Initial-boundary value problems for nearly incompressible vector fields, and applications to the Keyfitz and Kranzer system
aa r X i v : . [ m a t h . A P ] O c t Initial-boundary value problems for nearly incompressiblevector fields, and applications to the Keyfitz and Kranzersystem
Anupam Pal Choudhury ∗ , Gianluca Crippa † , Laura V. Spinolo ‡ Abstract
We establish existence and uniqueness results for initial boundary value problems withnearly incompressible vector fields. We then apply our results to establish well-posedness ofthe initial-boundary value problem for the Keyfitz and Kranzer system of conservation lawsin several space dimensions.
The Keyfitz and Kranzer system is a system of conservation laws in several space dimensions thatwas introduced in [24] and takes the form ∂ t U + d X i =1 ∂ x i ( f i ( | U | ) U ) = 0 . The unknown is U : R d → R N and | U | denotes its modulus. Also, for every i = 1 , . . . , d thefunction f i : R → R N is smooth. In this work we establish existence and uniqueness results forthe initial-boundary value problem associated to (1.1).The well-posedness of the Cauchy problem associated to (1.1) was established by Ambrosio,Bouchut and De Lellis in [2, 6] by relying on a strategy suggested by Bressan in [12]. Note that theresults in [2, 6] are one of the very few well-posedness results that apply to systems of conservationlaws in several spaces dimensions. Indeed, establishing either existence or uniqueness for a generalsystem of conservation laws in several space dimensions is presently a completely open problem,see [18, 27, 28] for an extended discussion on this topic.The basic idea underpinning the argument in [2, 6] is that (1.1) can be (formally) writtenas the coupling between a scalar conservation law and a transport equation with very irregularcoefficients. The scalar conservation law is solved by using the foundamental work by Kruˇzkov [25],while the transport equation is handled by relying on Ambrosio’s celebrated extension of theDiPerna-Lions’ well-posendess theory, see [1] and [21], respectively, and [3, 20] for an overview.Note, however, that Ambrosio’s theory [1] does not directly apply to (1.1) owing to a lack ofcontrol on the divergence of the vector fields. In order to tackle this issue, a theory of nearlyincompressible vector fields was developed, see [19] for an extended discussion. Since we will needit in the following, we recall the definition here. Definition 1.1.
Let Ω ⊆ R d be an open set and T > . We say that a vector field b ∈ L ∞ ((0 , T ) × Ω; R d ) is nearly incompressible if there are a density function ρ ∈ L ∞ ((0 , T ) × Ω) and a constant C > such that ∗ APC: Departement Mathematik und Informatik, Universit¨at Basel, Spiegelgasse 1, CH-4051 Basel, Switzerland.Email: [email protected] † GC: Departement Mathematik und Informatik, Universit¨at Basel, Spiegelgasse 1, CH-4051 Basel, Switzerland.Email: [email protected] ‡ LVS: IMATI-CNR, via Ferrata 1, I-27100 Pavia, Italy. Email: [email protected] . ≤ ρ ≤ C, L d +1 − a.e. in (0 , T ) × Ω , andii. the equation ∂ t ρ + div( ρb ) = 0 (1.1) holds in the sense of distributions in (0 , T ) × Ω . The analysis in [2, 6, 19] ensures that, if b ∈ L ∞ ((0 , T ) × R d ; R d ) ∩ BV ((0 , T ) × R d ; R d ) is anearly incompressible vector field with density ρ ∈ BV ((0 , T ) × R d ), then the Cauchy problem (cid:26) ∂ t [ ρu ] + div[ ρbu ] = 0 in (0 , T ) × R d u = u at t = 0is well-posed for every initial datum u ∈ L ∞ ( R d ). This result is pivotal to the proof of thewell-posedness of the Cauchy problem for the Keyfitz and Kranzer system (1.1). See also [4]for applications of nearly incompressible vector fields to the so-called chromatography system ofconservation laws. Note, furthermore, that here and in the following we denote by BV the spaceof functions with bounded variation , see [7] for an extended introduction.The present paper aims at extending the analysis in [2, 6, 19] to the case of initial-boundaryvalue problems. First, we establish the well-posedness of initial-boundary value problems with BV , nearly incompressible vector fields, see Theorem 1.2 below for the precise statement. Indoing so, we rely on well-posedness results for continuity and transport equations with weaklydifferentiable vector fields established in [16], see also [17] for related results. Next, we discuss theapplications to the Keyfitz and Kranzer system (1.1).We now provide a more precise description of our results concerning nearly incompressiblevector fields. We fix an open, bounded set Ω and a nearly incompressible vector field b withdensity ρ and we consider the initial-boundary value problem ∂ t [ ρu ] + div[ ρub ] = 0 in (0 , T ) × Ω u = u at t = 0 u = g on Γ − , (1.2)where Γ − is the part of the boundary (0 , T ) × ∂ Ω where the characteristic lines of the vectorfield ρb are inward pointing . Note that, in general, if b and ρ are only weakly differentiable, onecannot expect that the solution u is a regular function. Since Γ − will in general be negligible,then assigning the value of u on Γ − is in general not possible. In § Theorem 1.2.
Let
T > and Ω ⊆ R d be an open, bounded set with C boundary. Also, let b ∈ BV ((0 , T ) × Ω; R d ) ∩ L ∞ ((0 , T ) × Ω; R d ) be a nearly incompressible vector field with density ρ ∈ BV ((0 , T ) × Ω) ∩ L ∞ ((0 , T ) × Ω) , see Definition 1.1. Further, assume that u ∈ L ∞ (Ω) and g ∈ L ∞ (Γ − ) .Then there is a distributional solution u ∈ L ∞ ((0 , T ) × Ω) to (1.2) satisfying the maximumprinciple k u k L ∞ ≤ max {k u k L ∞ , k g k L ∞ } . (1.3) Also, if u , u ∈ L ∞ ((0 , T ) × Ω) are two different distributional solutions of the same initial-boundary value problem, then ρu = ρu a.e. in (0 , T ) × Ω . Note that the reason why we do not exactly obtain uniqueness of the function u is because ρ can attain the value 0. If ρ is bounded away from 0, then the distributional solution u of (1.2)is unique. Also, we refer to [9, 11, 16, 17, 22, 26] for related results on the well-posedness ofinitial-boundary value problems for continuity and transport equation with weakly differentiablevector fields.In § aper outline In § § § § § § Notation
For the reader’s convenience, we collect here the main notation used in the present paper. • div: the divergence, computed with respect to the x variable only. • Div: the complete divergence, i.e. the divergence computed with respect to the ( t, x ) vari-ables. • Tr(
B, ∂
Λ): the normal trace of the bounded, measure-divergence vector field B on theboundary of the set Λ, see § • ( ρu ) , ρ : the initial datum of the functions ρu and ρ , see Lemma 3.1 and Remark 3.2 . • T ( f ): the trace of the BV function f , see Theorem 2.6. • H s : the s -dimensional Hausdorff measure. • f | E : the restriction of the function f to the set E . • µ x E : the restriction of the measure µ to the measurable set E . • a.e. : almost everywhere. • | µ | : the total variation of the measure µ . • a · b : the (Euclidean) scalar product between a and b . • E : the characteristic function of the measurable set E . • Γ , Γ − , Γ + , Γ : see (3.6). • ~n : the outward pointing, unit normal vector to Γ. In this section, we briefly recall some notions and results that shall be used in the sequel.First, we discuss the notion of normal trace for weakly differentiable vector fields, see [5, 8,13, 14]. Our presentation here closely follows that of [5]. Let Λ ⊆ R N be an open set and let usdenote by M ∞ (Λ), the family of bounded, measure-divergence vector fields. The space M ∞ (Λ),therefore, consists of bounded functions B ∈ L ∞ (Λ; R N ) such that the distributional divergenceof B (denoted by Div B ) is a locally bounded Radon measure on Λ.The normal trace of B ∈ M ∞ (Λ) on the boundary ∂ Λ can be defined as follows.
Definition 2.1.
Let Λ ⊆ R N be an open and bounded set with Lipschitz continuous boundary andlet B ∈ M ∞ (Λ) . The normal trace of B on ∂ Λ is a distribution defined by the identity D Tr(
B, ∂ Λ) , ψ E = Z Λ ∇ ψ · B dy + Z Λ ψ d (Div B ) , ∀ ψ ∈ C ∞ c ( R N ) . (2.1) Here
Div B denotes the distributional divergence of B and is a bounded Radon measure on Λ . B is a smooth vector field, then Tr( B, ∂
Λ) = B · ~n , where ~n denotes the outward pointing, unit normal vector to ∂ Λ.Note, furthermore, that the analysis in [5] shows that the normal trace distribution satisfiesthe following properties.(a) The normal trace distribution is induced by an L ∞ function on ∂ Λ, which we shall continueto refer to as Tr(
B, ∂
Λ). The bounded function Tr(
B, ∂
Λ) satisfies k Tr(
B, ∂ Λ) k L ∞ ( ∂ Λ) ≤ k B k L ∞ (Λ) . (b) Let Σ be a Borel set contained in ∂ Λ ∩ ∂ Λ , and let ~n = ~n on Σ (here ~n , ~n denote theoutward pointing, unit normal vectors to ∂ Λ , ∂ Λ respectively). ThenTr( B, ∂ Λ ) = Tr( B, ∂ Λ ) H N − -a.e. on Σ. (2.2)In the following we will use several times the following renormalization result, which was estab-lished in [5]. Theorem 2.2.
Let B ∈ BV (Λ; R N ) ∩ L ∞ (Λ; R N ) and w ∈ L ∞ (Λ) be such that Div( wB ) is aRadon measure. If Λ ′ ⊂⊂ Λ is an open set with bounded and Lipschitz continuous boundary and h ∈ C ( R ) , then Tr( h ( w ) B, ∂ Λ ′ ) = h (cid:18) Tr( wB, ∂ Λ ′ )Tr( B, ∂ Λ ′ ) (cid:19) Tr(
B, ∂ Λ ′ ) H N − -a.e. on ∂ Λ ′ ,where the ratio Tr( wB, ∂ Λ ′ )Tr( B, ∂ Λ ′ ) is arbitrarily defined at points where the trace Tr(
B, ∂ Λ ′ ) vanishes. We can now introduce the notion of normal trace on a general bounded, Lipschitz continuous,oriented hypersurface Σ ⊆ R N in the following manner. Since Σ is oriented, an orientation of thenormal vector ~n Σ is given. We can then find a domain Λ ⊆ R N such that Σ ⊆ ∂ Λ and thenormal vectors ~n Σ , ~n coincide. Using (2.2), we can then defineTr − ( B, Σ) := Tr(
B, ∂ Λ ) . Similarly, if Λ ⊆ R N is an open set satisfying Σ ⊆ ∂ Λ , and ~n = − ~n Σ , we can defineTr + ( B, Σ) := − Tr(
B, ∂ Λ ) . Furthermore we have the formula(Div B ) x Σ = (cid:16) Tr + ( B, Σ) − Tr − ( B, Σ) (cid:17) H N − x Σ . Thus Tr + and Tr − coincide H N − − a.e. on Σ if and only if Σ is a (Div B )-negligible set.We next recall some results from [5] concerning space continuity. Definition 2.3.
A family of oriented surfaces { Σ r } r ∈ I ⊆ R N (where I ⊆ R is an open interval)is called a family of graphs if there exist • a bounded open set D ⊆ R N − , • a Lipschitz function f : D → R , • a system of coordinates ( x , · · · , x N ) such that the following holds true: For each r ∈ I , we can write Σ r = (cid:8) ( x , · · · , x N ) : f ( x , · · · , x N − ) − x N = r (cid:9) , (2.3) and the orientation of Σ r is determined by the normal ( −∇ f, p |∇ f | .
4e now quote a space continuity result.
Theorem 2.4 (see [5]) . Let B ∈ M ∞ ( R N ) and let { Σ r } r ∈ I be a family of graphs as above. Fora fixed r ∈ I , let us define the functions α , α r : D → R as α ( x , · · · , x N − ) := Tr − ( B, Σ r )( x , · · · , x N − , f ( x , · · · , x N − ) − r ) , and α r ( x , · · · , x N − ) := Tr + ( B, Σ r )( x , · · · , x N − , f ( x , · · · , x N − ) − r ) . (2.4) Then α r ∗ ⇀ α weakly ∗ in L ∞ ( D, L N − x D ) as r → r +0 . We will also need the following result, which was originally established in [16].
Lemma 2.5.
Let Λ ⊆ R N be an open and bounded set with bounded and Lipschitz continuousboundary and let B belong to M ∞ (Λ) . Then the vector field ˜ B ( z ) := (cid:26) B ( z ) z ∈ Λ0 otherwisebelongs to M ∞ ( R N ) . We conclude by recalling some results concerning traces of BV functions and we refer to [7, §
3] for a more extended discussion.
Theorem 2.6.
Let Λ ⊆ R N be an open and bounded set with bounded and Lipschitz continuousboundary. There exists a bounded linear mapping T : BV (Λ) → L ( ∂ Λ; H N − ) (2.5) such that T ( f ) = f | ∂ Λ if f is continuous up to the boundary. Also, Z Λ ∇ ψ · f dy = − Z Λ ψ d (Div f ) + Z ∂ Λ ψ T f · ~n d H N − , (2.6) for all f ∈ BV (Λ) and ψ ∈ C ∞ c ( R N ) . In the above expression, ~n denotes the outward pointing,unit normal vector to ∂ Λ . By comparing (2.1) and (2.6) we conclude thatTr( f, ∂
Λ) = T ( f ) · ~n, for every f ∈ BV (Λ) . (2.7)By combining Theorems 3.9 and 3.88 in [7] we get the following result. Theorem 2.7 ([7]) . Assume Λ ⊆ R N is an open set with bounded and Lipschitz continuousboundary. If f ∈ BV (Λ; R m ) , then there is a sequence { ˜ f m } ⊆ C ∞ (Λ) such that ˜ f m → f strongly in L (Λ; R m ) , T ( ˜ f m ) → T ( f ) strongly in L ( ∂ Λ; R m ) . (2.8) Also, we can choose ˜ f m in such a way that • ˜ f m ≥ if f ≥ , • if f ∈ L ∞ (Λ; R m ) , then k ˜ f m k L ∞ ≤ k f k L ∞ . (2.9)A sketch of the proof of Theorem 2.7 is provided in § Distributional formulation of the problem
In this section, we follow [11, 16] and we provide the distributional formulation of the problem(1.2). We first establish a preliminary result.
Lemma 3.1.
Let Ω ⊆ R d be an open bounded set with C boundary and let T > . We assumethat b ∈ L ∞ ((0 , T ) × Ω; R d ) is a nearly incompressible vector field with density ρ ∈ L ∞ ((0 , T ) × Ω) ,see Definition 1.1. If u ∈ L ∞ ((0 , T ) × Ω) satisfies Z T Z Ω ρu ( ∂ t φ + b · ∇ φ ) dxdt = 0 , ∀ φ ∈ C ∞ c ((0 , T ) × Ω) , (3.1) then there are two unique functions, which we henceforth denote by Tr( ρub ) ∈ L ∞ ((0 , T ) × ∂ Ω) and ( ρu ) ∈ L ∞ (Ω) , that satisfy Z T Z Ω ρu ( ∂ t ψ + b ·∇ ψ ) dxdt = Z T Z ∂ Ω Tr( ρub ) ψ d H d − dt − Z Ω ψ (0 , · )( ρu ) dx, ∀ ψ ∈ C ∞ c ([0 , T ) × R d ) . (3.2) Also, we have the bounds k Tr( ρub ) k L ∞ ((0 ,T ) × ∂ Ω) , k ( ρu ) k L ∞ (Ω) ≤ max {k ρu k L ∞ ((0 ,T ) × Ω) ; k ρub k L ∞ ((0 ,T ) × Ω) } . (3.3) Proof.
First of all, let us note that the uniqueness of such functions follow from the liberty inchoosing the test functions ψ . Therefore it is enough to discuss the existence of the functions withthe above properties. Let us define B ( t, x ) := (cid:26) ( uρ, uρb ) ( t, x ) ∈ (0 , T ) × Ω0 elsewhere in R d +1 . (3.4)Then B ∈ L ∞ ( R d +1 ) and from (3.1), it also follows that (cid:2) Div B x (0 , T ) × Ω (cid:3) = 0. We can nowapply Lemma 2.5 with Λ = (0 , T ) × Ω to conclude that B ∈ M ∞ ( R d +1 ) . Hence B induces theexistence of normal trace on ∂ Λ. LetTr( ρub ) := Tr(
B, ∂ Λ) (cid:12)(cid:12)(cid:12) (0 ,T ) × ∂ Ω , ( ρu ) := − Tr(
B, ∂ Λ) (cid:12)(cid:12)(cid:12) { }× Ω . The identity (3.2) then follows from (2.1) by virtue of the fact that Div B = 0 in (0 , T ) × Ω. Remark 3.2.
We define the vector field P := ( ρ, ρb ) and we point out that P ∈ L ∞ ((0 , T ) × Ω; R d +1 ) since ρ and b are both bounded functions. By introducing the same extension as in (3.4) and usingthe fact that Z T Z Ω ρ ( ∂ t φ + b · ∇ φ ) dxdt = 0 , ∀ φ ∈ C ∞ c ((0 , T ) × Ω) , we can argue as in the proof of the above lemma to establish the existence of unique functions Tr( ρb ) ∈ L ∞ ((0 , T ) × ∂ Ω) and ρ ∈ L ∞ (Ω) defined as Tr( ρb ) := Tr( P, ∂ Λ) (cid:12)(cid:12)(cid:12) (0 ,T ) × ∂ Ω , ρ := − Tr(
P, ∂ Λ) (cid:12)(cid:12)(cid:12) { }× Ω . In this way, we can give a meaning to the normal trace
Tr( ρb ) and to the initial datum ρ . Also,we have the bounds k Tr( ρb ) k L ∞ ((0 ,T ) × ∂ Ω) , k ρ k L ∞ (Ω) ≤ max {k ρ k L ∞ ((0 ,T ) × Ω) ; k ρb k L ∞ ((0 ,T ) × Ω) } . (3.5)We can now introduce the distributional formulation to the problem (1.2) by using Lemma3.1. We introduce the following notation:Γ := (0 , T ) × ∂ Ω , Γ − := { ( t, x ) ∈ Γ : Tr( ρb )( t, x ) < } , Γ + := { ( t, x ) ∈ Γ : Tr( ρb )( t, x ) > } , Γ := { ( t, x ) ∈ Γ : Tr( ρb )( t, x ) = 0 } . (3.6)6 efinition 3.3. Let Ω ⊆ R d be an open bounded set with C boundary and let T > . Let b ∈ L ∞ ((0 , T ) × Ω; R d ) be a nearly incompressible vector field with density ρ , see Definition 1.1.Fix u ∈ L ∞ (Ω) and g ∈ L ∞ (Γ − ) . We say that a function u ∈ L ∞ ((0 , T ) × Ω) is a distributionalsolution of (1.2) if the following conditions are satisfied:i. u satisfies (3.1) ;ii. ( ρu ) = uρ ;iii. Tr( ρub ) = g Tr( ρb ) on the set Γ − . In this section we establish the existence part of Theorem 1.2, namely we prove the existence offunctions u ∈ L ∞ ((0 , T ) × Ω) and w ∈ L ∞ (Γ ∪ Γ + ) such that for every ψ ∈ C ∞ c ([0 , T ) × R d ), Z T Z Ω ρu ( ∂ t ψ + b ·∇ ψ ) dxdt = Z Γ − g Tr( ρb ) ψ d H d − dt + Z Γ + ∪ Γ Tr( ρb ) ψw d H d − dt − Z Ω ρ u ψ (0 , · ) dx. (4.1)We proceed as follows: first, in § § In this section we rely on the analysis in [19, § , T ) × Ω and we recall that by assumption ρ ∈ BV (Λ) ∩ L ∞ (Λ) . We applyTheorem 2.7 and we select a sequence { ˜ ρ m } ⊆ C ∞ (Λ) satisfying (2.8) and (2.9). Next, we set ρ m := 1 m + ˜ ρ m ≥ m . (4.2)We then apply Theorem 2.7 to the function bρ and we set b m := g ( bρ ) m ρ m . (4.3)Owing to Theorem 2.7 we have ρ m → ρ strongly in L ((0 , T ) × Ω) , b m ρ m → bρ strongly in L ((0 , T ) × Ω; R d ) . (4.4)and, by using the identity (2.7),Tr( ρ m ) → Tr( ρ ) strongly in L (Γ) , Tr( ρ m b m ) → Tr( ρb ) strongly in L (Γ) ,ρ m → ρ strongly in L (Ω) . (4.5)Note, furthermore, that k Tr( b m ρ m ) k L ∞ (3.5) ≤ k b m ρ m k L ∞ (2.9) ≤ k bρ k L ∞ . (4.6)In the following, we will use the notationΓ − m := (cid:8) ( t, x ) ∈ Γ : Tr( ρ m b m ) < (cid:9) , Γ + m := (cid:8) ( t, x ) ∈ Γ : Tr( ρ m b m ) > (cid:9) (4.7)Finally, we extend the function g to the whole Γ by setting it equal to 0 outside Γ − and weconstruct two sequences { g m } ⊆ C (Γ) and { u m } ⊆ C ∞ (Ω) such that g m → g strongly in L (Γ) , u m → u strongly in L (Ω) (4.8)7nd k g m k L ∞ ≤ k g k L ∞ , k u m k L ∞ ≤ k u k L ∞ . (4.9)We can now define the function u m as the solution of the initial-boundary value problem ∂ t u m + b m · ∇ u m = 0 on (0 , T ) × Ω u m = u m at t = 0 u m = g m on ˜Γ − m , (4.10)where ˜Γ − m is the subset of Γ such that the characteristic lines of b m starting at a point in ˜Γ − m areentering (0 , T ) × Ω. We recall (4.7) and we point out thatΓ − m ⊆ ˜Γ − m ⊆ (cid:8) ( t, x ) ∈ Γ : b m · ~n ≤ (cid:9) . In the previous expression, ~n denotes as the outward pointing, unit normal vector to ∂ Ω. By usingthe classical method of characteristics (see also [9]) we establish the existence of a solution u m satisfying k u m k ∞ ≤ max {k u m k ∞ , k g m k ∞ } (4.9) ≤ max {k u k ∞ , k g k ∞ } . (4.11)We now introduce the function h m by setting h m := ∂ t ρ m + div( b m ρ m ) (4.12)and by using the equation at the first line of (4.10) we get that ∂ t ( ρ m u m ) + div( b m ρ m u m ) = h m u m . Owing to the Gauss-Green formula, this implies that, for every ψ ∈ C ∞ c ([0 , T ) × R d ), Z T Z Ω ρ m u m [ ∂ t ψ + b m · ∇ ψ ] dxdt + Z T Z Ω h m u m ψ dxdt = − Z Ω ψ (0 , x ) ρ m u m dx − Z T Z ∂ Ω ψu m ρ m b m · ~n d H d − dt = − Z Ω ψ (0 , x ) ρ m u m dx − Z T Z ∂ Ω Γ − m g m ψ Tr( ρ m b m ) d H d − dt − Z T Z ∂ Ω Γ + m u m ψ Tr( ρ m b m ) d H d − dt. (4.13)In the above expression, we have used the notation introduced in (4.7) and the fact that Tr( ρ m b m ) =0 on Γ \ (Γ − m ∪ Γ + m ). Owing to the uniform bound (4.11), there are a subsequence of { u m } (which, to simplify notation,we do not relabel) and a function u ∈ L ∞ ((0 , T ) × Ω such that u m ∗ ⇀ u weakly ∗ in L ∞ ((0 , T ) × Ω). (4.14)The goal of this paragraph is to show that the function u in (4.14) is a distributional solutionof (1.2) by passing to the limit in (4.13). We first introduce a technical lemma. Lemma 4.1.
We can construct the approximating sequences { ρ m } and { b m } in such a way thatthe sequence { h m } defined as in (4.12) satisfies h m → strongly in L ((0 , T ) × Ω) . (4.15)The proof of Lemma 4.1 is deferred to § emma 4.2. Assume that
Tr( ρ m b m ) → Tr( ρb ) strongly in L (Γ) . (4.16) Let Γ − m and Γ + m as in (4.7) and Γ − and Γ + as in (3.6) , respectively. Then, up to subsequences, Γ − m → Γ − + Γ ′ strongly in L (Γ) (4.17) and Γ + m → Γ + + Γ ′′ strongly in L (Γ) , (4.18) where Γ ′ and Γ ′′ are (possibly empty) measurable sets satisfying Γ ′ , Γ ′′ ⊆ Γ . (4.19) Proof of Lemma 4.2.
Owing to (4.16) we have that, up to subsequences, the sequence { Tr( ρ m b m ) } satisfies Tr( ρ m b m )( t, x ) → Tr( ρb )( t, x ) , for L ⊗ H d − -almost every ( t, x ) ∈ Γ . Owing to the Lebesgue Dominated Convergence Theorem, this implies (4.17) and (4.18).We can now pass to the limit in all the terms in (4.13). First, by combining (4.4), (4.11), (4.14)and (4.15) we get that Z T Z Ω ρ m u m [ ∂ t ψ + b m · ∇ ψ ] dxdt + Z T Z Ω h m u m ψ dxdt → Z T Z Ω ρu [ ∂ t ψ + b · ∇ ψ ] dxdt, (4.20)for every ψ ∈ C ∞ c ([0 , T ) × R d ). Also, by combining the second line of (4.5) with (4.8) and (4.9)we arrive at Z Ω ψ (0 , x ) ρ m u m dx → Z Ω ψ (0 , x ) ρ u dx, (4.21)for every ψ ∈ C ∞ c ([0 , T ) × R d ) . Next, we combine (4.5), (4.8), (4.9), (4.17), (4.19) and the factthat Tr( ρb ) = 0 on Γ to get that Z T Z ∂ Ω Γ − m g m ψ Tr( ρ m b m ) d H d − dt → Z T Z ∂ Ω Γ − gψ Tr( ρb ) d H d − dt = Z T Z Γ − gψ Tr( ρb ) d H d − dt, (4.22)for every ψ ∈ C ∞ c ([0 , T ) × Ω; R d ). We are left with the last term in (4.13): first, we denote by u m | Γ the restriction of u m to Γ. Since u m is a smooth function, then k u m | Γ k L ∞ (Γ) ≤ k u m k L ∞ ((0 ,T ) × Ω) (4.11) ≤ max (cid:8) k ¯ u k L ∞ , k ¯ g k L ∞ (cid:9) and hence there is a function w ∈ L ∞ (Γ) such that, up to subsequences, u m | Γ ∗ ⇀ w weakly ∗ in L ∞ (Γ) . (4.23)By combining (4.5), (4.18), (4.23) and the fact that Tr( ρb ) = 0 on Γ we get that Z T Z ∂ Ω Γ + m u m ψ Tr( ρ m b m ) d H d − dt → Z T Z ∂ Ω Γ + wψ Tr( ρb ) d H d − dt = Z Γ + ∪ Γ wψ Tr( ρb ) d H d − dt. (4.24)By combining (4.20), (4.21), (4.22) and (4.24) we get that u satisfies (4.1) and this establishesexistence of a distributional solution of (1.2). 9 .3 Proof of Lemma 4.1 To ensure that (4.15) holds we use the same approximation `a la
Meyers-Serrin as in [7, pp.122-123]. We now recall some details of the construction. First, we fix a countable family of open sets (cid:8) Λ h (cid:9) such thati. Λ h is compactly contained in Λ, for every h ;ii. (cid:8) Λ h (cid:9) is a covering of Λ, namely ∞ [ h =1 Λ h = Λ;iii. every point in Λ is contained in at most 4 sets Λ h .Next, we consider a partition of unity associated to (cid:8) Λ h (cid:9) , namely a countably family of smooth,nonnegative functions { ζ h } such thativ. we have ∞ X h =1 ζ h ≡ h >
0, the support of ζ h is contained in Λ h .Finally, we fix a convolution kernel η : R d +1 → R + and we define η ε by setting η ε ( z ) := 1 ε d +1 η (cid:16) zε (cid:17) For every m > h > ε mh in such a way that ( ρζ h ) ∗ η ε mh is supported in Λ h and furthermore Z T Z Ω | ρζ h − ( ρζ h ) ∗ η ε mh | + | ρ ∂ t ζ h − ( ρ ∂ t ζ h ) ∗ η ε mh | + | ρb ·∇ ζ h − ( ρb ·∇ ζ h ) ∗ η ε mh | dxdt ≤ − h m . (4.26)We then define ˜ ρ m by setting ˜ ρ m := ∞ X h =1 ( ρζ h ) ∗ η ε mh . (4.27)The function ( e ρb ) m is defined analogously. Next, we proceed as in [7, p.123] and we point out that h m (4.12) = ∂ t ρ m + div( ρ m b m ) (4.12) = ∞ X h =1 ( ∂ t ρζ h ) ∗ η ε mh + ∞ X h =1 (div( ρb ) ζ h ) ∗ η ε mh | {z } =0 by (1.1) + ∞ X h =1 ( ρ ∂ t ζ h ) ∗ η ε mh + ∞ X h =1 ( ρb · ∇ ζ h ) ∗ η ε mh (4.25) = ∞ X h =1 ( ρ ∂ t ζ h ) ∗ η ε mh − ρ ∞ X h =1 ∂ t ζ h + ∞ X h =1 ( ρb · ∇ ζ h ) ∗ η ε mh − ρb · ∞ X h =1 ∇ ζ h By using (4.26) we then get that Z T Z Ω | h m | dxdt ≤ ∞ X h =1 − h m = 1 m and this establishes (4.15). 10 Proof Theorem 1.2: comparison principle and uniqueness
In this section we complete the proof of Theorem 1.2. More precisely, we establish the followingcomparison principle.
Lemma 5.1.
Let Ω , b and ρ as in the statement of Theorem 1.2. Assume u and u ∈ L ∞ ((0 , T ) × Ω) are distributional solutions (in the sense of Definition 3.3) of the initial-boundary value prob-lem (1.2) corresponding to the initial and boundary data u ∈ L ∞ (Ω) , g ∈ L ∞ (Γ − ) and u ∈ L ∞ (Ω) , g ∈ L ∞ (Γ − ) , respectively. If u ≥ u and g ≥ g , then ρu ≥ ρu a.e. in (0 , T ) × Ω . (5.1)Note that the uniqueness of ρu , where u is a distributional solution of the initial-boundaryvalue problem (1.2), immediately follows from the above result. Proof of Lemma 5.1.
Let us define the function˜ β ( u ) = (cid:26) u u ≥ u < . In what follows, we shall prove that the identity ρ ˜ β ( u − u ) = 0 holds almost everywhere,whence the comparison principle follows. To see this, we proceed as described below. First, wepoint out that, since the equation at the first line of (1.2) is linear, then u − u is a distributionalsolution of the initial boundary value problem with data u − u , g − g . In particular, for every ψ ∈ C ∞ c ([0 , T ) × R d ) we have Z T Z Ω ρ ( u − u )( ∂ t ψ + b ·∇ ψ ) dxdt = Z T Z ∂ Ω [Tr( ρu b ) − Tr( ρu b )] ψ d H d − dt − Z Ω ψ (0 , · ) ρ ( u − u ) dx (5.2)and Tr( ρu b ) = g Tr( ρb ) , Tr( ρu b ) = g Tr( ρb ) on Γ − . (5.3)Note that (5.2) implies that Z T Z Ω ρ ( u − u )( ∂ t φ + b · ∇ φ ) dxdt = 0 , ∀ φ ∈ C ∞ c ((0 , T ) × Ω) . (5.4)By using [19, Lemma 5.10] (renormalization property inside the domain), we get Z T Z Ω ρ ˜ β ( u − u )( ∂ t φ + b · ∇ φ ) dxdt = 0 , ∀ φ ∈ C ∞ c ((0 , T ) × Ω) . (5.5)We next apply Lemma 3.1 to the function ˜ β ( u − u ) to infer that there are bounded functionsTr( ρ ˜ β ( u − u ) b ) and ( ρ ˜ β ( u − u )) such that, for every ψ ∈ C ∞ c ([0 , T ) × R d ) , we have Z T Z Ω ρ ˜ β ( u − u )( ∂ t ψ + b ·∇ ψ ) dxdt = Z T Z ∂ Ω Tr( ρ ˜ β ( u − u ) b ) ψ d H d − dt − Z Ω ψ (0 , · )( ρ ˜ β ( u − u )) dx. (5.6)We recall (5.2) and we apply Lemma 2.2 (trace renormalization property) with w = u − u , h = ˜ β , B = ( ρ, ρb ), Λ = R d +1 and Λ ′ = (0 , T ) × Ω. We recall that the vector field P is defined by setting P := ( ρ, ρb ) and we get( ρ ˜ β ( u − u )) = − Tr( ˜ β ( u − u ) P, ∂ Λ ′ ) (cid:12)(cid:12)(cid:12) { }× Ω = − ˜ β ( ρ ( u − u )) Tr(
P, ∂ Λ ′ ) (cid:12)(cid:12)(cid:12) { }× Ω Tr(
P, ∂ Λ ′ ) (cid:12)(cid:12)(cid:12) { }× Ω = − ˜ β (cid:18) ρ ( u − u ) ρ (cid:19) ρ = 0 , since u ≥ u (5.7)11ndTr( ρ ˜ β ( u − u ) b ) = Tr( ˜ β ( u − u ) ρ, ∂ Λ ′ ) (cid:12)(cid:12)(cid:12) (0 ,T ) × ∂ Ω = ˜ β Tr(( u − u ) ρ, ∂ Λ ′ ) (cid:12)(cid:12)(cid:12) (0 ,T ) × ∂ Ω Tr(
P, ∂ Λ ′ ) (cid:12)(cid:12)(cid:12) (0 ,T ) × ∂ Ω Tr(
P, ∂ Λ ′ ) (cid:12)(cid:12)(cid:12) (0 ,T ) × ∂ Ω = ˜ β (cid:18) Tr( ρ ( u − u ) b )Tr( ρb ) (cid:19) Tr( ρb ) . By recalling (5.3) and the inequality ¯ g ≥ ¯ g , we conclude thatTr( ρ ˜ β ( u − u ) b ) = 0 on Γ − and, since ˜ β ≥
0, that Tr( ρ ˜ β ( u − u ) b ) ≥ ν ∈ C ∞ c ( R d ) in such a way that ν ≡ ∂ t ν + b · ∇ ν = 0 on (0 , T ) × Ω. (5.9)Next we choose a sequence of functions χ n ∈ C ∞ c ([0 , + ∞ )) that satisfy χ n ≡ , ¯ t ] , χ n ≡ t + 1 n , + ∞ ) , χ ′ n ≤ , and we define ψ n ( t, x ) := χ n ( t ) ν ( x ) , ( t, x ) ∈ [0 , T ) × R d . Note that ψ is smooth, non-negative and compactly supported in [0 , T ) × R d . By combining theidentities (5.6), (5.7) and the inequality (5.8), we get0 ≤ Z T Z Ω ρ ˜ β ( u − u )[ ∂ t ( χ n ν ) + b · ∇ ( χ n ν )] dxdt = Z T Z Ω νρ ˜ β ( u − u ) χ ′ n dxdt + Z T Z Ω χ n ρ ˜ β ( u − u )( ∂ t ν + b · ∇ ν ) dxdt (5.9) = Z T Z Ω νρ χ ′ n ˜ β ( u − u ) dxdt. Passing to the limit as n → + ∞ and noting that χ ′ n → − δ ¯ t as n → ∞ in the sense of distributionsand recalling that ν ≡ Z Ω ρ (¯ t, · ) ˜ β ( u − u )(¯ t, · ) ≤ . Since the above inequality is true for arbitrary ¯ t ∈ [0 , T ], we can conclude that ρ ˜ β ( u − u ) = 0 , for almost every ( t, x ) ⇒ ρu ≥ ρu , for almost every ( t, x ) . (5.10)This concludes the proof of Lemma 5.1. In this section, we discuss some qualitative properties of solutions of the initial-boundary valueproblem (1.2). First, we establish Theorem 6.1, which establishes (weak) stability of solutionswith respect to perturbations in the vector fields and the data. Theorem 6.2 implies that, understronger hypotheses, we can establish strong stability. Finally, Theorem 6.3 establishes spacecontinuity properties. 12 heorem 6.1.
Let
T > and let Ω ⊆ R d be an open and bounded set with C boundary. Assumethat b n , b ∈ BV ((0 , T ) × Ω; R d ) ∩ L ∞ ((0 , T ) × Ω; R d ) , ρ n , ρ ∈ BV ((0 , T ) × Ω) ∩ L ∞ ([0 , T ) × Ω) satisfy ∂ t ρ n + div( b n ρ n ) = 0 ,∂ t ρ + div( bρ ) = 0 , (6.1) in the sense of distributions on (0 , T ) × Ω . Assume furthermore that ≤ ρ n , ρ ≤ C and k b n k ∞ is uniformly bounded , (6.2)( b n , ρ n ) −−−−→ n →∞ ( b, ρ ) strongly in L ((0 , T ) × Ω; R d +1 ) , (6.3) ρ n −−−−→ n →∞ ρ strongly in L (Ω) , (6.4)Tr( ρ n b n ) −−−−→ n →∞ Tr( ρb ) strongly in L (Γ) , (6.5) Let u n ∈ L ∞ ((0 , T ) × Ω) be a distributional solution (in the sense of Definition 3.3) of the initial-boundary value problem ∂ t ( ρ n u n ) + div( ρ n u n b n ) = 0 in (0 , T ) × Ω u n = u n at t = 0 u n = g n on Γ − n (6.6) and u ∈ L ∞ ((0 , T ) × Ω) be a distributional solution of the equation ∂ t ( ρu ) + div( ρub ) = 0 in (0 , T ) × Ω u = u at t = 0 u = g on Γ − . (6.7) If u m , ¯ u ∈ L ∞ (Ω) and g n , ¯ g ∈ L ∞ (Γ) satisfy u n ∗ ⇀ u weak- ∗ in L ∞ (Ω) , (6.8) g n ∗ ⇀ g weak- ∗ in L ∞ (Γ) , (6.9) then ρ n u n ∗ ⇀ ρu weak-* in L ∞ ((0 , T ) × Ω) (6.10) and
Tr( ρ n u n b n ) ∗ ⇀ Tr( ρub ) weak-* in L ∞ (Γ) . (6.11)Note that in the statement of the above theorem g m and g are functions defined on the wholeΓ, although the values of ρ m u m and ρu are only determined by their values on Γ − m and Γ − ,respectively. Proof.
We proceed according to the following steps.
Step 1: we apply Theorem 1.2 and we infer that the function ρ n u n satisfying (6.6) is unique.Also, without loss of generality, we can redefine the function u n on the set { ρ n = 0 } in such away that u n satisfies the maximum principle (1.3). Owing to (6.11), the sequences k u m k L ∞ and k g m k L ∞ are both uniformly bounded and by the maximum principle so is k u m k L ∞ . Also, bycombining (3.3) and (6.2) we infer that the sequence k Tr( ρ n b n u n ) k ∞ is also uniformly bounded.We conclude that, up to subsequences (which we do not label to simplify the notation), we have u n ∗ ⇀ r weak-* in L ∞ ((0 , T ) × Ω) , Tr( ρ n u n b n ) ∗ ⇀ r weak-* in L ∞ (Γ) (6.12)13or some r ∈ L ∞ ((0 , T ) × Ω) and r ∈ L ∞ (Γ). By using (3.1) and (3.2), we get that Z T Z Ω ρr ( ∂ t φ + b · ∇ φ ) dxdt = 0 , ∀ φ ∈ C ∞ c ((0 , T ) × Ω) , (6.13)and Z T Z Ω ρr ( ∂ t ψ + b ∇ ψ ) dxdt = Z T Z ∂ Ω r ψ d H d − dt − Z Ω ψ (0 , · ) ρ u dx, ∀ ψ ∈ C ∞ c ([0 , T ) × R d ) . (6.14)From Lemma 3.1, it also follows that r = Tr( ρr b ) . (6.15)Assume for the time being that we have established the equality r = g Tr( ρb ) , on Γ − , (6.16)then by recalling (6.15) and the uniqueness part in Theorem 1.2 we conclude that r = ρu and r = Tr( ρbu ). Owing to (6.12), this concludes the proof of the theorem. Step 2: we establish (6.16). First, we decompose Tr( ρ m u m b m ) asTr( ρ n u n b n ) = Tr( ρ n u n b n ) Γ − n + Tr( ρ n u n b n ) Γ + n + Tr( ρ n u n b n ) Γ n = g n Tr( ρ n b n ) Γ − n + Tr( ρ n u n b n ) Γ + n + Tr( ρ n u n b n ) Γ n , (6.17)where Γ − n , Γ + n and Γ n are defined as in (3.6). By using Lemma 2.2 (trace renormalization), onecould actually prove that the last term in the above expression is actually 0. This is actuallynot needed here. Indeed, it suffices to recall (6.5) and Lemma 4.2 and point out that by combin-ing (4.17) and (4.18) we get Γ n → Γ − Γ ′ − Γ ′′ . (6.18)Next, we recall that the sequence k Tr( ρ n u n b n ) k L ∞ is uniformly bounded owing to the uniformbounds on k ρ n k L ∞ and k u n k L ∞ . By recalling (6.9), we conclude that g n Tr( ρ n b n ) Γ − n ∗ ⇀ g Tr( ρb ) (cid:16) Γ − + Γ ′ (cid:17) weak-* in L ∞ (Γ). (6.19)By recalling that Γ ′ ⊆ Γ we get that Tr( ρb ) Γ ′ = 0. We now pass to the weak star limit in (6.17)and using (4.17), (4.18), (6.12), (6.9) and (6.19) we get r = g Tr( ρb ) Γ − + r (cid:16) Γ + + Γ ′ (cid:17) + r (cid:16) Γ − Γ ′ − Γ ′′ (cid:17) , (6.20)which owing to the propertiesΓ − ∩ Γ = ∅ , Γ − ∩ Γ ′ = ∅ , Γ − ∩ Γ ′′ = ∅ implies (6.16). This concludes the proof Theorem 6.1. Theorem 6.2.
Under the same assumptions as in Theorem 6.1, if we furthermore assume that u n −−−−→ n →∞ u strongly in L (Ω) , (6.21) g n −−−−→ n →∞ g strongly in L (Γ) , (6.22) then we get ρ n u n −−−−→ n →∞ ρu strongly in L ((0 , T ) × Ω) , Tr( ρ n u n b n ) −−−−→ n →∞ Tr( ρub ) strongly in L (Γ) . (6.23)14 roof. First, we point out that the first equation in (6.11) implies that ρ n u m ⇀ρu weakly in L ((0 , T ) × Ω) . (6.24)Next, by using Lemma 2.2 (trace-renormalization property), we get that ρ m u n and ρu satisfy (inthe sense of distributions) ∂ t ( ρ n u n ) + div( ρ n u n b n ) = 0 in (0 , T ) × Ω u n = u n at t = 0 u n = g n on Γ − n , and ∂ t ( ρu ) + div( ρu b ) = 0 in (0 , T ) × Ω u = u at t = 0 u = g on Γ − , respectively. Also, by combinig (6.8),(6.9), (6.21) and (6.22), we get that u n ∗ ⇀ u weak- ∗ in L ∞ (Ω) , g n ∗ ⇀ g weak- ∗ in L ∞ (Γ)and by applying Theorem 6.1 to ρ m u m we conclude that ρ m u m ∗ ⇀ ρu weak- ∗ in L ∞ ((0 , T ) × Ω)and that Tr( ρ n u n b n ) ∗ ⇀ Tr( ρu b ) weak- ∗ in L ∞ (Γ) . (6.25)Since the sequence k ρ m k L ∞ is uniformly bounded, then by recalling (6.3) we get ρ m u m ∗ ⇀ ρ u weak- ∗ in L ∞ ((0 , T ) × Ω)and hence ρ m u m ⇀ρ u weakly in L ((0 , T ) × Ω) . (6.26)By combining (6.24) and (6.26) we get that ρ m u m −→ ρu strongly in L ((0 , T ) × Ω) and thisimplies the first convergence in (6.23).Next, we establish the second convergence in L ((0 , T ) × Ω). Since Γ is a set of finite measure,from (6.11) and (6.25) we can infer thatTr( ρ n u n b n ) ⇀ Tr( ρub ) weakly in L (Γ) , Tr( ρ n u n b n ) ⇀ Tr( ρu b ) weakly in L (Γ) . (6.27)By using the uniform bounds for k Tr( ρ n b n ) k ∞ , we infer from the L convergence of Tr( ρ n b n ) toTr( ρb ) that Tr( ρ n b n ) −−−−→ n →∞ Tr( ρb ) strongly in L (Γ) . (6.28)Next, we apply Lemma 2.2 (trace renormalization property) and we get that[Tr( ρ n u n b n )] = (cid:20) Tr( ρ n u n b n )Tr( ρ n b n ) (cid:21) [Tr( ρ n b n )] = Tr( ρ n u n b n )Tr( ρ n b n )and [Tr( ρub )] = (cid:20) Tr( ρub )Tr( ρb ) (cid:21) [Tr( ρb )] = Tr( ρu b )Tr( ρb ) . From (6.27) and (6.28), we can then conclude that[Tr( ρ n u n b n )] ⇀ [Tr( ρub )] weakly in L (Γ) , (6.29)and by recalling (6.27) the second convergence in (6.23) follows.15inally, we establish space-continuity properties of the vector field ( ρu, ρub ) similar to thoseestablished in [11, 16]. Theorem 6.3.
Under the same assumptions as in Theorem 1.2, let P be the vector field P :=( ρ, ρb ) , u be a distributional solution of (1.2) and { Σ r } r ∈ I ⊆ R d be a family of graphs as inDefinition 2.3. Also, fix r ∈ I and let γ , γ r : (0 , T ) × D → R be defined by γ ( t, x , · · · , x d − ) := Tr − ( uP, (0 , T ) × Σ r )( t, x , · · · , x d − , f ( x , · · · , x d − ) − r ) ,γ r ( t, x , · · · , x d − ) := Tr + ( uP, (0 , T ) × Σ r )( t, x , · · · , x d − , f ( x , · · · , x d − ) − r ) . (6.30) Then γ r → γ strongly in L ((0 , T ) × D ) as r → r +0 . The proof of the above result follows the same strategy as the proof of [16, Proposition 3.5]and is therefore omitted.
In this section, we consider the initial-boundary value problem for the Keyfitz and Kranzer sys-tem [24] of conservation laws in several space dimensions, namely ∂ t U + d X i =1 ∂ x i ( f i ( | U | ) U ) = 0 in (0 , T ) × Ω U = U at t = 0 U = U b on Γ . (7.1)Note that, in general, we cannot expect that the boundary datum is pointwise attained on thewhole boundary Γ. We come back to this point in the following.We follow the same approach as in [2, 6, 12, 20] and we formally split the equation at the firstline of (7.1) as the coupling between a scalar conservation law and a linear transport equation.More precisely, we set F := ( f , · · · , f d ) and we point out that the modulus ρ := | U | formallysolves the initial-boundary value problem ∂ t ρ + div( F ( ρ ) ρ ) = 0 in (0 , T ) × Ω ρ = | U | at t = 0 ρ = | U b | on Γ . (7.2)We follow [10, 15, 28] and we extend notion of entropy admissible solution (see [25]) to initialboundary value problems. Definition 7.1.
A function ρ ∈ L ∞ ((0 , T ) × Ω) ∩ BV ((0 , T ) × Ω) is an entropy solution of (7.2) if for all k ∈ R , Z T Z Ω n | ρ ( t, x ) − k | ∂ t ψ + sgn( ρ − k )[ F ( ρ ) − F ( k )] · ∇ ψ o dxdt + Z Ω | ρ − k | ψ (0 , · ) dx − Z T Z ∂ Ω sgn( | U b | ( t, x ) − k ) [ F ( T ( ρ )) − F ( k )] · ~n ψ dxdt ≥ , for any positive test function ψ ∈ C ∞ c ([0 , T ) × R d ; R + ) . In the above expression T ( ρ ) denotes thetrace of the function ρ on the boundary Γ and ~n is the outward pointing, unit normal vector to Γ . Existence and uniqueness results for entropy admissible solutions of the above systems wereobtained by Bardos, le Roux and N´ed´elec [10] by extending the analysis by Kruˇzkov to initial-boundary value problems (see also [15, 28] for a more recent discussion). Note, however, that one16annot expect that the boundary value | U b | is pointwise attained on the whole boundary Γ, seeagain [10, 15, 28] for a more extended discussion.Next, we introduce the equation for the angular part of the solution of (7.1). We recall that,if | U b | and | U | are of bounded variation, then so is ρ and hence the trace of F ( ρ ) ρ on Γ is welldefined. As usual, we denote it by T ( F ( ρ ) ρ ). In particular, we can introduce the setΓ − := (cid:8) ( t, x ) ∈ Γ : T ( F ( ρ ) ρ ) · ~n < (cid:9) , where as usual ~n denotes the outward pointing, unit normal vector to Γ. We consider the vector θ = ( θ , · · · , θ N ) and we impose ∂ t ( ρθ ) + div( F ( ρ ) ρθ ) = 0 in (0 , T ) × Ω θ = U | U | at t = 0 θ = U b | U b | on Γ − , (7.3)where the ratios U / | U | and U b / | U b | are defined to be an arbitrary unit vector when | U | = 0and | U b | = 0, respectively. Note that the product U = θρ formally satisfies the equation at thefirst line of (7.1). We now extend the notion of renormalized entropy solution given in [2, 6, 20]to initial-boundary value problems. Definition 7.2.
A renormalized entropy solution of (7.1) is a function U ∈ L ∞ ((0 , T ) × Ω; R N ) such that U = ρθ , where • ρ = | U | and ρ is an entropy admissible solution of (7.2) . • θ = ( θ , . . . , θ N ) is a distributional solution, in the sense of Definition 3.3, of (7.3) . Some remarks are here in order. First, we can repeat the proof of [19, Proposition 5.7] andconclude that, under fairly general assumptions, any renormalized entropy solution is an entropysolution. More precisely, let us fix a renormalized entropy solution U and an entropy-entropy fluxpair ( η, Q ), namely a couple of functions η : R N → R , Q : R N → R d such that ∇ ηDf i = ∇ Q i , for every i = 1 , . . . , d .Assume that L (cid:8) r ∈ R : ( f ) ′ ( r ) = · · · = ( f d ) ′ ( r ) = 0 (cid:9) = 0 . By arguing as in [19] we conclude that, if η is convex, then Z T Z Ω η ( U ) ∂ t φ + Q ( U ) · ∇ φ dxdt ≥ entropy-entropy flux pair ( η, Q ) and for every nonnegative test function φ ∈ C ∞ c ((0 , T ) × Ω). Second, we point out that, as the Bardos, le Roux and N´ed´elec [10] solutions of scalar initial-boundary value problems, renormalized entropy solutions of the Keyfitz and Kranzer system donot, in general pointwise attain the boundary datum U on the whole boundary Γ.We now state our well-posedness result. Theorem 7.3.
Assume Ω is a bounded open set with C boundary. Also, assume that U ∈ L ∞ (Ω; R N ) and U b ∈ L ∞ (Γ; R N ) satisfy | U | ∈ BV (Ω) , | U b | ∈ BV (Γ) . Then there is a uniquerenormalized entropy solution of (7.1) that satisfies U ∈ L ∞ ((0 , T ) × Ω; R N ) . roof. We first establish existence, next uniqueness.
Existence: first, we point out that the results in [10, 15, 28] imply that there is an entropyadmissible solution of (7.2) satisfying ρ ∈ L ∞ ((0 , T ) × Ω) ∩ BV ((0 , T ) × Ω) . Also, ρ satisfies themaximum principle, namely 0 ≤ ρ ≤ max (cid:8) k U k L ∞ , k U b k L ∞ (cid:9) . (7.4)For every j = 1 , . . . , N we consider the initial-boundary value problem ∂ t ( ρθ j ) + div( F ( ρ ) ρθ j ) = 0 in (0 , T ) × Ω θ j = U j | U | at t = 0 θ j = U bj | U b | on Γ − , (7.5)where U j and U bj is the j -th component of U and U b , respectively. The existence of a distribu-tional solution θ j follows from the existence part in Theorem 1.2.We now set U := ρθ , where θ = ( θ , . . . , θ N ). To conclude the existence part we are left to showthat | U | = ρ . To this end, we point out that, by combining [19, Lemma 5.10] (renormalizationproperty inside the domain) with Theorem 2.2 (trace renormalization property) and by arguingas in §
5, we conclude that, for every j = 1 , . . . , N , θ j is a distributional solution, in the sense ofDefinition 3.3, of the initial-boundary value problem ∂ t ( ρθ j ) + div( F ( ρ ) ρθ j ) = 0 in (0 , T ) × Ω θ j = U j | U | at t = 0 θ = U bj | U b | on Γ − . By adding from 1 to N , we conclude that | θ | is a distributional solution of ∂ t ( ρ | θ | ) + div( F ( ρ ) ρ | θ | ) = 0 in (0 , T ) × Ω θ j = 1 at t = 0 θ = 1 on Γ − . By recalling the equation at the first line of (7.2) we infer that | θ | = 1 is a solution of the aboveinitial-boundary value problem. By the uniqueness part of Theorem 1.2, we then deduce that ρ | θ | = ρ and this concludes the proof of the existence part. Uniqueness: assume U and U are two renormalized entropy solutions, in the sense of Defini-tion 7.2, of the initial-boundary value problem (7.1). Then ρ := | U | and ρ := | U | are twoentropy admissible solutions of the initial-boundary value problem (7.2) and hence ρ = ρ . Byapplying the uniqueness part of Theorem 1.2 to the initial-boundary value problem (7.5), for every j = 1 , . . . , N , we can then conclude that U = U . Acknowledgments
This paper has been written while APC was a postdoctoral fellow at the University of Baselsupported by a “Swiss Government Excellence Scholarship” funded by the State Secretariat forEducation, Research and Innovation (SERI). APC would like to thank the SERI for the supportand the Department of Mathematics and Computer Science of the University of Basel for the kindhospitality. GC was partially supported by the Swiss National Science Foundation (Grant 156112).LVS is a member of the GNAMPA group of INdAM (“Istituto Nazionale di Alta Matematica”).Also, she would like to thank the Department of Mathematics and Computer Science of theUniversity of Basel for the kind hospitality during her visit, during which part of this work wasdone. 18 eferences [1] L. Ambrosio, Transport equation and Cauchy problem for BV vector fields,
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