN ) (cid:1) Z (cid:0) αλ ε ( uN ) (cid:1) := R (cid:0) αλ ε ( uN ) (cid:1) . Now let h be the inverse function of R to obtain h (cid:0) R (cid:0) αλ ε ( uN ) (cid:1)(cid:1) = αλ ε ( uN ) ⇒ λ ε ( uN ) h (cid:0) E ˜ ν Nα [ η ( u )] (cid:1) = α ⇔ E ˜ ν Nα [ η ( u )] = m α ( uN ) , where m α is a steady-state solution to ∂ t ρ + ∂ x ( λ ε ( x ) h ( ρ )) = 0 . (4.7)Furthermore, it follows that E ˜ ν Nα [ g ( η ( u ))] = h (cid:16) m α ( uN ) (cid:17) . From now on, we set µ Nm α ( η ) = Y u ν m α ( uN ) ( η ( u )) := Y u ˜ ν Nλ ε ( uN ) h ( m α ( un )) ( η ( u )) . (4.8)The important attribute of the ZRP with nonconstant speed-parameter is that the product measure µ Nm α ( η ) is invariant under the generator N L Nε , i.e., Z L Nε ( f ( η )) dµ Nm α ( η ) = 0 . (4.9) As a reasonable initial distribution, we choose the product measure µ N ( η ) associated to a boundeddensity profile defined as follows: For a bounded density profile ρ ≥
0, the probability thatparticles at time t = 0 are distributed with configuration η is equal to µ N ( η ) := Y u Z ( h ( ρ u,N ) /λ ε ( uN )) ( h ( ρ u,N )) η ( u ) ( λ ε ( uN )) η ( u ) g ( η ( u ))! , (4.10)where ρ u,N ≥ N →∞ R | ρ [ Nx ] ,N − ρ ( x ) | dx = 0 for [ N x ] as the integerpart of
N x . With this definition, we say that a sequence of probability measures µ N is associatedto a density profile ρ ≥ N →∞ h µ N ( η ) ; (cid:12)(cid:12) N X u J ( uN ) η ( u ) − Z J ( x ) ρ ( x ) dx (cid:12)(cid:12) i = 0 for every test function J. Furthermore, let µ Nt = S Nt ∗ µ N , (4.11)where S Nt = e tNL Nε is the semigroup corresponding to the generator N L Nε . Then the attractivenessfor two initial measures µ Nρ and µ Nω with profiles ρ t and ω t , respectively, implies that µ Nρ ≤ µ Nω ⇒ µ Nρ t ≤ µ Nω t is satisfied by the assumption that g is a nondecreasing function. Moreover, it is easy to provethat µ ρ ≤ µ ω if ρ ≤ ω . It then follows by attractiveness that, for any constant α such that m α ( x ) ≥ ρ ( x ), we obtain that the inequality µ N ≤ µ Nm α implies S Nt µ N ≤ S Nt µ Nm α = µ Nm α . (4.12)Since our initial distribution has a bounded density profile, then the density profile remains boundedat later time t .The goal in proving the hydrodynamic limit of a ZRP is that, if we start from a configuration η distributed with an initial measure µ N associated to the bounded density profile ρ , then theconfiguration η t at later time t is distributed with the measure µ Nt defined by (4.11) and havingdensity profile ρ ( t, · ), where ρ is the solution of the Cauchy problem (4.1)–(4.2) in the sense ofDefinition 2.1. In other words, our main theorem in this section is the following. Theorem . Let η t be an attractive ZRP with (4.3) initially distributed by the measure µ N associated to a boundeddensity profile ρ : R R + as defined in (4.10) . Let ε = ε ( N ) = N − σ , σ ∈ (0 , . Then, at latertime t , lim N →∞ h µ Nt ( η ); (cid:12)(cid:12)(cid:12) N X u J ( uN ) η t ( u ) − Z J ( x ) ρ ( t, x ) dx (cid:12)(cid:12)(cid:12) i = 0 (4.13) for any test function J : R → R , where ρ is the unique solution of the Cauchy problem (4.1) – (4.2) in the sense of Definition . To achieve this, we have to establish an entropy inequality in the sense of Definition 2.1 atmicroscopic level. This will be done in § ε = ε ( N ) = N − σ , σ ∈ (0 , η t , we may define the empirical measure viewed as a randommeasure on R by χ Nt ( x ) := 1 N X u η t ( u ) δ uN ( x ) . (4.14)Then h χ Nt ( · ) , J ( · ) i = N P u J ( uN ) η t ( u ), and we can rewrite (4.13) bylim N →∞ h µ Nt ( η ); (cid:12)(cid:12) h χ Nt ( · ) , J ( · ) i − Z J ( x ) ρ ( t, x ) dx (cid:12)(cid:12) i = 0 . (4.15) YPERBOLIC CONSERVATION LAWS AND HYDRODYNAMIC LIMIT FOR PARTICLE SYSTEMS 15
The entropy inequality at microscopic level.
The following proposition is essential to-wards the hydrodynamic limit.
Proposition ε = N − σ with σ ∈ (0 ,
1) as N → ∞ ) .Let m εα be the steady-state solutions of (3.2) as defined in (1.3) with F ε ( x, ρ ) = λ ε ( x ) h ( ρ ) . Let η t be the ZRP generated by N L Nε defined by (4.4) and initially distributed by the measure µ N definedby (4.10) . Let η l ( u ) be the average density of particles in large microscopic boxes of size l + 1 andcentered at u : η l ( u ) := 12 l + 1 X | u − v |≤ l η ( v ) . Then, for every test function J : R → R + , lim l →∞ lim N →∞ µ Nt n Z t N X u (cid:16) ∂ s J ( s, uN ) (cid:12)(cid:12) η ls ( u ) − m εα ( uN ) (cid:12)(cid:12) + ∂ x J ( s, uN ) (cid:12)(cid:12) λ ε ( uN ) h ( η ls ( u )) − α (cid:12)(cid:12)(cid:17) ds + 1 N X u J (0 , uN ) (cid:12)(cid:12) η l ( u ) − m εα ( uN ) (cid:12)(cid:12) ≥ − δ o = 1 . (4.16)Inequality (4.16) is the entropy inequality (2.3) with ρ replaced by the average density of particlesin the microscopic boxes of length 2 l + 1. To prove the microscopic entropy inequality, we considerthe coupled process ( η t , ξ t ) generated by N ¯ L Nε , where ¯ L Nε is defined by¯ L Nε f ( η, ξ ) = X u,v p ( v − u ) λ ε ( uN ) min { g ( η ( u )) , g ( ξ ( u )) } ( f ( η u,v , ξ u,v ) − f ( η, ξ ))+ X u,v p ( v − u ) λ ε ( uN ) { g ( η ( u )) − g ( ξ ( u )) } + ( f ( η u,v , ξ ) − f ( η, ξ ))+ X u,v p ( v − u ) λ ε ( uN ) { g ( ξ ( u )) − g ( η ( u )) } + ( f ( η, ξ u,v ) − f ( η, ξ )) . (4.17)Furthermore, denote the initial distribution of ( η t , ξ t ) by ¯ µ N = µ N × µ Nm εα , where µ N is the initialmeasure with density profile ρ defined by (4.10) and µ Nm εα denotes the invariant measure as definedin (4.8).Then, to prove Proposition 4.1, it suffices to prove the following proposition. Proposition . Let ( η t , ξ t ) be the coupled process, starting from ¯ µ N , generated by N ¯ L Nε as definedby (4.17) . Let ¯ µ Nt = ¯ S Nt ∗ ¯ µ N , where ¯ S Nt is the semigroup corresponding to the generator N ¯ L Nε .Then, for every test function J : R → R + and every ε = N − σ with σ ∈ (0 , , lim l →∞ lim N →∞ ¯ µ Nt (cid:26)Z T N X u n ∂ s J ( s, uN ) (cid:12)(cid:12) η ls ( u ) − ξ ls ( u ) (cid:12)(cid:12) + ∂ x J ( s, uN ) λ ε ( uN ) (cid:12)(cid:12) h ( η ls ( u )) − h ( ξ ls ( u )) (cid:12)(cid:12)o ds + 1 N X u J (0 , uN ) (cid:12)(cid:12) η l ( u ) − ξ l ( u ) (cid:12)(cid:12) ≥ − δ (cid:27) = 1 . Recall that a microscopic entropy inequality leading to the Kruzkov entropy inequality hasbeen proved in [10] for the process of PdM with nonconstant but continuous speed-parameter λ ε . Since there does not exist an invariant product measure for a PdM in general such that E µ Nmεα [ ξ ( u )] = m εα ( uN ), to replace the process ξ by the process m εα ( uN ), one has to apply therelative entropy method of Yau [23]. In our case of a space-dependent ZRP, the invariant product measure is available so that wecan approximate the steady-state solution m εα by a process ξ distributed by the invariant measure µ Nm εα for any α ∈ (0 , ∞ ). Then, Proposition 4.1 indeed directly follows from Proposition 4.2.4.3. Proof of Proposition 4.2.
We split the proof in three steps.
Step 1: Lower bound for the martingale.
For a test function J with compact support in R ,define by M Jt the martingale vanishing at time t = 0: M Jt = 1 N X u J ( t, uN ) | η t ( u ) − ξ t ( u ) | − N X u J (0 , uN ) | η ( u ) − ξ ( u ) |− Z t ( ∂ s + N ¯ L Nε ) (cid:0) N X u J ( s, uN ) | η s ( u ) − ξ s ( u ) | (cid:1) ds. Since J has compact support, then, for t large enough, M Jt = − N X u J (0 , uN ) | η ( u ) − ξ ( u ) | − Z t ( ∂ s + N ¯ L Nε ) (cid:16) N X u J ( s, uN ) | η s ( u ) − ξ s ( u ) | (cid:17) ds. We now calculate¯ L Nε | η ( u ) − ξ ( u ) | = X v,w p ( w − v ) λ ε ( vN ) n min { g ( η ( v )) , g ( ξ ( v )) } (cid:0) | η v,w ( u ) − ξ v,w ( u ) | − | η ( u ) − ξ ( u ) | (cid:1) + { g ( η ( v )) − g ( ξ ( v )) } + (cid:0) | η v,w ( u ) − ξ ( u ) | − | η ( u ) − ξ ( u ) | (cid:1) + { g ( ξ ( v )) − g ( η ( v )) } + (cid:0) | η ( u ) − ξ v,w ( u ) | − | η ( u ) − ξ ( u ) | (cid:1)o = X v (cid:0) − G u,v ( η, ξ ) (cid:1)(cid:16) − p ( v − u ) λ ε ( uN ) | g ( η ( u )) − g ( ξ ( u )) | + p ( u − v ) λ ε ( vN ) | g ( η ( v )) − g ( ξ ( v )) | (cid:17) − X v G u,v ( η, ξ ) (cid:16) p ( v − u ) λ ε ( uN ) | g ( η ( u )) − g ( ξ ( u )) | + p ( u − v ) λ ε ( vN ) | g ( η ( v )) − g ( ξ ( v )) | (cid:17) , (4.18)where G u,v is the indicator function that equals to 1 if η and ξ are not ordered, i.e., G u,v ( η, ξ ) = { η ( u ) < ξ ( u ); η ( v ) > ξ ( v ) } + { η ( u ) > ξ ( u ); η ( v ) < ξ ( v ) } . Notice that the second sum is nonpositive. Therefore, plugging in the last expression in themartingale M Jt and then interchange u and v in the last term, we can bound the martingale belowby − N X u J (0 , uN ) | η ( u ) − ξ ( u ) | − Z t N X u ∂ s J ( s, uN ) | η s ( u ) − ξ s ( u ) | ds + Z t X u,v (cid:0) J ( s, uN ) − J ( s, vN ) (cid:1) p ( v − u ) (cid:0) − G u,v ( η s , ξ s ) (cid:1) λ ε ( uN ) | g ( η s ( u )) − g ( ξ s ( u )) | ds. Since the transition probability p is of finite range, i.e. p ( z ) = 0 if | z | > r for some r , then (cid:16) J ( s, uN ) − J ( s, vN ) (cid:17) p ( v − u ) = − N ( v − u ) p ( v − u ) ∂ x J ( s, uN ) + O ( 1 N ) . YPERBOLIC CONSERVATION LAWS AND HYDRODYNAMIC LIMIT FOR PARTICLE SYSTEMS 17
With v = u + y , it then follows that the martingale is bounded below by − Z t N X u n ∂ s J ( s, uN ) (cid:12)(cid:12) η s ( u ) − ξ s ( u ) (cid:12)(cid:12) + ∂ x J ( s, uN ) λ ε ( uN ) τ u (cid:0) X y yp ( y )(1 − G ,y ) (cid:1)(cid:12)(cid:12) g ( η s (0)) − g ( ξ s (0)) (cid:12)(cid:12)o ds − N X u J (0 , uN ) | η ( u ) − ξ ( u ) | + O ( 1 N ) . Step 2:
We show lim N →∞ E ¯ µ Nt (cid:2) (cid:0) M Jt (cid:1) (cid:3) = 0 . (4.19)Recall that N Jt := ( M Jt ) − Z t (cid:16) N ¯ L Nε ( A J ( s, η, ξ )) − A J ( s, η, ξ ) N ¯ L Nε ( A J ( s, η, ξ )) (cid:17) ds is a martingale vanishing at time t = 0, where A J is defined by A J ( t, η, ξ ) = 1 N X u J ( t, uN ) | η t ( u ) − ξ t ( u ) | . Then, by definition, E ¯ µ Ns (cid:2) N Js (cid:3) = 0 for all 0 ≤ s ≤ t . Thus, it suffices to show that the expectationof the integral term of N Jt converges to zero as N → ∞ . In order to prove this, we first find that,by careful calculation, N ¯ L Nε ( A J ( s, η, ξ )) − N A J ( s, η, ξ ) ¯ L Nε ( A J ( s, η, ξ ))= X v,w p ( w − v ) N λ ε ( vN ) n | g ( η s ( v )) − g ( ξ s ( v )) | N (cid:0) − G v,w ( η s , ξ s ) (cid:1)(cid:0) J ( s, wN ) − J ( s, vN ) (cid:1) + | g ( ξ s ( v )) − g ( η s ( v )) | N G v,w ( η s , ξ s ) (cid:0) J ( s, vN ) + J ( s, wN ) (cid:1) o . Since J is a smooth function, the first term of this expression is less O ( g ( CN ) N ) for some constant C depending on the total initial mass and therefore converges to zero as N → ∞ by (4.3). For thesecond term, we know that ( J ( s, vN ) + J ( s, wN )) ≤ k J k ∞ , which implies N ¯ L Nε ( A J ( s, η, ξ )) − N A J ( s, η, ξ ) ¯ L Nε ( A J ( s, η, ξ ))= O ( g ( CN ) N ) + 4 k J k ∞ N X v,w G v,w ( η s , ξ s ) p ( w − v ) λ ε ( vN ) | g ( ξ s ( v )) − g ( η s ( v )) | . Then, to conclude the proof of (4.19), it suffices to show E ¯ µ Nt h Z t (cid:0) X v,w G v,w ( η s , ξ s ) p ( w − v ) λ ε ( vN ) | g ( ξ s ( v )) − g ( η s ( v )) | (cid:1) ds i = O (1) . (4.20)For this, we use the martingale M Jt vanishing at 0 with J ≡
1, that is, M t := 1 N X u | η t ( u ) − ξ t ( u ) | − N X u | η ( u ) − ξ ( u ) | − Z t N X u N ¯ L Nε | η s ( u ) − ξ s ( u ) | ds. By (4.18), the integral term of the martingale is equal to Z t N X u,v N G u,v ( η s , ξ s ) p ( v − u ) λ ε ( uN ) | g ( η s ( u )) − g ( ξ s ( u )) | ds, by interchanging u and v in some terms. Then we find E ¯ µ Nt h Z t X u,v G u,v ( η s , ξ s ) p ( v − u ) λ ε ( uN ) | g ( η s ( u )) − g ( ξ s ( u )) | ds i = E ¯ µ Nt h Z t N X u | η ( u ) − ξ ( u ) | ds i − E ¯ µ Nt h Z t N X u | η t ( u ) − ξ t ( u ) | ds i ≤ E ¯ µ Nt h Z t N X u | η ( u ) − ξ ( u ) | ds i . Since we assumed that both marginals of ¯ µ Nt are bounded, (4.20) follows, which leads to (4.19).With the result of Step 1 and (4.19) and using the Chebichev inequality, we obtain¯ µ Nt n N X u J (0 , uN ) | η ( u ) − ξ ( u ) | + Z t N X u (cid:8) ∂ s J ( s, uN ) (cid:12)(cid:12) η s ( u ) − ξ s ( u ) (cid:12)(cid:12) + ∂ x J ( s, uN ) λ ε ( uN ) τ u (cid:0) X y yp ( y )(1 − G ,y )( η, ξ ) (cid:1)(cid:12)(cid:12) g ( η s (0)) − g ( ξ s (0)) (cid:12)(cid:12)(cid:9) ds + O ( 1 N ) < − δ o ≤ ¯ µ Nt (cid:8) M Jt > δ (cid:9) ≤ ¯ µ Nt (cid:8)(cid:12)(cid:12) M Jt (cid:12)(cid:12) > δ (cid:9) ≤ δ E ¯ µ Nt (cid:2) (cid:0) M Jt (cid:1) (cid:3) , (4.21)which converges to 0 as N → ∞ , for all δ > Step 3.
We next use the following summation by parts formula:
For any bounded function a of η ( · ) with a (0) = 0 and for any smooth test function J : R → R , we obtain that, for any L > , N X | u |≤ LN J ( uN ) a ( η ( u )) = 1 N l + 1) X | u |≤ LN J ( uN ) X | u − v |≤ l a ( η ( v )) + O ( l k J k Lip N ) . (4.22)Since we restrict ε = N − σ , σ ∈ (0 , k λ ε k Lip ≤ C/ε = CN σ and O ( l k λ ε k Lip N ) = O ( lN − σ ) → N → ∞ so that we can use this summation by parts formula (4.22) to replace inequality (4.21)by lim l →∞ lim N →∞ ¯ µ Nt n N X u J (0 , uN ) 12 l + 1 X | z − u |≤ l | η ( z ) − ξ ( z ) | + Z t N X u ∂ s J ( s, uN ) 12 l + 1 X | z − u |≤ l | η s ( z ) − ξ s ( z ) | ds + Z t N X u ∂ x J ( s, uN ) λ ε ( uN ) 12 l + 1 × X | z − u |≤ l τ z (cid:0) X y yp ( y )(1 − G ,y )( η s , ξ s ) (cid:1)(cid:12)(cid:12) g ( η s (0)) − g ( ξ s (0)) (cid:12)(cid:12) ds < − δ o = 0 . (4.23)Notice that, in (4.23), since J is a positive function, by the triangle inequality, we can remove thesum inside the absolute value in the first line. Following the same argument as in [10, 21] (also [9]),since we first set ε = N σ , independent of λ ε ( x ), we can obtain the following one block estimates :lim l →∞ lim N →∞ E ¯ µ Nt n Z t N X u (cid:12)(cid:12)(cid:12) l + 1 X | u − z |≤ l | η s ( z ) − ξ s ( z ) | − | η ls ( u ) − ξ ls ( u ) | (cid:12)(cid:12)(cid:12) ds o = 0 , (4.24) YPERBOLIC CONSERVATION LAWS AND HYDRODYNAMIC LIMIT FOR PARTICLE SYSTEMS 19 and lim l →∞ lim N →∞ E ¯ µ Nt n Z t N X u τ u (cid:12)(cid:12)(cid:12) l + 1 X | z |≤ l τ z (cid:0) X y yp ( y )(1 − G ,y )( η s , ξ s ) (cid:1)(cid:12)(cid:12) g ( η s (0)) − g ( ξ s (0)) (cid:12)(cid:12) − (cid:12)(cid:12) h ( η ls (0)) − h ( ξ ls (0)) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ds o = 0 . (4.25)Combining (4.23) with (4.24)–(4.25), we complete the proof of Proposition 4.2.4.4. Existence of measure-valued entropy solutions.
In this section, we prove that Theorem4.1 implies the existence of a measure-valued entropy solution associated to the configuration η t .We recall the empirical measure χ Nt ( x ) associated to a configuration η t in (4.14). We define theYoung measures associated to η t as follows: π N,lt ( x, k ) := 1 N X u δ uN ( x ) δ η lt ( u ) ( k ) , (4.26)which implies h π N,lt ; J i = N P u J ( uN , η lt ( u )) for any J ∈ C ( R × R + ). If E is the configurationspace, then these two measures are finite positive measures on E and, for any J ∈ C ( R ), they arerelated by the formula h π N,lt ; kJ ( x ) i ≈ h χ Nt ( · ); J ( · ) i . (4.27)Notice that, since there are jumps, the probability measure µ Nt defined by (4.11) must be definedon the Skorohod space D [(0 , ∞ ) , E ], which is the space of right continuous functions with left limitstaking values in E . Then, using the one to one correspondence between the configuration η t andthe empirical measure χ Nt ( · ), the law of χ N with respect to µ Nt will give us a probability measure Q N on the Skorohod space D [(0 , ∞ ) , M + ( R )], for the space M + ( R ) of finite positive measures on R endowed with the weak topology.In the same way, we can associate a probability measure ˜ Q N,l on the space D [(0 , ∞ ) , M + ( R )].With these definitions, we can state the main theorem of this section as follows. Theorem . Let ( µ N ) N ≥ be a sequence ofprobability measures, as defined by (4.10) , associated to a bounded density profile ρ : R → R + .Then the sequence ˜ Q N,l converges, as N → ∞ first and l → ∞ second, to the probability measure ˜ Q concentrated on the measure-valued entropy solution π t,x in the sense of Definition .Proof. In order to be allowed to take the limit points Q and ˜ Q of Q N and ˜ Q N,l respectively, wemust know that the sequences are tight (weakly relatively compact). If Q N,l is weakly relativelycompact, we can take ˜ Q l as a limit point if N → ∞ for each l . Denote by ˜ Q a limit point of˜ Q N,l if N → ∞ first and l → ∞ second. Therefore, the proof consists in two main steps: Thefirst is to show that ˜ Q N,l is weakly relatively compact and the second is to show the uniquenessof limit points. The key point in the proof is that these can be achieved independent of thechoice of mollification λ ε of the discontinuous speed-parameter λ with our choice of the notion ofmeasure-valued entropy solutions.These can be achieved by following exactly the standard argument in [10, 21, 16] since it requiresonly the uniform boundedness of λ ε in the proof. That is, we can conclude the following: Let µ Nt be a measure defined by (4.11). Then(i) The sequence Q N defined above is tight in D [(0 , ∞ ) , M + ( R )] and all its limit points Q are concentrated on weakly continuous paths χ ( t, · );(ii) Similarly, the sequence ˜ Q N,l is tight in D [(0 , ∞ ) , M + ( R × R + )] and all its limit points ˜ Q are concentrated on weakly continuous paths π ( t, · , · ); (iii) For every t ≥ π ( t, x, k ) := π t ( x, k ) is absolutely continuous with respect to the Lebesguemeasure on R , ˜ Q a.s.. That is, ˜ Q a.s. π t ( x, k ) = π t,x ( k ) ⊗ dx ; (4.28)(iv) For every t ∈ [0 , T ], π t,x ( k ) is compactly supported, that is, there exists k > π t,x ([0 , k ] c ) = 0 ∀ ( t, x ) ∈ [0 , T ] × R . (v) π t,x is a measure-valued entropy solution in the sense of Definition 2.2 for any α ∈ [ M , ∞ ),i.e., ∂ t h π t,x ; | k − m α ( x ) |i + ∂ x h π t,x ; | h ( k ) λ ( x ) − α |i ≤ R in the sense of distributions for any α ∈ [ M , ∞ ) or α ∈ ( −∞ , M ].The last result follows from the entropy inequality at microscopic level in Theorem 4.1. Indeed,in terms of the Young measures, the expression (4.16) of Proposition 4.1:lim l →∞ lim N →∞ µ Nt n Z ∞ N X u (cid:8) ∂ t H ( t, uN ) (cid:12)(cid:12) η lt ( u ) − m α ( uN ) (cid:12)(cid:12) + ∂ x H ( t, uN ) (cid:12)(cid:12) λ ( uN ) h ( η lt ( u )) − α (cid:12)(cid:12)(cid:9) dt ≥ − δ o = 1can be restated aslim l →∞ lim N →∞ ˜ Q N,l n Z T (cid:0) h π t ( x, k ); | k − m α ( x ) | ∂ t H ( t, x ) i + h π t ( x, k ); | λ ( x ) h ( k ) − α | ∂ x H ( t, x ) i (cid:1) dt ≥ − δ o = 1 , for every smooth function H : (0 , T ) × R → R + with compact support, any α ∈ [ M , ∞ ) or α ∈ ( −∞ , M ], and any δ >
0. Since ˜ Q is a weak limit point concentrated on absolutely continuousmeasures and since we already proved that π t,x is concentrated on a compact set (and thereforethe integrand is a bounded function), we obtain from the last expression that˜ Q n Z T Z (cid:16) h π t,x ; | k − m α ( x ) |i ∂ t H ( t, x ) + h π t,x ; | λ ( x ) h ( k ) − α |i ∂ x H ( t, x ) (cid:17) dxdt ≥ − δ o = 1 . Letting δ →
0, we have that ˜ Q a.s. (4.29) holds on (0 , T ) × R in the sense of distributions forevery α ∈ [0 , ∞ ). This proves the uniqueness of ˜ Q and thereby concludes the proof of Proposition4.2. (cid:3) Then Theorem 4.1 follows immediately from this result since the measure-valued entropy solu-tion reduces to the Dirac mass concentrated on the unique entropy solution ρ ( t, x ) of (4.1)–(4.2)as we noticed in § Acknowledgments . Gui-Qiang Chen’s research was supported in part by the National ScienceFoundation under Grants DMS-0505473, DMS-0426172, DMS-0244473, and an Alexandre vonHumboldt Foundation Fellowship. The research of Nadine Even and Christian Klingenberg wassupport in part by a German DAAD grant (PPP USA 2005/2006). The first author would liketo thank Professor Willi J¨ager for stimulating discussions and warm hospitality during his visit atthe University of Heidelberg (Germany).
YPERBOLIC CONSERVATION LAWS AND HYDRODYNAMIC LIMIT FOR PARTICLE SYSTEMS 21
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E-mail address : [email protected] N. Even, Department of Mathematics, University of W¨urzburg, Am Hubland, D–97074 W¨urzburg,Germany
E-mail address : [email protected] C. Klingenberg, Department of Mathematics, University of W¨urzburg, Am Hubland, D–97074 W¨urzburg,Germany
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